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1

Analysis of a multiphase free boundary problem

Abdelouahab Ahlem, Bensid Sabri

Opuscula Mathematica

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2024

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Vol. 44, no. 5

631--649

EN

In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution.

2

On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems

Lyaghfouri A.

Journal of Applied Analysis

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2008

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Vol. 14, nr 2

165-181

EN

In this article we prove local interior and boundary Lipschitz continuity of the solutions of a general class of elliptic free boundary problems in divergence form.

3

Differential inequalities for general fluid motions bounded by a free surface

Zadrzyńska E., Zajączkowski W. M.

Journal of Applied Analysis

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2004

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Vol. 10, nr 1

131-168

EN

We consider a motion of a viscous compressible heat conducting fluid of a fixed mass bounded by a free surface. For a local solution of equations describing such a motion we derive some energy-type inequalities which are necessary to prove the global existence of solutions.

4

Free boundary of journal bearing with spiral groove in HDD

Lu P., Xiao D., Zhou G.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, no 3

859-874

EN

In a self-sealing journal bearing with spiral grooves, the bearing gap is not actually fully filled with lubricant. Specially at the seal interfaces, the so-called free boundary between the lubricant and ambient air is formed. A free boundary does not only influence the load-capacity and stability of a bearing, more importantly, it affects the bearing dynamic sealing capability. In this paper, an analytical model and numerical procedure is developed to investigate the free boundary of a journal bearing with spiral grooves. The simulation results are discussed on how the bearing parameters may affect free boundary and its impact on the bearing leakage. The approach establishes a base for precise calculation of performance parameters and optimization design of a journal bearing with spiral grooves in HDD.

5

Approximation of a Solidification Problem

Aboulaich R., Haggouch I., Souissi A.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 4

921-955

EN

A two-dimensional Stefan problem is usually introduced as a model of solidification, melting or sublimation phenomena. The two-phase Stefan problem has been studied as a direct problem, where the free boundary separating the two regions is eliminated using a variational inequality (Baiocci, 1977; Baiocchi et al., 1973; Rodrigues, 1980; Saguez, 1980; Srunk and Friedman, 1994), the enthalpy function (Ciavaldini, 1972; Lions, 1969; Nochetto et al.., 1991; Saguez, 1980), or a control problem (El Bagdouri, 1987; Peneau, 1995; Saguez, 1980). In the present work, we provide a new formulation leading to a shape optimization problem. For a semidiscretization in time, we consider an Euler scheme. Under some restrictions related to stability conditions, we prove an L^2-rate of convergence of order 1 for the temperature. In the last part, we study the existence of an optimal shape, compute the shape gradient, and suggest a numerical algorithm to approximate the free boundary. The numerical results obtained show that this method is more efficient compared with the others.

6

Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting capillary fluids

Zadrzyńska E., Zajączkowski W. M.

Journal of Applied Analysis

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2000

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Vol. 6, nr 2

227-250

EN

In the paper the motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. We prove the local existence and uniqueness of a solution to a problem describing such a motion in anisotropic Sobolev-Slobodetskii spaces. This solution is such that the velocity and temperature belong to (wzór), and density to (wzór).

7

Chemical attack in free boundary domains

Clain S.

Journal of Applied Analysis

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1999

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Vol. 5, nr 1

35-58

EN

This paper presents a mathematical model for a chemical process used to machine cristal as glass or silica. A short physical description is presented from which we draw the mathematical model. We obtain a coupled parabolic equations system on a free boundary domain with a non-linear condition on the boundary. The existence and the uniqueness is proved in the one-dimensional case.