Osmotically driven flows and their relation to ... - Niels Bohr Institutet

The University of Copenhagen The Niels Bohr Institute

Osmotically driven flows and their relation to sugar transport in plants

By

K˚ are Hartvig Jensen Supervisors: Tomas Bohr and Mogens Høgh Jensen

Thesis submittet for the degree of Candidatus Scientiarum in Physics The thesis corresponds to 60 ECTS points

Copenhagen, June 15, 2007

Acknowledgments • I would like to thank my supervisors Tomas Bohr and Mogens Høgh Jensen for their support and many valuable words of advice. • I would like to thank Emmanuelle Rio, Rasmus Hansen and Cristophe Clanet for useful discussions and Erik Hansen and Frederik Bundgaard for technical assistance. • I would like to thank Teis O. Schnipper and Thomas R. N. Jansson for proofreading. • I would like to thank Anders P. Andersen and the rest of the lunch-club for being such good company. • Also, I would like to thank Novo Nordisk, and in particular Morten Colding-Jørgensen, for financial support. • Finally, I would like to thank my parents and Signe for supporting me through all this.

3

Resume Emnet for denne specialeafhandling er osmotisk drevne strømninger, og disses relation til sukkertransport i planter. Strømninger drevet af osmose findes i s˚ a godt som alle levende væsener, men det er dog i planter, at de mest interessante – set fra et fluid mekanisk synspunkt – findes. Her har naturen designeta et nætværk af rørligende celler, kaldet phloem-kanaler, til transport af sukker. Inde i disse rør, som typisk m˚ aler 20 µm i diameter og 1 mm i længden, strømmer en opløsning af sukker og vand fra steder hvor sukker produceres, sædvanligvis i bladene, til steder hvor planten har brug for sukker til at vokse, f.eks. i skud og frugter. Mekanismen bag denne transport af sukker har dog været genstand for en del debat, primært i de biologiske forskningsmiljøer hvor man traditionelt ikke har lagt stor vægt p˚ a de strømningsmekaniske aspekter bag processen. For at belyse dette emne, har jeg valgt at undersøge strømningerne i en enkelt phloem-kanal. Jeg har s˚ aledes udført eksperimenter vhja. en forsøgsopstilling best˚ aende af et membranrør omgivet af et vandreservoir, der skal modellere hhv. phloem-kanalen og det omkringliggende væv. Mine eksperimentelle resultater viser, at hvis sukker til at begynde med er koncentreret i den ene ende af røret, vil osmose skabe vandindstrømning der hvor sukkerkoncentrationen er højest, og p˚ a den m˚ ade flytte sukkerfronten fra den ene ende til den anden. Hastigheden hvormed fronten bevæger sig aftager efterh˚ anden som sukkeret nærmer sig den anden ende af røret. Desuden viser resultaterne ogs˚ a, at hastigheden afhænger af middelkoncentrationen af sukker inde i røret, s˚ aledes at fronten bevæger sig hurtigere ved høje koncentrationer end ved lave. For at forst˚ a mekanismen bag denne process har jeg udledt bevægelsesligninger for osmotisk drevne strømninger i tynde rør. Bevægelsesligningerne, der best˚ ar af to, koblede, partielle differentialligninger, der afhænger af to ¯ karakteriserer hhv. dimensionsløse parametre. Disse parametre, M og D, vigtigheden af aksiel strømningsmodstand set i forhold til membranens permeabilitet, og vigtigheden af diffusiv sukkertransport set i forhold til den a

Øjensynligt p˚ a en yderst intelligent m˚ ade.

5

advektive sukkertransport. ¯ = 0, kan bevægelsesI det mest enkle tilfælde, svarende til M = D ligningerne disse løses analytisk. Resultatet heraf viser at en sukkerfront startende i en ende af et rør bevæger sig med en eksponentielt aftagende hastighed mod den anden ende, og, at hastigheden hvormed fronten bevæger sig afhænger af middelkoncentrationen af sukker inde i røret. Disse resultater har vist sig at være i god kvalitativ s˚ avel som kvantatitativ overensstemmelse med de eksperimentelle data. Udgangspunktet for de eksperimentelle undersøgelser var strømningen i en enkelt phloem-kanal, og anvendes resultaterne p˚ a disse f˚ as en typisk strømningshastighed p˚ a 7 mh−1 , hvor man til sammenligning typisk observerer 0.5-2 mh−1 i planter. At mine resultater forudsiger en for stor hastighed er imidlertid ikke overraskende, idet antagelsen om at M er meget lille kun er gyldig i planter over korte afstande. Desuden er transport af sukker i planter en ligevægtsprocess, og vil s˚ aledes foreg˚ a langsommere end bevægelsen af en sukkerfront. De fremtidige perspektiver for forskning i osmotisk drevne strømninger g˚ ar primært p˚ a en undersøgelse af disse i forsøgsopstillinger hvor diameteren af membranrøret er sammenlignlig med de 20 µm som findes i træer. Dette kan f.eks. realiseres ved at benytte mikrokanaler, der de seneste ˚ ar har været meget benyttet, bla. til undersøgelse af strømninger i smalle kanaler.

K˚ are Hartvig Jensen, 15 juni 2007.

6

Contents Acknowledgments

3

Resume

5

List of symbols

11

I

13

Introduction to osmotically driven flows

1 Introduction 1.1 Introduction to osmotically driven flows . . . . . . . 1.1.1 Objectives . . . . . . . . . . . . . . . . . . . 1.2 Plant Physiology for Pedestrians . . . . . . . . . . . 1.2.1 The sugar transporting tissue: The Phloem 1.3 The M¨ unch hypothesis . . . . . . . . . . . . . . . . 1.3.1 Recent work on osmotically driven flows . .

II

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Setup, Methods and Data Processing

2 Setup, methods and data processing 2.1 Introduction . . . . . . . . . . . . . . . . . . . 2.2 Setup I . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Fastening the membrane . . . . . . . . 2.2.2 Tracking the front position . . . . . . . 2.3 Setup II . . . . . . . . . . . . . . . . . . . . . 2.4 Pros and cons of Setups I and II . . . . . . . . 2.5 Materials and Equipment . . . . . . . . . . . . 2.5.1 Chemicals . . . . . . . . . . . . . . . . 2.5.2 Membrane . . . . . . . . . . . . . . . . 2.6 Experimental techniques . . . . . . . . . . . . 2.6.1 Measuring the membrane permeability

. . . . . . . . . . .

15 15 16 17 19 21 23

27 . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

29 29 30 32 33 35 40 41 41 42 42 42

8

CONTENTS 2.6.2

III

Measuring the elastic properties of the membrane . . . 45

Experimental Results

3 Experimental Results 3.1 Setup I . . . . . . . . . . . . . . 3.2 Setup II . . . . . . . . . . . . . 3.2.1 The effects of diffusion . 3.3 Membrane properties . . . . . . 3.3.1 Elasticity . . . . . . . . 3.3.2 Permeability . . . . . . . 3.4 Dextran properties . . . . . . . 3.4.1 Osmotic strength . . . . 3.4.2 Diffusion coefficient . . . 3.5 The buckling instability . . . . 3.6 The mushroom cloud instability

IV

47 . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

Theory

4 Modelling osmotically driven flows 4.1 The thermodynamics of osmosis . . . . . . . . . . . . . . . . 4.1.1 Derivation of the van’t Hoff equation . . . . . . . . . 4.2 Derivation of the flow equations . . . . . . . . . . . . . . . . 4.2.1 Volume conservation . . . . . . . . . . . . . . . . . . 4.2.2 Mass conservation . . . . . . . . . . . . . . . . . . . . 4.3 Non-dimensionalization of the flow equations . . . . . . . . . ¯ . . . . . . . . . . . . . . . 4.3.1 The parameters M and D 4.3.2 Boundary and initial conditions . . . . . . . . . . . . 4.3.3 Conservation laws . . . . . . . . . . . . . . . . . . . . 4.4 Generalization of the equations of motion to non-cylindrical geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Corrections to the equations of motion . . . . . . . . . . . . 4.5.1 The Unstirred Layer . . . . . . . . . . . . . . . . . .

49 49 53 53 56 56 57 58 58 58 60 61

63 . . . . . . . . .

65 65 66 69 70 71 72 74 74 75

. 76 . 78 . 78

5 Analytical solutions of flow equations 83 5.1 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . 83 ¯ = 0 . . . . . . . . . . . . . . . . . . . . . 84 5.2 Solution for M = D 5.2.1 Case 1 : Boundary and initial conditions for the closed tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

CONTENTS 5.2.2 5.2.3 5.2.4 5.2.5

9 Solution . . . . . . . . Case 2: Boundary and tube . . . . . . . . . . Solution . . . . . . . . Summary . . . . . . .

. . . . . . . . . . . . . . initial conditions for the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . open . . . . . . . . .

6 Numerical solution of the flow equations 6.1 Transformation of the flow equations . . . . . . . . . . . 6.1.1 Summary . . . . . . . . . . . . . . . . . . . . . . 6.2 Numerical implementation . . . . . . . . . . . . . . . . . 6.2.1 Solution scheme . . . . . . . . . . . . . . . . . . . 6.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . 6.3.1 Initial conditions . . . . . . . . . . . . . . . . . . 6.3.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Comparison between the analytical and numerical lutions . . . . . . . . . . . . . . . . . . . . . . . .

V

. . . . . . . . . . . . . . so. .

Data analysis, Discussion and Conclusion

7 Data Analysis and Discussion 7.1 Movement of the sugar front . . . . 7.2 The shape of the sugar front . . . . 7.3 Summary . . . . . . . . . . . . . . 7.4 Application of the results to plants

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. 85 . 89 . 89 . 91

. . . . . . .

93 94 96 97 97 99 99 100

. 100

103 . . . .

. . . .

. . . .

. . . .

105 . 105 . 108 . 111 . 111

8 Conclusion and outlook 113 8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Bibliography

120

VI

121

Appendices

A Setup and Methods A.1 PC . . . . . . . . . . A.2 Camera . . . . . . . A.3 Pressure transducer . A.4 Calibrating the prism

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

123 123 123 123 126

10

CONTENTS

B Numerical simulations

129

C Numerical Code

143

D Error analysis 147 D.1 Uncertainty in measurements . . . . . . . . . . . . . . . . . . 147 D.2 Propagation of uncertainties . . . . . . . . . . . . . . . . . . . 147 D.3 Fit to a straight line . . . . . . . . . . . . . . . . . . . . . . . 148 E Riemann’s method of characteristics 149 E.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 F Estimation of Lp from Pouiseille’s law of friction

153

G Diffusion properties of the red dye

155

List of Symbols

11

List of Symbols Symbol

Description

Unit

A C c c′ c0 cI cm cb D D(x, ξ) ¯ D E G G(x, ξ) g J L Lp l M N n p p0 Q r R r0 r1 r2 S s T

Membrane tube crossection area Non-dimensional concentration Concentration Concentration Characteristic concentration scale Dimensionless concentration scale Concentration at membrane interface Bulk concentration Diffusion coefficient − ∂G(x,ξ) ∂ξ Ratio of diffusive to advective flux Young’s Modulus Distance from prism to screen Green’s function Thickness of prism Flux Length of the membrane tube Membrane conductivity/permeability Initial height of sugar front Ratio of axial to membrane flow resistance Normal to light beam trajectory Index of refraction Pressure Characteristic pressure scale Volume flux Radius of the membrane tube Radius of curvature Equilibrium tube radius Deflection angle Deflection angle Membrane tube perimeter Side length of prism Absolute temperature

m2 M (moles/Liter) M M M M m2 s−1

Pa m

ms−1 m ms−1 Pa−1 m

Pa Pa m3 s−1 m m m rad rad m m K

continued on next page

12

List of Symbols continued from previous page

Symbol t t0 tc U u u0 V X x Z α β δ ∆1 ∆2 ε η Θ1 Θ2 ρ τ µ γ

Description Time Characteristic timescale Diversion timescale Non-dimensional axial flow velocity Axial flow velocity Characteristic axial flow velocity scale Voltage Non-dimensional axial tube coordinate Axial tube coordinate Digital output from Arduino board Radial velocity gradient Unstirred layer parameter Unstirred layer thickness Horizontal deflection in setup II Vertical deflection in setup II Parameter controlling sugar front width Dynamic viscosity Deflection angle Deflection angle Radial tube coordinate Non-dimensional time Chemical potential cm /cb

Unit s s s ms−1 ms−1 V m s−1 m m m Pa s rad rad m J

List of Physical constantsb Symbol kb R DSuc

b

Description Boltzmann constant The gas constant Diffusion coefficient for sucrose

From [Atkins, 1978]

Value 1.38 × 10−23 8.31 × 103 4.6 × 10−10

Unit JK−1 LPa K−1 mol−1 m2 s−1

Part I Introduction to osmotically driven flows

Chapter 1 Introduction In this chapter, I will start by giving an introduction to osmotically driven flows and explain why these are interesting to study. I will then discuss the objectives of this thesis and lay out a road map for the following chapters.

1.1

Introduction to osmotically driven flows

Flows driven by osmosis occurs in many places in nature, the prime example being the transport of water across cell walls occurring in virtually all living creatures. From a fluid mechanics point of view, however, it is in plants that we find the most interesting examples. Here, nature has devised an intricate network of pipe-like cell structures, collectively called the phloem, responsible for transporting sugar from the leaves to places of growth and storage. In these pipes, it is believed, osmosis creates a bulk flow of water and sugar directed from high to low sugar concentrations. This process, however, is not well understood on the quantitative level. The current belief [Nobel, 1999, Knoblauch and van Bel, 1998], based on the pioneering work of Ernst M¨ unch in the 1920’ies [M¨ unch, 1930], is that the motion in the phloem is purely passive, due to the osmotic pressures that builds up relatively to the neighboring tissue as a response to the loading and unloading of sugar in different parts of the plant. Such flows are called called Osmotically Driven Pressure Flows [Thompson and Holbrook, 2003a], Osmotically Driven Volume Flows [Eschrich et al., 1972], or simply osmotically driven flows. This mechanism of sugar trans-location is much more efficient than diffusion, since the osmotic pressure differences caused by different sugar concentrations in the phloem creates a bulk flow of sugar dissolved in water directed from high to low concentrations, in accordance with the basic

16

Introduction

needs of a the plant. It is, however, not clear how well this mechanism is able to account for the sugar transport on the quantitative level. Since the sieve tubes elements that make up the phloem are living cells, the picture can indeed be much more complicated. Evidence of targeted transport exists [Taiz and Zeiger, 2002], and while it is probable that transport from a leaf to a nearby shoot can be described convincingly by the M¨ unch hypothesis, it is unclear whether it is adequate to describe the transport of sugar from e.g. from a leaf to the root by this mechanism. For longer stretches active transport might be important across the sieve plates separating the sieve cell elements e.g. by the relay hypothesis [Canny, 1977, Huber, 1953]. To make progress on these issues, I believe that it is of importance to learn more about the fundamental properties of osmotically driven flows. Such flows have not, to my knowledge, been given sufficient attention in the fluid dynamics literature – neither experimentally nor theoretically.

1.1.1

Objectives

The objectives of this thesis is to obtain a thorough experimental and theoretical understanding of the fluid mechanical properties of osmotically driven flows. Since the interest in these flows are motivated by processes in plants, these will be the starting point of this investigation. We begin by reviewing some fundamental aspects of plant physiology, including previous work on osmotically driven flows. Then, I will motivate and describe the experimental setups used in my experimental work, and present the data acquired from these. Next, I will derive the equations of motion for osmotically driven flows, and analyse these. Finally, I will compare the experimental results to the model and and discuss their applicability to transport of sugar in plants.

