On Steady Flow through a Channel Consisting of an Uneven Wall and a Plane Wall : Part 1. Case of No Relative Motion in Two Walls

Lubrication theory analysis of the permeability of rough-walled fractures

“…Although this approximation has been frequently used, quantitative estimates of its accuracy have not been available. By examining the higher-order solutions to the Navier-Stokes equa tions for flow through a sinusoidally-varying channel developed by Hasegawa and Izuchi, 11 we have been able to make some comments regarding this question. Deviations from the permeability predicted under the lubrication approximation seem to become appreciable only when the spatial wavelength of the dominant roughness component becomes on the order of (or smaller than) the amplitude of that roughness.…”

Hydraulic conductivity of rock fractures

“…The zeroth-order solution, corresponding to Re=O and E=O, is identical to the corresponding solution of the lubrication equation for this geometry. Hasegawa and Izuchi (1983) Reynolds number will be smaller than that due to nonzero &>A. Hence Re < 1 seems to be a conservative criterion for the lubrication equation to provide a reasonable approximation to the Navier-Stokes equations, for this particular problem.…”

“…The velocity components u, and u, are nonzero, and are functions of x and z. Following the standard procedure of regular perturbations, Hasegawa and Izuchi (1983) essentially assumed that ux and u, could be expanded as power series in Re and E, inserted these expansions into the Navier-Stokes equations, and then equated the coefficients of each power of Re and E to zero. This approach reduces the nonlinear Navier-Stokes equations to a sequence of linear equations.…”

Conductances between confined rough walls

“…Nevertheless, most of these theoretical investigations were interested in the weakly perturbed limit of large constrictions for which the amplitude of variations is small when compared to the mean diameter. [7][8][9]16,17 On the other hand, some of them bore on the opposite situation of small constrictions, for which the macroscopic behavior of the flow can display singularities, as first shown by Richardson. 10,15,[18][19][20][21] Hence, in this case, it is interesting to develop analytical solutions that can be useful either to match numerical boundary conditions, or to directly provide a good approximation for the macroscopic hydrodynamical behavior.…”