Shear deformation of partially molten rock in laboratory experiments causes the emergence of melt-enriched sheets (bands in cross-section) that are aligned at about 15-20 • to the shear plane. Deformation and deviatoric stress also cause the coherent alignment of pores at the grain scale. This leads to a melt-preferred orientation that may, in turn, give rise to an anisotropic permeability. Here we develop a simple, general model of anisotropic permeability in partially molten rocks. We use linearised analysis and nonlinear numerical solutions to investigate its behaviour under simple-shear deformation. In particular, we consider implications of the model for the emergence and angle of melt-rich bands. Anisotropic permeability affects the angle of bands and, in a certain parameter regime, it can give rise to low angles consistent with experiments. However, the conditions required for this regime have a narrow range and seem unlikely to be entirely met by experiments. Anisotropic permeability may nonetheless affect melt transport and the behaviour of partially molten rocks in Earth's mantle. arXiv:1505.00559v2 [physics.geo-ph] 5 Sep 2015 whether the low angles could arise as a consequence of a permeability that is directionally dependent (i.e. anisotropic).Anisotropic permeability could be a consequence of another empirically known feature of partially molten rocks subjected to deviatoric stress: the microstructural alignment of interconnected pockets of melt between solid grains. This is called melt-preferred orientation (MPO) and has been observed in many laboratory studies [e.g. Bussod and Christie, 1991, Daines and Kohlstedt, 1997, Takei, 2010. The alignment may be attributed to the instantaneous state of deviatoric stress [Daines and Kohlstedt, 1997, Takei andHoltzman, 2009a], or to the combined effects of finite strain, lattice-preferred orientation, and anisotropic surface energy of olivine grains [Bussod and Christie, 1991, Daines and Kohlstedt, 1997, Jung and Waff, 1998; it is likely some combination of the two. Since the Darcian permeability of partially molten rocks arises from the shape and interconnectedness of melt pockets at the grain scale [e.g. Bear, 1972, Scheidegger, 1974, it is reasonable to assume that the anisotropic alignment of pores between grains leads to anisotropy in permeability. Daines and Kohlstedt [1997] estimated this anisotropy as a function of differential stress and found that permeability in the direction parallel to the maximum compressive stress σ 1 was enhanced by a factor of up to 15 over that parallel to the direction of maximum tensile stress. This is consistent with a theoretical model for anisotropy of permeability due to MPO by Hier-Majumder [2011].Since both melt-banding at the macroscopic scale and melt-preferred orientation at the microscopic scale emerge under the same physical conditions, it is logical to ask whether their dynamics are linked. In particular, the question we address here is whether the low angle of high-porosity bands observed in experiments [see s...

When viscous fluid in a corner is disturbed, eddies can form in the absence of inertia. Examples of flow configurations in which this motion occurs include flow through an abrupt contraction and over a cavity. Six decades ago, Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18) calculated the slow viscous flow of Newtonian fluids in sharp corners, detailing his eponymous ‘Moffatt eddies’. In this study we examine corner flows of viscoplastic materials, a class of non-Newtonian fluids which exhibit solid-like behaviour for stresses below a yield stress. Specifically, we consider a Bingham fluid, for which the material is perfectly rigid at stresses below the yield stress. While a static unyielded plug forms at the tip of the corner, eddies analogous to those found by Moffatt can also form. We examine these viscoplastic eddies numerically, by computing finite element solutions using the augmented-Lagrangian method, and analytically, by employing a viscoplastic boundary layer formulation and scaling arguments. We measure the depth of the static plug as a function of the Bingham number (dimensionless yield stress), show that the process of a new eddy forming as the Bingham number is decreased is driven by the pressure in the yielded fluid adjacent to the static plug, and provide a heuristic argument for the critical Bingham number at which this occurs.

Abstract

The incompressible motion of viscoplastic fluid between two semi-infinite rigid plates, hinged at their ends and rotating towards one another at constant angular velocity, generates self-similar flow fields because there is no externally imposed length scale in the absence of inertia. The magnitude of the strain rate scales with the angular velocity of the plates and the dimensionless deviatoric stresses are functions only of the polar angle and a dimensionless measure of the yield stress; they are independent of the radial distance from the corner. These flows feature unyielded regions adjacent to the boundaries for sufficiently large angles between the plates. Moreover, when the dimensionless yield stress is large, there are viscoplastic boundary layers that are attached to the boundary or the plug, the asymptotic structures of which are constructed.