1.2 Plant Physiology for Pedestrians

1.2

17

Plant Physiology for Pedestrians

Current estimates place the number of plant species at around 3 × 105 , not including the ones that have gone extinct through the ages [Niklas, 1992]. They display a staggering diversity in shapes and sizes, ranging from the micrometer sized unicellular aquatic algae, to the current record holder in size, the 115.55 m tall Hyperion [Conifers.org, 2007], a Coast Redwood found in Redwood National Park, California, as seen on the right. The formal definition of a plant is that it is an eukaryotic photoautotroph [Niklas, 1992]. Eukaryotic means that it is a cell (or collection of cells, as is most often the case with plants) with a variety of complex internal organs. Autotroph means that it can produce its metabolic energy from inorganic materials, such as carbon dioxide and water, and photo indicates that it does so by photosynthesis. Photosynthesis is the process in which water and carbon dioxide is converted into oxygen and carbon hydrates in the presence of chlorophyll and light energy. In a very simplified manner, this process can be expressed by the chemical reaction [Cotterill, 2002] light

6 H2 O + 6 CO2 −−−−−−→ H2 O + C6 H12 O6 . chlorophyll

(1.1)

In this way, the photon energy absorbed by the plant is converted into the chemical bonds of glucose (C6 H12 O6 ) while expending carbon dioxide and water. The glucose that comes out of this reaction and its derivatives is the main source of energy for plants, and these have developed highly specialized organs to facilitate the transport of sugars. These organs are collectively called the phloem, and we will discuss them in details shortly. However, we begin by looking at a typical cross section of a plant stem shown in figure (1.1). As shown in the figure, there are four major types of tissue in a typical plant stem; the Xylem, the Phloem, the Pith, the Cortex and the Epidermis. The xylem – the large, blue tubes – consists of water-filled, tubular cells joined together at the ends to form a network running along the entire length of the tree. The primary role of the xylem is to conduct water and nutrients from the roots to the leaves, but it also plays a major role in keeping the

18

Introduction

Figure 1.1: Schematics of a typical plant stem cross section. The xylem and phloem are responsible for conducting water and nutrients to and from different parts of the plant. The Cortex and Epidermis protects the stem from mechanical injuries while the pith acts as storage. Image courtesy of http://www.life.uiuc.edu/plantbio/.

1.2 Plant Physiology for Pedestrians

19

Figure 1.2: (A). A detailed view of phloem sieve elements (SE), and their companion cells (CC). Separating the sieve elements are sieve plates (SP). From [Knoblauch and van Bel, 1998]. (B) Close up of a sieve plate. Part of the sieve plate is covered by material which accumulates at the sieve plates when the sieve tube is damaged, effectively acting as a plug. From http://www.biocel.versailles.inra.fr/phloem

plant upright [Niklas, 1992]. Typically, the xylem is filled with water and since its walls are porous, the xylem and the tissue surrounding it acts as a water reservoir for other parts the plant. The phloem – the smaller, green tubes – consists of tubular cells filled with a mixture of sugar and water called phloem sap. Like the xylem, it also forms a network running from the leaves down to the roots. The primary role of the phloem is to conduct the products of photosynthesis from places of production to places of growth and storage. Finally, the pith acts as storage organs, and the cortex and epidermis protects the stem from mechanical injuries.

1.2.1

The sugar transporting tissue: The Phloem

A more detailed view of the phloem tissue can be seen in figure (1.2, A), which shows a cross-section of recorded under a microscope. The conducting tubular cells, called sieve elements, vary in size but typically measure about 10 µm in diameter and 1 mm in length and typical flow velocities inside the sieve elements are 0.5 − 2 mh−1 (see table (1.1)). The sieve elements are separated by sieve plates, as shown in figure (1.2, B). They measure a few µm in thickness and are covered by holes, which are plugged if the sieve element is mechanically damaged or heated, thereby preventing the valuable content of the sieve elements from potentially leaking from plant [Knoblauch and van Bel, 1998]. An important property of the sieve elements is, that their walls consists of semipermeable membranes. These membranes facilitates osmotic interac-

20

Introduction

Quantity

Magnitude

Ref.

Radius [µm]

4.5 (Fava bean), 4 (Winter squash), 6–25

†, ∗, ||

Length [mm]

0.09 (Fava bean), 0.1–3

Flow velocity [mh

−1

]

0.5–1, 0.2–2

Elastic Modulus [MPa]

17, 5.6–7.4 (Ash)

Permeability [10−11 ms−1 Pa−1 ]

5,1.1 (Zitella translucence)

Sucrose concentration [M]

0.3–0.9

†, ||

†, ||

•, h

•, i ∗

Table 1.1: Characteristic properties of phloem sieve elements. References: † [Knoblauch and van Bel, 1998], ∗ [Taiz and Zeiger, 2002], || [Nobel, 1999], • [Thompson and Holbrook, 2003a], h [Niklas, 1992], i [Eschrich et al., 1972].

tion between the inside of the phloem tubes and the surrounding tissue, by allowing water, but not sugar to pass, thereby supplying the driving force for the flow inside the phloem tubes. This mechanism will be discussed in detail below. The flow of sugar in the phloem takes place from sugar sources to sugar sinks, in accordance with the needs of the plant. Sugar sources are parts of the plant producing a surplus of sugar, e.g. mature leaves. Also, most plants have storage organs that can play the role of sugar sources, especially during the early parts of the growth season. Sugar sinks are parts of the plant that do not produce enough sugar to supply their own needs for growth or storage. Examples include roots, immature leaves and developing fruits. The content of the phloem sap is dominated by sugars, which at around 90% of the total mass transported, are at the top of the list in most plants. Among the sugars, sucrose is by far the most abundant, and it is typical to observe concentrations between 0.3 and 0.9 M, the first occurring in small plants and the latter in tall trees. Other than that, amino acids, proteins and a handful of ions are also found in the phloem sap. The composition of the sap exhibits daily and seasonal variations, and also depend on where in the plant it is measured [Taiz and Zeiger, 2002, Niklas, 1992]. We will now the driving force behind the flow inside the phloem, known as the “M¨ unch Hypothesis”.

1.3 The M¨ unch hypothesis

21

Figure 1.3: On the left the convectionist Ernst M¨ unch (1876 – 1946) facing his opponent, the diffusionist Julius von Sachs (1832 – 1897). Images courtesy of http://www.forst.tu-dresden.de/ and http://www.biologie.uni-hamburg.de.

1.3

The M¨ unch hypothesis

The story of osmotically driven flows begins in 1837 with the discovery of the phloem sieve elements by the German professor of forest sciences, Theodor Hartig. Later, in 1858, he postulated that the principal movement of sugar in plants occur in the sieve elements, and while this result was widely accepted, the mechanism behind the movement of sugar was the topic of much controversy. The researchers fell into two groups; (1) the diffusionists who attributed the movement to diffusion aided by an active process occurring in the tissue surrounding the phloem, and (2) the convectionists who pictured the flow of solution as being driven by a pressure gradient caused by osmosis [Huber, 1953, Canny, 1973, Canny, 1977]. The diffusionists had many supporters, among others Julius von Sachs whose influence was so strong that it took more than 50 years before experimental evidence was strong enough to debunk this hypothesis. The evidence was clear: The rates of sugar translocation observed was several million times faster than predicted by diffusion thus requiring unrealistically large amounts of energy for the active processes. In 1930 the German professor of botanics Ernst M¨ unch, published the book titled Die Stoffbewegungen in der Pflanze [M¨ unch, 1930], in which he gathered much of the experimental evidence against the diffusion hypothesis. He argued strongly for the convection hypothesis, which since then has been known as the M¨ unch hypothesis. Today, M¨ unch’s hypothesis is the largely accepted model for the transport of sugar in plants [Nobel, 1999, Knoblauch and van Bel, 1998]. The M¨ unch hypothesis, which is sketched in figure (1.4) states the following: In the source tissues, sugar is produced and actively loaded into the sieve elements. This accumulation of sugar inside the sieve elements lowers

22

Introduction

Water Reservoir

Source

Sieve elements Sink

Figure 1.4: The M¨ unch hypothesis for sugar transport in plants [Taiz and Zeiger, 2002]. Sugar is produced in the leaves on the left and are transported actively into the sieve elements via the dashed red arrow. As the sugar enters the sieve element, the chemical potential of the water inside is lowered compared to the surrounding tissue (large blue box), thereby creating a flux of water into the sieve element (blue arrows). This flow of water into the sieve elements in turn creates a bulk flow of sugar and water towards the sugar sink on the right (purple arrows). At the sink, active unloading of sugar takes place, raising the chemical potential of the water inside the sieve elements resulting in a flow of water out of the sieve elements. The sugar concentration inside the sieve elements is represented by different shades of gray, going from high concentrations on the left to low on the right.


23

the chemical potential of the sugar-water solution. By osmosis, this in turn creates a flux of water into the phloem sieve elements from the surrounding tissue thereby creating a bulk flow along the chain of connected sieve elements. At the receiving end of the translocation pathway, sugar is unloaded, and thus the chemical potential is raised thereby creating an out-flux of water. M¨ unch formulated this hypothesis in the following way: Bei Konzentrationsunterschieden erfolgt eine Strömung von Lösnung in der Richtung der abnehmenden Konzentration.a ([M¨ unch, 1930], p. 9) The most important word in this quote is Strömung, by which M¨ unch meant a bulk flow of sugar and water, as opposed to a diffusive movement of sugar alone.

1.3.1

Recent work on osmotically driven flows

Since M¨ unch, much work has been done on osmotically driven flows, both experimentally and theoretically [Canny, 1977, Bel, 2003]. Most of the experimental work has been done on real plants, with very little attention given to the actual fluid dynamical properties of the flow. Lately much attention has also been given to computer simulations of the flow, although only using simple 1-D models [Thompson and Holbrook, 2003a, Hölttä, 2006]. Of the work done on the fluid mechanics of osmotically driven flows, a few papers stand out. The work of Alexander Lang Alexander Lang, [Lang, 1973], build experiments which demonstrated that osmosis could create considerable bulk flows in narrow tubes. His setup, consisted of a long, narrow membrane tube submerged in a water bath. At one end, a 0.2M sucrose solution was pumped in at a regular rate of 0.05 ml min−1 , and at the other end the tube was open at atmospheric pressure. At regular intervals along the membrane tube, 6 measurement stations were placed that enabled him to measure local sucrose concentration and pressure. Lang however, focused most of his attention on the steady-state case in which sugar was constantly loaded into the tube, thus ignoring the transient behavior of the flow, which is interesting from a fluid dynamics perspective. a

In my translation: “A difference in concentration implies a flow of solution in the direction of the lowest concentration”.

24

Introduction

The work of Walter Eschrich Trying to investigate the flow inside a single phloem sieve element, Walter Eschrich and colleagues build experiments to investigate the dynamics of a moving sugar front inside a membrane tube [Eschrich et al., 1972]. Their setup, sketched in figure (1.5, A), consisted of a membrane tube inside a water-filled glass tube. At the beginning of the experiment, a sugar solution was introduced into one end of the tube, which was then closed at both ends. The movement of the sugar front was then observed, with the results shown in figure (1.5, B) for three different sugar concentrations. They observed that the stronger the sucrose solution, the faster the front traveled. Also, they found that the velocity of the sugar front decayed exponentially in time as the front approached the far end of the tube. Quantitatively, these results are summarized in table 1.2 which shows the decay times, texp 0 , for the three different sucrose concentrations. Eschrich and colleagues showed, that if the flow is driven according to the Munch hypothesis, the velocity of the front is given by t L front , (1.2) u (t) = exp − t0 t0 where L is the length of the tube and t0 is the decay time given by r ttheory = . 0 2Lp Π

(1.3)

Here, r is the radius of the pipe, Lp is the conductivity of the membrane tube and Π is the osmotic pressureb of the sugar solution inside the tube. In their derivation of these results they assumed, that the tube was inelastic, ie. that the radius was constant, and that the pressure inside the tube rose to Π instantaneously as the experiment started. With values of Lp = 3.2 × 10−12 ms−1 Pa−1 and r = 3.5 mm given in the paper, one gets the values in listed theory in table 1.2 as ttheory . The ratio texp takes on values from 6 to 12 0 0 /t0 indicating that the front in the experiments move much slower than predicted by the M¨ unch hypothesis. Thus, the experiments can not be said to be in good quantitative agreement with theory. This, however is what Eschrich claims, attributing the errors to ”. . .experimental difficulties encountered and the theoretical approximations made. . .” ([Eschrich et al., 1972], p. 295.) and finally stating, that ”we therefore conclude that our theoretical explanation of solution flow in tubular semipermeable membranes is essentially correct” (same page). While this may be true, they do not dwell on the reasons for the discrepancies between theory and experiment. b

The osmotic pressure of a sugar solution is discussed in section 4.1.


25

Figure 1.5: (A): Setup used by Walter Eschrich and colleagues in [Eschrich et al., 1972]. (B): The movement of sugar fronts observed. The numbers 0.5, 1.0 and 1.5 M corresponds to different sucrose solution concentrations. Both are from the original paper. Sucrose concentration:

0.5 M

1M

1.5 M

Π [bar]

2.4

4.8

7.2

texp [104 s] 0 ttheory [104 0 exp theory t0 /t0

1.4

1.3

0.67

0.23

0.11

0.076

6.1

12

8.8

s]

Table 1.2

26

Introduction

Looking at equation (1.3) one sees that the decay time depends on the tube radius, conductance and the osmotic pressure of the sugar solution inside. Accordingly, one would need to determine these separately to compare equation (1.3) to the experiments. Nonetheless, Eschrich and colleagues only measures the static value of r and cite previous measurements of the two latter. While the static value of r is straightforward to determine, one would strictly need to know the elastic properties of the tube to be certain that radius of the tube did not change significantly during the experiment. Also, the pressure inside the tube, which they assume to be constant and equal to the osmotic pressure of the sugar solution, should be measured continuously as the sugar front moves. They state that, while using a mercury manometer in a preliminary experiment, they observe a ”. . .buildup of turgor pressure which then remains constant” (same pagec ). The validity of this result, however, is highly doubt-full since they observe leaks in their setups after 100-150 minutes, the typical running time of an experiment. This seems to indicate a slow, gradual pressure rise up to critical threshold value of the equipment, thus invalidating the assumption of a constant pressure inside the tube altogether. I have tried to correct the errors made by Eschrich et. al. in my experiments, which will be described in the next chapter.

c

Turgor pressure is the hydrostatic pressure inside the tube.

Part II Setup, Methods and Data Processing

Chapter 2 Setup, methods and data processing In the first part of this chapter I will motivate and describe the two setups I have used for studying osmotically driven flows. Then, I will discuss some of the materials and equipment, and finally some of the experimental used. For further details about the technical equipment, please consult appendix A.

