PAPER
www.rsc.org/loc | Lab on a Chip
Continuous flow separation of particles within an asymmetric microfluidic device Xunli Zhang,a Jon M. Cooper,b Paul B. Monaghanb and Stephen J. Haswell*a Received 27th October 2005, Accepted 2nd March 2006 First published as an Advance Article on the web 13th March 2006 DOI: 10.1039/b515272k A microfluidic based device has been developed for the continuous separation of polymer microspheres, taking advantage of the flow characteristics of systems. The chip consists of an asymmetric cavity with variable channel width which enables continuous amplification of the particle separation for different size particles within the laminar flow profile. The process has been examined by varying the sample inlet position, the sample to media flow rate ratio, and the total flow rate. This technique can be applied for manipulating both microscale biological and colloidal particles within microfluidic systems.
1. Introduction The development of miniaturized micro chemical systems based on the so-called ‘‘Lab-on-a-Chip’’ concept has witnessed an explosive growth over the last decade.1–3 Such micro systems represent the ability to ‘‘shrink’’ conventional bench chemical systems to a size of a few square centimetres with major advantages of speed, performance, integration, portability, sample/solvent quantity, automation, hazard control, and cost. These merits are important for a variety of applications in analytical chemistry, biochemistry, clinical diagnosis, medical chemistry and industrial chemistry.4,5 New research fields of microfluidics, microreaction technology, micromixing, microheat exchanging and microseparation have now spun off from conventional chemistry and chemical technology. As a result, numerous micro-total-analysis-systems (m-TAS) and micro reactor systems have been developed, and more are still under investigation.3 Sorting and separating of microparticles and biological cells has been an area of particular interest to a number of industrial and academic research groups as it represents an important step in many chemical and biological processes. A range of techniques have been reported for the separation of such particles which have included mechanical sieving, sedimentation, and so-called elution-based separation.6–9 The mechanical sieving and sedimentation methods are generally used to separate relatively large particles (.1 mm), whilst elution-based techniques allow separation of particles in a smaller size range. Examples of those techniques include hydrodynamic chromatography, size exclusion chromatography and electrophoretic techniques. In addition, field-flow fractionation (FFF) which is based on the introduction of an external force field has attracted some recent interest in the field of microfluidics. Since this method was pioneered by Giddings in 1960’s,7 it has been used to separate a range of a
Department of Chemistry, The University of Hull, Hull, United Kingdom HU6 7RX. E-mail:
[email protected]; Fax: +44 (0) 1482 466410; Tel: +44 (0) 1482 465469 b Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, United Kingdom G12 8QQ
This journal is ß The Royal Society of Chemistry 2006
samples including polymers, biological macromolecules, synthetic and environmental particles, and living cells.8 This technique has been proved being efficient and also flexible for different reagents by choosing appropriate external force fields, and new methods based on the same principle are still being developed.6 In recent years, alongside the continuing development of miniaturized total analytical systems (m-TAS) and micro reactor systems, effort has been made to separate micro-litre sample amounts of suspensions of particles and cells in Lab-on-a-Chip micro devices.9–15 Most of the studies involved the use of external force fields including dielectric (polarisability),9 acoustic,10 magnetic,11 thermal,12 optical13,14 and centrifugal15 forces. Clearly, the requirement of external forces increases the complexity of the device and may limit the application for some specific reagents such as biological samples. Consequently, researchers have been paying attention to the development of physical methods (without external forces) by making use of the laminar flow profile inside microchannels.16–19 In these systems, particles are carried by the laminar flow stream and the designed flow profile along the microchannel can be used to enhance the separation further. In this study, we develop a new format of microfluidic based device for the continuous separation of microspheres. The chip consists of an asymmetric cavity with a variable channel width which enables continuous amplification of the separation of particles in different sizes along the laminar flow profile. The main design and operation parameters of the process have been examined.
