Porous Gravity Currents of Non-Newtonian Fluids within Confining Boundaries

Abstract

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 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. 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∼ tF2 where F2 is a scalar depending on (i) the time exponent of the volume of fluid in the current, α, (ii) the geometry of the channel, parameterized by β and (iii) the exponent n of the rheological equation. It is found that for a critical value αc = n/(n + 1), F2 is independent on the shape of the channel; for α < αc, F2 is a decreasing function of β; the reverse is true for α > αc. Upon comparing results with free-surface viscous flow in open channels, it is found that: (i) the same expression for αc holds; (ii) the exponent F2 increases or decreases monotonically with β, while for the triangular section (β = 1) in open channels, a maximum or minimum value of F2 is attained for α < αc and α > αc, respectively.