Unsteady Shearing Flows

Abstract

In this chapter, we shall begin by studying the initial value problem of the start-up of the flow of a Bingham fluid in a channel from rest due to a constant applied pressure gradient. Further, the analytic solution is compared with a numerical approximation. One notes that this problem cannot be solved through the Laplace transform. It is shown that the bounds on the velocity and its derivatives can be obtained using maximum principles. Later on, a summary of the initiation of a Couette flow, followed by that in a pipe of circular cross-section is provided. In the unsteady viscoplastic flow problems, the yield surface propagates into the fluid with a finite speed. Due to the incompressibility of the fluid, the motion of the yield surface is lateral to itself. And, across the propagating yield surface, it turns out that the velocity and the acceleration are both continuous across the yield surface, while the derivative of the acceleration, known as the jerk, suffers a jump. These aspects of the kinematics have a corresponding impact on the shear stress and its temporal and spatial derivatives. These matters are fully explored through the theory of singular surfaces in motion.