2. Experimental 2.1. Materials Water was purified by reverse osmosis and by passage through a Milli-Q Reagent Water System (Millipore, Watford, UK). Polystyrene latex microspheres with diameters of 10 mm and 25 mm were manufactured by Alfa Aesar, A Johnson Matthey Company (Ward Hill, MA, USA), and were supplied as 2.5 wt% dispersion in water. HISTOPAQUE1-1119, supplied by Aldrich (Poole, Dorset, UK), was a solution containing Lab Chip, 2006, 6, 561–566 | 561
polysucrose and sodium diatrizoate, adjusted to a density of 1.119 g cm23. This medium was generally used to separate a range of viable mononuclear cells from the plasma by centrifugation which facilitates the mononuclear cells to form a distinct layer at the plasma-HISTOPAQUE1 interface.20 In this study Histopaque1-1119 was used to adjust the fluid density to minimize the microparticles sedimentation during flowing. The fluid media was prepared by mixing water with Histopaque1-1119, which was calculated to produce a desired density of 1.05 g cm23 to match the polymer particle density. 2.2. Microdevice design and fabrication Fig. 1 shows the outline of the chip design. The channel network consists of an asymmetric cavity with 3 inlet and 3 outlet channels. The key parameters for the chip design are shown in the figure. The depth for all the channels was 68 mm. The microchip was fabricated according to published procedures21 with minor adaptations. The chip consisted of a polymer sheet with microchannels fabricated on the surface and a glass sheet to act as a cover plate. To fabricate the channels a silicon mould was first constructed onto which the polydimethylsiloxane (PDMS) was poured and cured. The silicon mould was fabricated using the inductively coupled plasma (ICP) STS2 process. A silicon wafer was first cleaned in acetone for 10 min with ultrasonication and then rinsed under a stream of de-ionised water for 5 min followed by drying under a stream of nitrogen. The wafer was then spin-coated with a layer of adhesion promoter (hexamethyldisilane, HMDS), followed by coating AZ4562 positive tone photoresist at 4000 rev min21 for 30 s resulting in about 6 mm thick polymer film. The wafer was baked at 90 uC for 120 s, and once cooled, the photoresist layer was patterned by a 25 s exposure to a photomask. The channel network patterns was thus transferred into he photoresist.
In all cases, the photomask, a negative photo film, was produced by drawing the channel designs using AutoCAD (Autodesk, Farnborough, Hampshire, UK) software and then transferring onto to a polyester film. After exposure, the wafer was developed in AZ400K developer solution (1 : 4 v/v in water) for approximately 2 min and rinsed in water. Once dried under a stream of nitrogen, the exposed silicon was etched under the following conditions: SF6, 130 sccm; C4F8, 85 sccm; O2, 10 sccm; APC, 74%; pressure, 33 mTorr; power, 600 W coil, 12 W platen; temperature, 20 uC, which gave an etch rate of 2.2 mm min21. After etching, the remaining photoresist was stripped in acetone, and then the silicon wafer was mounted onto to a glass support to improve its robustness. Finally, the silicon mould was silanized by placing the wafer in a petri dish with a few drops of HMDS overnight in order to aid the removal of the moulded PDMS sheet. In order to make a moulded PDMS sheet, PDMS (Sylgard 184 Silicone Elastomer, manufactured by Dow Corning, Midland, Michigan, USA) was mixed in a 1 : 10 v/v ratio of curing agent to base oligomers and poured over the silicon mould. This was left for 1 h to allow the sample to degas, and was then baked in an oven at 75 uC for 2 h. Once cooled the PDMS structure was peeled away from the mould. Fluid access holes were then punched through the PDMS sheet (thickness: 4 mm) with a modified gauge 21 syringe needle, resulting in interconnect holes with a diameter of approximately 500 mm. The chips were then cleaned in ethanol with ultrasonication for 5 min followed by a rinse in water and drying under a stream of nitrogen. The cross section of the micro channels produced using this method has been found to be nearly rectangular.21 To seal the microfluidic structure, a glass cover plate (thickness: 1.5 mm) was bonded to the PDMS sheet. Before bonding both the PDMS and glass were exposed in the ICP
Fig. 1 Micrograph of the chip design and the channel network which is fabricated on a PDMS sheet with 6 interconnect holes, and the PDMS sheet is bonded to a glass cover.
