One-dimensional stability of dissipative couette flow

Abstract

Non-stationary one-dimensional flow, and especially the one-dimensional stability of a stationary Couette flow of a viscous incompressible fluid is considered, taking into account dissipative heat, under the assumption that the viscosity decreases fairly rapidly, e.g., exponentially, as the temperature incrases. It is shown that when the fluid is very viscous, the non-stationary plane problem can be reduced to a non-stationary problem of heat transfer in media with heat sources depending non-linearly on temperature. The dependence of the heat sources on temperature in the latter problem differs substantially for different types of boundary conditions in the initial problem. If a tangential stress is specified at the boundary, then the density of the heat sources will depend on temperature locally (such a problem was studied earlier in /1–6/. When the velocities of the boundary planes are given, the density of heat sources will depend on the temperature distribution as a whole, over the volume. As regards the stationary flow inspected here for stability, it does not always exist, nor is it unique /7–14/. We can utilize the results of /1, 2, 15–19/ by reducing the problem of the existence and uniqueness of such flow to the stationary problem of the temperature distribution in media with heat sources depending non-linearly on temparture.