2.1

Introduction

In the previous chapter, we have seen that plants transport sugar through tubular cells called phloem sieve elements. The driving mechanism behind the bulk flow inside the sieve elements is believed to be osmosis. The high concentration of sugar near the sugar sources lowers the chemical potential of the water in the phloem, resulting in a flux of water into the sieve elements. This influx of water in turn creates a bulk flow of the water-sugar solution away from the sources towards the sinks in agreement with the needs of the plant. Two basic properties of a sieve element is that it is a water-filled tubular cell whose surface is covered by a semipermeable membrane, and that it is surrounded by a water reservoir that allows water to flow in and out of the sieve element as a result of hydrostatic and osmotic pressure differences across the membrane. The aim of my experimental work has been to investigate the flow inside a single such sieve element. To do this, we consider a closed, tube-like container with walls made of semipermeable membrane. The tube is in osmotic contact with a water reservoir, and at the start of an experiment, sugar is deposited at one end of the tube, which is otherwise filled with water. The movement of this “sugar

30

Setup, methods and data processing

front” inside the tube is observed, and if the M¨ unch hypothesis is correct the sugar will move from one end to the other, driven by osmosis. As discussed in section 1.3.1 the previous experimental work made on this type of flow leave several questions unanswered. To make progress on these questions, I have studied the problem using two different experimental setups. The first, Setup I, is based on the design used by Walter Eschrich and colleagues. By fitting a pressure transducer which allows the pressure to be measured dynamically I have improved their setup. Also, I have measured the conductivity of the membrane and the osmotic pressures of the sugar solutions used directly. The second setup, Setup II, is a new design using a laser refraction technique for tracking the sugar front directly. These new and improved experimental setups should enable us to get a better, quantitative understanding of the flow inside a sieve element. The technical details of the two setups as well as their pros and cons are discussed below.

2.2

Setup I

Setup I, shown in figure (2.1), consisted of a 30 cm long, 30 mm wide, glass tube in which a semipermeable membrane tube of equal length and a diameter of 10 mm was inserted (details of the membrane is discussed below). At one end, the membrane tube was fitted over a glass stopcock equipped with a rubber stopper. At the other end, the membrane tube was fitted over a brass cylinder equipped with a screw thread to accommodate a pressure transducer for measuring the pressure inside. At the lower end of the stopcock a piece of silicone tubing was attached such that it could be easily connected to a disposable syringe. After filling the 30 mm wide glass tube with water, water was pressed into the semipermeable tube with the syringe. Care was taken that no air bubbles were stuck inside the tube. For introducing the sugar solution (details of the solution is discussed below) into the semipermeable tube, a syringe was filled with the solution and then attached to the lower end of the stopcock which was kept closed. After fitting the syringe, the stopcock was opened and the syringe piston was very slowly pressed in until a suitable part of the tube had been filled with the solution. Care was taken to avoid any mixing between the sugar solution and the water already present in the semipermeable tube. Finally the stopcock was closed, and the pressure transducer (see appendix A.3) was fitted to the brass cylinder, tightening it securely. To track the movement of the sugar solution – which was completely transparent – it was mixed with a red dye and we thus assumed that the sugar and dye travelled together. For a discussion of the diffusion properties

2.2 Setup I

31

-pt 10 mm

-bc

(A)

(B)

ss wr

spm -gt

30 mm

spm -gt

L 30 cm

wr

l

ss -rs -sc

-ds Figure 2.1: (A): Side, and (B): top view of Setup I as described in the text. Abbreviations: spm semipermeable memebrane tube; gt glass tube; pt pressure transducer; sc stopcock; ds disposable syringe; bc brass cylinder; rs rubber stopper; L length of membrane tube; l sugar front height; wr water reservoir.

32


of the red dye, please consult appendix G. To track the motion of the sugar-dye front, data was recorded by taking pictures of the the membrane tube at intervals of 15 min using a digital camera. Details of how the motion of the sugar-dye solution was derived from the images is discussed below in section 2.2.2.

2.2.1

Fastening the membrane

Fastening the membrane tube to the end of the stopcock and the brass cylinder proved to be a major challenge. The pressure inside the membrane tube could become relatively large, up to 2 bar in our experiments, which is comparable to the pressure inside a standard car tyre. Therefore, great care has to be taken to avoid leaks, which would ruin the experiments by allowing sugar to escape across the membrane as well as preventing the build-up of pressure inside the tube. This problem was particularly pronounced since typical time-spans for an experiment was a few days.

(A) spm

(B)

(C) spm

spm sct

st

st

st

ber sg

gt

gt

gt

Figure 2.2: Membrane fittings. (A): A small piece of silicone tube is stuffed into the end of the semipermeable membrane tube (spm). Sewing thread is then wound around the overlapping area, and the entire thing is gently mingled into the glass tube. As the pressure inside the spm rises, the silicone is tube pushed against the glass tube giving a reasonably tight fit. (B): The spm is fitted over the end of the glass tube, and sewing thread is wound around the overlapping area. (C): The spm is fitted over the brass extension ring and the silicone glue. Sewing thread is wound around the overlapping area, giving a very tight joint. Abbreviations: gt glass tube; sct silicone tube; st sewing tread; ber brass extention ring; sg silicone glue.

2.2 Setup I

33

To test for leaks in the membrane tube and fittings, the 30 mm wide glass tube was filled with water, and the pressure transducer was mounted. The membrane tube was then filled with air by a disposable syringe until a pressure of ∼ 2 bar had been reached. Then, the tube fittings were inspected visually for leaks which were very clearly visible in the form of air bubbles. Also, the pressure inside the tube was measured using the pressure transducer, but this was observed to drop slowly even in the absence of leaks, presumably because a pressure of 2 bar is sufficient to overcome the capillary forces normally preventing air from escaping through the membrane wall. We have tried several approaches to keep the membrane joints tight, as sketched in figure (2.2). The one which proved to be most successful was to fit a brass extension ring on the brass cylinder and on the end of the stopcock, with a diameter slightly larger than that of the membrane tube, as shown in (C). Behind this extension ring, silicon glue (Henkel Technologies, Transparent, Sanitær og Byg) was deposited and left to dry. When preparing an experiment, the membrane tube was fitted over the brass extension ring, and pulled back such that it would terminate near the end of the silicone glue layer. Sewing thread was then wound tightly around the part of membrane tube covering the brass ring and the silicon glue layer, giving a very tight joint.

2.2.2

Tracking the front position

After running an experiment with setup I, the raw data we had acquired consisted of a series of pictures. One such picture is shown in in figure (2.3, A), showing the sugar solution mixed with red dye at the bottom of the tube. To track the position of the sugar-dye front, the images was treated as shown in figure (2.3, A-C). First, the image was first imported into Matlaba as (A), and then cropped to shown only the membrane tube, (B). Simultaneously, it was filtered to give the highest contrast for obtaining a well-defined front position. To find the front position, a vertical line running along the center of the membrane tube was picked out, shown as a white line in (B). Along this line, the color intensity was found, shown as the black curve in (C). Finally, the gradient of the color intensity was found – shown in (C) as the red curve – and the front position was defined to be the the position of the maximum in intensity gradient.

a

Commercially available scientific software package. See http://www.mathworks.com

34


Figure 2.3: Data processesing in setup I. (A): Raw RGB data image showing the sugar solution mixed with red dye inside the membrane tube. (B): The image from (A) has been cropped and filtered. (C): The solid black curve is the intensity of color in (B) taken along the white vertical line. The red curve is the absolute value of the gradient of the intensity. The front position is taken to be where this curve has its maximum value (at approximately 200 pixels). The black dot in (B) is the position of the front found in this manner. The peak in the intensity gradient near 920 pixels is due to the membrane fitting, and was ignored.

2.3 Setup II

2.3

35

Setup II

Setup II, shown in figure (2.4), consisted of a hollow isosceles glass prism and a Plexiglas cuboid in osmotic contact through a membrane. When preparing an experiment, a piece of membrane was fitted at the interface between the prism and cuboid. These two parts where then glued together, and the setup was left to dry. After drying for 24 hours, the prism was filled half-way with a sugar solution and pure water was carefully deposited on top of the sugar solution to create a sharp sugar front. Then, the cuboid was filled with water, and a lid was placed on top of the prism. Osmotic interaction across the membrane would then push the sugar-water interface upwards, thus creating an osmotically driven flow. To track the time evolution of the the sugar front inside the prism, we used the refraction of a laser sheet passing through it. The laser sheet was generated by shining a laser beam, generated by a Melles Griot 3.1 mW laser, through a glass rod. When passing through the prism, light would deviate depending on the local index of refraction. The index of refraction varies linearly with sugar concentration and thus by looking at the refracted laser sheet projected onto a screen, we were able to reconstruct the concentration profile inside the prism. A camera recorded images of the screen at regular intervals, and figure (2.5, A) shows an example of an image produced in this way. The deflection at the bottom of the image corresponds to a high sugar concentration inside the lower part of the prism, and the vertical deflection is due to a strong gradient in index of refraction near the sugar front. Tracking the sugar front To reconstruct the concentration profile inside the prism from the image in figure (2.5, A), we will now discuss the the path taken by a beam of light as it travels through the prism. As the beam passes through the prism, it gets deflected due to variations in index of refraction of the fluid inside the prism relative to the surrounding air. To determine the horizontal deflection of a light beam passing through the prism, we consider the situation sketched in figure (2.6, A). The deflection ∆1 along the y axis is given by ∆1 = G tan Θ1

(2.1)

where G is the orthogonal distance from the prism to the screen, and Θ1 is the deflection angle as shown in the figure. To find Θ1 we notice that r1 − Θ1 = Ψ,

(2.2)

36


(A) sg

5 cm

Top view Prism-

z

7 cm Water reservoir y

sg Side view Screen

(B)

Water - Semipermeable membrane

Water

Las

Water reservoir

er s hee t

30 cm

Sugar solution

Sugar solution

x

Semipermeable membrane Water reservoir z y

x

y sg

Figure 2.4: (A): Sketch of Setup II. The laser sheet (red arrows) pass through the prism and are deflected and projected onto a screen as shown in (B). See further details in the text. Abbreviations: sg silicone glue.

2.3 Setup II

37

(B)

(A)

∆2

∆2

∆1

∆1

(C)

n

n0+n1

n0 l

x

Figure 2.5: Data processing in Setup II. (A): Raw data image. (B): Filtered data image. (C): Index of refraction inside the prism giving the red curve in (B) cf. equation (2.5) and (2.10)

where Ψ is the prism angle. The deflection angle, r1 , is given by the SnellDescartes law [Hecht, 2002] n sin = Ψ sin r1 ,

(2.3)

where n is the index of refraction of the fluid inside the prism. In the experiments, Θ1 is typically small so n sin Ψ = sin r1 = sin(Ψ + Θ1 ) ≃ sin Ψ + Θ1 cos Ψ.

(2.4)

so the horizontal deflection ∆1 ≃ GΘ1 = G (n − 1) tan Ψ.

(2.5)

38

Setup, methods and data processing z

(A)

(B) z Δ1

Screen

y x

Screen

Δ2 Θ2

Θ1 G

r1 Prism

Prism

n

n

g

r r2 Ψ

Figure 2.6: Deflection of the laser beam as i passes through the prism.

. To determine the vertical deflection consider the situation sketched in figure (2.6, B). According to Fermat’s principle, light travels along the path that can be traversed in the least possible time. Consequently, as the light passes through the prism, it travels in a circular arc with a radius of curvature, R, given by 1 ∇n = N· , (2.6) R n where N is the normal to the trajectory of the light beam [Landau and Lifshitz, 1984]. The vertical deflection is given by ∆2 = G tan Θ2

(2.7)

sin Θ2 = n sin r2

(2.8)

where the angle Θ2 is given by

Since r2 is small, it is clear from the figure that sin r2 ≃ (2.6) g ∂n g = . sin r2 ≃ R n ∂x

g , R

so from equation (2.9)

2.3 Setup II

39

In the limit where r2 and Θ2 are small we get for the vertical deflection ∆2 = Gg

∂n , ∂x

(2.10)

where we have assumed that ∇n is constant as the beam traverses the prism. Having obtained ∆1 and ∆2 we now have to deduce n(x) from the projected image. We will do this by assuming that n(x) has a generic, sigmoid shape ! 1 n(x) = n0 + n1 1 − (2.11) 1 + exp − x−l ǫ

where the constants n0 and n1 controls the magnitude of n and l and ǫ controls the position and steepness of the front. A plot of this function can be seen in figure (2.5, C) The procedure for obtaining n(x) was as follows. First, the raw data image was loaded into Matlab as shown in figure (2.5, A). Then, the image was filtered to show only regions of high light intensity, shown as the black dots in (B). Then, a guess of the the form (2.11) was made, and the deflections ∆1 and ∆2 was calculated from equations (2.5) and (2.10), shown as the red curve in (B). Finally, an optimization of the parameters was made using Matlabs fminsearch engine, thereby giving the n(x) of the form (2.11) best able to reproduce the image seen in (A). Generally, the assumption that n was of the form (2.11) gave very good fits, as can be seen in figure (2.5, B). To determine the concentration c(x) inside the prism from n(x), the prism was calibrated (see appendix A.4) giving the relation n = 1.333 + 0.002c.

(2.12)

40

2.4


Pros and cons of Setups I and II

We will now discuss the pros and cons of the two setups. Setup I Since setup I was build as an improved version of a previous setup, it’s main advantages was that it was fitted with a pressure transducer to measure the pressure inside the membrane tube while the front moved. The downside of setup I, however, was that the tracking of the sugar front occurred indirectly through the tracking of a colored dye which was assumed to be traveling along with the flow. Setup I thus tells us nothing about the dynamics of the shape of the sugar front. Finally, the typical flow velocities observed was in the order of 1 cm/h, so an experiment took approximately one day to complete. Setup II The great advantage of setup II compared to setup I was that that it enabled us to track the sugar directly via a laser refraction technique. However, one could easily imagine an asymmetry in the distribution of sugar across a crosssection of the prism arising because the membrane only covered one side of the prism. Nonetheless, setup II gave a much better picture of the front as it evolved than setup I did. Besides the possible asymmetry in sugar distribution, the downside of setup II was mainly its somewhat cumbersome design. Because of the way it was first constructed, setup II had to be glued together before each experimental run. Then, after the experimental run was completed it had to be taken apart by cutting away the glue used and then wrestling the two parts from one-another. This procedure meant that it was very difficult to keep tight, and that after running for a few days the setup began to leak. Thus, while setup II was well suited for investigating the evolution of the shape of the sugar front it was less suited for tracking the long-term motion of the front. Finally, the typical flow velocities observed was 0.3 mm/h, which is more than thirty times slower than setup I. Summary To make the best of the two setups, I therefore used setup I for tracking the long-term motion of sugar front and setup II for tracking the evolution of the shape of the sugar front itself.

