On A Two-Phase Stefan Problem with Nonlinear Thermal Coefficients ^{∗}

Adriana C. BRIOZZO (1) and Domingo A. TARZIA (1) (2)

(1) Depto. Matemática, FCE,Universidad Austral, Paraguay 1950, S2000FZF Rosario, ARGENTINA. (2) CONICET, ARGENTINA. E-mail: Adriana.Briozzo@fce.austral.edu.ar; Domingo.Tarzia@fce.austral.edu.ar

Abstract

We review some recent results concerning a heat conduction problem with a particular nonlinear thermal coefficients in both solid and liquid phases for a semiinfinite material x > 0, with phase change temperature T_{1}, an initial temperature T_{2} (> T_{1}) and a heat flux of the type q(t) = \frac{q_{0}}{\sqrt{t}} imposed on the fixed face x = 0.

We determine necessary and/or sufficient conditions on the parameters of the problem in order to obtain an instantaneous nonlinear two-phase Stefan problem (solidification process).We also give the corresponding explicit solution.

Resumen: Se da una revisión de algunos resultados recientes que conciernen a una ecuación del calor con particulares coeficientes térmicos no lineales en ambas fases sólida y líquida de un cuerpo semi-infinito x > 0, con una temperatura de cambio de fase T_{1}, una temperatura inicial T_{2} (> T_{1}) y un flujo de calor del tipo q(t) = \frac{q_{0}}{\sqrt{t}} que se impone en el borde fijo x = 0. Se determinan condiciones necesarias y suficientes sobre los parámetros del problema con el objetivo de obtener un problema no lineal instántaneo de Stefan a dos fases (proceso de solidificación). Se da también la correspondiente solución explícita.

Key words: Stefan problem, Instantaneous phase-change problem, Free boundary problem, Nonlinear thermal coefficients, Explicit solution.

Palabras claves: Problema de Stefan, Problema de cambio de fase instántaneo, Problemas de frontera libre, Coeficientes térmicos no lineales, Solución explícita.

AMS Subject classification: 35R35, 80A22, 35C05.

1. INTRODUCTION We consider the two-phase Stefan problem (solidification process)[Ta2] with nonlinear

thermal coefficients for a semi-infinite region x > 0 with phase change temperature T_{1} , an initial temperature T_{2} > T_{1} and an imposed heat flux of the type q(t) = \frac{q_{0}}{\sqrt{t}} , (q_{0} > 0) on the fixed face x = 0. For t > 0 we are going to determine, if there exist, the temperature distribution u(x, t) and the free boundary x = y(t), where

∗MAT - Serie A, 5 (2001), 1-10.

u(x, t) = \{ \begin{matrix}u_{1}(x, t) < T_{1} & 0 < x < y(t) \\ T_{1} & x = y(t) \\ u_{2}(x, t) > T_{1} & x > y(t) \end{matrix} ,  (1.1)

which must verify the following conditions

C_{1}(u_{1})\frac{∂u_{1}}{∂t} = \frac{∂}{∂x} [K_{1}(u_{1})\frac{∂u_{1}}{∂x} ] , 0 < x < y(t) , t > 0 ,  (1.2)

C_{2}(u_{2})\frac{∂u_{2}}{∂t} = \frac{∂}{∂x} [K_{2}(u_{2})\frac{∂u_{2}}{∂x} ] , x > y(t) , t > 0 ,  (1.3)
y(0) = 0  (1.4)
u_{2}(x, 0) = T_{2} > T_{1} , x > 0 ,  (1.5)
u_{1}(y(t), t) = u_{2}(y(t), t) = T_{1} , t > 0 ,  (1.6)
K_{1}(u_{1})\frac{∂u_{1}}{∂x} − K_{2}(u_{2})\frac{∂u_{2}}{∂x} = Ly′(t) , on x = y(t) , t > 0 ,  (1.7)
K_{1}(u_{1}(0, t))\frac{∂u_{1}}{∂x} (0, t) = \frac{q_{0}}{\sqrt{t}} , t > 0 ,  (1.8)

where x : spatial coordinate, t : time, u_{i}(x, t) : temperature distribution for phase i , T_{1} : phase-change or freezing temperature, T_{2} : initial temperature, L : volumetric latent heat , C_{i}(u_{i}) : volumetric heat capacity for phase i, K_{i}(u_{i}) : thermal conductivity for phase i , y(t) : free boundary (solid-liquid interface) at time t, q_{0} : positive given constant which characterizes the heat flux on the fixed face, i = 1 : solid phase , i = 2 : liquid phase . We assume that the volumetric heat capacity and the thermal conductivity for each

phase i (i = 1, 2 ) are relationed as follow :

