The method of the ‘carry over’ of integral transforms in non-linear mass and heat transfer problems

Polyanin, Andrei D.; Дильман, В. В.

doi:10.1016/0017-9310(90)90151-j

Cited by 11 publications

(9 citation statements)

References 2 publications

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“…This kind of heat flux condition was also considered in several papers, e.g. Barber [2], Coelho Pinheiro [7], Polyanin, Dil'man [16], Rogers [18].…”

Section: Introductionmentioning

confidence: 95%

Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases

Briozzo

Natale

Tarzia

2007

Journal of Mathematical Analysis and Applications

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A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x = 0, a constant temperature or a heat flux of the form −q 0 / √ t (q 0 > 0). The internal heat source functions are given byare functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, a 2 j is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x = 0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where β j (j = 1, 2) are of exponential type given by β j (x) = exp(−(x + d j ) 2 ) with x and d j ∈ R is also studied in details for both boundary temperature conditions at x = 0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x = 0 is considered; a similar result is obtained for a heat flux condition imposed on x = 0 if an inequality for parameter q 0 is satisfied.

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“…This kind of heat flux condition was also considered in several papers, e.g. Barber [2], Coelho Pinheiro [7], Polyanin, Dil'man [16], Rogers [18].…”

Section: Introductionmentioning

confidence: 95%

Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases

Briozzo

Natale

Tarzia

2007

Journal of Mathematical Analysis and Applications

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“…12 Briozzo-Natale-Tarzia, "An Explicit Solution..." MAT-Serie A, 8 (2004), [11][12][13][14][15][16][17][18][19] In [9] the one-phase Lam ¶ e-Clapeyron-Stefan problem [7] with a particular type of sources was studied and a generalized Lam ¶ e-Clapeyron explicit solution was obtained. Moreover, necessary and su±cient conditions were given in order to characterize the source term which provides a unique solution.…”

Section: Introductionmentioning

confidence: 99%

“…This kind of heat°ux condition was also considered in several papers, e.g. [2], [11], [12] and [15].…”

Section: Introductionmentioning

confidence: 99%

An explicit solution for a two-phase Stefan problem with a similarity exponential heat sources

Briozzo¹,

Natale²,

Tarzia³

2004

MAT

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A two-phase Stefan problem with heat source terms in both liquid and solid phases for a semi-in¯nite phase-change material is considered. The internal heat source functions are given by g j (x; t) = (¡1) j+1 ½l t exp ³ ¡( x 2a j p t + d j ) 2´( j = 1 solid phase; j = 2 liquid phase), ½ is the mass density, l is the fusion latent heat by unit of mass; a 2 j is the di®usion coe±cient, x is spatial variable, t is the temporal variable and d j 2 R. A similarity solution is obtained for any data when a temperature boundary condition is imposed at the¯xed face x = 0; when a°ux condition of the type ¡q 0 = p t (q 0 > 0) is imposed on x = 0 then there exists a similarity solution if and only if a restriction on q 0 is satis¯ed.

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“…This kind of heat°ux condition (3) was also considered in numerous papers, e.g. [2,10,22]. Other problems in this subject are [4,19,23,24].…”

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confidence: 96%

“…Natale-Tarzia, "An Integral Equation..." MAT-Serie A, 7 (2004), [15][16][17][18][19][20][21][22][23][24] The mathematical formulation of our free boundary (fusion process) problem consists in determining the evolution of the moving phase separation x = s(t) and the temperature distribution µ = µ(x; t) satisfying the conditions…”

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confidence: 99%

An integral equation in order to solve a one-phase Stefan problem with nonlinear thermal conductivity

Natale¹,

Tarzia²

2004

MAT

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María F. Natale -Domingo A. Tarzia, "An integral equation in order to solve a one-phase Stefan problem with nonlinear thermal conductivity", 15-24.María C. Sanziel -Domingo A. Tarzia, "Optimization on the heat flux in a mixed elliptic problem with temperature constraints", 25-30.Claudia M. Gariboldi -Domingo A. Tarzia, "A new proof of the convergence of distributed optimal controls on the internal energy in mixed elliptic problems", 31-42.

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The method of the ‘carry over’ of integral transforms in non-linear mass and heat transfer problems

Cited by 11 publications

References 2 publications

Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases

Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases

An explicit solution for a two-phase Stefan problem with a similarity exponential heat sources

An integral equation in order to solve a one-phase Stefan problem with nonlinear thermal conductivity

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