2.5 Materials and Equipment

2.5 2.5.1

41

Materials and Equipment Chemicals

Sugar The sugar type used was a dextran (Sigma-Aldrigde, type D4624) with an average molecular weight of 17.5 kDab . Dextran is a complex, branched polysaccharide made of many glucose molecules joined into chains of varying lengths. Dextran comes in the form of a white powder, which can be easily dissolved in water. The choice of dextran, and not sucrose, as the sugar used in the experiments was motivated by the fact that we had to choose a semipermeable membrane with a pore-size much smaller than the size of the sugar. Otherwise, the sugar would pass unhindered across the membrane. The pore size is directly related to the permeability, Lp , of the membrane (see appendix (F)). Thus, if we had chosen sucrose, which has a molecular weight of 342 Da, the permeability would have become unacceptably low. In the rest of this thesis, all references to “sugar” will be to dextran unless otherwise stated. Dyes To track the sugar front position in setup I, red dye (Flachsmann Scandinavia, Rød Frugtfarve, type 123000) was added to the sugar solution. This dye consisted of a aqueous mixture of the food additives E-124 and E-131 with molecular weights of 539 Da and 1159 Da respectively [PubChem-Database, 2007]. Even though the molecular weights are below the Molecular weight cut-off (MWCO, discussed below) of the membrane, the red dye were not observed to leak through the membrane. This however, was observed when using another type of dye, Metylene blue, which has a molecular weight of 320 Da [PubChem-Database, 2007]. Preparing the sugar-dye solution To prepare the sugar-dye solution, an appropriate amount dextran was weighed off using an laboratory weight. The dextran was then dissolved in water and stirred thoroughly until the solution became transparent. For use in setup I, two or three drops of red dye was finally added to the solution. b

1 Da = 1 g/mol.

42


2.5.2

Membrane

The membrane used was a semipermeable dialysis membrane tube (Spectra/Por Biotech Cellulose Ester (CE) dialysis membrane, type) with radius 5 mm, thickness 60 µm and MWCO of 3.5 kDa. The MWCO is defined as the solute size that is retained by at least 90% by the membrane. However, since a solute’s permeability is also dependent upon molecular shape, degree of hydration, ionic charge and polarity, a good rule of thumb is, that in order to ensure that no molecules passes through the membrane, one should choose a membrane with a MWCO such that the molecular weight of the molecules is at least two times as large as the MWCO. Our choice of a 3.5 kDa membrane for use with a 17.5 kDa molecule is therefore on the safe side. Preparing the membrane To prepare the membrane tube, a 30 cm long piece was soaked in demineralized water and left for 1 hour. Then it was carefully mounted on one end of the glass stopcock and on the brass cylinder as described in section (2.2). The process of fitting the membrane took about 10 minutes, and it was sprayed with water regularly to prevent it from drying out.

2.6 2.6.1

Experimental techniques Measuring the membrane permeability

The permeability of a membrane, Lp , is the volume of water flowing across a unit area of membrane per second, given a hydrostatic pressure difference of one Pascal across the membrane. Thus, for a membrane of area A and pressure drop p one gets a pressure driven volume flow of Q = −ALp p.

(2.13)

If, on the other hand, there is a concentration difference of a solute dissolved in water across the membrane, one gets a volume flow driven by osmosis Q = ALp Π,

(2.14)

where Π is the osmotic pressure difference across the membranec . To measure the permeability of the membrane, we used two different methods. One, using hydrostatic pressure, and one using osmotic pressure. c

See section 4.1 about osmosis.

2.6 Experimental techniques

43

Procedure I To measure Lp using a hydrostatic pressure difference, we used the following approach: Setup I was prepared in the usual way, except that the membrane tube was was filled with pure water until an internal pressure of 0.3 bar had been reached. Then, at t = 0, a small volume of water (0.1-1 mL) was injected into the tube using a syringe. This increase in volume inside the tube created a pressure difference between the inside of the membrane tube and the surrounding air, thus resulting in a net flux of water out across the membrane. After injection, we waited until a time t = t′ when pressure inside the membrane tube was again 0.3 bar. At this point all the water we introduced into the tube at t = 0 had flown across the membrane. All along, we measured the pressure inside the tube at a rate of 1 Hz. A plot of the pressure versus time can be seen in figure (2.7). To see how this may be used to measure the permeability of the membrane, consider the following. In terms of the volume inside the membrane tube, equation (2.13) be-

Pressure [105 Pa]

1.5

1

0.5

0 0

100

200

300

400 500 Time [s]

600

700

800

Figure 2.7: Plot showing pressure vs. time data used for determining Lp .

44


comes

∂V = −Lp Ap. ∂t If we integrate this equation with respect to time, we get Z

t′

0

∂V dt = −ALp ∂t

Z

t′

p dt,

V (t ) − V (0) = −ALp

(2.16)

0

or ′

(2.15)

Z

t′

p dt.

(2.17)

0

V (0)−V (t′ ) is the volume introduced into the membrane tube by the syringe. Rearranging, this leads to the following expression for Lp Lp =

V (0) − V (t′ ) . R t′ A 0 p dt

(2.18)

The integral in the denominator is found numerically from the measured values of p. Procedure II To measure Lp using an osmotic pressure difference we used a Pasteur pipette with a diameter of 7 mm at the bottom and 1 mm at the top. The large diameter part was filled with a sugar solution of known osmotic pressure, and the bottom of the pipette was closed with the membrane. The bottom end of the pipette was then put in a water bath, and the rise of the water column in the small diameter part of the tube was measured with time. The dextran solution used had an osmotic pressure of ∼ 0.7 × 105 Pa, so the hydrostatic pressure difference across the membrane arising because of the elevated water column was negligible. Equation (2.14) leads to ∂h A0 = Lp Π, ∂t A1

(2.19)

is the time derivative of the water column height and A0 and A1 where ∂h ∂t are the radii of the large and small parts of the pipette, respectively. Thus, the expression for Lp becomes Lp =

A1 ∂h 1 . A0 ∂t Π

(2.20)

2.6 Experimental techniques

2.6.2

45

Measuring the elastic properties of the membrane

When a thin-walled tube is put under internal pressure, the radius increases proportionally to the pressure [Lautrup, 2004] r = r0 +

r02 p , dE

(2.21)

where r0 is the equilibrium radius, d is the wall thickness, p is the internal pressure and E is Young’s modulus, characterising the elastic properties of the tube. To determine E, we measured the relation between r and p in the following manner: Setup I was prepared in the usual manner, except that the membrane tube was filled with water until the pressure inside was atmospheric. Then, using a syringe, volumes of 0.1, 0.2, . . . , 1 mL water was injected. The maximum pressure inside the tube was noted immediately after the injection. Thus, knowing the volume at different pressures we could determine r(p), and thus E from equation (2.21).

Part III Experimental Results

Chapter 3 Experimental Results In this chapter the experimental data acquired using the techniques described in the previous chapter will be presented. We will begin by discussing the motion and shape of the sugar front and then discuss the properties of the semipermeable membrane and the dextran solutions. Finally, we will briefly touch upon two interesting phenomena not directly related to our investigation of osmotically driven flows; the buckling instability and the mushroom cloud instability. Following the discussion in section 2.4, Setup I has been used to track the sugar front position in time and setup II to study the evolution of the shape of the front.

3.1

Setup I

Using setup I, as described in section 2.2, the motion of the sugar front was investigated for solutions of varying sugar concentration. An example of a set of data is shown in figure (3.1). The top, (A), are the raw images, which after processing gives (B) showing the position of the sugar front, xf , as a function of time. The errorbars on xf are estimated to be ±1 mm, but are to small to be seen. These were found by visual inspection of the front while the experiment were running. Finally, (C) shows the pressure inside the tube as a function of time. At first, a linear motion of the front is observed with at front velocity of ∼ 1 cm/h. This then succeeded by a decrease in the front velocity as the front approaches the end of the tube. The pressure is seen to rise rapidly during the first hour before settling to a constant value, indicated by a red, dashed line. This constant value is taken to be the osmotic pressure, Π, of the sugar solution. Looking at (A), one observes that diffusion has the effect of dispersing the front slightly as time passes. Below the front, the concentration seems to be uniform throughout the cross-section of the tube, and there is no indication

50


20

(B)

f

x [cm]

15 10 5 0

0

2

4

6

8

10

12

t [h] 1

(C)

p [bar]

0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

12

t [h]

Figure 3.1: Example of data collected with setup I for a sugar solution of concentration c¯ = 6.8 mM. The length of the membrane tube was 20.6 cm and at t = 0 h, the front was situated 4.8 cm above the tube entrance. (A): Time series of pictures taken of the membrane tube with the sugar dye solution inside. Time increases from left to right in steps of 30 minutes. (B): Plot of the front position versus time obtained from the images above. (C): Plot of the pressure inside the tube versus time. The red dashed line is the osmotic pressure of the solution, taken to be the average of the pressure from t = 2 h until the end of the experiment.

3.1 Setup I

51

of large boundary layers forming near the membrane wallsa . Similar experiments with different sugar concentrations were made and a plot of the results can be seen in figure (3.2). Qualitatively both the motion of the front and of the pressure follow the same pattern as in figure (3.1). One notices that the speed with which the fronts move are related to the mean sugar concentration inside the membrane tube, with the high concentration solutions moving faster than the low concentration ones. The reason why 2 seems to be moving slower than 1 is that experiment 2 was conducted in a shorter tube than 1, thereby decreasing the characteristic velocity as we shall see later.

a

That such boundary layers can form is discussed in section 3.6.

52


25 20 xf [cm]

(A)

1

3

4

2

5

15 10 5 0

0

10

20

30

40 t [h]

50

60

70

80

0.8

(B)

5 p [bar]

0.6 0.4

4 3

0.2

2

1 0

0

Experiment

10

20

30

40 t [h]

50

60

70

80

1

2

3

4

5

c¯ [mM]

1.5±0.3

2.10±0.03

2.4±0.2

4.2±0.7

6.8±0.1

Π [bar]

0.14 ± 0.02

0.15±0.01

0.31±0.03

0.39±0.01

0.68±0.02

28.5

20.8

28.5

28.5

20.6

4.9

3.7

6.6

6.5

4.8

L [cm] l [cm]

Figure 3.2: Data collected using setup I for different sugar concentration given as c¯ in the table. Π is the osmotic pressure indicated by the red, dashed lines in (B). L is the length of the membrane tube and l is the initial front height, which has been subtracted from the front positions in (A). (A): Time evolution of the front position. (B): Time evolution of the pressure inside the membrane tubes.

3.2 Setup II

3.2

53

Setup II

Figure (3.3) shows the data collected using setup II. At the top, a time series of picture is depicted showing the refracted laser-light projected onto a screen. Comparing the upper and lower parts of each picture, one generally observes a deflection to the right at the bottom, corresponding to a high sugar concentration at the bottom of the prism. In the intermediate region one sees a dip in the refracted light, corresponding to a strong concentration gradient, cf. equation (2.10). The dip gradually flattens while it advances upwards, representing a sugar front which advances while it is broadened by diffusion. This process can be seen directly in figure (3.3, B), which shows the time evolution of the sugar concentration obtained from the images, as discussed in section 2.3. Starting from a steep concentration profile, we see that the front moves forward while it flattens. Below, in (C), the time evolution of the concentration gradient is depicted clearly showing a peak which broadens while it moves forward. Finally, in (D), the position of the sugar front as a function of time is shown. The errorbars on xf are ±1 mm, found as discussed below. The front moves with a velocity of ∼ 0.3 mm/h, more than thirty times slower than observed in setup I. While the results from setup I clearly showed the front velocity decaying in time, it is difficult to tell whether this is the case in setup II.

3.2.1

The effects of diffusion

To study the effects of diffusion on the flow separately, an experiment was made with setup II, in which the membrane separating the two compartments was replaced by a plastic sheet. The experiment was then prepared in the usual way, and the motion of the front recorded. The results of this is shown in figure (3.4). Starting from a steep concentration gradient, we observe that the front flattens but otherwise does not move much. Comparing figures (3.3) and (3.4) we observe, that while the front moves 2 cm due to osmosis in 72 hours, it does not move at all in 140 hours due to diffusion. Thus, while diffusion has a flattening effect, it plays little role in the forward motion of the front. Since the front did note move due to diffusion, the fluctuations in the front position seen in (3.4, D) gives a measure of the uncertaing of a single measurement of the front position. Taking the standard deviation of the fluctuations gives an uncertainty of ±1 mm, as shown in (3.3, D)

54


c [Mm]

15

(B)

10 5 0

0

2

4

6 x [cm]

8

10

12

|dc/dx| [mM/cm]

15

(C)

10 5 0

0

2

4

6 x [cm]

8

10

12

xf [cm]

7

(D)

6 5 4

0

10

20

30

40 t [h]

50

60

70

80

Figure 3.3: Data from setup II. (A): Time series of data images, with time increasing from left to right in intervals of 4 hours. The images show the time evolution of the sugar front as it progresses upwards . The strong dip in the first images corresponds to a very steep sugar concentration gradient (cf. equation (2.10)). As time progresses, this gradient is flattened while the front moves upwards. (B): Time evolution (from black to light gray) of the sugar concentration profile derived from (A). (C): Time evolution (from black to light gray) of the concentration gradient. (D): Time evolution of the sugar front position. The position of the front is defined as the position at which the concentration gradient had a maximum.

3.2 Setup II

55

15

c [mM]

(B) 10 5 0

0

2

4

6 x [cm]

8

10

12

|dc/dx| [mM/cm]

15

(C) 10 5 0

0

2

4

6 x [cm]

8

10

12

7

f

x [cm]

(D) 6 5 4

0

20

40

60

80

100

120

140

t [h]

Figure 3.4: Data from setup II showing the effects of diffusion. (A): Time series of images captured using setup II, with time increasing from left to right in intervals of 8 hours. (B): Time evolution (from black to light gray) of the sugar concentration profile derived from (A). (C): Time evolution (from black to light gray) of the concentration gradient. (D): Time evolution of the sugar front position.

56


3.3

Membrane properties

3.3.1

Elasticity

Figure (3.5) shows the dependence of the membrane tube radius on internal pressure, determined as described in section 2.6.2. One notices, that the relative increase in radius is generally very small, eg. ∼1.3% at an internal pressure of 1 bar. At pressures below 1.2 bar, the radius depends linearly on pressure, in good agreement with equation (2.21). A linear fit (red dashed line), gives the value E = 0.66 ± 0.01 GPa (3.1) for Young’s modulus from equation (2.21). A similar material, polyethylene – the material grocery bags are made of – has a Young’s modulus of 0.4 − 1.3 GPa [Kaye and Laby, 1995].

1.6 1.4

p [105 Pa]

1.2 1 0.8 0.6 0.4 0.2 0 0

0.005

0.01

(r−r0)/r0

0.015

0.02

0.025

Figure 3.5: Plot showing the dependence of the membrane tube radius on internal pressure. Red dashed line is a linear fit to equation (2.21), which assumes that the membrane tube is linearly elastic. This assumption breaks down for p > 1.2 bar.

3.3 Membrane properties

3.3.2

57

Permeability

Using the two techniques described in section 2.6.1, the permeablilty Lp of the semipermeable membrane tube was determined using a hydrostatic and an osmotic pressure differences across the membrane. Using the hydrostatic method a value of Lp = 1.9 ± 0.1 × 10−12 ms−1 Pa−1

(3.2)

was found. When using osmotic pressure method, Lp was found to be Lp = 1.8 ± 0.2 × 10−12 ms−1 Pa−1 .

(3.3)

Comparing these results, we will use the value Lp = 1.8 ± 0.2 × 10−12 ms−1 Pa−1

(3.4)

in our further work. This is an order of magnitude smaller than in plants, but since the membrane is much thicker this was expected. In appendix F, an estimate on Lp is given using Poiseuille’s law of friction which gives a value of order 10−12 ms−1 Pa−1 , in good agreement with our experimental results.

58


3.4

Dextran properties

3.4.1

Osmotic strength

Figure (3.6) shows a plot of the relation between dextran concentration and osmotic pressure from the table in figure (3.2).

Osmotic pressure, Π [bar]

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

6

7

c [mM]

Figure 3.6: Plot showing the relation between dextran concentration and osmotic pressure.