C_{i}(u_{i}) = \frac{K_{i}(u_{i})c_{0}}{k_{0}a_{i}^{2} [b_{i} − \frac{1}{k_{0}} ^{∫_{0}^{\frac{u_{i}−T_{1}}{T_{2}−T_{1}}}} K_{i} (T_{1} + (T_{2} − T_{1})z) dz]^{2}}  (1.9)

with the assumption given by

\frac{1}{k_{0} (T_{2} − T_{1})} T_{2} ; ∫_{T_{1}} K_{2} (z) dz < b_{2}  (1.10)

where a_{i}, b_{i} (i = 1, 2) are positive constants and k_{0}, c_{0} are scales for the thermal conductivity and volumetric heat capacity respectively. The heat flux condition of the type (1.8) was firstly considered in [Ta1] where an inequality for the coefficient q_{0} was found in order to have an instantaneous two-phase Stefan problem with constant thermal coefficients, for both solid and liquid phases. Other problems in this direction are given by [BrTa1, HiHa, NaTa, Ro1, Ro2, SoWiAl]. The nonlinear relations (1.9) follows from the solidification of iron on a cooper base [TrBr]. Furthermore, these relations imply that the material is of the Storm’s type, that is to say [BrNaTa, HiHa, NaTa, Ro2, St]

\frac{1}{\sqrt{K_{i}(u_{i})C_{i}(u_{i})}} \frac{d}{dT} (log \sqrt{\frac{C_{i}(u_{i})}{K_{i}(u_{i})}}) = \frac{a_{i}}{\sqrt{c_{0}k_{0}}(T_{2} − T_{1})} = const. , i = 1, 2

The goal of this paper is to determine which conditions on the parameters of the problem (in particular q_{0}) must be satisfied in order to have an instantaneous phase-change process.

In Section 2 we consider the associated nonlinear heat conduction problem corresponding to the initial liquid temperature T_{2} and the heat flux condition on x = 0 of the type \frac{q_{0}}{\sqrt{t}} for t > 0. The nonlinear condition between the thermal conductivity heat capacity is supposed to be of the type (1.9). We give a necessary condition for the heat flux input coefficient q_{0}, i.e.

q_{0} > \frac{\sqrt{c_{0}k_{0}}(T_{2} − T_{1})}{a_{2}}Q^{−1} ((k_{0}b_{2}(T_{2} − T_{1}))^{−1} T_{2} ; ∫_{T_{1}} K_{2} (z) dz)  (1.11)

in order to obtain an instantaneous change phase process, where Q is the real function defined by

Q(x) = \sqrt{π}x exp(x^{2})(1 − erf(x)) , x > 0,  (1.12)

with the properties Q(0) = 0, Q(+∞) = 1, Q′(x) > 0, ∀x > 0, where the error function is given by

erf(x) = \frac{2}{\sqrt{π}} x ; ∫_{0} exp(−w^{2})dw .  (1.13)

In the Section 3 we consider the nonlinear two phase Stefan problem (1.2) − (1.8) and we prove that it admits a similarity solution if the condition (1.11) for the coefficient q_{0} obtained in Section 2 is verified. Some of these results are proved in [BrTa2].

2.- A NONLINEAR HEAT CONDUCTION PROBLEM AND ITS INSTAN-TANEOUS PHASE-CHANGE PROCESS.

We consider a semi-infinite slab x ≥ 0 of a material that freezes at temperature T_{1}. We suppose that it is initially hot at the uniform temperature T_{2} > T_{1} and it has nonlinear heat transfer coefficients. However, what happens if a heat flux of the type \frac{q_{0}}{\sqrt{t}} is imposed at x = 0 ? Our interest is found relations among data in order to obtain an instantaneous phase change process, that is the temperature of the material at x = 0 must be less than T_{1} for all positive time. Then, we consider the following nonlinear heat conduction problem corresponding to the initial phase (liquid phase) given by

C_{2}(u)\frac{∂u}{∂t} = \frac{∂}{∂x} [K_{2}(u)\frac{∂u}{∂x}] , x > 0 , t > 0 ,  (2.1)

u(x, 0) = T_{2}, x > 0 ,  (2.2)
K_{2}(u(0, t))\frac{∂u}{∂x} (0, t) = \frac{q_{0}}{\sqrt{t}} , t > 0 ,  (2.3)

where K_{2} and C_{2} have the relation (1.9).

