SINGULARITIES AT FLOW SEPARATION POINTS

• Published in: Quarterly journal of mechanics and applied mathematics 1973-01, Vol.26 (2), p.153-172
• Main Author: TAYLER, A. B.
• Format: Article
• Language: English
• ISSN: 0033-5614; 1464-3855
• DOI: 10.1093/qjmam/26.2.153

This paper discusses in detail possible local solutions for the flow in the neighbourhood of the intersection of a free boundary and a rigid boundary. In general only two-dimensional flows are considered and asymptotic expansions for the flow variables and free boundary shape in terms of distance from the flow separation point are obtained. Possible separation angles between the free and fixed boundary are also obtained. For the general unsteady inviscid flow situation with no surface tension five possible regimes can occur depending on the motion of the rigid boundary and the external body forces. Only one regime is properly unsteady with a separation angle which varies in time. In quasi-steady situations with fixed separation angles the free boundary curvatures are singular and the possible forms of these singularities are obtained. These results are applied to the water entry problem and show that for the entry of a two-dimensional wedge, where the self-similar flow is obtained as the large-time limit of an initial-value problem, the separation angle of the self-similar flow must be less than or equal to ⅕π. For the self-similar entry of a cone this regime is not possible and the separation angle can only be zero, ½π, or π. For the general steady viscous flow situation with no surface tension the separation angle is shown to be π and the free boundary has a square root curvature singularity. The inertia terms in the Navier—Stokes equations do not appear in the first two terms of the asymptotic expansion for the stream function. The effects of introducing a small surface tension are also considered.