The melting process with moving boundary for mixture consisting of two fluids

Abstract

In this paper, we investigated the melting process with moving boundary (MB) for the solid material occupying the finite region 0≤y*≤l. The melting material is a mixture consisting of two fluids, with constant thermal conductivities are constants. The melting process is on a slab, and the temperature T at the fixed boundary y*=0 is exponentially proportional to time. The MB can be determined by the coordinate transformation ξ=y∗S∗(t∗) first proposed by Landau[1], where, S*(t*) is the position of the moving boundary. We solved the second order two coupled partial differential equations which describe this problem by using the finite-difference scheme[2] which was first applied to a finite-difference scheme by Crank[3, 4]. Using the finite-difference we can transform the partial differential equations to a system of non-linear algebraic equations. The system of non-linear algebraic equations can be linearized by using (Newton's linearization methods). This system is solved with an initial guess Sn+1,0 where, Sn+1,0 is the moving boundary of the zero level (e.g by Taylor's projection method from the two previous time levels) to find Tn+1,1 where, Tn+1,1 is the temperature of the first level, and hence, Sn+1,1 from both of the MB conditions and the scheme aterated until convergence.