Then the question that follows is: Which conditions must be satisfied for the parameters q_{0}, T_{1}, T_{2}, K_{2} and C_{2} in order to have that the temperature u(0, t) < T_{1,} ∀ t > 0 ? If the answer is affirmative we can assure [SoWiAl, Ta1, TaTu] that the phase-change is instantaneous. Next, we are going to calculate the explicit solution to the problem (2.1) − (2.3), for the liquid phase and, we are going to demonstrate that this solution is constant in (0, t) for all t. Then we can answer affirmatively to the previous question by considering that the condition (1.10) is assumed. In order to obtain that explicit solution for the problem (2.1) − (2.3) we define the new variables and parameters

\{ \begin{matrix}x_{∗} = x\sqrt{\frac{c_{0}}{k_{0}t_{s}}} , & t_{∗} = \frac{t}{t_{s}} \\ u_{∗}(x_{∗}, t_{∗}) = \frac{u(x, t) − T_{1}}{T_{2} − T_{1}} > 0 \\ K_{2∗}(u_{∗}) = \frac{K_{2}(u)}{k_{0}} , & C_{2∗}(u_{∗}) = \frac{C_{2}(u)}{c_{0}}, \end{matrix}  (2.4)

where t_{s} is a time scale. Following [BrTrAv], we consider the Kirchhoff transformation given by

η(x_{∗,}t_{∗}) = μ (u_{∗}(x_{∗}, t_{∗})) =u_{∗}(x_{∗},t_{∗}) ; ∫_{0} K_{2∗} (z) dz  (2.5)

where

μ (Ψ) = Ψ ; ∫_{0} K_{2∗} (z) dz  (2.6)

Next, we define the new variables

\{ \begin{matrix}χ(x_{∗}, t_{∗}) =∫^{x_{∗}} ; 0 \frac{1}{a_{2}(b_{2} − η (z, t_{∗}))}dz , x_{∗} > 0 , t_{∗} > 0 \\ τ = t_{∗} \\ \overline{μ}(χ, τ) = η(x_{∗}, t_{∗}) , χ > 0 , τ > 0 . \end{matrix}  (2.7)

Now we are in condition in order to assume a similarity solution of the type

g(ϕ) = \frac{\overline{μ}(χ, τ)}{θ} , ϕ = \frac{χ}{2\sqrt{τ}} , with θ = 1 ; ∫_{0} K_{2∗} (z) dz,  (2.8)

then our problem reduce to the following conditions

2(ϕ + λ)g′(ϕ) + g′′(ϕ) = 0 , ϕ > 0 ,  (2.9)
g(+∞) = 1 ,  (2.10)

g′(0) = \frac{2λ}{θ} (b_{2} − θg(0))  (2.11)

for the unknown function g, with

q_{0∗} = \frac{q_{0}}{\sqrt{c_{0}k_{0}}(T_{2} − T_{1})} , λ = a_{2}q_{0∗}.  (2.12)

The solution to the equation (2.9) and conditions (2.10)-(2.11) is given by

g(ϕ) = A [erf(ϕ + λ) − erf(λ)] + B , ϕ > 0  (2.13)

where constants A and B are given by the following expressions

A = \begin{matrix}λ & (b_{2} − θ) & \sqrt{π} \\ θ & [exp(−λ^{2}) − \sqrt{π}λ(1 − erf(λ))] \end{matrix} , B = 1 − \frac{λ (b_{2} − θ) \sqrt{π}(1 − erf(λ))}{θ [exp(−λ^{2}) − \sqrt{π}λ(1 − erf(λ))]} .

(2.14)

Then, we obtain the following result.

