Porous gravity currents of non-Newtonian fluids within confining boundaries

Motion of non-Newtonian gravity currents in horizontal impermeable channels filled with a porous material is investigated theoretically and experimentally. A constant or time-variable volume of fluid, characterized rheologically by the Ostwald-de Waele constitutive equation, is released from a point source into a channel of uniform cross-section, whose boundary height is described by a monomial relationship. The mathematical problem is formulated and solved at the Darcy scale coupling the local mass balance equation with a modified Darcy’s law, taking into account the nonlinearity of the rheological equation. The differential problem admits a self-similar solution describing the propagation of the current and its shape. The resulting non-linear ODE is integrated numerically in the general case; for the release of a constant volume, a closed-form analytical solution is derived. Earlier results for Newtonian currents inside confining boundaries and power-law currents in two-dimensional geometry are generalized. The experiments were conducted in a transparent channel of semi-circular cross-section filled with uniform size glass ballotini. A syringe pump or a Mariotte bottle allowed the injection of a constant flux of shear-thinning fluid. The position of the current front, recorded by a photo camera, was generally in a good agreement with the theory. The propagation of the current is described by L∼ t^F2 where F2 is a scalar depending on (i) the time exponent of the volume of fluid in the current, \alpha, (ii) the geometry of the channel, parameterized by \beta and (iii) the exponent n of the rheological equation. It is found that for a critical value \alpha_c = n/(n+1), F2 is independent on the shape of the channel; for \alpha < \alpha_c, F2 is a decreasing function of \beta; the reverse is true for \alpha > \alpha_c. Upon comparing results with free-surface viscous flow in open channels, it is found that: (i) the same expression for \alpha_c holds; (ii) the exponent F2 increases or decreases monotonically with \beta, while for the triangular section (\beta = 1) in open channels, a maximum or minimum value of F2 is attained for \alpha < \alpha_c and \alpha > \alpha_c, respectively.