Behavior and asymptotic stability of thermomechanical processes (Q1083302)

This paper is concerned with a qualitative investigation of one- dimensional adiabatic shearing flows of incompressible Newtonian fluids with temperature-dependent viscosity. The layer of fluid is subjected to a time-dependent body force and appropriate initial conditions. One of its faces is free of stress while a time-periodic or steady tangential velocity is imparted on the other boundary. The heat is generated by viscous dissipation. The viscosity coefficient is assumed to satisfy one of the two inequalities, one corresponding to gases and the other to liquids. Employing some integral identities and differential and integral inequalities which are valid in certain function spaces the author proves that (i) if the boundary velocity is periodic (or steady with a bounded body force) then the shearing flow of a viscoelastic solid is well behaved for all times (ii) provided that the initial viscosity is sufficiently large the shearing flow of a viscoelastic gas under an oscillatory body force and a time-periodic boundary velocity behaves well for all times with bounded stress, velocity and velocity gradient (in case of steady boundary velocity this result becomes valid for an arbitrary initial viscosity), (iii) the flow of the fluid due to a steady boundary velocity converges asymptotically to a steady rigid motion at time-independent temperature.