Theorem 1 The parametric solution to the problem (2.1)-(2.3) is given by

u(x, t) = T_{1} + (T_{2} − T_{1})μ^{−1} (θA (erf(\frac{χ}{2\sqrt{τ}} + λ) − erf(λ)) + θB) ,  (2.15)

with

\{ \begin{matrix}x = a_{2}\sqrt{\frac{k_{0}t_{s}τ}{c_{0}}} \{(b_{2} − B + A erf(λ)) χ − 2A\sqrt{τ} [(\frac{χ}{2\sqrt{τ}} + λ) erf(\frac{χ}{2\sqrt{τ}} + λ) \\ +\frac{1}{\sqrt{π}} exp (−(\frac{χ}{2\sqrt{τ}} + λ)^{2}) − λ erf (λ) − \frac{1}{\sqrt{π}} exp (−λ^{2})]\} , χ > 0 , τ > 0. \\ t = t_{s}τ , τ > 0 \end{matrix} (2.16)

where A, B are defined in (2.14). Moreover, we have that

u(0, t) < T_{1,} ∀ t > 0 ⟺ q_{0} satisfies (1.11).  (2.17)

Proof.- See [BrTa2]. ■

3.- EXPLICIT SOLUTION FOR THE INSTANTANEOUS TWO-PHASE STEFAN PROCESS WITH NONLINEAR THERMAL COEFFICIENTS.

From now on we will consider the problem (1.2)-(1.8) and we will prove that it is well posed for t > 0 when data satisfy condition (1.11). In order to obtain the explicit solution corresponding to the problem (1.2) − (1.8) we will consider the same kind of transformations used for problem (2.1) − (2.3) and we define the new variables and parameters

\{ \begin{matrix}x_{∗} = x\sqrt{\frac{c_{0}}{k_{0}t_{s}}} , & t_{∗} = \frac{t}{t_{s}} \\ u_{1∗}(x_{∗}, t_{∗}) = \frac{u_{1}(x, t) − T_{1}}{T_{2} − T_{1}} < 0 , & u_{2∗}(x_{∗}, t_{∗}) = \frac{u_{2}(x, t) − T_{1}}{T_{2} − T_{1}} > 0 \\ K_{1∗}(u_{1∗}) = \frac{K_{1}(u_{1})}{k_{0}} , & K_{2∗}(u_{2∗}) = \frac{K_{2}(u_{2})}{k_{0}} \\ C_{1∗}(u_{1∗}) = \frac{C_{1}(u_{1})}{c_{0}} , & C_{2∗}(u_{2∗}) = \frac{C_{2}(u_{2})}{c_{0}} \\ L_{∗} = \frac{L}{c_{0}(T_{2} − T_{1})} , & y_{∗}(t_{∗}) = y(t)\sqrt{\frac{c_{0}}{k_{0}t_{s}}} . \end{matrix}  (3.1)

Following [BrTrAv], we consider the Kirchhoff transformation given by

η_{i}(x_{∗,}t_{∗}) = μ_{i} (u_{i∗}(x_{∗}, t_{∗})) = ∫^{u_{i∗}(x_{∗},t_{∗})} ; 0 K_{i∗}(z)dz , i = 1, 2  (3.2)

where

μ_{i} (Ψ) =∫^{Ψ} ; 0 K_{i∗}(z)dz , i = 1, 2 .  (3.3)

Next, we define the new variables through the Storm transformation given by [KnPh, St]

\{ \begin{matrix}χ_{1}(x_{∗}, t_{∗}) =∫^{x_{∗}} ; 0 \frac{1}{a_{1}(b_{1} − η_{1} (z, t_{∗}))}dz , 0 < x_{∗} < y_{∗}(t_{∗}) , \\ χ_{2}(x_{∗}, t_{∗}) = ∫^{x_{∗}} ; y_{∗}(t_{∗}) \frac{1}{a_{2}(b_{2} − η_{2} (z, t_{∗}))}dz , x_{∗} > y_{∗}(t_{∗}) \\ τ = t_{∗} \end{matrix}  (3.4)

and

\overline{μ_{i}}(χ_{i}, τ) = η_{i}(x_{∗}, t_{∗}) , i = 1, 2 .  (3.5)

Then, the free boundary is now given by

S(τ) = χ_{1}(y_{∗}(τ), τ) = ∫^{y_{∗}(τ)} ; 0 \frac{1}{a_{1}(b_{1} − η_{1}(z, τ))}dz.  (3.6)

Owing to the condition on the free boundary and following [TrBr] we have that the interface between the two phases must move as

y_{∗}(t_{∗}) = δ\sqrt{t_{∗}} ,  (3.7)

and the flux of η_{2} on the free boundary takes the explicit form :