562 | Lab Chip, 2006, 6, 561–566
This journal is ß The Royal Society of Chemistry 2006
system for 10 s (power, 100 W coil, 5 W platen; 100% O2, 100 sccm; APC, 91. 5%; pressure, 80 mTorr). After exposure, the two surfaces were brought into contact and were allowed to rest to enable bonding to proceed, and the device was ready for use. 2.3. Instrumentation and image analysis An Axiovert S100 inverted microscope (Carl Zeiss, Welwyn Garden City, Hertfordshire, UK) using transmission optics coupled with a monochrome CCD digital camera (Hamamatsu C4742-95-12NRB) was used to obtain both conventional micrographs and digital videos of the microchip which was placed horizontally on the microscope stage above the objective lens. Thus, the look direction in all the images obtained from this microscope is from the bottom of the microfluidic chip. The microscopic video images were then analyzed using IrfanView software. By tracking a particle in successive video images a velocity vector was obtained. Two KDS 200 syringe pumps (KD Scientific Inc., Holliston, MA, USA) were used to deliver the media and the sample containing microparticles. PEEK polymer tubing and connectors (Upchurch Scientific Inc., Oak Harbor, WA, USA) were used for plumbing to link the chip and the syringes.
3. Results and discussion 3.1. Flow characterisation Whilst flow characteristics within channels with given channel widths under pressure driven flow has been intensively studied, the flow pattern and velocity profile within an asymmetric cavity still need to be characterised in order to demonstrate the separation principle and optimise the operation conditions. This was carried out by flowing media containing particles within the cavity at a given flow rate, and then tracking the flowing particles in successive video images taken at given time intervals (100 ms). Fig. 2 shows the polymer microspheres (dia. 10 mm) flowing in the asymmetric cavity when the particles were carried by media and introduced via the middle inlet, with no additional fluid flow from either the left or right inlet. The input flow rate
was maintained at 1.375 ml min21. It can be seen that the particles were dispersed evenly within the media, in which they were flowing. To visualise the flow pattern five particles (marked in Fig. 2 (a)) were tracked as they progressed within the cavity producing results illustrated by Fig. 2 (b). It should be noted that the co-ordinate shown in the figure (and thereafter) are in accordance with that in the original digital image where the origin is at the top-left corner. From Fig. 2 (b) it can be seen that, as expected, the flow pattern in general follows the cavity shape. Specifically, when the particles were flowing in the x direction, the distance between them was increased across the y direction. This observation is indicative that particles can be separated within the asymmetric cavity based on the nature of the laminar flow field, given the appropriate particle position at the cavity entrance. The unique feature of this separation is its continuous amplification along the cavity, which suggests that a high resolution of separation can be reached. Focused on the five particles chosen, the behaviour of the flow velocity within the cavity was also examined, and a flow velocity profile was obtained by locating each particle at a given time. Fig. 2 (c) shows that when the five particles are assumed to start flowing at the same time and position, and the time interval between the dashed lines is 100 ms. The main features of the flow velocity profile can be summarized as follows: (i) The flow velocity decreases along the cavity, which is reflected by the reducing distance between the dashed lines. This decrease in the linear flow velocity is mainly due to the increase in the cross section area as it becomes wider along the flow direction; (ii) A deformed symmetric parabolic flow velocity profile was created under pressure driven flow conditions, where the fluid/wall boundary velocity tends to be zero and the velocity maxima is obtained in the channel centre. This is generally in line with the observation obtained in a symmetric cross section channel.22 However, the two sidewalls in the asymmetric cavity are neither symmetric nor in parallel, and it is this irregularity that results in the parabolic flow velocity profile to be deformed;
Fig. 2 Polymer microspheres (dia. 10 mm) flowing, carried by the media within the asymmetric cavity (a) where five particles are chosen, marked with different symbols accordingly for obtaining the flow pattern (b), and the flow velocity profile (c) by tracking particles in successive video images. Two solid curves define the cavity area.
This journal is ß The Royal Society of Chemistry 2006
Lab Chip, 2006, 6, 561–566 | 563
Fig. 3 The principle of the particle separation.