Fitting to a straight line through zero gives: Π = (0.10 ± 0.01) bar mM−1 c,

(3.5)

where Π has units of bar and c is measured in mM. This result is in good agreement with the value of 0.11 found by Jonsson [Jonsson, 1986].

3.4.2

Diffusion coefficient

To determine the diffusion constant of the dextran used in the experiments, the data from figure (3.4) was used. In figure (3.7, left) the experimental data has been plotted again. On the right is the result of running a numerical simulation of the diffusion equation ∂c ∂2c =D 2 (3.6) ∂t ∂x with the initial condition taken to be the data at t = 0. Fitting the diffusion constant D, we get that DDex = 6.9 ± 0.5 × 10−11 m2 s−1 .

(3.7)

3.4 Dextran properties

59

Data

Model

(A)

c [mM]

10 5 0

(D)

10 5

2

3

4

5

6

7

0

2

3

4

5

6

7

x [cm] dc/dx [mM/cm]

10

10

(B)

(E)

5

0

5

2

3

4

5

6

7

0

2

3

4

5

6

7

x [cm] 6

6

(C)

x [cm]

5 4

4

3

3

2

0

50

100

(F)

5

2

0

50

100

t [h]

Figure 3.7: Data and model used to obtain the diffusion constant for Dextran. (A): Time evolution (from black to light gray) of the concentrations profile. (B): Time evolution of the concentration gradient. (C): Sugar front position derived from (B) as the position of the maximum in concentration gradient. (D-F): Same as (A-C), but from numerical simulation of equation (3.6) starting with the initial condition from (A).

To compare, ordinary sugar (sucrose) has a diffusion coefficient of 4.6 × 10−10 m2 s−1 , but since dextran is a much larger molecule we expect a smaller value, cf. the Stokes-Einstein relation [Atkins, 1978]. See appendix G for a discussion of the diffusion properties of the red dye.

60


Figure 3.8: Buckling of the membrane tube under internal pressure with time advancing from left to right. The mean concentration of sucrose inside the tube was c¯ = 0.5 M.

3.5

The buckling instability

When conducting an experiment using sucrose rather than dextran in setup I, we observed an interesting buckling instability shown in figure (3.8). First, the front advanced in usual manner until the height shown the first image on the left had been reached. Then, the tube began to expand and then buckle. First towards the left and then, when blocked by the outer glass tube, as a s-shape. Also, in the last few images, sugar and blue dye is seen to leak from the joint near the bottom. Qualitatively, this behaviour can be explained by an increase in internal pressure as the front nears the end of the tube. As the pressure increases the tube expands both radially and axially. Then, when a certain critical pressure has been reached, the forces acting on the tube from the membrane fittings becomes larger than some critical value, and the tube begins to buckle. This behavior is very similar to that seen when a column become subject to a compressive force. At first, it is compressed, but when a certain critical threshold force is reached it start to buckle. For a column of length L, the famous Euler criterion states that the critical force is Fc ∝

EI , L2

(3.8)

3.6 The mushroom cloud instability

61

Figure 3.9: Images clearly showing a mushroom cloud forming above the front. Time between each images is 15 min.

where E is Young’s modulus and I is the moment of inertia of the column [Love, 1944]. Unfortunately, at the time the experiment shown in figure (3.8) was conducted, the pressure transducer had not yet been fitted to setup I. Thus there is no direct evidence of a gradual pressure rise before the buckling occurs.

3.6

The mushroom cloud instability

As discussed in section 3.1, no significant concentration boundary layers could be observed directly in the experiments done with setup I. When using sucrose rather than dextran, however, an interesting phenomena was observed, shown in figure (3.9). Near the front, a mushroom cloud formed travelling along with the bulk flow at a speed of ∼ 5 cm/h. Thus, while the concentration seems to be uniform throughout the cross-section of the tube near the bottom of the images, this is not the case near the front. The mechanism behind the the formation of the mushroom cloud is not fully understood, but one could speculate that the mechanism could be related to the SaffmanTaylor instability [Saffman and Taylor, 1958] or to the Rayleigh-Taylor instability [Chandrasekhar, 1981]. However, the fact that they do not form in the experiments conducted with dextran is probably due to the fact that there, the bulk flow was typically five times slower, thereby giving diffusion enough time to cancel any radial differences in concentration.

Part IV Theory

Chapter 4 Modelling osmotically driven flows In this chapter, I will present a model to describe the experimental findings discussed in the previous chapter. First, we will consider the thermodynamics of osmosis and then derive the equations of motion for osmotically driven flows in narrow, cylindrical tubes. The equations of motion have been derived several times before (e.g. in [Thompson and Holbrook, 2003a]), although they have not been solved and analysed in an as general manner as we will do. At the end of the chapter we will discuss the generalisation of the equations of motion to non-cylindrical geometries. Also, we will discuss some corrections to the equations which will prove useful when comparing the experimental data to theory.

4.1

The thermodynamics of osmosis

Osmosis is the tendency of water to move across a semipermeable membrane from a region of high chemical potential to an area of low chemical potential, up a solute concentration gradient [Schultz, 1980]. To demonstrate this consider first two compartments separated by a rigid barrier. Let compartment 1 contain n1 moles of gas in volume V1 and compartment 2 n2 moles of the same gas in volume V2 . The ideal gas law then states, that p1 V1 = RT n1 p2 V2 = RT n2 . (4.1) In terms of concentration this can be expressed as p1 = RT c1

p2 = RT c2

(4.2)

66

Modelling osmotically driven flows

Therefore, the pressure difference felt by pressure transducers inserted into the two compartments will be p = p1 − p2 = RT (c1 − c2 ) = RT c.

(4.3)

If the barrier is porous, gas will diffuse from the compartment with the higher concentration (pressure) to that with the lower concentration (pressure), until p = c = 0. Semipermable membrane

Piston Capillary tube

Compartment 1

Compartment 2

ci,1

ci,2

Figure 4.1: Adapted from [Schultz, 1980].

4.1.1

Derivation of the van’t Hoff equation

Consider now the system illustrated in figure (4.1), where compartment 1 contains an aqueous solution of a solute i at a concentration ci,1 and compartment 2 contains a solution of the same solute at a higher concentration ci,2 . The membrane separating the two compartments is assumed to be impermeable to the solute, but freely permeable to water. To the piston in compartment 2, we now apply a pressure sufficient to prevent any change in volume in either compartment – or in other words – to keep the system in equilibrium. This could be done by fitting a capillary tube to compartment 1, as shown in the figure. Since a change in volume can only come about as a result of water flow, the pressure applied must be sufficient to cancel the driving force for water flow. The driving force for water flow is the difference in chemical potential of the water across the membrane, ∆µw = µw,2 − µw,1, where µw is the chemical potential of water. The chemical potential for water can be written as [Atkins, 1978, Schultz, 1980] µw = (µw )0 + v¯w p + RT log(xw )

(4.4)

4.1 The thermodynamics of osmosis

67

where (µw )0 is a reference value, v¯w is the partial molar volume of water (≃ 18 cm3 /mole), P is pressure, R and T is the gass constant and temperature respectively. xw is the mole fraction of water defined by nw (4.5) xw = nw + ni where nw is the number of water molecules and nw + ni is the total number of molecules in the solution. The difference in chemical potential between compartment 1 and 2 is then ∆µw = v¯w (p2 − p1 ) + RT (log xw,2 − log xw,1 ) (4.6) ni,1 ni,2 = v¯w (p2 − p1 ) − RT log 1 + − log 1 + . (4.7) nw,1 nw,2 In dilute solutions nw ≫ ni , so the logarithm log(1 + x) ≃ x for small x. Assuming equlibrium, i.e. that ∆µ = 0, we get that ni,2 ni,1 v¯w (p2 − p1 ) = RT − . (4.8) nw,2 nw,1 Since the volume of the compartments is Vj ≃ v¯w nw,j , j = 1, 2, we finally obtain the results p2 − p1 = RT (ci,2 − ci,1 ) = RT c ≡ Π,

(4.9)

which is the the van’t Hoff equation for the osmotic pressure, Π, as a function of concentration [Atkins, 1978]. The result is completely analogous to equation (4.3) which was derived for perfect gasses. In this case, the pressure has to be applied to the compartment with the more concentrated solution which is the compartment with the lowest concentration of water. It is therefore useful to think of osmosis as an excluded-volume effect, where the presence of solute molecules effectively lowers the local concentration of water. It is important to emphasize that the system we have discussed above is in equilibrium, and that the pressure Π is the pressure that must be applied to compartment 2 to precisely offset the difference in chemical potential across the membrane. If the piston is released, such that p1 = p2 , water will flow at an initial rate of J = Lp RT c, (4.10) where Lp is the hydraulic conductivity of the membrane. Under more general condition, the flow rate will be given by J = Lp (RT c − p),

(4.11)

where c is the concentration difference and p is the hydrostatic pressure difference across the membrane.

68


Corrections to the van’t Hoff equation In general the van’t Hoff equation for Π(c) given in equation (4.9) only holds for small, ideal, molecules such as sucrose [Michel, 1972]. For larger molecules, such as dextran, the proportionality between Π and c still holds for dilute solutions, but the coefficient of proportionality has to be derived from empirical relations such as the one found in the previous chapter: Π = (0.10 ± 0.01) bar mM−1 c,

(4.12)

for dextran. For simplicity, we will assume that the relation Π = RT c holds when deriving the equations of motion, but when comparing the experimental data to theory we will use the empirically found expression for Π.

4.2 Derivation of the flow equations

69

Water reservoir Semipermeable membrane

S

Sugar Water flux r

A

xi-1

Δx

xi

xi+1

Figure 4.2: Sketch of the tube

4.2

Derivation of the flow equations

We will now derive the equations of motion for osmotically driven flows in cylindrical tubes, with the geometry of setup I in mind. Later, in section 4.4, we shall see that under certain conditions the equations are also valid in other geometries, such as the triangular geometry of setup II. We consider a tube of length L and radius r, as shown in figure (4.2). The tube has a constant cross section of area A = πr 2 and perimeter S = 2πr and its walls are made of a semipermeable membrane with permeability Lp . Inside the tube is a solution of sugar and water of concentration, c. The tube is surrounded by a water reservoir, modelling the water surrounding the membrane tube in setup I. We shall assume that L ≫ r and that the radial component of the flow velocity inside the tube is much smaller than the axial component, as is indeed the case in setup I. With these assumptions, we will model the flow in the spirit of lubrication theory and consider only a single, axial velocity component u(x, t) [Lautrup, 2004]. Also, we will assume that the concentration, c, is independent of the radial position, ρ. To justify the assumption on c, we must show that no significant radial variations in concentration occurs. ∂c where D is the A characteristic radial flux due to molecular diffusion is D ∂ρ ∂c diffusion constant, and ∂ρ is a characteristic radial concentration gradient. Similarly, a characteristic radial flux due to advection is Jc, where J is the flux across the membrane due to osmosis and c is the concentration. The condition for a homogeneous distribution of sugar in the radial direction is then that Lp RT cr Jc ≪ 1. (4.13) φ = ∂c ≃ D D ∂ρ Inserting numbers from table 1.1, we see that φ ≃ 0.2 in plants. Hence, radial

70


diffusion disperses the sugar homogeneously over a cross-section, justifying the assumption of a 1D sugar concentration c(x, t). However, for setup I, φ ≃ 6 and for setup II, φ ≃ 20. Consequently, our laboratory experiment does not fulfil the condition φ ≪ 1. This said, it can be argued that the axial advection facilitates radial mixing, mainly through Taylor dispersion [Taylor, 1953], and that error in considering our experiments as essentially 1D is therefore acceptable. The concentration boundary layers which form near the membrane due to the influx of water will be discussed in section 4.5.1.

4.2.1

Volume conservation

Let us now consider the equation of volume conservation by looking at a small section of tube between xi−1 and xi . The volume flux into the section due to advection is A(ui−1 − ui), (4.14) where the axial flow velocities are taken to be ui−1 and ui at xi−1 and xi , respectively. The volume flux inwards across the membrane due to osmosis (cf. equation 4.11) is JS∆x = S∆xLp (RT c − p).

(4.15)

Assuming conservation of volume, we get that A(ui−1 − ui) + S∆xLp (RT c − p) = 0

(4.16)

Letting ∆x → 0 this becomes A ∂u = Lp (RT c − p). S ∂x

(4.17)

The cross-section area to perimeter ratio reduces to 2r , so we finally get r ∂u = Lp (RT c − p) 2 ∂x

(4.18)

Elimination of p – The Stokes approximation To eliminate p from equation (4.18) one can choose several approaches. Thompson [Thompson and Holbrook, 2003a] eliminates pressure by allowing the tube to have a varying radius r = r0 exp(kp),

(4.19)

4.2 Derivation of the flow equations

71

where r0 is th equilibrium radius of the tube, and k is a constant that depends on the elastic properties of the tube. They show, however, that allowing the radius to vary in this has very little effect on their numerical results. Also, as shown in section 2.6.2 the variation in tube radius in our experiments is very small, of the order of a few percent. Therefore, we will choose a different approach to eliminate p, based on the fact that the flow velocity inside the tube is very small. For these very slow flows the Stokes approximation is a good approximation [Lautrup, 2004]. In this way, we get r 2 ∂p u=− , (4.20) 8η ∂x where η is the viscosity of the solution, typically ∼ 1.5 × 10−3 Pa s in our experiments. Differentiating equation (4.18) with respect to x and inserting the result from equation (4.20) we get for the conservation of water that RT

4.2.2

∂c r ∂ 2 u 8η = − 2 u. ∂x 2Lp ∂x2 r

(4.21)

Mass conservation

Let us consider the equation of sugar conservation. Again, there are two contributions to the equation of sugar conservation; advection and diffusion. Advection The number of sugar particles advected into the control volume pr. second from the left is Aui−1 ci−1 , (4.22) and out from the right Aui ci .

(4.23)

To get the change in mol/m3 pr. second we have to divide the sum of these by the control volume size, A∆x, A

ui−1ci−1 − uici A∆x

(4.24)

Letting ∆x → 0 the advective contribution to the equation of sugar conservation becomes ∂cA ∂uc =− . (4.25) ∂t ∂x

72


Diffusion Fick’s first law states, that the current of particles due to differences in concentration is proportional to the concentration gradient JD = D

∂c ∂x

(4.26)

where D is the diffusion constant with units m2 /s [Cotterill, 2002]. The number of particles that diffuses into the control volume pr. second from the left is then ∂c(xi−1 ) AJD (xi−1 ) = AD (4.27) ∂x and out from the right −AJD (xi ) = −AD

∂c(xi ) ∂x

(4.28)

Again, to get the change in mol/m3 pr. second due to diffusion we have to divide the sum of these by the control volume size A

i) D ∂c(x∂xi−1 ) − D ∂c(x ∂x A∆x

(4.29)

Letting ∆x → 0, we get ∂ ∂cD = ∂t ∂x

∂c ∂2c D = D 2. ∂x ∂x

(4.30)

Combining equation (4.25) and (4.30) we finally get the equation for solute conservation ∂2c ∂c ∂uc + = D 2. (4.31) ∂t ∂x ∂x

4.3

Non-dimensionalization of the flow equations

The final form of the flow equations are RT

r ∂ 2 u 8η ∂c = − 2 u. ∂x 2Lp ∂x2 r

(4.32)

∂c ∂uc ∂2c + = D 2. ∂t ∂x ∂x

(4.33)

4.3 Non-dimensionalization of the flow equations

73

To non-dimensionalize equations (4.32) and (4.33), we introduce the following scaling c = c0 C, u = u0 U, x = LX, t = t0 τ. L has been chosen such that the spatial domain is now the unit interval X ∈ [0, 1] and u0 = L/t0 . Choosing further t0 =

r , 2Lp RT c0

M=

16ηL2 Lp r3

Dr ¯ = D = , and D u0 L 2RT c0 L2 Lp

(4.34)

and inserting in (4.32-4.33), we get the non-dimensional flow equations. ∂2U ∂C − MU = , 2 ∂X ∂X

(4.35)

2 ∂C ∂UC ¯∂ C. + =D ∂τ ∂X ∂X 2

(4.36)

Going back to the original notation X → x,

U → u,

C → c,

τ →t

we finally obtain ∂2u ∂c − Mu = , 2 ∂x ∂x

(4.37)

2 ∂c ∂uc ¯ ∂ c. + =D ∂t ∂x ∂x2

(4.38)

This set of equations are equivalent to those of Thompson and Holbrook [Thompson and Holbrook, 2003a, Thompson and Holbrook, 2003b], except that we have eliminated pressure to get the term −Mu. The pressure is given by (cf. equation (4.18)) ∂c ∂ 2 u r ∂p = RT − (4.39) ∂x ∂x ∂x2 2Lp in dimensional variables. When scaling p by p0 = ∂p ∂c ∂ 2 u M = + ∂x ∂x ∂x2 in nondimensional variables.

