Shallow viscoplastic flow on an inclined plane

Abstract

Evolving viscoplastic flows upon slopes are an important idealization of many flows in a variety of geophysical situations where yield stress is thought to play a role. For such models, asymptotic expansions suitable for slowly moving shallow fluid layers (lubrication theory) reduce the governing equations to a simpler problem in terms of the fluid thickness. We consider the version of the theory for fluids described by the Herschel-Bulkley constitutive law, and provide a variety of solutions to the reduced equation, both numerical and analytical. For extruded inclined domes, we derive the characteristic temporal behaviour of measures of the dome's dimensions.

The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

“…and When S > 0 and in the limiting case of the Newtonian (n = 1 and Bi = 0) and power-law (Bi = 0) fluids, this evolution equation does not admit similarity or other exact solutions and so must be integrated numerically; when S = 0, similarity solutions can be worked out [41]. Fig.…”

“…An alternative approach is lubrication theory, which takes its roots in Reynolds' pioneering work. The theory is based on an approximation to the governing equations for shallow slopes and thin low-inertia flows through an asymptotic expansion in the aspect ratio ε = H * /L * , with 0377 H * and L * being the flow-depth and length scales, respectively [13,24,[33][34][35][36][37][38][39][40][41]. As pointed out by Balmforth et al [40], this theory can be extended to steep slopes by changing the scaling that underpins the asymptotic reduction of the local equations.…”

Experimental investigation of the spreading of viscoplastic fluids on inclined planes

“…While much of the earlier work has focused on time-dependent flows of viscous fluids over a rigid boundary [14][15][16][17], a growing attention has been paid to the corresponding problem with viscoplastic fluids from the theoretical point of view [18][19][20][21][22][23][24][25][26][27][28][29]. On rare occasions, exact or asymptotic analytical solutions to the governing equations can be worked out [10,21,[30][31][32][33][34], but most of the time, solutions must be computed numerically using flow-depth averaged equations of motion (the equivalent of the shallow-water equations in hydraulics) [35][36][37], nonlinear diffusion equations when inertial terms are negligible [19,21], or the full set of equations of motion (using a finite-element approach or smooth-particle-hydrodynamics techniques).…”

“…On rare occasions, exact or asymptotic analytical solutions to the governing equations can be worked out [10,21,[30][31][32][33][34], but most of the time, solutions must be computed numerically using flow-depth averaged equations of motion (the equivalent of the shallow-water equations in hydraulics) [35][36][37], nonlinear diffusion equations when inertial terms are negligible [19,21], or the full set of equations of motion (using a finite-element approach or smooth-particle-hydrodynamics techniques). Surprisingly, despite the substantial interest in the spreading of viscoplastic fluids, there have been to date very few experimental investigations reporting the flow behavior of a finite volume of viscoplastic fluids down a surface.…”

Viscoplastic dam breaks and the Bostwick consistometer

“…The theory is based on a shallow, slow approximation to the governing equations (e.g. [2,3,13]) which offers a compact description to explore the slump of a viscoplastic fluid described by the Herschel-Bulkley constitutive law. In the shallow, slow approximation, the fluid flow is controlled by the vertical shear stress and the side walls of the channel play no role; thus, the dam break becomes equivalent to the release of a two-dimensional sheet of fluid.…”