Young [Philos. Trans. (1805)] derived an equation for the contact angle between a liquid–gas interface and a solid boundary, but doubts have been raised about its validity. This issue is reexamined on the basis of a new integral equation for the interface [J. B. Keller and G. J. Merchant, J. Stat. Phys. 63, 1039 (1991)]. The equation is solved asymptotically by the method of matched asymptotic expansions for small values of the range of intermolecular forces divided by a typical macroscopic length. The leading term in the outer expansion satisfies the Young–Laplace partial differential equation for the interface. The leading term in the boundary-layer expansion satisfies a simplified integral equation. Matching the solutions of these two equations shows that the slope angle at the solid boundary, of the leading term in the outer expansion, is indeed given by the Young equation. Numerical solutions of the boundary-layer integral equation are presented to show how the interface varies near the solid boundary.
Plane stagnation-point flow is modulated in the free stream so that the velocity components are proportional to KH + K cosωt. Similarity solutions of the Navier-Stokes equations are examined using high-frequency asymptotics for K and KH of unit order. Special attention is focused on the steady streaming generated in this flow with strongly non-parallel streamlines. For small modulation amplitude K [les ] KH, unique self-similar streaming flows exist. For large modulation amplitude K > KH, if (K/ω) (K/KH) [ges ] 1.661 no self-similar streaming is possible, while if 4/3 [les ] (K/ω) (K/KH) [les ] 1.661, then multiple steady solutions occur.
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