An analytical solution for the laminar flow over a flat plate with similarity preserving suction

Abstract

An exact analytical solution is presented for the laminar boundary-layer flow over a semi-infinite flat plate subjected to a type of similarity preserving suction. The solution is developed for the case of a plate immersed in either a uniform compressible stream with viscosity proportional to temperature or a uniform incompressible stream with constant viscosity. The problem is formulated in Crocco's variables. It is described by a second-order, non-linear, ordinary differential (and singular) boundary-value problem for the shear stress as a function of the velocity in the boundary layer. A unique solution is shown to exist and to possess a power series representation for all magnitudes of suction. The series is constructed explicitly and provides a transcendental equation for the shear stress at the plate (the important skin friction) which can be solved to any desired accuracy. Examples of upper and lower bounds for the wall shear are presented for several magnitudes of suction and confirm the reasonable accuracy of results obtained heretofore only by numerical solutions of the problem. In addition to the intrinsic value of the technique developed, it can be the basis of accurate checks for the numerical solution of more complex problems.