Viscous free surface flow

F. M. Orr and L. E. Scriven, Rimming Flow Numerical Simulation of Steady, Viscous, Free-surface Flow with Surface Tension, J. Fluid Meek, 84, 145-165 (1978). 

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

Kistler, S. 1983, The Fluid Mechanics of Curtain Coating and Related Viscous Free Surface Flows with Contact Lines, Ph.D. Thesis, University of Minnesota. 

Capillary number Ca py a viscous force surface-tension force Two-phase flows, free surface flows... 

Free-surface flow is a limiting case of flow with interfaces, in which the treatment of one of the phases is simplified. For instance, for some cases of gas-liquid flow, we may consider the pressure pgas in the gas to depend only on time and not on space and the viscous stresses in the gas to be negligible. For such flows the jump condition formulation must be used, since the bulk momentum equation breaks down. The jump conditions become boundary conditions on the border of the liquid domain [245, 183[ ... 

Dimitrovova Z, Advani SG. Analysis and characterization of relative permeability and capillary pressure for free surface flow of a viscous fluid across an array of aligned cylindrical fibers. Journal of Colloid and Interface Science, 2002 245(2) 325-337. DOI http //dx.doi.org/10.1006/jcis.2001.8003. 

Nonlinear dynamics and breakup of free-surface flows is rewieved in the recent paper by Eggers (1997). The thin film rupture is considered also by Ida and Miksis (1996) and these authors [Ida and Miksis (1995)] are considered also the dynamics of a lamella in a capillary tube. A wavy free-surface flow of a viscous film down a cylinder is considered by A.L. Frenkel (1993). 

Nickell R E,Tanner R I and Caswell B (1974), The solution of viscous incompressible jet and free-surface flows using flnite-element methods ,/oitmaZ of Fluid Mechanics, 65,189-206. 

An important quantity determining the nature of the bubble flow considered by Yang et al. is surface tension, which often plays a dominant role in free-surface micro flows. However, viscous forces are also important in many cases. Hence the ratio of the viscous force and the surface tension force ... 

Harlow, F. H., Welsh, J. E., Numerical calculation of time dependent viscous incompressible flow with free surface,... 

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. 

The velocity distribution equation (27) indicates that in the absence of surface tension effects the maximum velocity in a film flowing in a flat channel of finite width should occur at the free surface of the film at the center of the channel. The surface velocity should then fall off to zero at the side walls. However, experimental observations have shown (BIO, H18, H19, F7) that the surface velocity does not follow this pattern but shows a marked increase as the wall is approached, falling to zero only within a very narrow zone immediately adjacent to the walls. The explanation of this behavior is simple because of surface tension forces, the liquid forms a meniscus near the side walls. Equation (12) shows that the surface velocity increases with the square of the local liquid depth, so the surface velocity increases sharply in the meniscus region until the side wall is approached so closely that the opposing viscous edge effect becomes dominant. 

F. H. Harlow and J. E. Welch, Numerical Calculation of Time Dependent Viscous Incompressible Flow of Fluid with Free Surface, Phys. Fluids, 8, 2182-2189 (1965). 

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. 

Bernoulli s general theorem applied to the field consisting of the upstream reservoir, the die and the free surface of the extruded rod, shows that [33] the head loss in the isochoric flow is the sum of two terms. The first term is the usual volume term, responsible for the pressure loss in classical fluid mechanics. For purely viscous materials, this term represents the power dissipated due to viscosity, in the whole volume of the flowing fluid. The second term is representative of the energy dissipated along the surface of the walls. Its value is... 

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. 

By (5.9.20), the no-slip and no-flow conditions hold on the hard surface, and a linear temperature distribution is maintained. Condition (5.9.21) says that the no-flow condition on the free surface and the condition of zero heat flux through the free surface must hold, and the balance of tangential thermocapillary and viscous stresses must be provided. Taking into account the quadratic dependence (5.9.19) of the surface tension on temperature, we rewrite the right-hand side of the last condition in (5.9.21) using the relation... 

The motion of plastic fluids with finite yield stress to has some qualitative specific features not possessed by nonlinearly viscous fluids. Let us consider a layer of a viscoplastic fluid on an inclined plane whose slope is gradually varied. It follows from (6.2.5) that, irrespective of the rheological properties of the medium, the tangential stress decreases across the film from its maximum value Tjnax = pgh sin a on the solid wall to zero on the free surface. Therefore, a flow in a film of a viscoplastic fluid can be initiated only when the tangential stress on the wall becomes equal to or larger than the yield stress to ... 

Orr E.M. Jr 1976. Numerical Simulation of Viscous Flow with a Free Surface (by Galerkin s Method with Finite Element Basis Eunctions). [LES]... 

To determine the critical thickness at which the film breakdown is possible, consider the forces acting on the film in the case of transversal flow. There is a dynamic pressure from the gas, which causes film accumulation in the rear part of the string. Forces of viscous friction at the wall and forces of surface tension at the free surface try to impede the breakdown of the film. Therefore, for the breakdown to happen, the following inequality should be satisfied ... 

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