8ηLu0 r2

this becomes (4.40)

74


4.3.1

¯ The parameters M and D

The parameter M corresponds to the ratio of axial to membrane flow resis¯ to the ratio of diffusive and advective solute flux. Thus, the tance and D longer the tube is the less important diffusion becomes and the more important the pressure gradient due to viscous effects become (cf. eq. (4.40)). ¯ in different situations can be seen in Values of the parameters M and D ¯ in plants are found table (4.1). The magnitude of the parameters M and D from the values given in table (1.1): r = 10 µm,

η = 1.5×10−3 Pas,

u0 = 2 mh−1 ,

Lp = 2×10−11 m(Pas)−1 .

¯ are small in both our experiments, and that for We observe, that M and D short distance transport in plants this is also the case. However, over lengthscales comparable to a small tree (L = 10 m) M is large, so in this case the pressure gradient is not negligible.

Setup I

M

¯ D

2 × 10−8

6 × 10−5

5 × 10−4

5 × 10−4

10−9

Setup II Single Sieve element (L = 1 mm) Leaf (L = 1 cm) Branch (L = 1 m) Small tree (L = 10 m)

5 × 10−2 5 × 102

5 × 104

2 × 10−2 5 × 10−5 5 × 10−7 5 × 10−8

¯ in various situations. Table 4.1: Values of the parameters M and D

4.3.2

Boundary and initial conditions

When solving equations (4.37) and (4.38) we generally have to choose two sets of boundary conditions on c and u and a set of initial conditions. The initial condition on c will prescribe some initial distribution of sugar inside the tube. Given this initial condition, and the boundary conditions on c and u, we wish to track the time evolution of the concentration and velocity fields. For equation (4.38) we will therefore choose an initial condition c(x, t = 0) = c0 (x), and some boundary condition on c and it’s derivatives.

(4.41)

4.3 Non-dimensionalization of the flow equations

75

In the case of a closed tube we would require zero diffusive flux at x = 0 and x = 1, or in other words ∂c(1, t) ∂c(0, t) = = 0. ∂x ∂x

(4.42)

Initial conditions on u will come from solving equation (4.37) itself ∂2u ∂c(x, 0) − Mu = 2 ∂x ∂x

(4.43)

while applying boundary condition on u. For the closed tube, these are u(0, t) = u(1, t) = 0.

4.3.3

(4.44)

Conservation laws

= ∂c(1) = 0 we If the pipe is closed, that is if u(0) = u(1) = 0 and ∂c(0) ∂x ∂x expect that the total amount of solute in the pipe will remain constant. This is equvivalent to Z 1 ∂ cr 2 dx = 0. (4.45) ∂t 0 To see that this is indeed the case, we multiply eq. (4.38) by r 2 and integrate from 0 to L 1 Z 1 1 ∂ 2 2 ∂c − ucr 2 0 = 0. (4.46) cr dx = Dr ∂t 0 ∂x 0

76

4.4


Generalization of the equations of motion to non-cylindrical geometries

When deriving equations (4.37-4.38) we have assumed that our system consisted of a cylindrical tube with semipermeable walls. The assumption of a cylindrical tube, however, have only been used in equation (4.18) where we assumed that the cross-section area to perimeter ratio was r A = S 2

(4.47)

and in equation (4.20) where we assumed the axial resistance in Stokes flow to be inversely proportional to the cross-section area 8 2π = 2 A r

(4.48)

¯ and These two factors are of purely geometrical nature and appear in M, D t0 as 2 8 16 M ∝ · 2 = 3, (4.49) r r r ¯∝r (4.50) D 2 and r t0 ∝ . (4.51) 2 Therefore, as long as the assumption of a 1D flow velocity and concentration holds inside the tube the equations of motion can be extended to include other geometries, e.g. triangular tubes as used in setup II, by replacing the ¯ and t0 as discussed above. geometric factors in M, D Finding the cross-section area to perimeter ratio is trivial, and the expression for the axial resistance in Stokes flow generally has the form u=

1 A ∂p η α ∂x

(4.52) 2

. For where Aα is a purely geometric factor, which for a cylindrical tube is πr 8π various pipe cross-sections Mortensen, Okkels and Bruus [Mortensen et al., 2005] has found α as a function of the dimensionless compactness C=

S2 . A

(4.53)

4.4 Generalization of the equations of motion to non-cylindrical geometries 77 Setup II To extend the equations of motion to setup II, consider the following. √ For a isosceles right triangle with two sides of length s and one of length 2s, we get that √ √ 2(2s + 2s)2 C= = 12 + 8 2 (4.54) s2 Mortensen et. al. showed that for pipes with triangular cross-sections √ 40 3 25 (4.55) α= C+ 17 17 so in our case

√ √ 300 200 2 40 3 + + ≃ 38.36. α= 17 17 17

(4.56)

Also,

A s √ . = S 4+2 2 ¯ and t0 we get Pluggin into the expressions for M, D √ ηL2 Lp 2)ηL2 Lp 38.36(8 + 4 √ = 153.44 3 , M II = s (2 + 2)s3 √ Ds (2 + 2)Ds II ¯ √ = D = 2RT c0 L2 Lp (4 + 2 2)2RT c0 L2 Lp and

(4.57)

(4.58)

(4.59)

√ s (2 + 2)s √ = = . (4.60) 2Lp RT c0 (4 + 2 2)Lp RT c0 √ The extra factors of 2 + 2 comes from the fact that the membrane only p covers one wall of length a, thereby scaling Lp down to 2+L√ . 2 tII 0

78

4.5 4.5.1


Corrections to the equations of motion The Unstirred Layer

When deriving the equations of motion for osmotically driven flows, we have assumed that the concentration inside the tube was a function of x and t only. However, the “real” concentration inside the tube will also depend on the radial position ρ in the form of a concentration boundary layer near the membrane, in the literature called an “Unstirred Layer” [Pedley, 1983]. Taking this into account, we will see that the concentration at the membrane cm = c(ρ = 0, x, t) will be lowered slightly compared to the bulk value cb , as shown in figure (4.3). This, in turn, results in a lower influx of water, ultimately causing the axial flow inside the tube to be slower than expected. To see this, let us look a little closer at the situation close to the membrane. As discussed in section 4.1, the flux across the membrane wall is J = Lp (RT cm − p),

(4.61)

where cm is the concentration at the boundary of the membrane on the inner side of the tube. If the concentration of sugar inside the tube is high enough we have that RT cm ≫ p, and we can ignore the hydrostatic pressure. In that case the flux across the membrane in steady state is given by Js = Lp RT cm ,

(4.62)

Close to the membrane, the concentration cm is lowered compared to the bulk value, cb , because sugar is advected away from the membrane by the influx c cb cm USL

J

0

δ

r

ρ

Membrane

Figure 4.3: The flux of water J across the membrane creates an unstirred layer (USL) of thickness δ in the immediate vicinity of the membrane, thereby lowering the concentration at the membrane to cm compared to the bulk value cb .

4.5 Corrections to the equations of motion

79

of water. The flux of sugar molecules away from the membrane is given by Js c,

(4.63)

where Js is the steady state flux and c = c(ρ) is the concentration. This tendency to dilute the solution close to the membrane is then in turn countered by diffusion which gives a flux ∂c (4.64) D ∂ρ in the opposite direction. In steady-state these two must balance, giving the relation ∂c Js = c (4.65) ∂ρ D which has the solution ′

c(ρ) = c exp

To find c′ we use the boundary conditions c(0) = cm

Js ρ . D

and c(δ) = cb ,

(4.66)

(4.67)

where δ is the thickness of the unstirred layer. Using equations (4.66) and (4.67) we get, that Js (ρ − δ) , (4.68) c = cb exp D so −Lp RT δ cm = cb exp cm . (4.69) D Now, we define γ as the ratio of the concentration at the membrane and the bulk concentration cm . (4.70) γ= cb Further, define β=

Lp RT cb δ D

(4.71)

Pluggin in, we finally obtain γ = exp(−βγ).

(4.72)

The solutions of this transcendental equation is shown in figure (4.4) and we notice that if β ≤ 0.01, then γ ≃ 1, and the unstirred layer has no effect on the bulk flow.

80


γ

1

0.5

0 −4 10

−2

0

10

10 β

2

10

4

10

Figure 4.4: Semilogarithmic plot of γ vs. β from equation (4.72). If β ≤ 0.01, γ ≃ 1, and the unstirred layer has no effect on the bulk flow.

In our experiments done with setup I, Lp , R, T , cb and D are known quantities, so to determine γ, we need to evaluate δ. Determination of δ for various gemoetries has been the focus of much work (see eg. [Pedley, 1983] for a comprehensive review). Pedley [Pedley, 1980a, Pedley, 1980b] found, that in the case of a steady Couette flow above a plane membrane (see figure (4.5)) δ is given by 1/3 DL (4.73) δ = 1.22 · α where L is the length of the membrane, and α is the velocity gradient at the membrane. Applying this expression to setup I, we get an unstirred layer thickness of δ = 2.2 mm (4.74) ρ u=αρ

USL

δ

x membrane

L

Figure 4.5: Couette flow above a membrane of length L. The water influx creates an unstirred layer of thickness δ. Downstream this thickness increases, lowering the water influx.

4.5 Corrections to the equations of motion

81

which is about half the tube radius and gives a value of β ≃ 1 and γ ≃ 0.5. For setup I, however, this value for δ, is unrealistically large. By visual inspection of the tube (see eg. figure (3.1)) one clearly sees that the unstirred layer thickness is not nearly as large as this. The reason for the discrepancy between the thickness calculated from equation (4.73) and the observed is, that this results was derived assuming that the osmotic flow across the membrane tube was negligible compared to the externally forced Couette flow running parallel to the membrane. In our experiment this is not the case, since the osmotic flow itself drives the flow parallel to the membrane. A more conservative estimate on δ is that it is a few percent of r, say δ = 0.1 mm. This gives a value of β ≃ 0.15 and γ ≃ 0.90, so in this case the unstirred layer lowers cm by 10 %. Because of the presence of the unstirred layer we thus expect that the flow inside the tube will be slightly slower than predicted by our 1D theory.

82


Chapter 5 Analytical solutions of flow equations In this chapter I will present analytical solutions to the equations of motion for osmotically driven flows in narrow tubes, derived in the previous chapter. ¯ ≃ 0 domain I will focus most of my Since the experiments are all in the M, D attention on this case, which can be solved analytically. In the next chapter, ¯ we will solve the equations numerically for non-zero values of M and D.

5.1

Solution methods

In the previous chapter we have seen, that the equations of motion for osmotically driven pipe flows are ∂2u ∂c − Mu = , 2 ∂x ∂x

(5.1)

2 ∂c ∂uc ¯ ∂ c. + =D ∂t ∂x ∂x2

(5.2)

and

This is a set of coupled partial differential equations which together with appropriate boundary and initial conditions gives the flow velocity, u, and the solute concentration, c, as a function of x and t. With currently available mathematical tools, it is not possible to obtain a general analytical solution ¯ = 0 we can solve the to these equations. However, in the case M = D equations analytically using Riemann’s method of characteristics. For an introduction to this method, please consult appendix E

84

5.2

Analytical solutions of flow equations

¯ =0 Solution for M = D

¯ = 0 the equations become In the limit M = 0 = D ∂2u ∂c = , 2 ∂x ∂x

(5.3)

∂c ∂uc + = 0. (5.4) ∂t ∂x As boundary conditions we will consider two different cases; one which corresponds to the experiments with a closed tube and another which corresponds to the case where the tube is open at one end. Although we have not examined the latter experimentally, we will include the analysis for completeness.

5.2.1

Case 1 : Boundary and initial conditions for the closed tube

As initial conditions for c, we will choose a Heaviside step function ( cI for 0 ≤ x ≤ λ, c(x, t = 0) = cI H(λ − x) 0 for λ < x ≤ 1,

(5.5)

where cI is some dimensionless concentration scale. The initial condition is shown in figure (5.1) As boundary conditions on c, we will choose ∂c(0, t) ∂c(1, t) = = 0. ∂x ∂x

(5.6)

u(0, t) = u(1, t) = 0.

(5.7)

And on u: To find the initial conditions on u we start by integrating equation (5.3) once with respect to x ∂u = c + F (t). (5.8) ∂x To find F (t) we integrate once more with respect to x u(1, t) − u(0, t) =

Z

1

c dx + F (t)

0

Z

1

dx.

(5.9)

0

Using the boundary condition on u then leads to F (t) = −¯ c(t),

(5.10)

¯ =0 5.2 Solution for M = D

85

c

cI

0 0

1

λ

x Figure 5.1: Initial condition for c. R1

c dx

where we have introduced the mean concentration c¯(t) = 0 1 . To see that c¯(t) is a constant in time, we notice that since u vanishes at the boundaries and the only transport of concentration occurs by advection. So, since the membrane is impermeable to sugar the mean concentration must be conserved. Thus ∂u c= + c¯. (5.11) ∂x This results enables us to find the initial condition on the velocity Z x Z x ′ ′ u(x, 0) = (c(x , 0) − c¯) dx = (c(x′ , 0) − λcI ) dx′ (5.12) 0 0 ( (cI − c¯)x for 0 ≤ x ≤ λ, (5.13) = c¯(1 − x) for λ < x ≤ 1, The initial velocity is piecewise linear in x, and has a maximum value umax = λcI (1 − λ) at x = λ,

(5.14)

as shown in figure 5.2.

