Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T05:45:00.610Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of pressure-driven flow in a deformable neo-Hookean channel

Published online by Cambridge University Press:  09 July 2010

GAURAV  and
V. SHANKAR
GAURAV
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
V. SHANKAR*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
*
†Email address for correspondence: vshankar@iitk.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (<1) for highly deformable walls. There are other types of instabilities that exist only due to wall deformability. In the absence of inertia (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

JFM classification

Waves/Free-surface Flows: Elastic waves Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 318 - 350
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Achenbach, D. J. 1973 Wave Propagation in Elastic Solids. North-Holland.Google Scholar
Beatty, M. F. & Zhou, Z. 1991 Universal motion for a class of viscoelastic materials of differential type. Contin. Mech. Thermodyn. 3, 169–191.Google Scholar
Benjamin, T. B. 1960 Effect of a flexible surface on boundary layer stability. J. Fluid Mech. 9, 513–532.Google Scholar
Boyd, J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.CrossRefGoogle Scholar
Chokshi, P. 2007 Studies in the stability of newtonian and viscoelastic flows past rigid and flexible surfaces. PhD thesis, Indian Institute of Science, Bangalore, India.Google Scholar
Choskshi, P. & Kumaran, V. 2008 Weakly nonlinear analysis of viscous instability in flow past a neo-Hookean surface. Phys. Rev. E 77, 056303.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Destarde, M. & Saccomandi, G. 2004 Finite-amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids. Wave Motion 40, 251–262.CrossRefGoogle Scholar
Drazin, P. G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of a inviscid fluid. In Advances in Applied Mechanics (ed. Chernyi, G. G.), vol. 9, pp. 1–90. Academic Press.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eggert, M. D. & Kumar, S. 2004 Observations of instability, hysterisis, and oscillation in low-Reynolds number flow past polymer gels. J. Colloid Interface Sci. 274, 234–242.Google Scholar
Fosdick, R. L. & Yu, J. H. 1996 Thermodynamics, stability and non-linear oscillations of viscoelastic solids. Part I. Differential type solids of second grade. Intl J. NonLinear Mech. 31, 495–516.CrossRefGoogle Scholar
Gad-el-Hak, M. 2003 Drag reduction using compliant walls. In IUTAM Symposium on Flow Past Highly Compliant Boundaries and in Collapsible Tubes (ed. Carpenter, P. W. & Pedley, T. J.), chap. 9, pp. 191–229. Kluwer Academic.CrossRefGoogle Scholar
Gajjar, J. S. B. & Sibanda, S. K. 1996 The hydrodynamic stability of channel flow with compliant boundaries. Theor. Comput. Fluid Dyn. 8, 105–129.Google Scholar
, Gaurav & Shankar, V. 2007 Stability of gravity-driven free-surface flow past a deformable solid layer at zero and finite Reynolds number. Phys. Fluids 19, 024105.CrossRefGoogle Scholar
, Gaurav & Shankar, V. 2009 Stability of fluid flow through deformable neo-Hookean tubes. J. Fluid Mech. 627, 291–322.CrossRefGoogle Scholar
Gkanis, V. & Kumar, S. 2003 Instability of creeping Couette flow past a neo-Hookean solid. Phys. Fluids 15, 2864–2471.CrossRefGoogle Scholar
Gkanis, V. & Kumar, S. 2005 Stability of pressure-driven creeping flows in channels lined with a nonlinear elastic solid. J. Fluid Mech. 524, 357–375.CrossRefGoogle Scholar
Green, C. H. & Ellen, C. H. 1972 The stability of plane Poiseuille flow between flexible walls. J. Fluid Mech. 51, 403–416.CrossRefGoogle Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Hains, F. D. & Price, J. F. 1962 Effect of flexible wall on stability of Poiseuille flow. Phys. Fluids 5, 365.Google Scholar
Holzapfel, G. A. 2000 Nonlinear Solid Mechanics. Wiley.Google Scholar
Huang, L. 1998 Reversal of the Bernoulli effect and channel flutter. J. Fluids Struct. 12, 131–151.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Goodreche, C. & Manneville, P.), chap. 2, pp. 95–118. Cambridge University Press.Google Scholar
Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes. J. Colloid Interface Sci. 71, 34–50.CrossRefGoogle Scholar
Kumaran, V. 1995 Stability of the viscous flow of a fluid through a flexible tube. J. Fluid Mech. 294, 259–281.CrossRefGoogle Scholar
Kumaran, V. 1998 Stability of fluid flow through a flexible tube at intermediate Reynolds number. J. Fluid Mech. 357, 123–140.Google Scholar
Kumaran, V., Fredrickson, G. H. & Pincus, P. 1994 Flow induced instability of the interface between a fluid and a gel at low Reynolds number. J. Phys. II France 4, 893–904.Google Scholar
Kumaran, V. & Muralikrishnan, R. 2000 Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys. Rev. Lett. 84, 3310–3313.CrossRefGoogle Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609–632.CrossRefGoogle Scholar
LaRose, P. G. & Grotberg, J. B. 1997 Flutter and long-wave instabilities in compliant channels conveying developing flows. J. Fluid Mech. 331, 37–58.CrossRefGoogle Scholar
Ma, Y. & Ng, C.-O. 2009 Wave propagation and induced steady streaming in viscous fluid contained in a prestressed viscoelastic tube. Phys. Fluids 21, 051901.CrossRefGoogle Scholar
Malvern, L. E. 1969 Introduction to the Mechanics of a Continuous Medium. Prentice-Hall.Google Scholar
Mandre, S. & Mahadevan, L. 2010 A generalized theory of viscous and inviscid flutter. Proc. R. Soc. A 466, 141–156.Google Scholar
McDonald, J. C. & Whitesides, G. M. 2002 Poly(dimethylsiloxane) as a material for fabricating microfluidic devices. Acc. Chem. Res. 35, 491–499.Google Scholar
Muralikrishnan, R. & Kumaran, V. 2002 Experimental study of the instability of viscous flow past a flexible surface. Phys. Fluids 14, 775–780.Google Scholar
Nagata, M. & Cole, T. R. 1999 On the stability of plane Poiseuille flow between compliant boundaries. In Proceedings of International Conference of Computational Methods and Experimental Measurements IX, Naples, Italy (ed. Carlomagno, G. M. & Brebbia, C. A.), pp. 231–240. WIT Press.Google Scholar
Rotenberry, J. M. & Saffman, P. G. 1990 Effect of compliant boundaries on the weakly nonlinear shear waves in a channel flow. SIAM J. Appl. Math. 50, 361–394.CrossRefGoogle Scholar
Schmid, P. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shankar, V. 2004 Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer. J. Non-Newton. Fluid Mech. 117, 163–182.CrossRefGoogle Scholar
Shankar, V. & Kumaran, V. 2000 Stability of fluid flow in a flexible tube to non-axisymmetric disturbances. J. Fluid Mech. 408, 291–314.CrossRefGoogle Scholar
Shankar, V. & Kumaran, V. 2001 Asymptotic analysis of wall modes in a flexible tube revisited. Eur. Phys. J. B 19, 607–622.CrossRefGoogle Scholar
Shankar, V. & Kumaran, V. 2002 Stability of wall modes in fluid flow past a flexible surface. Phys. Fluids 14, 2324–2338.Google Scholar
Shrivastava, A., Cussler, E. L. & Kumar, S. 2008 Mass transfer enhancement due to a soft elastic boundary. Chem. Engng Sci. 63, 4302–4305.Google Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026.Google Scholar
Srivatsan, L. & Kumaran, V. 1997 Stability of the interface between a fluid and gel. J. Phys. II. France 7, 947–963.Google Scholar
Stewart, P. S., Walters, S. L. & Jensen, O. E. 2010 Local instabilities of flow in a flexible channel: asymmetric flutter driven by a weak critical layer. Phys. Fluids 22, 031902.CrossRefGoogle Scholar
Thomas, M. S., Clift, J. M., Millare, B. & Vullev, V. I. 2010 Print-and-peel fabricated passive micromixers. Langmuir 26, 2951–2957.Google Scholar
Verma, M. K. S, Ganneboyina, S. R., Rakshith, R. V. & Ghatak, A. 2008 Three-dimensional multihelical microfluidic mixers for rapid mixing of liquids. Langmuir 24, 2248–2251.CrossRefGoogle ScholarPubMed
Weideman, J. A. & Reddy, S. C. 2000 A Matlab differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519.Google Scholar
Yeo, K. S. 1988 The stability of boundary layer flow over single and multilayer viscoelastic walls. J. Fluid Mech. 196, 359–408.Google Scholar
Yeo, K. S., Khoo, B. C. & Chong, W. K. 1994 The linear stability of boundary layer flow over compliant walls: the effects of wall mean state induced by flow loading. J. Fluids Struct. 8, 529–551.Google Scholar