Gravity currents of viscous fluids in a vertically widening and converging fracture

We are investigating flows in the viscous-buoyancy balance regime in a converging channel with an upward increase in the width, with the gap of the channel varying according to a x(k)z(r) power function, being x and z the horizontal and vertical coordinate, respectively, and with 0 < k < 1 and 0 < r < 1 in order to be consistent with the model. The fluid rheology is described according to the Ostwald-de Waele model, with a power-law relationship between shear stress and shear rate and with application for shear-thinning, shear-thickening and, as a special case, Newtonian fluids. While for the case of flow in the direction of widening of the horizontal channel, a self-similar solution of the first kind can be expected, for flow toward the origin, with channel narrowing horizontally, the solution is self-similar of the second kind, with the space and a reduced time coupled in a self-similar independent variable but with an unknown parameter of the transformation group that makes the differential problem invariant. The solution is found in the phase plane by numerical integration of the paths connecting pairs of singular points, separately for the pre-closure phase, in which the current front advances invading the channel, and for the post-closure or leveling phase, in which the fluid has reached the origin and the front no longer propagates, while the level progressively increases canceling the pressure gradient. Integration is performed with a trial and error procedure by modifying the unknown parameter, generally named eigenvalue and specifically critical eigenvalue when the path has been successfully integrated. The overall effect of an increasing permeability upward is that of an increase in the front speed, with the current profile also becoming locally steeper. The effect of an increase in the fluid-behavior index is mixed, as it reduces the speed of the front but still increases the steepness of the local current profile. In any case, the model implies that the eigenvalue tends to infinity for k -> 1 even in the presence of an increase in the vertical permeability (r> 0).