Free-boundary problems

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). 

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. 

The solution of (2.3.25) is explicitly constructed as that of the following free boundary problem... 

L. Rubinstein, Free boundary problem for a nonlinear system of parabolic equations, including one with reversed time, Ann. Mat. Pura Appl., 135 (1983), pp. 29-42. 

In the moving-boundary problems treated above, it was assumed that the interface retained its basic initial shape as it moved. It is important to realize that such problems are a subset of a much wider class of problems known as free-boundary problems, in which the boundary is allowed to change its shape as a function of time [2]. A mathematically correct solution for the motion of a boundary of a fixed ideal shape is no guarantee that it is physically realistic. 

Ham further shows that the free-boundary problem, starting with a precipitate particle of negligible dimensions, is not unique, since an arbitrary spheroid will grow at constant eccentricity, its dimensions being... 

Here A is some positive constant. The free boundary problem is now reduced to solving a functional equation for the boundary, expressing the fact that in the domain formed by extension past the free boundary the temperature and its gradient vanish ... 

Ryskin, G., and Leal, L. G., Numerical solution of free-boundary problems in fluid mechanics. Part I. The finite difference technique. J. Fluid Mech. 148,1 (1984a). 

Rudnicki, W.R., and M. Niezgddka, Phase Transitions in Cocoa Butter, in Proceedings of International Conference on Free Boundary Problems, Theory and Application, Gakuto International Series, Mathematical Sciences and Applications,Vol. 14, edited by N. Kenmochi, Gakkotosho, Tokyo, Japan, 2000, pp. 409-417. 

C. M. Elliott, Free boundary problems theory and applications, Boston, 1983, pp. 505-511. 

Short, M.B., Baygents, J.C., Beck, J.W., Stone, D.A., Toomey, R.S. Goldstein, R.E. (2005) Stalactite growth as a free-boundary problem a geometric law and its platonic ideal. Physical Review Letters 94, 018501(4). 

Stuhmiller JH (1977) The Influence of interfacial pressure forces on the character of two-phase flow model equations. Int J Multiphase Flow 3 551-560 Sussman M, Smereka P, Osher S (1994) A Level-Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J Comput Phys 114 146-159. Sussman M, Smereka P (1997) Axisymmetric free boundary problems. J Fluid Mech 341 269-294. 

Finally, we cannot overlook the development of computational tools for the solution of problems in fluid mechanics and transport processes. Methods of increasing sophistication have been developed that now enable quantitative solutions of some of the most complicated and vexing problems at least over limited parameter regimes, including direct numerical simulation of turbulent flows so-called free-boundary problems that typically involve large interface or boundary deformations induced by flow and methods to solve flow problems for complex fluids, which are typically characterized by viscoelastic constitutive equations and complicated flow behavior. 

The first two terms match the result obtained earlier by means of the expansion (6-149) and (6 151) applied to Eq. (6-148). The latter is, however, computed only to terms of 0(Ca/s3), and thus does not contain the third term in (6 185). The effort involved in obtaining (6 185) by means of the domain perturbation technique is, however, greater than the analysis to obtain the result (6-148) by means of the thin-film approach, and the latter does not make any a priori restriction on the shape function h. These observations suggest that the thin-film approach is both simpler and more powerful for this particular class of problems. It should be emphasized, however, that the domain perturbation technique can sometimes yield results when no other approach will work, and it has proven to be an invaluable tool in obtaining analytic solutions for a wide variety of free-boundary problems, both in fluid mechanics and other subjects. 

G. Ryskin and L.G. Leal, Numerical Solution of Free-Boundary Problems in Fluid Mechanics. Part 2. Bouyancy-Driven Motion of Gas Bubble Through a Quiescent Liquid, J. Fluid Mech, 148 (1984). 

