Boundary-layer similarity flows driven by a power-law shear over a permeable plane surface

Abstract

The boundary-layer similarity flow driven over a semi-infinite permeable flat plate by a power-law shear with asymptotic velocity profile U ∞(y)=βy α(y→∞,β>0) is considered in the presence of lateral suction or injection of the fluid (y denotes the coordinate normal to the plate). The analytically tractable cases α=−2/3 and α=−1/2 are examined in detail. It is shown that while for α=−2/3 the adjustment of the flow over an impermeable plate to the power-law shear is not possible, in the permeable cases the presence of suction allows for a family of boundary-layer solutions with the proper algebraic decay. The value of the skin friction corresponding to this family of solutions is given by the parameter s=9β3/(4f w ), where f w denotes the suction parameter. In the limiting case of a vanishing suction and a properly vanishing value of the parameter β (such that s=finite), this family of algebraically decaying solutions goes over into the (exponentially decaying) Glauert-jet. In the case α=−1/2, solutions showing the proper algebraic decay were found both for suction ( f w > 0) and injection ( f w <0) in the whole range −∞<f w <+∞. In this case the skin friction parameters s=2β2/3 is independent of the suction/injection parameter f w .