A. Viviani

Recent developments in vapour crystal growth fluid dynamics

Abstract This paper deals with possible sources of convection arising from non-linear irreversible thermodynamic effects (non-Navier—Stokes fluid dynamics) and slip boundary conditions during typical processes of crystal growth from vapour phase. Attention is focused on the role played by thermal (Burnett) stresses and side-wall temperature creep in single component gases, but the extension and generalization to the more general case of binary mixtures is also indicated. A rigorous non-dimensional order of magnitude analysis is performed to compare these effects to the vapour transport mechanisms usually considered in vapour crystal growth fluid dynamics, e.g. buoyancy and Stefan-Nusselt flow; new characteristic velocities, lengths and corresponding non-dimensional numbers are introduced and discussed, and a priori conditions for the existence and characterization of all possible flow regimes are formulated. Then, a quantitative analysis of natural, thermal stress, and thermal creep convection is given by considering a simplified geometrical configuration, for which an analytical solution is derived.

Double-diffusive boundary layers along vertical free surfaces

Abstract This paper deals with double-diffusive (or thermosolutal) combined free convection, i.e. free convection due to buoyant forces (natural convection) and surface tension gradients (Marangoni convection), which are generated by volume differences and surface gradients of temperature and solute concentration. Attention is focused on boundary layers that form along a vertical liquid-gas interface, when the appropriately defined non-dimensional characteristic transport numbers are large enough, in problems of thermosolutal natural and Marangoni convection, such as buoyancy and surface tension driven flows in differentially heated open cavities and liquid bridges. Classes of similar solutions are derived for each class of convection on the basis of a rigorous order of magnitude analysis. Velocity, temperature and concentration profiles are reported in the similarity plane; flow and transport properties at the liquid-gas interface (interfacial velocity, heat and mass transfer bulk coefficients) are obtained for a wide range of Prandtl and Schmidt numbers and different values of the similarity parameter.

Fluid dynamic modelling of crystal growth from vapour

Abstract Navier-Stokes equations along with no-slip boundary conditions have been widely used, up till now, in modelling vapour crystal growth fluidynamics. In this paper attention is focused on new types of free convection which occur in a gas or a vapour when viscous stresses, due to velocity gradients, are of the same order of magnitude as stresses due to temperature and/or concentration gradients (thermal and/or solutal-stress convection). In this case linear phenomenological relations, based on the classical principles of non-equilibrium thermodynamics, for the diffusive fluxes of momentum, energy and species concentration (conventional Stokes, Fourier, Fick laws; Navier-Stokes fluidynamics) become inadequate and a more general theory must be formulated to account for thermal stresses convection, included in the second-order approximation for the gas-dynamic equations (Burnett equations), and side-wall gas creep, induced by the slip boundary condition in the Knudsen layer. This work deals with the above mentioned gas-kinetic phenomena, presented in the framework of non-equilibrium thermodynamics. Both phenomenological (macroscopic) and kinetic (microscopic) points of view are considered together with the question of the coexistence of these approaches. The Burnett equations are written in non-dimensional form, and a-priori criteria of the non-dimensional order of magnitude analysis lead to the identification of new characteristic velocities and corresponding non-dimensional numbers; new classes of free convection (thermal stress and thermal creep convection) are discussed. The final part of the work is devoted to the evaluation of the orders of magnitude of these new terms during typical crystal growth experiments on Earth and under microgravity, and to the quantitative analysis of thermal stress convection in a simplified geometrical configuration.

Numerical simulation of non-Navier-Stokes fluid-dynamics in vapour crystal growth

Abstract In recent theoretical developments, non-linear irreversible thermodynamic effects, like Burnett stresses and slip boundary conditions, have been recognized as playing an important role under typical conditions encountered during vapour crystal growth in microgravity, and new classes of convection and flow regimes have been identified via a rigourous non-dimensional order of magnitude analysis. In this paper we present a numerical quantitative analysis of these regimes for geometrical configurations of interest in vapour crystal growth. The results are presented for different combinations of the non-dimensional characteristic parameters, ranging from microgravity to earth environment conditions. In each case we give isotherms, stream lines and velocity profiles, and compare the results with the corresponding solution of the Navier-Stokes approximation.

Numerical simulation of surface tension driven flows in liquid bridges

Abstract The paper deals with surface tension driven flows, induced by imposed temperature differences, in axisymmetric liquid bridges. This configuration is connected to processes of crystal growth from melt by the floating zone method. The field equations are solved via a two-dimensional, unsteady, finite difference numerical code in terms of vorticity, stream function and temperature. The Poisson equation for the pressure field is solved, by using Brileys correction, and the free surface shape is computed under the assumption of small capillary number. The flow field structure and behaviour are discussed in terms of non-dimensional characteristic numbers and the aspect ratio A = R L , with R and L, respectively, the radius and the length of the bridge. Stream lines, isotherms, velocity profiles, surface shapes are given and discussed. In particular, we report the maximum of the surface velocity, the maximum of the surface deformation and the bulk heat transfer coefficient (Nusselt number) as functions of the aspect ratio.