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

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## 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.

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## References

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