Viscous free surface flow problems

Particular advantages of the finite element method for viscous free surface flow problems are physical boundary conditions... 

From the brief discussion above it is apparent that the flow of viscous liquids in the form of thin films is usually accompanied by various phenomena, such as waves at the free surface. These waves greatly complicate any attempt to give a general theoretical treatment of the film flow problem Keulegan (Kl4) considers that certain types of wavy motion are the most complex phenomena that exist in fluid motion. However, by making various simplifying assumptions it is possible to derive a number of relationships which are of great utility, since they describe the limits to which the flow behavior should tend as the assumptions are approached in practice. 

Welch, J. E., Harlow, F. H., Shannon, J. P, and Daly, B. J., The MAC Method A Computing Technique for Solving Viscous Incompressible Transient Fluid Flow Problems Involving Free Surfaces. Los Alamos Scientific Laboratory Report LA-3425,1965. 

The two boundaries at z = 0 and d can be either rigid walls or a free surface. In fact, we do not need to be so restrictive in our description of the problem. One or both of the boundaries may also be at infinity. The function U(z ) can be considered as an arbitrary function of z. The equations governing U7 and the linear disturbance flow are the dimensional Navier Stokes equations, (12-1) and (12-2), but, in this case, with the viscous terms neglected. 

Given its efficiency, BEM has been applied to a variety of problems involving large deformations of a free surface. Several solutions have been developed for problems related to the nonlinear evolution of water waves, [9-11] and for problems related to nonlinear deformations of both viscous and inviscid drops [12,13]. BEM has been applied to several applications of creeping (Stokes) flows in liquid columns [14-16] and in annular layers [16]. Inviscid solutions have also been obtained for both infinite [16] and finite-length [13] liquid jet problems, as well as for dripping flows [13], fountains [13], and fluid sloshing problems [17]. 

These approaches do not provide the necessary spatial resolution, but, as their advocates claim, reproduce the flow structures observed experimentally. Indeed, the coherent structures in the developed turbulent flow are known to weakly depend on the Reynolds number. However, in the crystal-growth problems one is interested first of all in the flow behavior near the crystal, the cmcible and the free surface since it determines the heat and mass transfer through boundaries. This situation resembles the numerous attempts made about two decades ago to simulate the separated viscous flows using Euler equations such computations could reproduce the overall flow structure if a separation point/line is defined by the geometrical singularity, but could not give the skin friction and the heat flux. 

The diffusion layer thickness is seen to grow as the cube root of the streamwise distance rather than the square root, as in unbounded flow past a surface. Moreover, the growth is independent of the kinematic viscosity and therefore of the Schmidt number, although it does depend on viscosity through Both of these conditions result from the inertia free character of the channel flow itself that is, viscous forces predominate, and the mass density p does not enter the problem. 

First, we consider the influence of the liquid s viscosity on the damping of plane capillary waves on deep water. Suppose the liquid has low viscosity so that viscous effects only manifest themselves inside a thin boundary layer near the interface. Hence, outside the boundary layer, the liquid flow is potential, and the potential is described by the Laplace equation, while the liquid flow near the surface is described by the boundary layer equations with the accompanying condition that the tangent viscous stress at the free interface must be zero. The solution of this problem can be found in [2]. The main difference from the case of a non-viscous liquid is the appearance of a coefficient of the form exp(—jSjt) in the... 

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