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.
Similar content being viewed by others
Abbreviations
- A L, An Ān, AS :
-
constants defined in equations (8), (26), (46) and (1) respectively
- B L, Bn, BS :
-
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 /n) L,S (X),n = 1,2,3:
-
initial temperatures defined in equation (22), temperature/T m
- H :
-
enthalpy/ρc STm
- k :
-
thermal diffusivity, m2 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)
- α:
-
defined by α2=kt d/L 2
- λ:
-
l/c STm
- ρ:
-
density which is equal in all the phases, kg/m3
- ΔX :
-
mesh size
- Δy :
-
time step for determining liquid/mush boundary
- L :
-
liquid region
- M :
-
mushy region
- S :
-
solid region
References
Crowley, A. B. and Ockendon, J. R., A Stefan problem with a non-monotone boundary.J. Inst. Maths. Applics. 20 (1977) 269–281.
Atthey, D. R., A finite difference scheme for melting problems.J. Inst. Maths. Applics. 13 (1974) 353–366.
Crank, J. and Gupta, R. S., A moving boundary problem arising from the diffusion of oxygen in absorbing tissue.J. Inst. Maths. Applics. 10 (1972) 19–33.
Lacey, A. A. and Shillor, M., The existence and stability of regions with superheating in the classical two-phase one-dimensional Stefan problem with heat sources.IMA Journal of Applied Mathematics 30 (1983) 215–230.
Primicerio, M., Mushy region in phase change problems. In: Gorenflo and Hoffmann (eds),Applied Non-Linear Functional Analysis, Frankfurt (Main): Lang (1982) pp. 251–269.
Atthey, D. R., D.Phil. thesis, University of Oxford (1972).
Lacey, A. A. and Tayler, A. B., A mushy region in Stefan problem.IMA Journal of Applied Mathematics 30 (1983) 303–313.
Ughi, M., A melting problem with a mushy region: Qualitative properties.IMA Journal of Applied Mathematics 33 (1984) 135–152.
Rogers, J. C. W., A free boundary problem as diffusion with non-linear absorption.J. Inst. Maths. Applics. 20 (1977) 261–268.
Hansen, E. and Hougaard, P., On a moving boundary problem from biomechanics.J. Inst. Maths. Applics. 13 (1974) 385–398.
Crank, J. and Gupta, R. S., A method for solving moving boundary problems in heat flow using cubic splines or polynomials.J. Inst. Maths. Applics. 10 (1972) 296–304.
Dahmardah, H. O. and Mayers, D. F., A Fourier-series solution of the Crank-Gupta equation.IMA Journal of Numerical Analysis 3 (1983) 81–85.
Carslaw, H. S. and Jaeger, J. C.,Conduction of Heat in Solids. 2nd edn. Oxford: Clarendon Press (1959).
Gupta, S. C., Axisymmetric solidification in a long cylindrical mold.Applied Scientific Research 42 (1985) 229–244.
Rubinstein, L. I., Crystallization of a binary alloy. In:The Stefan Problem. American Mathematical Society (1967) pp. 52–60.
Fasano, A. and Primicerio, M., General free boundary problems for the heat equation.J. Math. Anal. Appl. I 57 (1977) 694–723.
Murray, W. D. and Landis, F., Numerical and machine solutions of transient heat conduction problems involving melting or freezing.ASME J. Heat Transfer 81 (1959) 106–112.
Gupta, S. C., Numerical and analytical solutions of one dimensional freezing of dilute binary alloys with coupled heat and mass transfer.Int. J. Heat Mass Transfer 33 (1990) 393–602.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gupta, S.C. Moving boundaries due to distributed sources in a slab. Appl. Sci. Res. 54, 137–160 (1995). https://doi.org/10.1007/BF00864370
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00864370