Stability of non-isothermal flow in channels—I. Temperature dependent Newtonian fluid without heat generation

Abstract

We study here stability of non-isothermal flow between two closely spaced, heat conducting, parallel flat plates forming a channel of length l, uniform depth h and infinite width. Very viscous fluid enters along x = 0 at temperature t 1⪢ w the plate temperature. We look for flow nonuniformity caused by coupling between the energy equation, which describes the heat transfer mechanism between fluid and channel walls, and the flow equation which includes the (exponential) temperature dependence of viscosity. The simplified model chosen for the flow assumes similarity profiles for velocity and temperature in the flow direction. The velocities are assumed non-zero in both directions parallel to the bounding plates and negligible in the direction perpendicular to the plates ( h ⪡ l). This is analogous to lubrication theory. The governing equations are taken to be ▪ where v is a two-dimensional mean velocity in the ( x, y)-plane, p( x,y) is pressure and T( x, y) temperature; H is a thermal transfer coefficient; C and b are rheological parameters of the fluid; p(0, y) and p( l,y) are constant, their difference being the relevant pressure drop. We show first that the system can be described in terms of two dimensionless parameters B = b( T 1− T w ) and Gz = V| Hl, V being the mean velocity, and that for steady flow (independent of y, with v = ( V, 0)) a plot of inlet pressure P vs V is multivalued for B sufficiently large. We then investigate the stability of this unidirectional flow to small disturbances of the form fn(x) exp ( iλ y+ω t), λ real. The resulting eigen-value problem for λ ( B, Gz, ω) is solved numerically. Results indicate that a unique neutral stability curve in the plane of ( B, Gz) can be obtained for ω = 0. This stability curve indicates that for B < 2·4, flow instability will not be observed for any Gz. A comparison between the multivalued curve in V( P) obtained for a unidirectional solution and the neutral stability curve is also presented.