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Stability of the flow of a fluid through a flexible tube at high Reynolds number

Published online by Cambridge University Press:  26 April 2006

V. Kumaran
V. Kumaran
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560, 012 India
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Abstract

The stability of the Hagen-Poiseuille flow of a Newtonian fluid in a tube of radius R surrounded by an incompressible viscoelastic medium of radius R < r < HR is analysed in the high Reynolds number regime. The dimensionless numbers that affect the fluid flow are the Reynolds number Re = (ρVR / η), the ratio of the viscosities of the wall and fluid ηr = (ηs/η), the ratio of radii H and the dimensionless velocity Γ = (ρV2/G)1/2. Here ρ is the density of the fluid, G is the coefficient of elasticity of the wall and Vis the maximum fluid velocity at the centre of the tube. In the high Reynolds number regime, an asymptotic expansion in the small parameter ε = (1/Re) is employed. In the leading approximation, the viscous effects are neglected and there is a balance between the inertial stresses in the fluid and the elastic stresses in the medium. There are multiple solutions for the leading-order growth rate do), all of which are imaginary, indicating that the fluctuations are neutrally stable, since there is no viscous dissipation of energy or transfer of energy from the mean flow to the fluctruations due to the Reynolds strees.

There is an O(ε1/2) correction to the growth rate, s(1), due to the presence of a wall layer of thickness ε1/2R where the viscous stresses are O(ε1/2) smaller than the inertial stresses. An energy balance analysis indicates that the transfer of energy from the mean flow to the fluctuations due to the Reynolds stress in the wall layer is exactly cancelled by an opposite transfer of equal magnitude due to the deformation work done at the interface, and there is no net transfer from the mean flow to the fluctuations. Consequently, the fluctuations are stabilized by the viscous dissipation in the wall layer, and the real part of s(1) is negative. However, there are certain values of Γ and wavenumber k where s(l) = 0. At these points, the wail layer amplitude becomes zero because the tangential velocity boundary condition is identically satisfied by the inviscid flow solution. The real part of the O(ε) correction to the growth rate s(2) turns out to be negative at these points, indicating a small stabilizing effect due to the dissipation in the bulk of the fluid and the wall material. It is found that the minimum value of s(2) increases ∝ (H − 1)−2 for (H − 1) [Lt ] 1 (thickness of wall much less than the tube radius), and decreases ∝ (H−4 for H [Gt ] 1. The damping rate for the inviscid modes is smaller than that for the viscous wall and centre modes in a rigid tube, which have been determined previously using a singular perturbation analysis. Therefore, these are the most unstable modes in the flow through a flexible tube.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 302 , 10 November 1995 , pp. 117 - 139
Copyright
© 1995 Cambridge University Press

