Analysis of stagnation point flow over a stretching/shrinking surface

Abstract

. In this article we analyze the boundary value problem governing stagnation-point fl ow of a fl uid with a power law outer fl ow over a surface moving with a speed proportional to the outer fl ow. The fl ow is characterized by two physical parameters; ε , which measures the stretching ( ε > 0) or shrinking ( ε < 0) of the sheet relative to the outer fl ow, and n > 0, the power law exponent. In the case of aiding fl ow ( ε > 0), where the (stretching) surface and the outer fl ow move in the same direction, we prove existence of a solution for all values of n . For opposing fl ow ( ε < 0), where the (shrinking) surface and the outer fl ow move in opposite directions, the situation is much more complicated. For − 1 < ε < 0 and all n we prove a solution exists. However, for ε (cid:2) − 1, we prove there exists a value, ε crit ( n ) (cid:2) − 1, such that no solutions exist for ε (cid:2) ε crit . For n = 1 / 7 and n = 1 / 3 we prove that ε crit = − 1. For other values of n , we derive bounds which illustrate the complicated nature of the existence/nonexistence boundary for opposing ( ε < 0) fl ows.