On the description of the motion of the yield surface in unsteady shearing flows of a Bingham fluid as a jerk wave

In the unsteady shearing flows of a Bingham fluid, the yield surface may move laterally into the fluid with a finite speed. The exact nature of this motion can be explained by assuming that it is a jerk wave. That is, the yield surface is a singular surface across which the velocity, the acceleration and the velocity gradient are continuous, whereas the jerk, which is the time derivative of the acceleration, the spatial gradient of the acceleration and the second gradient of the velocity all suffer jumps. Simultaneously, across this singular surface, the shear stress, its time derivative and its gradient are continuous, while the corresponding temporal and spatial gradients of second order suffer jumps, with Hadamard's Lemma defining the speed of propagation of the jerk wave. These theoretical assumptions are found to hold true in a shearing flow of a Bingham fluid in an unbounded domain, studied by Sekimoto. It is further shown that the same kinematical and dynamical conditions explain the movement of yield surfaces in the shearing flows of all viscoplastic fluids.