Visco-plastic models of isothermal lava domes

Abstract

The dynamics of expanding domes of isothermal lava are studied by treating the lava as a viscoplastic material with the Herschel-Bulkley constitutive law. Thin-layer theory is developed for radially symmetric extrusions onto horizontal plates. This provides an evolution equation for the thickness of the fluid that can be used to model expanding isothermal lava domes. Numerical and analytical solutions are derived that explore the effects of yield stress, shear thinning and basal sliding on the dome evolution. The results are briefly compared with an experimental study. It is found that it is difficult to unravel the combined effects of shear thinning and yield stress; this may prove important to studies that attempt to infer yield stress from morphology of flowing lava.

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

“…Field data and comparison with historical events have not settled this controversial issue [19][20][21][22][23] since traces left by debris flows could be interpreted using viscoplastic theory, whereas other clues argue in favor of a Coulomb behavior. The same difficulties arise in the rheology of lava, with an additional degree of complexity induced by temperature and phase changes [6,[24][25][26][27].…”

“…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).…”

Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case

“…Furthermore, this model was derived through an asymptotic expansion where the slope is supposed to be small (α 1) and the norm of the gradient of b(x) is small ( ∇ x b 1). But it is worth noting that this model is also valid for a slope α = 0 (horizontal bottom), which is not generally the case for other models proposed in the literature (see for example [3], [18]). Another interesting feature of the model is that in the case of a plane horizontal slope (α = 0) and with a vanishing yield stress (τ y = 0), we recover a viscous shallow water system which has the same structure as the one derived by Gerbeau and Perthame in [21] (note that the hypothesis of friction at the bottom, instead of a no-slip condition is a key point in this degeneracy to [21]).…”