\frac{∂η_{2}}{∂x_{∗}} (y_{∗}(t_{∗}), t_{∗}) = \frac{γ}{\sqrt{t_{∗}}} ,  (3.8)

where the positive constants δ and γ must be determined. Now the free boundary S(τ) may be expressed in terms of the transformed coordinates as follows

S(τ) = 2(Λ_{1} − λ_{1})\sqrt{τ} , τ > 0 , Λ_{1} > λ_{1} > 0  (3.9)

where

λ_{1} = a_{1}q_{0∗} ,  Λ_{1} = a_{1}γ + \frac{δ}{2} [\frac{1}{a_{1}b_{1}} + a_{1}L_{∗}] .  (3.10)

Then, the problem take the following form

\frac{∂\overline{μ_{1}}}{∂τ} = \frac{∂^{2}\overline{μ_{1}}}{∂χ_{1}^{2}} + \begin{matrix}λ & 1 \\ √ & \overline{τ} \end{matrix} \frac{∂\overline{μ_{1}}}{∂χ_{1}} , 0 < χ_{1} < S(τ) , τ > 0 ,  (3.11)

\frac{∂\overline{μ_{2}}}{∂τ} = \frac{∂^{2}\overline{μ_{2}}}{∂χ_{2}^{2}} + \begin{matrix}λ & 2 \\ √ & \overline{τ} \end{matrix} \frac{∂\overline{μ_{2}}}{∂χ_{2}} , χ_{2} > 0 , τ > 0 ,  (3.12)
\overline{μ_{2}}(χ_{2}, 0) = θ_{2} , χ_{2} > 0  (3.13)
\overline{μ_{1}}(S(τ), τ) = \overline{μ_{2}}(0, τ) = 0 , τ > 0 ,  (3.14)
\frac{∂\overline{μ_{1}}}{∂χ_{1}} (S(τ), τ)\frac{1}{a_{1}b_{1}} − \frac{∂\overline{μ_{2}}}{∂χ_{2}} (0, τ)\frac{1}{a_{2}b_{2}} = L_{∗} \frac{δ}{2\sqrt{τ}} , , τ > 0 ,  (3.15)
\frac{∂\overline{μ_{1}}}{∂χ_{1}} (0, τ) = \begin{matrix}λ & 1 & (b_{1−}\overline{μ_{1}}(0, τ)) \\ √ & \overline{τ} \end{matrix} , τ > 0 ,  (3.16)

where

λ_{2} = a_{2}γ + \frac{δ}{2a_{2}b_{2}} .  (3.17)

Now we can remark that problem (3.11)-(3.16) is a two-phase Stefan problem with convective terms in both heat equations and a convective boundary condition on the fixed face. From (3.10) and (3.17) we have for the unknowns γ and δ the following relations

γ = \frac{Λ_{1}a_{1}b_{1} − a_{2}b_{2}λ_{2} [1 + a_{1}^{2}b_{1}L_{∗}]}{a_{1}^{2}b_{1} (1 − a_{2}^{2}b_{2}L_{∗}) − a_{2}^{2}b_{2}} , δ = (λ_{2} − a_{2}γ) 2a_{2}b_{2}.  (3.18)

If we assume a similarity solution for the problem (3.11)-(3.17) of the following type

\{ \begin{matrix}g_{1}(ϕ_{1}) = \overline{μ_{1}}(χ_{1}, τ) , ϕ_{1} = \frac{χ_{1}}{2\sqrt{τ}} , \\ g_{2}(ϕ_{2}) = \frac{\overline{μ_{2}}(χ_{2}, τ)}{θ_{2}} , ϕ_{2} = \frac{χ_{2}}{2\sqrt{τ}} , θ_{2} = 1 ; ∫_{0} K_{2∗} (z) dz \end{matrix}  (3.19)

then it reduces to the problem (3.20)-(3.25) given by

2(ϕ_{1} + λ_{1})g′_{1}(ϕ_{1}) + g′′_{1}(ϕ_{1}) = 0 , 0 < ϕ_{1} < Λ_{1} − λ_{1} ,  (3.20)
2(ϕ_{2} + λ_{2})g′_{2}(ϕ_{2}) + g′′_{2}(ϕ_{2}) = 0 , 0 < ϕ_{2} ,  (3.21)
g_{2}(+∞) = 1 ,  (3.22)
g_{1}(Λ_{1} − λ_{1}) = g_{2}(0) = 0  (3.23)
g′_{1}(0) = 2q_{0∗}a_{1} (b_{1} − g_{1}(0))  (3.24)
\frac{g1^{′}(Λ_{1} − λ_{1})}{a_{1}b_{1}} − \frac{g2^{′}(0)θ_{2}}{a_{2}b_{2}} = L_{∗}δ,  (3.25)

