A layer of yield-stress material on a flat plate that moves suddenly

Abstract

The flow of an unbounded Newtonian fluid above a flat solid plate, set into motion suddenly with uniform velocity, is a canonical problem investigated by Stokes (On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc., vol. 9, 1851) and Rayleigh (Lond. Edinb. Dubl. Philos. Mag. J. Sci., vol. 21, issue 126, 1911, pp. 697–711). We tackle the same problem but with the fluid replaced by a yield-stress material of finite height with a free surface; the latter ensures both yielded (fluid-like) and rigid behaviour. The interface between the yielded and rigid regions – the so-called ‘yield surface’ – always emerges from the plate at a rate faster than that for momentum diffusion. The rigid region, adjacent to the free surface, is accelerated by the constant yield stress acting at this yield surface. As the velocity of this region increases following start-up, its acceleration climaxes and subsequently diminishes. In this latter period, the thickness of the rigid region increases owing to relaxation of the velocity gradients and the stress. The yield surface eventually collides with the plate in finite time, after which the entire material moves in concert with the plate as a rigid body. Analytical and numerical results are presented that can form the basis for practical applications. This includes the development of a ‘rheological microscope’ to directly detect and measure the yield stress. The analysis focuses on exploring the flow physics of a Bingham material. This is shown to be similar to that of a general Herschel–Bulkley material with shear-thinning or shear-thickening rheology.

The motion of a layer of yield-stress material on an oscillating plate

“…(2009) and Hinton et al. (2022); for simplicity, this will henceforth be referred to as the ‘arbitrary- numerical solution’. The velocity field is then computed via the momentum equation (2.5).…”

“…These equations constitute a Stefan problem with an unknown free boundary , which can be solved numerically for any value of ; see Hinton et al. (2022).…”

“…The creation and destruction of yield surfaces is a dynamic process, and the yield surfaces can move through the material, driven by inertia (Burgess & Wilson 1997; Huilgol 2004). Hinton, Collis & Sader (2022) studied the dynamics of the viscoplastic Stokes’ first problem where a yield-stress material sits atop a solid plate that moves suddenly. Unlike the Newtonian case, in which Stokes’ first and second problems can be related via a linear transformation in time, the inherent nonlinearity generated by the yield stress of a viscoplastic material means that solutions for these two settings must be sought independently.…”

Clogged corners

“…Balmforth & Craster 1999), Stokes’ first and second problems (impulsive lateral motion, or oscillation, of one boundary, respectively; e.g. Balmforth, Forterre & Pouliquen 2009; Hinton, Collis & Sader 2022), flow around a falling cylinder (e.g. Tokpavi, Magnin & Jay 2008) and slender-body theory (e.g.…”

“…Balmforth & Craster 1999), Stokes' first and second problems (impulsive lateral motion, or oscillation, of one boundary, respectively; e.g. Balmforth, Forterre & Pouliquen 2009;Hinton, Collis & Sader 2022) The flow of material past a corner is, of course, a commonplace occurrence. Materials with a yield stress tend to become clogged in the vicinity of corners where the stress is insufficient to yield them; similar clogging occurs near other geometrical features like side branches and local obstructions or expansions.…”