Numerical solution of a Stefan-like problem in Bingham rheology
Introduction
Many physical as well as geodynamical processes connected with heat flow and diffusion involving phase-change phenomena give rise to free boundary problems for parabolic partial differential equations of the multiphase problem of Stefan-like type.
The phenomenon of the physical reality is highly non-linear and it includes bistability, which means that the temperature may not determine the phase. The classical mathematical model of phase transitions (melting and solidification) is known as the Stefan problem. A zone in which components of both phases—solid and liquid—are occurred at the same time is called mushy zone. At the macroscopic scale, such a set is characterized by a liquid and solid fractions cL(1−fS)+cSfS=cLfL+cSfS, where fS and fL represent the relative portion of solid (visco-plastic) and liquid (strongly visco-plastic with low threshold of plasticity) phases and satisfy fL+fS=1, where these ratios are volumic.
Two-phase Stefan problems with convection in the liquid phase are investigated, e.g. in [2], [3], [7], [14], [15], etc. In these problems, the rheology is taken as the Newtonian rheology often times in the Boussinesq approximation. From the optimal approximation point of view as well as from the high performance computation possibility point of view the Bingham rheology is useful, as for the case if the threshold of plasticity ĝ is equal to zero, then we have the usual case of Newtonian rheology and if the threshold of plasticity ĝ tends to infinity then the medium behaves as absolute rigid. Between them we can model all types of visco-plastic materials (as the threshold of plasticity ĝ is determined by the Mises-type relation). Such model problems facilitate to study also the mushy zone [10], [11], [12]. This paper is devoted to the numerical analysis of a Stefan-like problem in Bingham rheology. The Kirchoff transformation and the formulation in enthalpy are used.
Section snippets
Formulation of the problem
Let be a bounded domain (of disjoint subdomains ) with a smooth boundary , where the boundary consists of two disjoint parts Γu and Γτ. Let I=(t0,t1) and let . We suppose that the components of are smooth enough to admit a normal almost everywhere. Moreover, we will assume incompressible materials so that , ι=1,…,m, where is the velocity vector. We will assume that Γι,s,ι=1,…,m,s=1,…,r denote N−1 dimensional open sets
Variational solution of the problem
Let us define the spaces
on on be their dual spaces, , on , where we denote .
Solving the problem variationally, we will assume [10], [11]:
- (A1)
is an open bounded domain (of disjoint subdomains ) with Lipschitzian boundary .
On the domain for functions
Numerical solution
Solving the problem numerically, the conforming finite elements for solving , and the non-conforming finite elements for solving (10) will be used.
Let us suppose that be approximated by a bounded polygonal domain . Let be a family of triangulations of such that , where h is the length of the greatest side of a triangle Th. We will assume that the family of triangulations is regular (see [5]). If , we denote by Ai the vertices of the triangle and by Bi the
Acknowledgements
This research was partially supported by the grant COPERNICUS-HIPERGEOS II-KIT 977006 and by the grant of the Ministry of Education, Youth and Sports of the Czech Republic No. OK-407.
References (16)
Nonlinear analysis of the generalized thermo-magneto-dynamic problem
J. Comput. Appl. Math.
(1995)On a coupled Stefan-like problem in thermo-visco-plastic rheology
J. Comput. Appl. Math.
(1997)Analysis of a coupled system of equations of a global geodynamic model of the Earth
J. Math. Comput. Simulat.
(1999)- et al.
On a numerical approach to Stefan-like problems
Numer. Math.
(1991) - P. Blanc, L. Gasser, J. Rappaz, Existence for a stationary model of binary alloy solidification, M2AM, Model. Math....
- et al.
The bi-dimensional Stefan problem with convection; the time dependent case
Partial Diff. Equations
(1983) - P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam,...
Analyse numérique d’un probléme de Stefan a deux phases par une méthode d’elements finis
SIAM J. Numer. Anal.
(1975)
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