(iii) The deformation of the parabolic flow profile was pronounced along the cavity, a fact that was likely to be related to the development state of the flow profile. Generally, when the fluid enters a channel or pipe it needs a certain time period to develop the flow field because of the ‘‘entrance effect’’23 mainly caused by the sudden change of the channel geometry. Once developed, a parabolic velocity profile is obtained in the case of a symmetric straight channel. As can be seen in the asymmetric cavity, the cavity geometry is continuous changing that the flow within the entire cavity is underdeveloped, or continuously developing. As a result, the parabolic feature becomes more significant with flowing towards the wider area. 3.2. Effects of particle sample inlet positions As discussed above, as the particles of different sizes contained in a stream of media were placed in appropriate positions in the flow field they can be separated continuously following the flow profile, thus the initial entry position of the particle is important for separation. Fig. 3 diagrammatically shows the principle of the separation at the entry section where a relatively narrow stream of media containing particle mixture is introduced along the left sidewall and the rest channel is occupied by media without particles. When the left side stream is narrow, all of the particles can be forced to flow along the
channel sidewall. In that case, particles in different sizes are passively placed in different distances between the spheres centre and the sidewall according to their diameters. This distance difference also forces the particles into different flowing tracks; the smaller being close to the sidewall and the larger, farther away, which consequently initialises the separation. While the laminar flow continues within the hornshape cavity the separation is further amplified. Fig. 4 (a) shows an example micrograph taken during the separation where the media containing particles was introduced via the left inlet at a flow rate of 1 ml min21 and the media without particles via the middle and right inlets at a total flow rate of 2.75 ml min21 in total. It can be seen that the particles were flowing within the laminar flow field in different tracks according to their sizes and were being separated continuously along the flow direction. It also shows that the flow pattern, flow velocity and separation effect are all agreed with that described and discussed previously for the expected observations. Yamada et al.17 has studied the particle separation based on a different principle within a chip with a suddenly enlarged channel linked to a much narrower neck, which suggested that the ‘‘pinched’’ entrance section is then key to the separation results. In contrast, in our study, it was not necessary for the entrance channel to be small as long as the fluid stream that contains the particle mixture was narrow enough to position the particles at a different distance from the sidewall. Based on the analysis and observation, the separation principle should also apply when the particle-containing media is introduced via the right inlet along the smaller curved sidewall. In this case, the flowing track which carries the smaller particles is expected to be close to the sidewall (Fig. 4 (b)). In contrast, when the particle-containing media enters the cavity via the middle inlet while the media without particles are flowing along both sides, this separation effect was lost (Fig. 4 (c)). To quantitatively compare the effect of the inlet positions of the particle sample, a relatively large number of particles were analysed statistically to obtain their position in y direction when they passed across the detection line, the dashed line
Fig. 4 Micrographs of the separation process when (a) the media with particles was introduced from the left inlet whilst the media without particles entered via both middle and right inlets, (b) the media with particles was introduced from the right inlet whilst the media without particles entered via both middle and left inlets, and (c) the media with particles was introduced from the middle inlet whilst the media without particles entered via both left and right inlets. The dashed line (x = 1277.3 mm) shows a reference for detection.
564 | Lab Chip, 2006, 6, 561–566
This journal is ß The Royal Society of Chemistry 2006
Table 1 Comparison of separation results for particles introduced via different inlets Left inlet a
b
Distance from upper sidewall /mm
Middle inlet c
d10
b
d25
Right inlet c
d10
d10b
d25
d25c
Mean/mm Range (¡)/mm
1227.7 1026.7 649.3 614.1 141.4 298.2 +169.9 +57.9 +106.5 +122.5 +113.0 +128.7 2159.0 2175.2 2235.2 2113.8 2110.6 2174.7 d 201.0 35.1 2156.7 Separation /mm a Particles distance from the small curved sidewall in y direction when passing across the reference line (x = 1277.3 mm) shown in Fig. 4. b Particles in a diameter of 10 mm. c Particles in a diameter of 25 mm. d Average distance between d10 and d25 particles when passing across the reference line.
