Nonlinear Boundary Value Problem for Concave Capillary Surfaces Occurring in Single Crystal Rod Growth from the Melt

Abstract

The boundary value problem z″=((ρ⋅g⋅z−p)/γ)[1+(z′)2]3/2−(1/r)⋅[1+(z′)2]⋅z′, r∈[r1, r0], z′(r1)=−tan⁡(π/2−αg), z′(r0)=−tan⁡αc, z(r0)=0, and z(r) is strictly decreasing on [r1,r0], is considered. Here, 0<r1<r0,  ρ,  g,â�����‰â€‰Î³,  p,  αc,  αg are constants having the following properties: ρ,  g,  γ are strictly positive and 0<π/2−αg<αc<π/2. Necessary or sufficient conditions are given in terms of p for the existence of concave solutions of the above nonlinear boundary value problem (NLBVP). Numerical illustration is given. This kind of results is useful in the experiment planning and technology design of single crystal rod growth from the melt by edge-defined film-fed growth (EFG) method. With this aim, this study was undertaken.