Slow spreading of a sheet of Bingham fluid on an inclined plane

Abstract

To study the dynamics of fluid mud with a high concentration of cohesive clay particles, we present a theory for a thin sheet of Bingham-plastic fluid flowing slowly on an inclined plane. The physics is discussed on the approximate basis of the lubrication theory. Because of the yield stress, the free surface need not be horizontal when the Bingham fluid is in static equilibrium, nor parallel to the plane bed when in steady flow. We then show that there is a variety of gravity currents that can advance at a constant speed and with the same profile. Experimental confirmation of one type is presented. By solving a nonlinear partial differential equation, transient flows due either to a steady upstream discharge or to the sudden release of a finite fluid mass on another fluid layer are studied. In the first case there is a mud front which ultimately propagates as a constant speed as a steady gravity current. In the second case, when the ambient layer is sufficiently shallow that there is no initial motion, the flow induced by the new fluid can terminate after the disturbance has travelled a finite distance. The extent of the final spread is examined. Disturbances due to an external pressure travelling parallel to the free surface are also examined. It is found in particular that a travelling localized pulse of pressure gradient not only generates a localized mud disturbance which travels along with the forcing pressure, but further leaves behind a permanent footprint.

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).…”

Slumps of viscoplastic fluids on slopes

“…Provided the yield stress is exceeded, then to leading order the down-slope velocity exhibits a shearing region close to the basal boundary, overridden by a plug-like region (see, for example [18]). At the yield surface between the two, the velocity field and shear stress are continuous.…”

“…Liu and Mei [18] studied the transient slump into a channel that has a pre-existing layer of fluid and they showed how the slump is progressively slowed and then finally arrested as the driving forces are no longer able to surmount the yield stress. Huang and Garcia [16] analysed the motion along an initially fluid-free channel: their calculations revealed that strong curvatures in the free surface are only found close to the propagating front and this permitted a simplified analysis of the motion in which the front was handled separately and matched to the interior.…”

“…Often these relatively small-scale experiments have employed clay dispersions and have measured the flow speed and runout length, following the release of a volume of material (see, for example [16,[18][19][20][21]). Liu and Mei [18] released mud into a sloping channel and showed that the profile of the material, once the flow had arrested, was quite similar to the theoretical predictions, for which the rheology of the mud was independently determined. Huang and Garcia [16] studied the transient problem by measuring the position of the front of the flow at various times after release and found that its position was reasonably well predicted by their theory.…”

“…The final arrested state in then constructed in Section 3. This has been formulated before (see, for example [2,18,19]). Here we write it in compact form, expressing the profile in terms of a Lambert-W function (see Appendix A).…”