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PHYSICS OF FLUIDS

VOLUME 13, NUMBER 1

JANUARY 2001

LETTERS The purpose of this Letters section is to provide rapid dissemination of important new results in the fields regularly covered by Physics of Fluids. Results of extended research should not be presented as a series of letters in place of comprehensive articles. Letters cannot exceed four printed pages in length, including space allowed for title, figures, tables, references and an abstract limited to about 100 words. There is a three-month time limit, from date of receipt to acceptance, for processing Letter manuscripts. Authors must also submit a brief statement justifying rapid publication in the Letters section.

Stabilization and destabilization of channel flow by location of viscosity-stratified fluid layer Balaji T. Ranganathan and Rama Govindarajana) Fluid Dynamics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India

共Received 18 February 2000; accepted 21 September 2000兲 The stability of the channel flow of two fluids of different viscosities with a mixed layer in between is demonstrated to be qualitatively different from both interface dominated flows and stratified flows. More important, this flow displays unexpected changes in stability when the mixed layer overlaps the critical layer of the disturbance: this feature can be exploited for flow control. When these layers are distinct, the flow is mildly destabilized when the less viscous fluid is in the outer region. When the layers overlap, however, there is an order of magnitude stabilization of the flow. The reverse occurs when the more viscous fluid is in the outer region. This behavior may be explained by the balance of stresses in the critical layer. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1329651兴

Expressions for the mean flow are presented here for the upper half of the channel. Under a parallel-flow approximation, the concentration in the mixed layer is a function of height. The viscosity ␮ m is here taken to be a cubic function of y:

The stability of the parallel flow of two fluids has been studied extensively.1–5 In most of these studies, the interface between the two fluids 共where the slope of the velocity profile is discontinuous兲 plays a dominant role in triggering instabilities. In many real flows, however, the fluids are at least slightly miscible, and this singularity is smoothed out into a thin mixed layer. The present study considers the symmetric flow through a channel 共half-width H, centerline velocity U c ) of two fluids of equal densities but different viscosities 共Fig. 1兲. The streamwise development of the flow due to the growth of the mixed layer is neglected, as are perturbations to the basic concentration 共which is exact when Pe⬅U c H/D⫽0, where D is an appropriate diffusivity兲. A real flow would not be parallel, especially if the mixed layer was created by a temperature difference between the outer and inner layers; nor would Pe be small, especially if the two fluids were distinct and the viscosity stratification was due to a concentration gradient. The effect of relaxing these assumptions will be presented elsewhere. It may be noted, however, that Wall and Wilson6 have shown that the stability of a stratified flow is highly insensitive to Pe 共the difference in their case between the critical Reynolds numbers for Pe⫽0 and Pe⫽105 is ⬍2%). The stability of the flow in Fig. 1 is shown here to be qualitatively different from both the interfacial instability of immiscible fluids and that of stratified fluids. Our main result is in Fig. 3, which shows a high degree of stabilization or destabilization for careful positioning of the viscosity-stratified layer.

␮ m ⫽1⫹

册

共1兲

but the stability is insensitive to the exact nature of the function. The subscripts m and o stand for mixed and outer fluid, respectively. The basic flow is obtained by enforcing that mean quantities and all relevant derivatives are continuous at the edges of the mixed layer, U⫽1⫺

Gy 2 , 2

U⫽U 共 p 兲 ⫺G

共2兲

y⭐p,

冕␮ y

p

y

dy,

p⭐y⭐q,

共3兲

m

where G is the streamwise pressure gradient, and U⫽

G 共 1⫺y 2 兲 , 2␮o

y⭓ p⫹q.

