Linear stability analysis of plane Couette flow with viscous heating

The linear stability of a plane Couette flow of Newtonian fluids with exponential dependence of viscosity on temperature according to the Arrhenius law and Nahme law is investigated. The modified Chebyshev polynomials of the second kind are used in Galerkin’s method to solve the eigenvalue problem. The combinations of Chebyshev polynomials to form the modified ones for satisfying the required boundary conditions, and their orthogonality conditions, are derived. It is found that an analysis that does not include enough terms of the trial functions may lead to erroneous results. The results show that instability occurs for fluids with highly temperature-sensitive viscosity in a range of the Brinkman numbers; when the temperature-sensitive viscosity is small enough, instability is not observed. Furthermore, only two kinds of modes, the viscous and inviscid ones, are found for the instability of the flow system. Also, the results indicate that the fluids obeying the Arrhenius-type model are more stable than those of the Nahme-type model if both are based on the same temperature-sensitive viscosity, reference viscosity, and temperature.