Book excerpt: We investigate in this thesis, the problem of stability of thermo-viscoelastic fluid flow between rotating coaxial cylinders. By using the thermo-viscoelastic constitutive equations given by Eringen, we reduce the equations of motion into a form suitable for stability analysis. The course of reduction which we follow yields some interesting intermediate results. The solution for the steady state couette flow problem is found. Interestingly enough, the velocity field for this problem is found to be identical with the classical viscous case and the case of a Reiner-Rivlin fluid, but the temperature and pressure fields are different. The non-dimensional forms of the equations of motion governing the couette flow of thermo-viscoelastic fluids in a heat reservoir are given. These equations are reduced further by considering a small gap between the cylinders and by imposing some physically reasonable mechanical and geometrical restrictions on the flow. This results in a secular equation which forms a characteristic value problem. The solution of the characteristic value problem has been obtained and this yields a criterion for stability in terms of a critical Taylor number. In general, the critical values of Taylor numbers are found to be higher than corresponding ones in classical hydrodynamic stability problems, which implies that thermo-viscoelastic fluids are more stable, in a couette flow, than classical viscous fluids under a similar situation. Comparing this result with existing investigations in non- Newtonian fluids we find that, like Bingham fluids, thermoviscoelastic fluids are more stable than viscous and Reiner-Rivlin fluids.