5.2.2

Solution

Using (5.11) in equation (5.4) gives ∂ ∂u ∂u ∂ u + c¯ + + c¯ = 0, ∂t ∂x ∂x ∂x

(5.15)

86


u

cI λ(1-λ)

0 0

1

λ

x Figure 5.2: Initial condition for u.

or, since

∂¯ c ∂t

= 0, ∂ ∂x

∂u +u ∂t

∂u + c¯ = 0. ∂x

(5.16)

Integrating, and using the boundary conditions on u, this becomes ∂u ∂u +u = −¯ cu. ∂t ∂x

(5.17)

Equation (5.17) is a damped Burgers equation [Gurbatov et al., 1991], which can be solve using Riemann’s method of characteristics. The characteristic equations are du = −¯ cu dt ∂x = u. ∂t

(5.18) (5.19)

Equation (5.18) has the solution u = u0 (ξ) exp(−¯ ct),

(5.20)

where the parametrization ξ(x, t) of the initial velocity has to be found from 1 x = ξ + u0 (ξ) (1 − exp(−¯ ct)) c¯

(5.21)


87

where ξ = x at t = 0. From equation (5.13), we have that ( (cI − c¯)ξ for 0 ≤ ξ ≤ λ, u0 (ξ) = c¯(1 − ξ) for λ < ξ ≤ 1. Then, solving for ξ(x, t) in (5.21) gives ( x ξ(x, t) =

1 1+ λ (1−λ)(1−exp(−¯ ct) x−1+exp(−¯ ct exp(−¯ ct)

for x ∈ I1

for x ∈ I2

(5.22)

(5.23)

where the intervals I1 and I2 are defined by I1 = [0, 1 − (1 − λ) exp(−¯ ct)],

(5.24)

I2 = [1 − (1 − λ) exp(−¯ ct), 1].

(5.25)

Finally, u(x, t) is calculated from equation (5.20) ( (c −¯c) exp(−¯ct) I , for x ∈ I1 1 u(x, t) = λ (1−λ)(1−exp(−¯ct)) c¯(1 − x), for x ∈ I2

(5.26)

We can now calculate the instantaneous sugar front position xf and velocity uf using the right boundary of I1 from equation (5.24) xf (t) = 1 − (1 − λ) exp(−¯ ct),

dxf = c¯(1 − λ) exp(−¯ ct). dt From equations (5.11) and (5.26) we can also find c(x, t) uf (t) =

c(x, t) =

c¯ H(xf − x), 1 − (1 − λ) exp(−¯ ct)

(5.27) (5.28)

(5.29)

These results show, that the front velocity decays exponentially and that the front position approaches 1 as t → ∞. Qualitatively this agrees well with what we observed in setup I. Also, we note that the concentration inside the tube is conserved in time. The results can be seen in figure 5.3, where the solutions (5.26),(5.29) and (5.27) have been plotted.

88


t=0

cI

(A) c

t=0.2 t=0.4 t=0.6

0

0

t=0.8 t=1

λ

1 x

cI λ(1-λ)

(B)

t=0 t=0.2

u

t=0.4 t=0.6 t=0.8 t=1

0

0

λ

1 x

1

xf

(C)

λ 0

0

1

2

3

t

Figure 5.3: Plots of the analytical solutions for concentration (A), velocity (B) and front position (C). The times indicated are in units of 1c¯ = cI1λ .


5.2.3

89

Case 2: Boundary and initial conditions for the open tube

Once again, as initial condition for c, we will choose a Heaviside step function ( cI for 0 ≤ x ≤ λ, (5.30) c(x, t = 0) = cI H(λ − x) 0 for λ < x ≤ 1, As boundary condition on c we will choose c(1, t) = 0.

(5.31)

The choice of this boundary condition on c implies that the solution only works as long as no sugar has left the tube. As boundary conditions on u we will choose ∂u(1, t) = 0. (5.32) u(0, t) = ∂x To find the initial conditions on u we start by integrating equation (5.3) once with respect to x ∂u = c + F (t). (5.33) ∂x From the boundary condition at x = 1 it is clear, that F (t) = 0, or that ∂u =c ∂x

(5.34)

Therefore, the choice of boundary condition is effectively the same as choosing p = 0 in equation (4.18). The initial condition on u is now particularly simple Z x u(x, 0) = c(x′ , 0) dx′ (5.35) 0 ( cI x for 0 ≤ x ≤ λ, (5.36) = cI λ for λ < x ≤ 1, The initial velocity is piecewise linear in x below x = λ, and has a constant value of cI λ above x = λ.

5.2.4

Solution

Inserting equation (5.34) in (5.4) gives ∂ ∂u ∂u = 0. +u ∂x ∂t ∂x

(5.37)

90


Integrating, and using the boundary condition on u at x = 0 we get ∂u ∂u +u = 0. ∂t ∂x

(5.38)

In this case, the characteristic equations are du = 0 dt ∂x = u, ∂t

(5.39) (5.40)

with solutions u(ξ, t) = u0 (ξ), x(ξ, t) = ξ + u0 (ξ)t.

(5.41) (5.42)

Using the initial condition on u we get the following expression for ξ(x, t) ( x for x ∈ J1 1+cI t (5.43) ξ(x, t) = x − cI λt for x ∈ J2 J1 = [0, λ(1 + cI t)],

(5.44)

J2 = [λ(1 + cI t), 1].

(5.45)

When inserting in equation (5.41), we get that ( cI x for 0 ≤ x ≤ λ(1 + cI t) 1+cI t u(x, t) = cI λ for λ(1 + cI t) ≤ x ≤ 1

(5.46)

so, the front positon xf is given by the right boundary of J1 xf = λ + λcI t

(5.47)

and the front velocity

dxf = cI λ. dt So, in this case, the front velocity is constant. uf =

(5.48)


5.2.5

91

Summary

A summary of the analytical solutions in both non-dimensional and dimensional variables are given in table (5.1).

Closed tube, non-dim Closed tube, dim Open tube, non-dim Open tube, dim

Front position, xf

Front velocity, uf

1 − (1 − λ) exp(−¯ ct)

c¯(1 − λ) exp(−¯ ct)

L − (L − l) exp(−t/t0 )

1/t0 (L − l) exp(−t/t0 )

l + (L/t0 )t

L/t0

λ + λcI t

λcI

Table 5.1: Analytical solutions of equation (5.1) and (5.2) in nonr dimensional(non-dim) and dimensional(dim) variables. t0 = 2Lp RT c0 , cf. page 73

.

Validity of the analytical solution The fact that the velocity of the sugar front goes to zero in the closed tube as t → ∞ implies that the movement after some time will go from being dominated by advection to being dominated by diffusion. To estimate this time, tc , consider the sketch in figure (5.4). As the sugar front moves, diffusion acts to smoothen the concentration gradient while convection translates the front. The thickness, d, of the front can be estimated from the fact, that these two processes are in local equilibrium D or since

∂c ∂x

≃

∂c = uf c ∂x

c d

d=

D uf

(5.49)

(5.50)

The solution will be valid, as long as the remaining distance for the front to travel is larger than d L − xf > d (5.51) Inserting xf from table (5.1) we get L − L + (L − l) exp(−t/t0 ) >

D , 1/t0 (L − l) exp(−t/t0 )

(5.52)

92


or

1 t < log t0 2

So, the critical time is given by

tc 1 = log t0 2

(L − l)2 t0 D

(L − l)2 t0 D

.

(5.53)

(5.54)

Inserting numbers from setup I, (L = 0.2 m, l = 0.05 m, t0 ≃ 5 × 104 s, D = 6.9 × 10−11 m2 s−1 ) we get tc ≃ 4. t0

(5.55)

Thus, the flow will only become dominated by diffusion once the front has reached 98% of L. c uf

x d

Figure 5.4: Sketch of sugar front with (dashed) and without diffusion (solid). Diffusion act to smoothen the front.

Chapter 6 Numerical solution of the flow equations ¯ we cannot solve the flow equations For non-zero values of M and D ∂c ∂2u − Mu = , 2 ∂x ∂x

(6.1)

2 ∂c ∂uc ¯ ∂ c, + =D ∂t ∂x ∂x2

(6.2)

and

analytically. Therefore, we will turn to numerical methods to characterize the behavior of the flow for different values of these parameters. The equations have been studied numerically before, but only in the steady-state case [Thompson and Holbrook, 2003a]. Thus, we will focus our attention on the transient case. In order to do this, we will recast the entire problem into one, integro-differential equation which can be easily solved numerically. The boundary conditions we will choose corresponds to the closed tube: u=0

at x = 0, x = 1

(6.3)

and ∂c ∂c = = 0, ∂x ∂x

at x = 0, x = 1

(6.4)

The initial conditions used in the numerical simulations will be discussed in section (6.3.1)

94

Numerical solution of the flow equations

0.25

G(x,ξ)

0.2

0.15

0.1

0.05

0 0

0.2

0.4

ξ

0.6

0.8

1

Figure 6.1: Green’s function G(x, ξ) given by equation (6.6) for a = 1 and x = 1/2 (solid), x = 1/4 (dashed) and x = 1/10 (bold).

6.1

Transformation of the flow equations

To recast equations (6.1) and (6.2) into a single integro-differential we begin by solving equation (6.1) using a Green’s function [Zwillinger, 1998]: u=

Z

0

1

G(x, ξ)

∂c dξ. ∂ξ

(6.5) ∂2 ∂x2

− M and the

( − sinh(a(1−x)) sinh aξ for ξ < x, a sinh a G(x, ξ) = ax − asinh sinh(a(1 − ξ)) for ξ > x, sinh a

(6.6)

The integral kernel, G(x, ξ), for the differential operator boundary conditions u(0, t) = u(1, t) = 0 is

Where a2 = M. G(x, ξ) satisfies the homogeneous version of (6.1) in both x and ξ (as long as they are different) and the boundary condition on u at 0 and 1. It is continuous when x → ξ, but the derivative is not dG(x, ξ) dG(x, ξ) − = 1. dx x=ξ+ dx x=ξ−

(6.7)

6.1 Transformation of the flow equations

95

By inserting in equation (6.5) we find u Z sinh(a(1 − x)) x u(x, t) = − sinh(aξ) a sinh a 0 Z sinh(ax) 1 sinh(a(1 − ξ)) − a sinh a x

∂c dξ ∂ξ t=t0 ∂c dξ ∂ξ

(6.8)

t=t0

This expression is an efficient way to calculate the velocity when the concentration is known for all x at some particular time t. To solve the complete equations, however, we need to combine this integral with the solution of (6.2). This can be done in the following way. Using partial integration in equation (6.5) we get Z 1 u= D(x, ξ)c(ξ) dξ

(6.9)

0

where ∂G(x, ξ) D(x, ξ) = − = ∂ξ

(

sinh(a(1−x)) cosh aξ sinh a sinh ax − sinh a cosh(a(1 −

for ξ < x, ξ)) for ξ > x.

(6.10)

We observe, that Z

0

1

D(x, ξ) dξ = G(1) − G(0) = 0.

(6.11)

This implies that we can replace c in equation (6.9) by c − c¯, where c¯ is the (constant) average value of c, and c − c¯ has zero mean since the tube is closed Z 1 u= D(x, ξ)(c − c¯) dξ. (6.12) 0

We now define

∂f = c − c¯ (6.13) ∂x Rx and choose f (0) = f (1) = 0 such that f (x) = 0 (c − c¯) dξ. Inserting in equation (6.2), we get Z 1 2 ∂ ∂f ∂ ∂ f ∂ ¯ c (6.14) D(x, ξ)(c − c¯) dξ = D + ∂x ∂t ∂x ∂x ∂x2 0 This implies that Z 1 2 ∂f ∂f ∂f ¯ ∂ f + F (t) + + c¯ dξ = D (6.15) D(x, ξ) ∂t ∂x ∂ξ ∂x2 0

96

Numerical solution of the flow equations 2

Letting x → 0 we observe that f (0) → 0, ∂∂xf2 → 0 and D(0, ξ) → 0 so we conclude, that F (t) = 0. Rearranging, we get that Z 1 Z 1 ∂f ∂2f ∂f ∂f ∂f ¯ + dξ = D 2 − c¯ dξ (6.16) D(x, ξ) D(x, ξ) ∂t ∂ξ ∂x ∂x ∂ξ 0 0 Using partial integration, this finally becomes Z 1 ∂f ∂D(x, ξ) ∂f + f (x) − f (ξ) dξ = ∂t ∂ξ ∂x 0 Z 1 ∂D(x, ξ) ∂2f ¯ D 2 − c¯ f (x) − f (ξ) dξ , ∂x ∂ξ 0

where

∂D(x, ξ) = ∂ξ

( −a sinh(a(1−x)) sinh aξ for ξ < x, sinh a ax sinh(a(1 − ξ)) for ξ > x. −a sinh sinh a

(6.17)

(6.18)

For small a this becomes

( a2 (1 − x)ξ, ∂D(x, ξ) → ∂ξ a2 (1 − ξ)x .

(6.19)

¯ →0 so in the limit M = a2 → 0, D

∂f ∂f +f = −¯ cf. ∂t ∂x which is equvivalent to equation (5.17).

6.1.1

(6.20)

Summary

To sum up, we have transformed the original flow equations (6.1)-(6.2), which were functions of velocity u(x, t) concentration c(x, t), into a single equation for the function f (x, t). Z 1 ∂f ∂f ∂2f ∂D(x, ξ) ¯ + f (x) − f (ξ) dξ + c¯ = D 2 (6.21) ∂t ∂ξ ∂x ∂x 0 where f satisfies

∂f = c − c¯, ∂x Z x f (x, t) = (c(ξ, t) − c¯) dξ,

(6.22) (6.23)

0

f (x, t) = 0, ∂ f (x, t) = 0, ∂x2

at x = 0, 1

(6.24)

at x = 0, 1.

(6.25)

2

6.2 Numerical implementation

6.2

97

Numerical implementation

Considering equation (6.21) from a numerics point of view we see it is a integro-differential equation with spatial derivatives of first and second order, spatial integrals of first order and one time derivative. Regarding the spatial derivatives, we make use of the standard second order scheme for the 1st and 2nd order derivatives [Press, 2001] fi−1 − fi+1 + O((∆x)2 ), (6.26) 2∆x fi−1 − 2fi + fi+1 + O((∆x)2 ). (6.27) fi′′ = 2∆x These schemes will be implemented using Matlab’s gradient and 4*del2 routines. For the spatial integration, we use the trapezoidal rule [Press, 2001] Z x0 +∆x ∆x f (x) dx = (f (x0 ) + f (x0 + ∆x)) + O((∆x)3 ). (6.28) 2 x0 fi′ =

This scheme will be implemented using the routine trapz in Matlab. For integrating equation (6.21) in time, we used Matlab’s build in function ode23t, which is based on an explicit Runge-Kutta algorithm. This solver is very well suited to moderately stiff problems such as ours.

6.2.1

Solution scheme

To ease the implementation of equation (6.21) we will introduce some notation. First we define Z 1 ∂D(x, ξ) f (ξ, t) dξ (6.29) P (x, t) = f (x, t) − ∂ξ 0 Equation (6.21) then becomes 2 ∂f ¯∂ f −P =D ∂t ∂x2

∂f . c¯ + ∂x

(6.30)

Thus, given an initial concentration distribution c0 we calculate f (x, 0) by Z x f (x, 0) = (c(ξ, 0) − c¯) dξ. (6.31) 0

2

and ∂∂xf2 using equations (6.26)-(6.28) At each time-step we evaluate P , ∂f ∂x for all x and propagate f in time by ode23t. An excerpt of the code for implementing this in Matlab is shown below. For the full solver code, please consult appendix C.

