Free boundary

The general class of free boundary flow problems can, however, be modelled using the volume of fluid (VOF) approach (Nichols et ai, 1980). The main concept in this technique is to solve, simultaneously with the governing flow equations, an additional equation that represents the unknown boundary. Three different versions of this method are described in the following sections. 

In a fixed two-dimensional Cartesian coordinate system, the continuity equation for a free boundary is expressed as... 

As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the model equations. The developed algorithm for the solution of the described model equations and updating of the free surface boundaries is as follows ... 

The predicted free boundary distributions within a chamber of 0.05 m radius for a fluid with the following physical parameters, rjo = 10 n = 0.25, b - 0.014,... 

We would like to stress at this point that the derivation of (1.36) and (1.38)-(1.39) is connected with the simulation of contact problems and therefore contains some assumptions of a mechanical character. This remark is concerned with the sign of the function p in the problem (1.36) and with the direction of the vector pi,P2,p) in the problem (1.38), (1.39). Note that the classical approach to contact problems is characterized by a given contact set (Galin, 1980 Kikuchi, Oden, 1988 Grigolyuk, Tolkachev, 1980). In contact problems considered in the book, the contact set is unknown, and we obtain the so called free boundary problems. Other free boundary problems can be found in (Hoffmann, Sprekels, 1990 Elliot, Ock-endon, 1982 Antontsev et ah, 1990 Kinderlehrer et ah, 1979 Antontsev et ah, 1992 Plotnikov, 1995). 

The stress free boundary condition (1.45) for crack surfaces implies... 

Baiocchi C., Capelo A. (1984) Variational and quasivariational inequalities. Applications to free boundary problems. Wiley, Chichester. 

Khludnev A. M. (1992) Contact viscoelastoplastic problem for a beam. In Free boundary problems in continuum mechanics. S.N.Antontsev, K.-H. Hoffmann, A.M.Khludnev (Eds.). Int. Series of Numerical Mathematics 106, Birkhauser Verlag, Basel, 159-166. 

Kinderlehrer D., Nirenberg L., Spruck J. (1979) Regularity in elliptic free boundary problems. II Equations of higher order. Ann. Scuola Norm. Sup. Pisa 6, 637-687. 

Plotnikov P.I. (1995) On a class of curves arising in a free boundary problem for Stokes flow. Siberian Math. J. 36 (3), 619-627. 

We have to stress that the analysed problems prove to be free boundary problems. Mathematically, the existence of free boundaries for the models concerned, as a rule, is due to the available inequality restrictions imposed on a solution. As to all contact problems, this is a nonpenetration condition of two bodies. The given condition is of a geometric nature and should be met for any constitutive law. The second class of restrictions is defined by the constitutive law and has a physical nature. Such restrictions are typical for elastoplastic models. Some problems of the elasticity theory discussed in the book have generally allowable variational formulation... 

J. C. Reginato, D. A. Tarzia, and A. Cantero, On the free-boundary problem for the Michealis-Menten absorption model for root growth 2. High concentrations. 

Water-miscible semisolids, such as some gels, in direct contact with an aqueous donor or receptor fluid represent a free boundary system, with transport occurring across a liquid/gel interface. 

This method can, in principle, be used to determine the transport characteristics of drugs dissolved or suspended in pharmaceutical gels. A potential problem with this method is the possibility for the gel to dissolve into the aqueous phase, with resulting disruption of the free boundary. These changing experimental conditions would lead to lack of precision for the experimental results and difficulty in interpreting them. 

Capillaries and tubes (free boundary method) Diffusion coefficient determination 1,2... 

Liquid/liquid stirred cell (free boundary method) Mass transport between immiscible phases 6-8... 

Rocking device with two- and three-phase cells (free boundary method) Nonemulsifying method for determining partition coefficient 9,10... 

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