Nonlinear boundary value problem for concave capillarysurfaces occurring in single crystal ribbon growth from the melt

摘要

In this paper the boundary value problem: or xi <.)c < xo { Z" = P.gz—P [1+ (2)2] 3/2z' (xi )— tan 2 (-7c — ag) (xo) = — tan a, z (x0) = 0 and z(x) is strictly decreasing on [xi, xo] is considered. Here: 0 < xl , g, y, p, etc, ag are constants having the following properties: p, g, y are strictlypositive and 0 < 71/2 — ag < ac < 11/2 . Necessary or sufficient conditions are given in terms of p for the existence of a concave solution of theabove nonlinear boundary value problem. The static stability of the concave solution is proven. Numericalexample is given. This kind of results can be useful in the experiment planning and technology design ofsingle crystal sheet growth from the melt. With this aim this study was undertaken.