The motion induced by the orthogonal stretching and shearing of a membrane beneath a quiescent fluid

The flow induced above an impermeable membrane undergoing orthogonal linear stretching and orthogonal linear shearing is investigated. For an exact solution of the Navier-Stokes equations, the orthogonal shearing motions must be related through the constant sigma = gamma delta, where gamma and delta are the dimensionless streamwise and transverse shear rates, respectively. The resulting similarity reduction leads to three nonlinearly coupled ordinary differential equations governed by sigma and the ratio of membrane stretch rates beta. All possible solutions of these equations are found either numerically or, in special cases, analytically. Features of the sigma = 0 solutions at beta = 0 and asymptotically as beta -> a are found to be in excellent agreement with numerical calculations. An aside calculation shows that orthogonal shearing in the absence of any plate stretching cannot exist. However, shearing in one coordinate direction is possible as long as the membrane stretches in at least one direction with the caveat that there exists uniform suction through a porous membrane.