Thermocapillary free boundaries in crystal growth

Abstract

In this paper a two-dimensional free boundary arising from the steady thermo-capillary flow in a viscous incompressible fluid is studied numerically. The problem is considered in the context of the open-boat crystal-growth technique. The motion of the fluid is governed by the Navier-Stokes equations coupled with the heat equation. The problem is solved numerically by a finite-element-method discretization. Three iterative methods are introduced for the computation of the free boundary. The non-dimensional form of the problem gives rise to the following characteristic parameters: Reynolds, Grashof, Prandtl, Marangoni, Bond, Ohnesorge, Biot numbers. The influence of these parameters on the flow field, the temperature distribution and the shape of the free boundary is studied.

Parallel adaptive solution of coupled Rayleigh–Bénard–Marangoni problems with the Navier‐slip

“…where the approximation in Equation (15) is justified by the same arguments as in the Boussinesqtype approximation discussed earlier; see also [58]. A powerful simplifying assumption frequently imposed on these problems is that the liquid/gas interface is flat (or nearly so).…”

“…[58,59]) where, as in the case of the Boussinesq description, we assume * /*T <0 is approximately constant over a wide range of temperatures. Using the model of surface-tension given by Equation (13), we observe that Equations (11) and (12) become…”

Numerical and Experimental Investigations of a Rotating Heat Pipe

“…In Table B.2. In order from top to bottom, the table displays: the average Nusselt number on the hot plate, the maximum Nusselt number on the hot plate followed by the η -location where that maximum occurs, the minimum Nusselt number on the hot plate followed by location, Simulations of a square side-heated cavity with free-surface (without evaporation) boundary conditions at the top surface were also compared to a reference case from the open literature (Cuvelier and Driessen 1986). In the reference calculation, sliding contact points are assumed at the side-walls with ( 0) deflection is also expected in the rotating heat pipe application.…”

“…In the reference calculation, sliding contact points are assumed at the side-walls with ( 0) deflection is also expected in the rotating heat pipe application. Cuvelier and Driessen (Cuvelier and Driessen 1986) show that the surface deflection is determined by the ratio of buoyancy forces to surface tension forces (the Bond number). High buoyancy forces (high accelerations) tend to flatten the interface.…”

An Inverse Finite Element Method for Pure and Binary Solidification Problems

“…Since a local energy balance must also be liquid, 2 c p ϭ c p (x) is the specific heat, and S 1 ʜ S 2 и и и ʜ maintained, the conditions at the interface are S k ϭ S is the boundary of the region ⍀, which includes ampoule, crystal, and melt. T k is the temperature applied L V и n ϭ c ٌ T c и n Ϫ m ٌ T и n for x ʦ ⍀ m ʝ ⍀ c , (4) to the S k boundary, Ͱ is a heat transfer coefficient, and…”