Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-28T12:49:51.577Z Has data issue: false hasContentIssue false

A dynamical instability due to fluid–wall coupling lowers the transition Reynolds number in the flow through a flexible tube

Published online by Cambridge University Press:  02 August 2011

M. K. S. Verma
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
V. Kumaran*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
*
Email address for correspondence: kumaran@chemeng.iisc.ernet.in

Abstract

A flow-induced instability in a tube with flexible walls is studied experimentally. Tubes of diameter 0.8 and 1.2 mm are cast in polydimethylsiloxane (PDMS) polymer gels, and the catalyst concentration in these gels is varied to obtain shear modulus in the range 17–550 kPa. A pressure drop between the inlet and outlet of the tube is used to drive fluid flow, and the friction factor is measured as a function of the Reynolds number . From these measurements, it is found that the laminar flow becomes unstable, and there is a transition to a more complicated flow profile, for Reynolds numbers as low as 500 for the softest gels used here. The nature of the curves is also qualitatively different from that in the flow past rigid tubes; in contrast to the discontinuous increase in the friction factor at transition in a rigid tube, it is found that there is a continuous increase in the friction factor from the laminar value of in a flexible tube. The onset of transition is also detected by a dye-stream method, where a stream of dye is injected into the centre of the tube. It is found that there is a continuous increase of the amplitude of perturbations at the onset of transition in a flexible tube, in contrast to the abrupt disruption of the dye stream at transition in a rigid tube. There are oscillations in the wall of the tube at the onset of transition, which is detected from the laser scattering off the walls of the tube. This indicates that the coupling between the fluid stresses and the elastic stresses in the wall results in an instability of the laminar flow.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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

1. Bazant, M. Z. & Squires, T. M. 2004 Induced-charge electro-kinetic phenomena: theory and microfluidic applications. Phys. Rev. Lett. 92, 066101.CrossRefGoogle Scholar
2. Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover Publications Inc.Google Scholar
3. Chokshi, P. P. & Kumaran, V. 2008 Weakly nonlinear analysis of viscous instability in flow past a neo-Hookean surface. Phys. Rev. E 77, 056303.CrossRefGoogle Scholar
4. Chokshi, P. P. & Kumaran, V. 2009 Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers. Phys. Fluids 21, 014109.CrossRefGoogle Scholar
5. Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
6. Eggert, M. D. & Kumar, S. 2004 Observations of instability, hysteresis, and oscillation in low-Reynolds-number flow past polymer gels. J. Colloid. Interface Sci. 278, 234242.CrossRefGoogle ScholarPubMed
7. Gaurav, & Shankar, V. 2009 Stability of fluid flow through deformable neo-Hookean tubes. J. Fluid Mech. 627, 291322.CrossRefGoogle Scholar
8. Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91, 244502.CrossRefGoogle Scholar
9. Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443, 5962.CrossRefGoogle ScholarPubMed
10. Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17R44.CrossRefGoogle Scholar
11. Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes. J. Colloid. Interface Sci. 71, 3950.CrossRefGoogle Scholar
12. Kumaran, V. 1995 Stability of the viscous flow of a fluid through a flexible tube. J. Fluid Mech. 294, 259281.CrossRefGoogle Scholar
13. Kumaran, V. 1996 Stability of an inviscid flow in a flexible tube. J. Fluid Mech. 320, 117.CrossRefGoogle Scholar
14. Kumaran, V. 1998a Stability of wall modes in a flexible tube. J. Fluid Mech. 362, 115.CrossRefGoogle Scholar
15. Kumaran, V. 1998b Stability of fluid flow in a flexible tube at intermediate Reynolds number. J. Fluid Mech. 357, 123140.CrossRefGoogle Scholar
16. Kumaran, V. 2003 Hydrodynamic stability of flow through flexible channels and tubes. In Flow Through Collapsible Tubes and Past Other Highly Compliant Surfaces (ed. Carpenter, P. W. & Pedley, T. J. ). Kluwer Academic.Google Scholar
17. Kumaran, V., Fredrickson, G. H. & Pincus, P. 1994 Flow induced instability at the interface between a fluid and a gel at low Reynolds number. J. Phys. France II 4, 893911.Google Scholar
18. Kumaran, V. & Muralikrishnan, R. 2000 Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys. Rev. Lett. 84, 33103313.CrossRefGoogle Scholar
19. Lee, J. N., Park, C. & Whitesides, G. M. 2003 Solvent compatibility of poly (dimethyl siloxane) based microfluidic devices. Analyt. Chem. 75, 65446553.CrossRefGoogle ScholarPubMed
20. Muralikrishnan, R. & Kumaran, V. 2002 Experimental study of the instability of the viscous flow past a flexible surface. Phys. Fluids 14, 775.CrossRefGoogle Scholar
21. Rands, C., Webb, B. W. & Maynes, D. 2006 Characterisation of transition to turbulence in microchannels. Intl J. Heat Mass Transfer 49, 29242930.CrossRefGoogle Scholar
22. Reynolds, O. 1893 An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. Phil. Trans. R. Soc. 174, 935.Google Scholar
23. Shankar, V. & Kumaran, V. 1999 Stability of non-parabolic flows in a flexible tube. J. Fluid Mech. 395, 211236.CrossRefGoogle Scholar
24. Shankar, V. & Kumaran, V. 2000 Stability of non-axisymmetric modes in a flexible tube. J. Fluid Mech. 407, 291314.CrossRefGoogle Scholar
25. Shankar, V. & Kumaran, V. 2001a Asymptotic analysis of wall modes in a flexible tube revisited. Eur. Phys. J. B 19, 607.CrossRefGoogle Scholar
26. Shankar, V. & Kumaran, V. 2001b Weakly nonlinear stability of viscous flow past a flexible surface. J. Fluid Mech. 434, 337354.CrossRefGoogle Scholar
27. Shankar, V. & Kumaran, V. 2002 Stability of wall modes in the flow past a flexible surface. Phys. Fluids 14, 2324.CrossRefGoogle Scholar
28. Sharp, K. V. & Adrian, A. J. 2004 Transition from laminar to turbulent flow in liquid filled microtubes. Exp. Fluids 741747.CrossRefGoogle Scholar
29. Shrivastava, A., Cussler, E. L. & Kumar, S. 2008 Mass transfer enhancement due to a soft elastic boundary. Chem. Engng Sci. 63, 43024305.CrossRefGoogle Scholar
30. Sibulkin, A. 1962 Transition from turbulent to laminar pipe flow. Phys. Fluids 5, 280284.CrossRefGoogle Scholar
31. Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A. & Whitesides, G. M. 2002 Chaotic mixer for microchannels. Nature 295, 647651.Google ScholarPubMed
32. Sutterby, J. L. 1965 Finite difference analysis of viscous laminar converging flow in conical tubes. Appl. Sci. Res. A 15, 241252.CrossRefGoogle Scholar
33. Thaokar, R. M., Shankar, V. & Kumaran, V. 2001 Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces. Eur. Phys. J. B 23, 533.CrossRefGoogle Scholar
34. Verma, M. K. S., Majumder, A. & Ghatak, A. 2006 Embedded template-assisted fabrication of complex microchannels in PDMS and design of a microfluidic adhesive. Langmuir 22, 1029110295.CrossRefGoogle ScholarPubMed
35. Yang, C., Grattoni, C. A., Muggeridge, A. H. & Zimmerman, R. W. 2000 A model for steady laminar flow through a deformable gel-coated channel. J. Colloid. Interface Sci. 266, 104111.Google Scholar