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

Abstract

In this paper the boundary value problem:

\begin{equation}\displaystyle\left\{
\begin{array}{l}{\displaystyle z"=\frac{\rho \cdot g\cdot z-p}{\gamma }\cdot \left[1+\left(z'\right)^{2} \right]^{{\raise0.7ex\hbox{$3$}\!\mathord{\left/{\vphantom{3 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ 2 $}}} \quad for\quad x_{1} \le x\le x_{0} \quad \quad \quad }
\\{\displaystyle z'\left(x_{1}\right)=-\tan \left(\frac{\pi}{2} -\alpha_{g} \right)}
\\{\displaystyle z'\left(x_{0}\right)=-\tan \, \alpha _{c} }
\\{\displaystyle z\left(x_{0} \right)=0\; {\rm and\; \; z(x)\; \; is\; strictly\; decreasing\; on}\; \left[{\rm x}_{{\rm 1}} ,\, x_{0} \right]{\rm \; \; \; }}
\end{array}
\right.\
\end{equation}
is considered.

Here: $0<x_{1} <x_{0} , \;\; \rho ,\; g, \gamma ,\; p,\; \alpha
_{c} ,\; \alpha _{g}$ are constants having the following
properties:$\; \rho ,\; g,\; \gamma $ $ $are strictly positive and
$0<{\raise0.7ex\hbox{$ \pi  $}\!\mathord{\left/{\vphantom{\pi
2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ 2 $}}
-\alpha _{g} <\alpha _{c} <{\raise0.7ex\hbox{$ \pi
$}\!\mathord{\left/{\vphantom{\pi
2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ 2 $}}\;
.$

Necessary or sufficient conditions are given in terms of $p$ for the existence of a concave solution of the above nonlinear boundary value problem. The static stability of the concave solution is proven. Numerical example is given. This kind of results can be useful in the experiment planning and technology design of single crystal sheet growth from the melt. With this aim this study was undertaken.