A mass transfer model has been developed for the pulse plating of copper into high aspect ratio sub-0.25 micron trenches and vias. Surface and concentration overpotentials coupled with the shape change due to the deposition on the sidewalls and the bottom of the tiench/via with time have been explicitly accounted for in the model. Important parameters have been identified and their physical significance described. The resulting model equations have been solved numerically as a coupled non-linear free boundary problem. A complete parametric analysis has been performed to study the effect of the important parameters on the step coverage and deposition rate. In addition, a linear analytical model has also been developed to obtain key physical trends in the system. 

Not all such flows, however, are linear—as, for example, in the case of non-Newtonian creeping flows around spherical particles (B4a, B4b, Cl, D3, F9, FIO, G5, L8, LIO, Rl, S2, SIO, T4, T7, W2, W3, W3a, W3b, W4, W5, W6, Zl). Similarly, owing to the unknown shape of the interface at the outset, free-boundary problems involving liquid droplets in nonuniform flows (Section II, C, 2, b) are intrinsically nonlinear despite the possible linearity of the equations of motion (and boundary conditions) inside and outside of the droplet. 

Serrin, J.B. (1952). Uniqueness theorems for two free boundary problems. American Journal of Mathematics 74(2) 492-506. 

From mathematical point of view, interface motion is equivalent to the solution of a free boundary problem. The question is to determine a solution for a scalar field (pressure, temperature,... 

In case (a), the same material particles remain on the surface during its motion, and the analysis concerns a film or layer, while in case (b) the analysis is applicable mainly to wave propagation problems and phase transition phenomena. Moreover, in a number of free boundary problems surfaces may be applied in modeling as well as in analysis. 

For closed tube-growth, numerous unsolved problems remain requiring further investigation. Quantitative experiments have shown that these tubes essentially conserve the volume of injected solutions within their expanding structures. In addition, semiquantitative models have been proposed by Thouvenel-Romans et al. [42] and Pantaleone et al. [38] to account for specific features of popping and budding tube growth, respectively. However, there are no quantitative models available today that address tube formation based on detailed reaction and transport processes. A major obstacle toward the study of such models relates to the involved free-boundary problems that affect fluid motion, transport, and possibly reaction rates in a nontrivial fashion. 

The free boundary problem becomes analytically tractable when the lubrication approximation is used. This removes technical but not basic difficulties, and therefore we will use this approximation in further discussion. The lubrication approximation, assumes the characteristic length scale in the vertical direction z (normal to the solid surface) to be much smaller than that in the horizontal (parallel) directions spanned by the 2D vector x. The approximation is applicable in a liquid film with a large aspect ratio, when the interface is weakly inclined and curved. The scaling is consistent if one assumes dz = 0(1), V = 0 y/S) < 1, where V is now the 2D gradient in the plane of the solid support. Then the continuity equation Eq. (39) requires that the vertical velocity v should be much smaller than the horizontal velocity (denoted by the 2D vector u), i.e. V oc y/Su. 

A common feature of the problems to be presented, is the presence of a separating interface between two fluid phases. As discussed in the Introduction the fluid dynamics problem is complicated by the unknown dynamics of the interface this motion must be determined as part of the solution, leading to very difficult nonlinear free boundary problems. It is useful, therefore, to seek models which are subsets of the full equations but which are, at the same time, of practical importance and derivable by rational asymptotic expansions, for instance. 

BRANDT A., C. W. CRYER (1983), "Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems". SIAM. J. Scl. Stat. Comput. vol 4, No. 4 pp 655-684. 

To summarize, the problem for the fully developed spiral is, to lowest order in 6, the following free boundary problem Find a function v(r,e), a function H(r), and constants w and y, satisfying... 

More on the mathematical side, an existence theory should be developed for the free boundary problem, not to speak of computational techniques for handling it. 

The boundary conditions are on the external system surface, a Dirichlet condition or a usual mass-transfer law on the grain-solution interface, a mass-balance equation of dissolution-growth D3C/8r=(l/V-C).dR/dt, characterizing a free-boundary problem. 

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