Applied Scientific Research

, Volume 54, Issue 2, pp 137–160 | Cite as

Moving boundaries due to distributed sources in a slab

  • S. C. Gupta
Article

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

AL, An Ān, AS

constants defined in equations (8), (26), (46) and (1) respectively

BL, Bn, BS

constants defined in equations (8), (73) and (1) respectively

c

specific heat, Jkg−1 °C−1

C1

constant defined in equation (4)

C2(V)

defined in equation (4)

D1,D2

constants defined in equation (15)

fL,S/n)(X),n = 1,2,3

initial temperatures defined in equation (22), temperature/Tm

H

enthalpy/ρcSTm

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)

S1(t)

solid/mush boundary,X =S1(t)

S2(y)

liquid/mush boundary,X =S2(y)

t

time/td

td

variable having dimensions of time,s

te

time at which mushy region disappears

t*

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

T

temperature/Tm

Tm

melting temperature, °C

V

time defined in equation (23)

X

x-coordinate/L

y

time defined in equation (71)

Greek Symbols

α

defined by α2=ktd/L2

λ

l/cSTm

ρ

density which is equal in all the phases, kg/m3

ΔX

mesh size

Δy

time step for determining liquid/mush boundary

Subscripts

L

liquid region

M

mushy region

S

solid region

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References

  1. 1.
    Crowley, A. B. and Ockendon, J. R., A Stefan problem with a non-monotone boundary.J. Inst. Maths. Applics. 20 (1977) 269–281.Google Scholar
  2. 2.
    Atthey, D. R., A finite difference scheme for melting problems.J. Inst. Maths. Applics. 13 (1974) 353–366.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    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.Google Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    Atthey, D. R., D.Phil. thesis, University of Oxford (1972).Google Scholar
  7. 7.
    Lacey, A. A. and Tayler, A. B., A mushy region in Stefan problem.IMA Journal of Applied Mathematics 30 (1983) 303–313.Google Scholar
  8. 8.
    Ughi, M., A melting problem with a mushy region: Qualitative properties.IMA Journal of Applied Mathematics 33 (1984) 135–152.Google Scholar
  9. 9.
    Rogers, J. C. W., A free boundary problem as diffusion with non-linear absorption.J. Inst. Maths. Applics. 20 (1977) 261–268.Google Scholar
  10. 10.
    Hansen, E. and Hougaard, P., On a moving boundary problem from biomechanics.J. Inst. Maths. Applics. 13 (1974) 385–398.Google Scholar
  11. 11.
    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.Google Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    Carslaw, H. S. and Jaeger, J. C.,Conduction of Heat in Solids. 2nd edn. Oxford: Clarendon Press (1959).Google Scholar
  14. 14.
    Gupta, S. C., Axisymmetric solidification in a long cylindrical mold.Applied Scientific Research 42 (1985) 229–244.Google Scholar
  15. 15.
    Rubinstein, L. I., Crystallization of a binary alloy. In:The Stefan Problem. American Mathematical Society (1967) pp. 52–60.Google Scholar
  16. 16.
    Fasano, A. and Primicerio, M., General free boundary problems for the heat equation.J. Math. Anal. Appl. I 57 (1977) 694–723.Google Scholar
  17. 17.
    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.Google Scholar
  18. 18.
    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.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • S. C. Gupta
    • 1
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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