Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid

Abstract

We consider linearized dynamics associated with step jumps in the velocity or displacement of the boundary of a fluid in a shearing motion. The discontinuity will propagate into the interior with a speed$C = \sqrt {{{G\left( 0 \right)} \mathord{\left/ {\vphantom {{G\left( 0 \right)} \rho }} \right. \kern-\nulldelimiterspace} \rho }}$ (ρ is the density) if the initial valuesG(0) andG′(0) of the fading memory kernels are bounded, 0 <G(0) < ∞, − ∞ <G′ (0) < 0. IfG(0) ≠ ∞ butG′(0) = − ∞, then the boundary of the support of the solution still propagates with the speedC. However, the solutions on both sides of the boundary match together in aC ^∞-fashion. IfG(0) ≠ ∞ butG′(0) = 0, the amplitude of the discontinuity will not damp as in a purely elastic fluid. IfG(0) = ∞, the step change is felt immediately throughout the fluid, without shocks, as in Navier-Stokes fluids. This same type of parabolic behavior can be achieved by a small Newtonian contribution added to the integral form of the stress but if this contribution is small, a smooth transition layer around the shock will propagate with the speedC. In the case of step displacement, from rest to rest, singular surfaces of infinite velocity can propagate into the interior with speed of propagationC. The singular surfaces undergo multiple reflections off bounding walls, but the final steady state reached asymptotically is in universal form independent of material.