Table 2 Comparison of separation results for different flow rate ratios (0.250 : 1–0.875 : 1) of media with particles to media without particles 0.250 : 1
0.364 : 1
0.500 : 1
0.667 : 1
0.875 : 1
Distance from upper sidewall/mm
d10
d25
d10
d25
d10
d25
d10
d25
d10
d25
Mean/mm Range (¡)/mm
1283.3 +136.7 2131.5 475.3
807.9 +62.8 290.5
1381.7 +169.9 2159.0 201.0
1180.8 +57.9 2175.2
1268.2 +98.3 261.4 123.1
1145.0 +33.0 224.5
1130.1 +338.5 2214.0 101.1
1029.0 +101.1 284.1
1097.3 +505.4 2254.6 83.0
1014.4 +125.3 279.0
Separation/mm
shown in Fig. 4. The results are summarized in Table 1, which shows that the best separation was obtained when the particle sample was introduced via the left inlet. An average distance of 201 mm between large and small particles was created when they passed across the reference line (x = 1277.3 mm). The right inlet introduction of the particle sample can also result in separation but to a much lesser extent. As discussed above, this was mainly due to the relatively smaller change of flow field compared to that near the big curved sidewall. No significant separation was found when the particle sample was delivered through the middle inlet. 3.3. Effects of varying flow rate ratio It has been demonstrated that a narrow stream of particlecontaining media is essential to initialise the separation by placing the particles in different flow tracks according their sizes. That can be created by changing the channel width of the asymmetric cavity entrance (which is impractical once the chip is fabricated). An operational way to reach this condition is to change the flow rate ratio, i.e., the flow rate of the media containing particles compared to the flow rate of the media without particles. This variation of flow rates in turn changes the effective width of the stream of particle-containing media. Table 2 summarizes the results obtained for a range of flow rate ratios between 0.250 : 1 and 0.875 : 1 when the particle sample was introduced via the left inlet and the media without particles was delivered via both middle and right inlets. The overall flow rate of the three streams was kept at a constant of 3.75 ml min21. The flow rate of the media containing particle sample was varied from 0.75 to 1.75 ml min21 while the other two inlets were controlled in a range of 3.0 to 2.0 ml min21 in total. The average separated distance between large and small particles was plotted as a function of flow rate ratio in Fig. 5. It can be seen that the separation was significantly affected by the flow rate ratio. In general, a better separation was obtained when the flow rate ratio was decreased. That effect was enhanced when the ratio was below 0.4 : 1, which was reflected This journal is ß The Royal Society of Chemistry 2006
on the dramatic increase in the average distance between the big particles and the smalls. By varying flow rate ratios, R, the width of the particle carrier stream, Wcarr, is altered according to the following equation. R Wchan (1) Wcarr ~ 1zR where Wchan is the entrance channel width of 87 mm. As can be seen from Fig. 5, there is a critical flow rate ratio of 0.4 : 1 which corresponds to the carrier stream width of 24.86 mm. This is in good agreement with the larger particle diameter of 25 mm, suggesting that a minimum particle-containing stream width, which is equal to the big particle diameter, should be maintained in order to obtain a separation. It was also observed that the flow rate ratio had a more significant effect on the small particles travelling distance in y direction. When the width of the particle-containing stream
Fig. 5 Average separated distance between big and small particles as a function of flow rate ratio.
Lab Chip, 2006, 6, 561–566 | 565
Table 3
Comparison of separation results for different total flow rates (2.80–5.60 ml min21) 2.80 ml min21
3.75 ml min21
5.60 ml min21
Distance from upper sidewall/mm
d10
d25
d10
d25
d10
d25
Mean/mm Range (¡)/mm
1168.0 +181.8 2137.6 158.3
1009.6 +59.1 259.1
1227.7 +169.9 2159.0 201.0
1026.7 +57.9 2175.2
1175.0 +225.8 2138.2 106.3
1068.7 +118.2 2143.7
Separation/mm
was increased, the small particles contained in that stream also spreaded in a wider area compared to the larger ones, which again was reflected on changes in both the small particles distance from the lower sidewall and their covering range (see the maximum and minimum ranges in Table 2). 3.4. Effects of total flow rates At a constant flow rate ratio of 0.364 : 1 (media with particles relative to media without particles) while the particle sample was introduced via the left inlet, the effect of varying total flow rate was also examined. The total flow rate was controlled at three levels of 2.80, 3.75 and 5.60 ml min21. The results are compared in Table 3. It can be seen that the influence of the total flow rate on the separation was insignificant, which indicated that the flow profile remained unaltered with the change of total flow rates under the experimental conditions. Thus, a higher total flow rate can be operated to obtain a high throughput without the cost of separation efficiency. This observation is in line with that obtained by Yamada et al.17 It was also interesting to note from Table 3 that the large particles were moving towards the large circle sidewall. This was reflected by an increase in the distance to the small circle sidewall, from 1009.7 to 1068.7 mm, when the total flow rate was increased from 2.813 to 5.625 ml min21. The change was likely due to the centrifugal effect with a higher velocity flow along the curved cavity which, however, still needs more investigation.
