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.
Solutions for the growth rate of perturbations in the locations of moving steps on a growing or evaporating crystal are presented. They are obtained by solving an equation derived by R. Ghez, H. G. Cohen, and J. B. Keller [J. Appl. Phys. 73, 3685 (1993)] based upon the Burton–Cabrera–Frank theory of crystal growth. They agree with the results derived via the adiabatic approximation when the dimensionless growth rate is small, which shows that those results are correct. However, when the growth rate is large the present exact results differ from those of the adiabatic approximation, as might be expected.
Merchant and Davis performed a linear stability analysis on a model for the directional solidification of a dilute binary alloy valid for all speeds; in their work, the temperature field is decoupled from the analysis. The analysis revealed that, in addition to the Mullins-Sekerka cellular mode, there is an oscillatory instability whose preferred wave number is zero. In this paper, the nonlinear interaction of these two modes in the vicinity of their simultaneous onset is analyzed. A pair of coupled Landau equations is obtained that determines the amplitudes of the modes. Any combination of primary bifurcations may occur. Secondary bifurcations also occur; there exist stable mixed modes and regions of bistability. The results are discussed in terms of experimentally relevant variables.
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