Boundary Conditions for Multi-region Method

Studies employing the multi-domain method usually consider solidification problems cooled from the bottom. There are primarily three approaches for treating boundary conditions between the mush-melt and solid-mush interfaces in the multi-domain approach,. The first model to be considered is that used by Worster [2] and Chen et al. [97] in which the Darcy s equation is employed as the momentum equation in the mushy layer and the Navier-Stokes equation as the momentum equation in the fluid layer and no-slip boundary condition is prescribed at the melt-mushy interface. 

Consider a mushy layer that lies above a static solid region and below a semiinfinite fluid region in a binary solution of concentration Coo, and temperature Too, and unidirectional solidification from below. Both the melt/mushy and mushy/solid interfaces move upwards with a constant velocity V. The mushy layer extends form z = 0 to z = h x, y. t). The boundary conditions at the Z - oo are. 

The boundary conditions at the melt/mushy interface Z = haie 

10 Numerical Modeling of Multiphase Flows in Materials Processing 

Since the temperature here is assumed to be the eutectic temperature Te and due to viscosity the non-penetration condition at the solid boundary is satisfied. 

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