The stability of plane Couette flow with viscous heating

Abstract

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.