The stability of plane Couette flow of a power-law fluid with viscous heating

Abstract

Linear stability analysis is carried out to examine the effect of shear thinning and shear thickening on the stability of plane Couette flow with viscous heating for a power-law fluid that obeys Arrhenius-type law. The relation between shear stress and shear rate is plotted, and the maximum shear stress that must be applied to the system is calculated for different values of the power-law index n. The results showed that the S-shaped curve characterizing the fluid flow that obeys Arrhenius-type law appears at smaller values of the activation energy parameter β, in contrast to the Newtonian case, as a result of shear thinning effect. The resulting eigenvalues are calculated using the Chebyshev collocation method with the QZ algorithm {QZ algorithm is used in solving the singular generalized eigenvalue problem [SIAM J. Numer. Anal. 10, 241 (1973)]}. It is found, for shear thinning/thickening fluid, that the instability occurs at lower/higher values of the Brinkman number Br than in the Newtonian case. Also, the results indicate, for both Newtonian and power-law fluids, that two modes of the instability occur: one termed an inviscid mode, and the second a coupled mode, not a viscous mode as conjectured in C. S. Yueh and C. I. Weng, “Linear stability analysis of plane Couette flow with viscous heating,” Phys. Fluids 8, 1802 (1996).