STEADY FLOW ON TO A CONVEYOR BELT - CAUSAL VISCOSITY AND SHEAR SHOCKS

In hydrodynamics, a shock occurs when a supersonic flow encounters a downstream obstacle and makes a sudden transition to a subsonic velocity. There is a shock rather than a smooth transition because information about the downstream boundary condition, carried by sound waves, does not reach the supersonic upstream fluid. We describe here analogous shear shocks which occur when information about the transverse velocity, i.e. the component of velocity in the plane of the shock front, is unable to penetrate upstream. This could be the case were a fluid flowing perpendicularly on to a conveyor belt about which, even in principle, the fluid far upstream had no knowledge. We consider a model of viscosity which has a finite propagation speed of shear information, and show that it produces two kinds of shear shock. A 'pure shear shock' corresponds to a transition from a superviscous to a subviscous state with no discontinuity in the velocity. The transverse velocity is continuous throughout this shock; however, the velocity profile acquires an extra degree of freedom on the downstream side which is used to accommodate the downstream boundary condition. A 'mixed shear shock' has a shear transition occurring at the same location as a normal adiabatic or radiative shock. The flow makes a sudden jump from a supersonic superviscous velocity to a subsonic subviscous state, and there is a discontinuity in both the normal and transverse components of the velocity. A discontinuity in the transverse velocity is not usually allowed in a regular shock, but it is permitted in a mixed shear shock because of the introduction of causality into the model of viscous interactions. We derive a generalized version of the Rankine-Hugoniot conditions for mixed shear shocks and present self-consistent numerical solutions to a model two-dimensional problem in which an axisymmetric radially infalling stream encounters a spinning star. The numerical examples illustrate various features of the pure and mixed shear shocks. The calculations are performed using a polytropic equation of state with adiabatic shocks, as well as an ideal gas equation of state with radiative shocks.