On the stability against shear waves of steady flows of non-linear viscoelastic fluids

Abstract

A shear-acceleration wave is a propagating singular surface across which the velocity vector and the normal component of the acceleration are continuous, while the tangential component $\dot{v}$ of the acceleration suffers a jump discontinuity [$\dot{v}$]. We here discuss plane-rectilinear shearing flows of general, non-linear, incompressible simple fluids with fading memory. Working within the framework of such planar motions, we derive a general and exact formula for the time-dependence of the amplitude a = [$\dot{v}$] of a shear-acceleration wave propagating into a region undergoing a steady but not necessarily homogeneous shearing flow. When this expression is specialized to the case in which the velocity gradient is constant in space ahead of the wave, it assumes a form familiar in the theory of longitudinal acceleration waves in compressible materials with fading memory (cf., e.g., Coleman & Gurtin 1965, equation (4.12)).

In earlier work (1965) we observed that a planar shear-acceleration wave cannot grow in amplitude if it is propagating into a fluid in a state of equilibrium. It is clear from our present results that if the fluid ahead of the wave is being sheared, |a(t)| not only increases, but can approach infinity in a finite time, provided a(0) is of proper sign and |a(0)| exceeds a certain critical amplitude. We expect this critical amplitude to decrease as the rate of shear ahead of the wave is increased.