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References

Benjamin, T. B. 1959 Shearing flow over a wavy boundary layer. J. Fluid. Mech. 6, 161–205.Google Scholar
Benjamin, T. B. 1963 The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid. Mech. 16, 436–450.Google Scholar
Carpenter P. W. 1990 Status of transition delay using compliant walls. Prog. Astro. Aero. 123, 79–113.Google Scholar
Carpenter. P. W. & Gajjar, J. G. S. 1990 A. general theory for two and three dimensional wall mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theor Comput. Fluid dyn. 1. 349–378.Google Scholar
Carpenter P. W. GARRAD 1985 The hydrodynamic stability of flows over Kramer-compliant surfaces. Part 1. Tollmien-Schlichting instabilities. J. Fluid. Mech. 155, 465–510.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability Dover.
Corcos, G. M. & Sellars, J. R., 1959 On the stability of fully developed flow in a pipe. J. Fluid. Mech. 5, 97–112.Google Scholar
Davey, A. & Drazin, P. G. 1969 The stability of Pioseuille flow in a pipe. J. Fluid. Mech. 36, 209–218.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Garg, V. K. & Rouleau, W. T. 1972 Linear spatial stability of pipe Pioseullle flow. J. Fluid. Mech. 54, 113–127.Google Scholar
Gill, A. E. 1965 On the behaviour of small disturbances to Poiseuille flow in a circular pipe.. J. Fluid Mech. 21 145–172.Google Scholar
Graebel, W. P. 1970 The stability of pipe flow. Part 1 Aymptoic analysis of small wave numbers. J. Fluid Mech. 43, 279–290.Google Scholar
Hansen, R. J. & Hunston, D. L. 1974 An experimental study of turbulent flows over a compliant surface. J. Sound vib. 34, 297–308.Google Scholar
Hansen, R. J. & Hunston, D. L. 1983 Fluid property effects on flow-generated waves on a compliant surface. J. Fluid Mech. 133, 161–177.Google Scholar
Harden, J. H., Pleiner, H. & Pincus, P. A. 1991 Hydrodynamic modes on concentrated polymer solutions and gels. J. Chem. Phvs. 94 5208–5217.Google Scholar
Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes.. J. Colloid Interface Sci. 71 34–50.Google Scholar
Kumaran, V. 1993 Surface modes on a polymer gel of finite thickness.. J. Chem. Phys. 98, 3429–3438.Google Scholar
Kumaran, V., 1995 Stability of the viscous flow of a fluid through a flexible tube.. J. Fluid Mech. 294, 259–281.Google Scholar
Kumaran, V., Fredrickson, G. H. & Pincus, P. 1994 Flow induced instability at the interface between a fluid anda gel at low Reynolds number. J. Phys. Paris 4, 93–904.Google Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over flexible surface. J. Fluid. Mech. 609–632.Google Scholar
Landu, L. D. & Lifhitz, E. M. 1989, Theory of Elasticity Pergamon.
Lin, C. C. 1945 a On the stability of two dimensional parallel flows, Part 1. Q. Appl. Maths 3, 117–142.Google Scholar
Lin, C. C. 1945 b On the stability of two dimensional parallel flows, Part 3. Q. Appl. Maths 3, 218–234.Google Scholar
Lin, C. C. 1945 c On the stability of two dimensional parallel flows, flows, Part 3. Q. Appl. Maths 3, 277–301.Google Scholar
Lin, C. C. 1966 The Theory of Hydrodynamic Stability Cambridge University Press.
Pekeris, C. L. 1948 Stability of the laminar flow through a pipe of circular cross-section to infinitesimal disturbances which are symmetrical about the axis of the pipe. Proc. US Natl. Acad. Sci. 34, 285–295.Google Scholar
Riley, J., Gad Fl Hak, M. & Metcalfe, R. W.1988 Compliant coatings.. Ann. Rev. Fluid. Mech. 20, 393–420.Google Scholar
Salwen, H & Grosch, C. E. 1972 The stability of poiseuille flow in a pipe of circular corss section. J. Fluid Mech. 54, 93–112.Google Scholar
Schlichting, H. 1933 Zur Entstechung der Turbulenz bie der Plattenstromung. Z. Agnew. Math. Mech. 13, 171–174.Google Scholar
Silberberg, A. 1987 Physico-chemical hydrodynamics in turbulent flows close to an interfce. Physico-chem. Hydrodyn. 9, 419–426.Google Scholar
Sreenivasan, K. R. & Ramshankar R. 1986 Transition intermittency in open flows, and intermittency routes to. Physica 23D, 246–258.Google Scholar
Tollmien W. 1929 Über die Entstehung der Turbulenz, 1 Mitt. Ges. Wiss. Göttingen, Math. Phys. Klasse. 21–44.Google Scholar
Wasow, W. 1953 Asymptoic solution of the differential of hydrodynamic stability in a domain containing a transtion point.. Ann. Maths. 58 222–252.Google Scholar
Wygnanski, I. & Champagne F. H. 1973 On transition in a pipe Part 1 The origin of puffs and slugs, and the flow in a turbulent slug. J. Fluid Mech. 59, 281–335.Google Scholar