Multiple similarity solutions in boundary-layer flow on a moving surface

Abstract

The similarity equations that arise when there is a power-law outer flow, characterized by the parameter $$\beta $$ , over a surface moving with the same power-law speed, described by the dimensionless parameter $$\lambda $$ , are considered. The critical values $$\lambda _c$$ of $$\lambda $$ are calculated in terms of $$\beta $$ , except in a range $$0.139 \lesssim \beta \lesssim 0.5$$ where there are no critical points. The behaviour of the solution with $$\lambda $$ for representative values of $$\beta $$ is examined, including cases where there are no critical points and one or two critical points leading to two and three solution branches. The asymptotic behaviour for large $$\lambda $$ is derived. For $$-2<\beta <-1$$ , the solution proceeds to large negative values of $$\lambda $$ with this asymptotic limit derived. Aiding flow, $$\lambda >0$$ , shows the existence of additional critical points, with a range $$-2.6583<\beta <-2$$ over which $$\lambda _c$$ takes all values both positive and negative. Relatively weak, $$\lambda =-0.5$$ , and stronger, $$\lambda =-5.0$$ , cases of opposing are treated. The weak case shows two disjoint sections of the solution. For the larger value of $$|\lambda |$$ , one section of the solution in which $$f''(0)$$ decreases monotonically as $$\beta $$ is increased and another section where there is a critical point with two solution branches is seen. In all the cases considered, the solution became singular as $$\beta \rightarrow -2$$ , this limit being discussed.