# Moving boundaries due to distributed sources in a slab

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## Abstract

Analytical and numerical solutions have been obtained for some moving boundary problems associated with Joule heating and distributed absorption of oxygen in tissues. Several questions have been examined which are concerned with the solutions of classical formulation of sharp melting front model and the classical enthalpy formulation in which solid, liquid and mushy regions are present. Thermal properties and heat sources in the solid and liquid regions have been taken as unequal. The short-time analytical solutions presented here provide useful information. An effective numerical scheme has been proposed which is accurate and simple.

### Key words

moving boundary mushy region weak enthalpy formulation oxygen-diffusion model### Nomenclature

*A*_{L}, A_{n}Ā_{n}, A_{S}constants defined in equations (8), (26), (46) and (1) respectively

*B*_{L}, B_{n}, B_{S}constants defined in equations (8), (73) and (1) respectively

*c*specific heat, Jkg

^{−1}°C^{−1}*C*_{1}constant defined in equation (4)

*C*_{2}(*V*)defined in equation (4)

*D*_{1},*D*_{2}constants defined in equation (15)

*f*_{L,S}^{/n)}(*X*),*n*= 1,2,3initial temperatures defined in equation (22), temperature/

*T*_{m}*H*enthalpy/ρ

*c*_{S}T_{m}*k*thermal diffusivity, m

^{2}s^{−1}*l*latent heat of fusion, Jkg

^{−1}*L*length of the slab, m

*N*total member of mesh points =

*N*+ 1*Q*heat source in the mushy region, equation (12)

*S*(*t*)sharp melting front,

*X*=*S*(*t*)*S*_{1}(*t*)solid/mush boundary,

*X*=*S*_{1}(*t*)*S*_{2}(*y*)liquid/mush boundary,

*X*=*S*_{2}(*y*)*t*time/

*t*_{d}*t*_{d}variable having dimensions of time,

*s**t*_{e}time at which mushy region disappears

*t**time at which liquid/mush boundary starts growing, equation (70)

*T*temperature/

*T*_{m}*T*_{m}melting temperature, °C

*V*time defined in equation (23)

*X**x*-coordinate/*L**y*time defined in equation (71)

### Greek Symbols

- α
defined by α

^{2}=*kt*_{d}/*L*^{2}- λ
*l*/*c*_{S}T_{m}- ρ
density which is equal in all the phases, kg/m

^{3}- Δ
*X* mesh size

- Δ
*y* time step for determining liquid/mush boundary

### Subscripts

*L*liquid region

*M*mushy region

*S*solid region

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### References

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