4. Conclusions A microdevice has been designed and fabricated with an asymmetric cavity. This device has been used to separate polymer microspheres of two sizes, 10 and 25 mm. The process is based on the microfluidic profile within the asymmetric cavity, which can amplify the separation in a continuous manner along the flow profile. The results showed that the introduction of the particle sample via the left inlet along the large circle sidewall gave the best separation. It was also found that a narrower stream width of particles gave a better separation which can be obtained by varying the flow rate ratio, but a minimum particle-containing stream width equal to the large particle diameter should be maintained in order to obtain a reasonable separation. No significant effect of total flow rate on the separation result was observed which however
566 | Lab Chip, 2006, 6, 561–566
suggested that a higher sample throughput is reachable. This technique can in principle be applied for manipulating both microscale biological and colloidal particles within microfluidic systems.
References 1 A. Manz, J. C. Fettinger, E. Verpoorte, H. Ludi, H. M. Widmer and D. J. Harrison, Trends Anal. Chem., 1991, 10, 144–149. 2 K. F. Jensen, Chem. Eng. Sci., 2001, 56, 293–303. 3 8th International Conference on Miniaturized Systems for Chemistry and Life Sciences (MicroTAS 2004), ed. T. Laurell, J. Nilsson, K. Jensen, D. J. Harrison and J. P. Kutter, Malmo¨, Sweden, September 26–30, 2004, Royal Society of Chemistry, Cambridge, UK, 2004. 4 P. D. I. Fletcher, S. J. Haswell, E. Pombo-Villar, B. H. Warrington, P. Watts, S. Y. F. Wong and X. Zhang, Tetrahedron, 2002, 58, 4735–4757. 5 Microsystem Technology in Chemistry and Life Sciences, ed. A. Manz and H. Becker, Springer, Berlin, 1998. 6 P. Vastamaki, M. Jussila and M.-L. Riekkola, Analyst, 2005, 130, 427–432. 7 J. C. Giddings, Sep. Sci., 1966, 1, 123–125. 8 Field-Flow Fractionation Handbook, ed. M. Schimpf, K. Caldwell and J. C. Giddings, John Wiley & Sons, Inc., New York, 2000. 9 J. Auerswald and H. F. Knapp, Microelectron. Eng., 2003, 67–68, 879–886. 10 T. Lilliehorn, U. Simu, M. Nilsson, M. Almqvist, T. Stepinski, T. Laurell, J. Nilsson and S. Johansson, Ultrasonics, 2005, 43, 293–303. 11 N. Pamme and A. Manz, Anal. Chem., 2004, 76, 7250–7256. 12 J. Janca, I. A. Ananieva, A. Y. Menshikova, T. G. Evseeva and J. Dupak, J. Chromatogr., A, 2004, 1046, 167–173. 13 M. P. MacDonald, S. Neale, L. Paterson, A. Richies, K. Dholakia and G. C. Spalding, J. Biol. Regul. Homeostatic Agents, 2004, 18, 200–205. 14 J. Kruger, K. Singh, A. O’Neill, C. Jackson, A. Morrison and P. O’Brien, J. Micromech. Microeng., 2002, 12, 486–494. 15 J. Kim, S. H. Jang, G. Jia, J. V. Zoval, N. A. Da Silva and M. J. Madou, Lab Chip, 2004, 4, 516–522. 16 J. Oakey, J. Allely and D. W. M. Marr, Biotechnol. Prog., 2002, 18, 1439–1442. 17 M. Yamada, M. Nakashima and M. Seki, Anal. Chem., 2004, 76, 5465–5471. 18 S. Ookawara, R. Higashi, D. Street and K. Ogawa, Chem. Eng. J., 2004, 101, 171–178. 19 J. Takagi, M. Yamada, M. Yasuda and M. Seki, Lab Chip, 2005, 5, 778–784. 20 M. Slifkin and R. Cumbie, J. Clin. Microbiol., 1992, 30, 2722–2724. 21 F. T. Docherty, P. B. Monaghan, R. Keir, D. Graham, W. E. Smith and J. M. Cooper, Chem. Commun., 2004, 118–119. 22 Y. J. Kang, C. Yang and X. Y. Huang, Microfluid Nanofluid, 2005, 1, 168–176. 23 J. O. Wilkes and S. G. Bike, Fluid Mechanics for Chemical Engineers, Prentice Hall PTR, Upper Saddle River, 1998.
This journal is ß The Royal Society of Chemistry 2006