共4兲

All quantities have been nondimensionalized using H and U c ; viscosities are scaled by the inner fluid viscosity. In view of Yih’s7 extension of Squire’s theorem to stratified fluids, we consider two-dimensional disturbances of the form ␺ ⫽ ␾ (y)exp关i␣ (x⫺ct) 兴 , where ␺ is the disturbance streamfunction. The stability of this flow is governed by the modified Orr-Sommerfeld equation

a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

1070-6631/2001/13(1)/1/3/$18.00

冋

3 共 ␮ o ⫺1 兲 ⫺2 共 y⫺ p 兲 3 ⫹ 共 y⫺ p 兲 2 , q2 3q

1

© 2001 American Institute of Physics

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Phys. Fluids, Vol. 13, No. 1, January 2001

B. T. Ranganathan and R. Govindarajan

FIG. 1. Schematic of flow.

i␣ 关共 ␾ ⬙ ⫺ ␣ 2 ␾ 兲共 U⫺c 兲 ⫺U ⬙ ␾ 兴 ⫽

1 关 ␮␾ i v ⫹2 ␮ ⬘ ␾ ⵮ ⫹ 共 ␮ ⬙ ⫺2 ␣ 2 ␮ 兲 ␾ ⬙ ⫺2 ␣ 2 ␮ ⬘ ␾ ⬘ R ⫹共 ␣ 2␮ ⬙⫹ ␣ 4␮ 兲␾ 兴

共5兲

with the boundary conditions

␾ 共 ⫾1 兲 ⫽ ␾ ⬘ 共 ⫾1 兲 ⫽0.

共6兲

FIG. 3. Dependence of critical Reynolds number on the width of the inner fluid; q⫽0.1 ( ␮ o is scaled by ␮ i ).

The Reynolds number is defined as R⬅ ␳ Uc H/ ␮ i , and the primes denote differentiation with respect to y. Equation 共5兲 is solved using a Chebychev collocation spectral method.8 The physical grid is clustered adequately in the mixed region. The effect of the thickness of the mixed layer is shown in Fig. 2. At low to moderate thickness, the behavior is independent of q and a more viscous fluid in the outer region stabilizes the fluid. When q⫽1, i.e., the situation is that of stratified flow;6 this trend is reversed. Since our focus is on a thin mixed layer, we fix q at 0.1 for the rest of this study. The dependence of the stability on the fraction p of the channel width occupied by the inner fluid is shown in Fig. 3. Stability characteristics show a marked change beyond a certain value of p. For ␮ o ⫽1.1, when the mixed layer extends from y⫽0.7 to y⫽0.8, there is a sudden drop in stability. It may be seen from Fig. 4 that the critical point of the disturbance lies within the mixed layer in this case, i.e., the critical layer 共whose thickness ⑀ ⬃R ⫺1/3) overlaps significantly with the layer of viscosity stratification. A similar situation exists for ␮ o ⫽0.9 when p⬎0.8.

where the disturbance eigenfunction has been expanded in powers of ⑀ as ␾ ⫽ 兺 k ⑀ k ␹ k , k⫽0,1,2, . . . as in Govindarajan and Narasimha.9 In Eq. 共7兲 alone, primes refer to differentiation with respect to the coordinate ␩ ⬅(y⫺y c )/ ⑀ . The derivatives of viscosity are nonzero in the mixed layer alone, while viscous terms in ␹ 0⵮ and ␹ ⬙0 are large only in the critical and wall layers. It is immediately obvious that if the critical layer 共similar arguments hold for the wall layer兲 is well separated from the mixed layer, the second and third terms in Eq. 共7兲 are negligible in the mixed layer and zero

FIG. 2. Dependence of critical Reynolds number on the thickness of the mixed layer. The dotted line is the value for a single fluid. Here p⫹q/2⫽0.5.

FIG. 4. Location of critical point. The region between the dotted lines is the mixed layer in this case.

The volte-face in the flow behavior can be better understood by analyzing the lowest order effects in the critical layer. At the Reynolds numbers under consideration, the critical layer thickness ⑀ is of the same order as the thickness q of the mixed layer. For O(1) differences in the two viscosities, the lowest order critical layer equation is

␮ ␹ i0v ⫹2 ␮ ⬘ ␹ 0⵮ ⫹ ␮ ⬙ ␹ 0⬙ ⫺i␣

冏

⳵U ␩ ␹ ⬙ ⫽0, ⳵y c 0


共7兲

Phys. Fluids, Vol. 13, No. 1, January 2001

Destabilization by location of viscosity-stratified layer

FIG. 6. Critical Reynolds number from the primary mode.

