The linear stability of channel flow of fluid with temperature-dependent viscosity

The linear stability of flow of fluid with temperature-dependent viscosity through a channel with heated walls is considered. The resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosity models. For all the viscosity models considered a non-uniform increase of the viscosity in the channel always stabilises the flow whereas a non-uniform decrease of the viscosity in the channel may either destabilise the flow or, more unexpectedly, stabilise the flow. We discuss our results in terms of three physical effects, namely bulk effects, velocity-profile shape effects and thin-layer effects.

[1]  C. Lin,et al.  On the stability of two-dimensional parallel flows. I. General theory , 1945 .

[2]  D.M Sloan.  Eigenfunctions of Systems of Linear Ordinary Differential Equations with Separated Boundary Conditions Using Riccati Transformations , 1977 .

[3]  Daniel D. Joseph,et al.  Couette flow of two fluids between concentric cylinders , 1985, Journal of Fluid Mechanics.

[4]  W. Lakin,et al.  Stokes multipliers for the Orr-Sommerfeld equation , 1970, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  H. Herwig,et al.  Stability of plane Poiseuille flow with temperature dependent viscosity , 1993 .

[6]  W. Lakin,et al.  Approximations to the eigenvalue relation for the Orr-Sommerfeld problem , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[7]  C C Lin,et al.  On the Stability of Two-Dimensional Parallel Flows. , 1944, Proceedings of the National Academy of Sciences of the United States of America.

[8]  C. Lin.  On the stability of two-dimensional parallel flows. II. Stability in an inviscid fluid , 1945 .

[9]  viscosity and density stratification in vertical Poiseuille flow , 1987 .

[10]  C. Grosch,et al.  The stability of steady and time-dependent plane Poiseuille flow , 1968, Journal of Fluid Mechanics.

[11]  M. Potter,et al.  Stability of Plane Poiseuille Flow with Heat Transfer , 1972 .

[12]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[13]  A. Hooper.  The stability of two superposed viscous fluids in a channel , 1989 .

[14]  D. Joseph,et al.  Stability of Parallel Flow between Concentric Cylinders , 1968 .

[15]  R. Helgason,et al.  A matrix method for ordinary differential eigenvalue problems , 1970 .

[16]  W. Tollmien.  Asymptotische Integration der Störungsdifferentialgleichung ebener laminarer Strömungen bei hohen Reynoldsschen Zahlen , 1947 .

[17]  W. Heisenberg.  Über Stabilität und Turbulenz von Flüssigkeitsströmen , 1924 .

[18]  A. Liakopoulos,et al.  The effect of variable viscosity on the interfacial stability of two‐layer Poiseuille flow , 1995 .

[19]  S. Orszag.  Accurate solution of the Orr–Sommerfeld stability equation , 1971, Journal of Fluid Mechanics.

[20]  H. Squire.  On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls , 1933 .

[21]  James Hardy Wilkinson,et al.  Kronecker''s canonical form and the QZ algorithm , 1979 .

[22]  M. Vinokur,et al.  On one-dimensional stretching functions for finite-difference calculations. [computational fluid dynamics] , 1983 .