Numerical solution of a Stefan-like problem in Bingham rheology

doi:10.1016/S0378-4754(02)00083-6

Mathematics and Computers in Simulation

Volume 61, Issues 3–6, 30 January 2003, Pages 271-281

https://doi.org/10.1016/S0378-4754(02)00083-6 Get rights and content

Abstract

Many physical as well as geodynamical and technological processes connected with heat and diffusion involving phase-change phenomena lead to solve free boundary problems of Stefan-like type. In this paper, a generalized two-phase Stefan-like problem is formulated and discussed. The method used is based on the weak formulation in enthalpy formulation, on Bingham rheology, on concentration of solid (visco-plastic) and liquid (strongly visco-plastic with very low threshold of plasticity) components and the finite element technique. The existence of the solution of the continuous problem and of its FE approximation are presented and discussed.

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 c_L(1−f_S)+c_Sf_S=c_Lf_L+c_Sf_S, where f_S and f_L represent the relative portion of solid (visco-plastic) and liquid (strongly visco-plastic with low threshold of plasticity) phases and satisfy f_L+f_S=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 $Ω=⋃_{ι=1}^{m} Ω^{ι} ⊂R^{N},N=2(3)$ be a bounded domain (of disjoint subdomains $Ω^{ι}$ ) with a smooth boundary $∂Ω$ , where the boundary $∂Ω$ consists of two disjoint parts Γ_u and Γ_τ. Let I=(t₀,t₁) and let $Ω_{t} =Ω×(t_{0},t),∂Ω_{t} =∂Ω×(t_{0},t),t∈I$ . We suppose that the components of $∂Ω$ are smooth enough to admit a normal $n$ almost everywhere. Moreover, we will assume incompressible materials so that $div u^{ι} =0$ , ι=1,…,m, where $u^{ι} =(u_{i}^{ι})$ is the velocity vector. We will assume that Γ^ι,s,ι=1,…,m,s=1,…,r denote N−1 dimensional open sets

Variational solution of the problem

Let us define the spaces

$^{1} V={z|z∈H^{1} (Ω),z=0$ on $| Γ_{τ}},^{2} V={w|w∈H^{1} (Ω)},V={v | v ∈[H^{1} (Ω)]^{2}, div v =0, v =0$ on $Γ_{u}},^{1} V′,^{2} V′,V′$ be their dual spaces, $^{1} W={z|z∈H^{1} (I;^{1} V),z(x,t_{0})=0},H(Ω)={v | v ∈[L^{2} (Ω)]^{2}, div v =0, v =0$ , on $Γ_{u}},W={v | v ∈L^{2} (I;V), v ′∈L^{2} (I;H), v (x,t_{0})=0},^{2} W={w|w∈H^{1} (I;^{2} V),w(x,t_{0})=0}$ , where we denote $v ′≡∂_{t} v ≡∂ v /∂t$ .

Solving the problem variationally, we will assume [10], [11]:

(A1)
$Ω=⋃_{ι=1}^{m} Ω^{ι} ⊂R^{2}$ is an open bounded domain (of disjoint subdomains $Ω^{ι}$ ) with Lipschitzian boundary $∂Ω=Γ_{u} ⋃Γ_{τ}$ .
On the domain $Ω$ for functions $T ̂^{ι} :R_{+}^{2} →R$

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 $Ω⊂R^{2}$ be approximated by a bounded polygonal domain $Ω_{h} ⊂R^{2}$ . Let ${T_{h}}$ be a family of triangulations $T_{h}$ of $Ω$ such that $Ω ̄ =⋃_{T_{h}∈T_{h}} T_{h}$ , where h is the length of the greatest side of a triangle T_h. We will assume that the family of triangulations $T_{h}$ is regular (see [5]). If $T_{h} ∈ T_{h}$ , we denote by A_i the vertices of the triangle and by B_i 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.

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Numerical solution of a Stefan-like problem in Bingham rheology

Abstract

Introduction

Section snippets

Formulation of the problem

Variational solution of the problem

Numerical solution

Acknowledgements

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Math. Comput. Simulat.

On a numerical approach to Stefan-like problems

Numer. Math.

The bi-dimensional Stefan problem with convection; the time dependent case

Partial Diff. Equations

Analyse numérique d’un probléme de Stefan a deux phases par une méthode d’elements finis

SIAM J. Numer. Anal.