FIG. 5. Critical Reynolds numbers of two different modes, p⫽0.2.

everywhere else, and a traditional stability scenario emerges, with the viscous fourth derivative term balancing the largest inviscid term in the critical layer. If the two layers overlap, however, the viscous effects are modified substantially— depending on the sign of the viscosity gradient, the second and third terms in 共7兲 either augment or cancel the effect of the first term. Going back to Eq. 共5兲, we see that the divergence of disturbance shear stress is balanced by the shear stress divergence created by normal variations in the mean viscosity. This results in a dramatic stabilization of the flow for ␮ ⬘ ⬍0 and a large destabilization for ␮ ⬘ ⬎0. 关In effect, a much larger or much smaller Reynolds number, respectively, is required to balance the left hand side of 共5兲.兴 The stability of the channel flow of a single fluid is dominated by a single mode. However, the above-mentioned results indicate that when ␮ o ⬎1, there may be two modes dominating the present flow: one of which 共the primary mode兲 is similar to that for single fluid flow. In the case where the critical layer of the primary mode is distinct from the mixed layer, another 共secondary兲 mode, whose phase speed equals the mean velocity within the mixed layer, may become important. In Fig. 5, where the channel width occupied by the inner fluid is about a quarter, it is shown that the secondary mode is more unstable for ␮ o greater than about 1.5. If p⬍0.7, the critical layer and the mixed layer are distinct, and the interplay of two modes is evident, with a crossover to the second mode at a particular value of ␮ o . For p⭓0.7, as expected, only a single mode is seen 共Fig. 6兲, since the critical point of the primary mode lies within the mixed layer. It is to be emphasized that the primary instabilities here are of growing TS modes, which exist in the flow of two immiscible fluids too, but in the latter case, the leading eigenmodes arise solely due to the existence of the interface and are qualitatively different. Detailed comparisons will be made elsewhere—we mention here that the present instability is a high Reynolds number phenomenon while a sharp interface can be unstable at any Reynolds number1,4,5 共de-

3

pending on the ratios of viscosity and layer thickness of the two fluids兲. The study is being extended to larger viscosity ratios. The results have immediate implications for passive control of the flow. For example, an extremely stable two-fluid flow may be obtained 共when the two viscosities are close to each other兲 by putting the more viscous fluid in the inner region and fixing the mass-flux ratio so that the mixed region overlaps the critical layer of the most unstable disturbance. The opposite—an early transition to turbulence—may be achieved by putting the more viscous fluid in the outer region in an appropriate ratio. Since the viscosity ratios involved are close to 1, the desired stratification could be achieved by the addition of either heat or relatively small quantities of another fluid. ACKNOWLEDGMENTS

This work is supported by the AR&DB, India. We are grateful to Dr. K. R. Sreenivas and Professor A. K. Prasad for sharing with us the findings of their two-fluid pipe-flow experiment. We thank Professor R. Narasimha for useful suggestions. 1

C. S. Yih, ‘‘Instability due to viscosity stratification,’’ J. Fluid Mech. 27, 337 共1967兲. 2 Y. Renardy, ‘‘Viscosity and density stratification in vertical Poiseuille flow,’’ Phys. Fluids 30, 1638 共1987兲. 3 D. D. Joseph and Y. Renardy, Fundamentals of Two-fluid Dynamics, Part I: Mathematical Theory and Applications 共Springer, Berlin, 1993兲. 4 P. Laure, H. Le Meur, Y. Demay, S. C. Saut, and S. Scotto, ‘‘Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids,’’ J. NonNewtonian Fluid Mech. 71, 1 共1997兲. 5 M. J. South and A. P. Hooper, ‘‘Linear growth in two-fluid plane Poiseuille flow,’’ J. Fluid Mech. 381, 121 共1999兲. 6 D. P. Wall and S. K. Wilson, ‘‘The linear stability of channel flow of fluid with temperature-dependent viscosity,’’ J. Fluid Mech. 323, 107 共1996兲. 7 C. S. Yih, Q. Appl. Math. 12, 434 共1955兲. 8 S. Srinivasan, M. Klika, M. H. Ludwig, and V. Vasanta Ram, ‘‘Spectral collocation method for solving some eigenvalue problems in fluid mechanics,’’ PD EA Report No. 9408, NAL, Bangalore, 1994. 9 R. Govindarajan and R. Narasimha, ‘‘A low-order parabolic theory for boundary-layer stability,’’ Phys. Fluids 11, 1449 共1999兲.