98


%P a r a meter s M=1e−5 D=1e−5 a=sqrt ( M ) % I n i t i a l c o n d i t i o n s on c c0=1−1./(1+exp(−(x−x0 ) / epsilon ) ) ; cbar=trapz ( x , c0 ) ; % I n i t i a l c o n d i t i o n s on f for i =2:Nx f0 ( i)=trapz ( x ( 1 : i ) , c0 ( 1 : i)−cbar ) ; end % Run s o l v e r [ t , f ] = ode23t ( @fderiv , tspan , f0 , [ ] , D , a , cbar , x , Nx ) ; % Functio n f d e r i v function df = fderiv ( t , f , D , a , cbar , x , Nx ) ; % calc P for i =2:Nx−2 P ( i)=f ( i)−(a/ sinh ( a ) ) ∗ ( trapz ( x ( 1 : i ) , sinh ( a∗(1−x ( i ) ) ) . . . . . ∗ f ( 1 : i ) . ∗ sinh ( a . ∗ x ( 1 : i ) ) ) +trapz ( x ( i +1: end ) , . . . sinh ( a .∗(1 −x ( i+1: end ) ) ) . ∗ f ( i +1: end ) . ∗ sinh ( a∗x ( i ) ) ) ) ; end %Ca lc d f df=D ∗4∗ del2 ( x , f)−P . ∗ ( cbar+gradient ( f , x ) ) ;

6.3 Numerical solutions

6.3

99

Numerical solutions

6.3.1

Initial conditions

As initial conditions on c for the numerical simulations we will choose a generic sigmoid step-function given by ! 1 (6.32) c(x) = cI 1 − 1 + exp − x−λ ε where cI is a dimensionless concentration scale and λ and ε controls the position and the steepness of the initial sugar front respectively. Examples are shown in figure (6.2)

1 0.8

c

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

x

Figure 6.2: Initial concentration from equation (6.32) with parameters cI = 1, λ = 0.2, ε = 5 × 10−3 (solid), ε = 2 × 10−2 (dashed), ε = 4 × 10−2 (dashed-dotted).

As standard initial conditions to be used in the numerical examples , we will choose cI = 1, λ = 0.2 and ε = 2 × 10−2 , (6.33) giving and average concentration of c¯ = 0.2

(6.34)

100

6.3.2


Solutions

Using the initial conditions discussed above, we have solved equation (6.17) ¯ In appendix B, a comprehensive for a wide range of values of M and D. catalogue of solutions can be found. Examples of the numerical solutions is shown in figure (6.3), where the time evolution of the sugar concentration (black) and velocity field (red) ¯ = 10−2 , 10−4 , 10−6 . inside tube is given for M = 10, 1, 10−8 and D Starting from a steep concentration front one generally observes that the front moves forward while it flattens, in good, qualitative, agreement with that observed in setup II. Also, we see that diffusion acts to smoothen the sugar front while it moves forward. This effect dominates the concentration ¯ while it is less prominent when D ¯ becomes smaller. flow for large values of D, As for M, one observes that the velocity is very damped for large M and that this effect diminishes as M decreases, corresponding to a lower axial flow resistance.

6.3.3

Comparison between the analytical and numerical solutions

¯ = A comparison between the full numerical solutions for M = 10−8 and D −6 ¯ 10 and the analytical solutions for M = D = 0 is shown in figure (6.4). The numerical solution are shown as solid lines and the analytical as dashed. For both c and u one observes, that there is very good agreement between the two. From the catalogue of numerical solutions in B, it can be seen that for ¯ ≤ 10−4 there is generally very little difference between M ≤ 10−1 and D ¯ = 0 and the full numerical the approximate analytical solution for M = D solution.

6.3 Numerical solutions

M=10

101

M=1

−8

M=10

Velocity

D=10

−6

D=10

−4

D=10

−2

Concentration

Figure 6.3: Plots showing the time evolution of the concentration (black curves) ¯ M decreases from left and velocity (red curves) for different values of M and D. ¯ to right and D decreases from top to bottom. The numerical solver was stopped once the concentration at x = 0.9 had reached 10% of the mean concentration, 1 and the time between each plot is 5¯ c (cf. eq (6.34) and (5.27)). See appendix B for a complete catalogue of solutions

102


1

t=0

Concentration

Velocity 0.2

M = 10−8 ¯ = 10−6 D

t=0

0.8

0.15 t=1 t=1

c

u

0.6

t=2

0.4

t=3

0.2

0.4

0.6 x

0.8

t=3 t=4 t=5

t=4t=5

0.05

0.2 0 0

t=2

0.1

1

0 0

0.2

0.4

0.6

0.8

1

x

Figure 6.4: Analytical (dashed) and numerical (solid) solutions for M = ¯ = 10−6 for the concentration (left) and the velocity (right). 108 , D

Part V Data analysis, Discussion and Conclusion

Chapter 7 Data Analysis and Discussion In chapter 3, I presented experimental results demonstrating osmotically driven flows in two different setups. Next, in chapter 4, I derived the equations of motion for these flows and analysed these. We will now consider how good qualitative and quantitative agreement there is between theory and experiment. First, we will compare the experimental data from setup I to the analytical solution derived in the limit ¯ = 0. Then, we will compare the data from setup II to the numerical M =D solutions and then summarize our results. Finally, we will discuss the applicability of our findings to sugar transport in plants.

7.1

Movement of the sugar front

Using the equations of motion derived in chapter 4, we found that in the ¯ =0 limit M = D xf = L − (L − l) exp(−t/t0 ), (7.1) for the movement of the sugar front in setup I. Here, L is the length of the membrane tube, l is the initial height of the sugar front and t0 =

r , 2Lp Π

(7.2)

is the decay time for the motion. The plot in figure (7.1, A) shows the relative front position L − xf , (7.3) L−l plotted against time for five different experiments with setup I. The numbers 1 − 5 indicates the osmotic pressure Π of the sugar solutions used, as

106

Data Analysis and Discussion

Experiment

1

texp 0 ttheory 0

[104 s]

M ¯ D c¯

[mM]

1.5

Π

[bar]

0.14 ±0.02

4

[10 s]

2

13.46 ±0.03 10.57 ±0.05 10

±2

9

±1

[10−8 ]

3

1.4

[10

3.7

±1

±0.7

−5

]

±0.1 ±0.3

6.5

±0.2

2.10 ±0.03 0.15 ±0.01

3

4

5

6.71 ±0.02 4.6

±0.7

5.40 ±0.01 3.6

±0.4

2.33 ±0.04 2.1

±0.2

3

±1

3

±1

1.4

±0.7

1.7 2.4

±0.1 ±0.2

0.31 ±0.03

1.3 4.2

±0.11 ±0.7

0.39 ±0.01

1.4 6.8

±0.1 ±0.1

0.68 ±0.02

Table 7.1: Data from five experiments conducted with setup I.

summarised in table (7.1), where 1 has the lowest and 5 the highest osmotic pressure. One clearly sees, that the relative front position approaches zero faster for high concentrations than for low. Also, we note that experiments 1 and 2 now fall in the right order, cf. the discussion on page 51. From the ¯ ∼ 10−5 , so it is reasonable to assume that table we see that M ∼ 10−8 and D ¯ = 0 is valid. we are in the domain where the analytical solution for M = D To test result from equation (7.1), the plot in figure (7.1, B) shows the logarithm of the relative front position plotted against time. For long stretches of time the curves are seen to approximately follow straight lines in good, qualitative agreement with theory. The red dashed lines are fits to equation (7.1), and we interpret the slopes as − t10 , the different values given in table 7.1 as texp 0 , together with the values predicted by equation (7.2), given as theory t0 . The theoretically and experimentally obtained values of t0 are in fairly generally being a little larger than good quantitative agreement, with texp 0 theory t0 implying that the observed motion of the sugar front is a little slower than expected. This, however, is not surprising, considering the number of simplifying assumptions made when deriving the theoretical result. In particular, the fact that we have assumed that there is no radial variation in neither velocity nor concentration would imply a slowing down of the flow through the formation of unstirred layers as discussed in section (4.5.1).

7.1 Movement of the sugar front

107

1

(A)

0.9 0.8 0.7

L − xf L− l

0.6 0.5 0.4 0.3 4

0.2

1

3 0.1 0

5 0

2

0.5

1

1.5

2

2.5

t [s]

5

x 10

0

(B)

L − xf L − l

¶

−0.5

−1

log

µ

−1.5 4 1

3

−2 5

−2.5

2 −3

0

0.5

1

1.5

t [s]

2

2.5 5

x 10

Figure 7.1: (A): Experimental data (black dots) and fits to equation (7.1) (red dashed lines) for the relative front position vs. time. (B): Semi-logarithmic version of (A)

108


7.2

The shape of the sugar front

The plot in figure (7.2, A) show the dynamics of the sugar front as observed with setup II with time increasing from black to light gray. We clearly see the front move from left to right while it gradually flattens. To test how well this moving flattening of the sugar front was reproduced by our model, we solved the equations of motion numerically starting with the initial ¯ = 0, The results are shown conditions from figure (7.2, A). For M = D as red curves in figure (7.2, B). While the front position are reproduced relatively well, the shape of the front is not, so diffusion must play a role. This can be seen in figure (7.2, C) which shows the result of a simulation with M = 10−9 , D = 6.9 × 10−11 m2 s−1 . Clearly, the model which includes diffusion reproduces the experimental data significantly better than the one which did not include diffusion. 15

(A)

10 5 0

0

2

4

6

8

10

12

15

(B)

10 5 0

0

2

4

6

8

10

12

c [mM]

15

(C)

10 5 0

0

2

4

6 x [cm]

8

10

12

Figure 7.2: (A): Experimental data from setup II showing the time evolution of the sugar front. ¯ = 0. (B): Results of solving the flow-equations numerically with M = D −9 −11 2 −1 (C): Numerical results for M = 10 , D = 6.9 × 10 m s .

7.2 The shape of the sugar front

109

To study the shape of the front in greater detail, consider the plots in figure (7.3, A-C). Here the gradient of the concentration curves from figure (7.2) are shown. In (A) we clearly see a peak moving from left to right while it gradually broadens and flattens. In (B) we also see the peak advancing, but the flattening and broadening is much less pronounced. In (C) we see that the model which includes diffusion reproduces the gradual broadening and flattening of the front very well. Finally, as can be seen in figure (7.4), the front position found from both (B) and (C) follow the experimental data closely.

15 10 5 0

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6 x [cm]

8

10

12

15 10 5

|dc/dx| [mM/cm]

0 15 10 5 0

Figure 7.3: (A): Experimental data from setup II showing the time evolution of the concentration gradient. ¯ = 0. (B): Results of solving the flow-equations numerically with M = D −9 −11 2 −1 (C): Numerical results for M = 10 , D = 6.9 × 10 m s .

110


6.6 6.4 6.2 6

5.6

f

x [cm]

5.8

5.4 5.2 5 4.8 4.6

0

10

20

30

40 t [h]

50

60

70

80

Figure 7.4: Front position as a function of time for the experimental data (black ¯ = 0 simulation (red stars) and the M = 10−9 , D = 6.9 × dots), the M = D −11 2 −1 10 m s simulation (blue crosses).

7.3 Summary

7.3

111

Summary

In the previous sections we have considered how well our model for osmotically driven flows were able to reproduce the experimentally observed data. Using the data for the motion of the front position from setup I, we saw good qualitative and quantitative agreement between the experiments and theory. Therefore, we conclude that our model is most likely correct, thereby confirming the results of Eschrich et. al. [Eschrich et al., 1972], although on a much firmer experimental ground. In our discussion of the shape of the sugar front, we also saw good agreement between theory and experiment.

7.4

Application of the results to plants

The source of inspiration for the experiments was the osmotically driven flow inside a phloem sieve element which is believed to be responsible for sugar transport in plants. We will now consider the applicability of our findings to ¯ for short and long distance plants. In table (7.2), relevant values of M and D transport of sugar in plants are shown. M

¯ D

2 × 10−8 6 × 10−5

Setup I

10−9

Setup II

2 × 10−2

Single Sieve element (L = 1 mm) 5 × 10−4 5 × 10−4 Leaf (L = 1 cm)

Branch (L = 1 m) Small tree (L = 10 m)

5 × 10−2 5 × 10−5 5 × 102 5 × 10−7 5 × 104 5 × 10−8

¯ in various situations. Table 7.2: Values of the parameters M and D

We see, that the transport of sugar over short distances (< 1 cm) fall ¯ = 0 is valid. Applying within the domain where the approximation of M = D our results to the flow inside a single sieve element (L=1 mm), we get a characteristic velocity (cf. page 73) of u0 =

L 2ΠLLp = = 7 mh−1 . t0 r

This is in the same order of magnitude as the observed velocities of 0.5 − 2 mh−1 (see table 1.1). The fact that we predict a larger velocity is not

112


surprising. First, as the length of the tube grows, so does M (as L2 ) thereby increasing the axial resistance significantly. Second, the transport of sugar in plants involves the steady state transport of sugar and not the movement of a sugar front from one end of the plant to the other. Further, the calculations takes no account of the resistance offered by the sieve plates which separate the sieve elements.

Chapter 8 Conclusion and outlook 8.1

Conclusion

In this thesis, I have presented the results of my experimental and theoretical work on osmotically driven flows. I have presented data from two different experimental setups, and have developed a model to describe these. I have shown good qualitative and quantitative agreement between theory and experiment, and have discussed the applicability of my results to sugar transport in plants. In my experimental work I have investigated osmotically driven flows using both well established and new techniques. My results show that a sugar front initially confined to one end of a semipermeable tube submerged in water, will move towards the other end with a velocity which decays exponentially as the front approaches the other end of the tube. The characteristic velocity of this motion have been shown to depend on the sugar concentration. Further, I have shown that the movement of the front is not due to diffusion, but rather to an osmotically driven bulk flow, although diffusion plays a role in determining the shape of the front. To explain the experimental result I have derived the equations of motion for osmotically driven flows in narrow tubes and have analysed these. The ¯ where M is equations depends on two non-dimensional quantities, M and D, ¯ the ratio of axial to membrane flow resistance and D is the ratio of diffusive to advective sugar flux. In both my experiments, as well as in some cases ¯ are small. In the case M = D ¯ = 0, I have solved the in plants, M and D equations of motion analytically. My results show, that a sugar front initially confined to one end of a tube moves with a exponentially decreasing velocity towards the far end of the tube with a decay time of t0 = 2Lrp Π . Here, r is the radius of the membrane tube, Lp is the permeability of the membrane and Π is the osmotic pressure of the sugar solution inside the tube. Comparing the experimental data and the theory, I have shown good

114

Conclusion and outlook

qualitative and quantitative agreement between the two, significantly better than in any previous experimental work. The interest in osmotically driven flows is motivated by the transport of sugar in plants, which is believed to be driven by this mechanism. I have discussed the applicability of my results to flows in “real” plants, and have argued that these may be used to describe short distance transport of sugar satisfactorily. Further, my results provide tools which can be used directly by biologist working in the field to investigate problems in plants. Finally, I have observed two very interesting phenomena: the buckling and mushroom cloud instabilities, both of which underlines the quintessence of science exploration: Looking answers we find new questions.

8.2

Outlook

Ultimately, the motivation for investigating osmotically driven flows comes from an interest in the processes behind sugar transport in plants. Therefore, since M is typically large in plants, experimental investigation of osmotically driven flows in this limit would be very interesting. To to this, one would need to use either a very long tube, a very permeable membrane, a very viscous fluid or a very narrow channel. The most practical approach would probably be to use a narrow channel, e.g. in the form of a micro channela . As an example, a 5 cm long, 100 µm wide channel gives a M of 4, so results obtained in this way would be applicable to medium range (