for the unknown functions g_{1} and g_{2} , and the unknown coefficients Λ_{1} and λ_{2}. The solution to the equations (3.20)-(3.21) and conditions (3.22)-(3.23) is given by

\{ \begin{matrix}g_{1}(ϕ_{1}) = b_{1} \frac{erf(ϕ_{1} + λ_{1}) − erf(Λ_{1})}{g̃(λ_{1}) − erf(Λ_{1})} , 0 < ϕ_{1} < Λ_{1} − λ_{1} \\ g_{2}(ϕ_{2}) = \frac{erf(ϕ_{2} + λ_{2}) − erf(λ_{2})}{1 − erf(λ_{2})} , 0 < ϕ_{2} \end{matrix}  (3.26)

where

g̃(z) = erf(z) + \frac{1}{\sqrt{π}} \frac{exp(−z^{2})}{z} = g(z, \frac{1}{\sqrt{π}})  (3.27)

and g(z, p) was defined in [BrNaTa] with the following useful properties

g̃(+∞) = 1 , g̃(0) = +∞ , g̃′ (z) < 0 , ∀ z > 0.  (3.28)

Then, the new unknowns coefficients Λ_{1} and λ_{2} must satisfy the following system of equations given by

\{ \begin{matrix}(i) λ_{2} = \frac{a_{1}b_{1}}{a_{2}b_{2} (1 + a_{1}^{2}b_{1}L_{∗})} [Λ_{1} − \overline{A_{1}}G(Λ_{1})] \\ (ii) G(Λ_{1}) = \frac{θ_{2}}{\sqrt{π}a_{1}b_{1}a_{2}b_{2}} F (λ_{2}) \end{matrix}  (3.29)

where

\overline{A_{1}} = a_{1}^{2}b_{1} (1 − a_{2}^{2}b_{2}L_{∗}) − a_{2}^{2}b_{2}  (3.30)

and

\{ \begin{matrix}G(z) = \frac{(1 + a_{1}^{2}b_{1}L_{∗}) exp(−z^{2})}{\sqrt{π}a_{1}^{2}b_{1} [g̃(λ_{1}) − erf(z)]} − L_{∗}z , z > 0 \\ F (z) = \frac{exp(−z^{2})}{1 − erf(z)} , z > 0 . \end{matrix}  (3.31)

In order to obtain the solution to our problem (3.29) we can define λ_{2} = λ_{2}(Λ_{1}) from (3.29(i)).

Taking into account that L_{∗} is from a physical point of view the inverse of the Stefan number we can obtain now the existence theorem of solution in order to have a instantantaneous phase change process for problem (1.2)-(1.8) as a function of this important physical number. Then we have the following result.

Lemma 2 If q_{0} satisfies the inequality (1.11), then the system of equations (3.29) admit a unique solution Λ̃_{1}, λ̃_{2} = λ_{2} \binom{}{Λ_{1}} when data satisfies

\frac{L}{c_{0}(T_{2} − T_{1})} = L_{∗} ≥ \frac{1}{a_{2}^{2}b_{2}} or 0 < \frac{L}{c_{0}(T_{2} − T_{1})} = L_{∗} ≤ max \{0, \frac{1}{a_{2}^{2}b_{2}} − \frac{1}{a_{1}^{2}b_{1}} \}

and at least one solution Λ̃_{1}, λ̃_{2} = λ_{2} \binom{}{Λ_{1}} when

max \{0, \frac{1}{a_{2}^{2}b_{2}} − \frac{1}{a_{1}^{2}b_{1}} \} < \frac{L}{c_{0}(T_{2} − T_{1})} < \frac{1}{a_{2}^{2}b_{2}} .

Proof.- It follows from the properties of functions λ_{2}, G and F obtained in [BrTa2]. ■

Therefore, we have obtained the following theorem in terms of the original data of the problem (1.2)-(1.8), that is:

Theorem 3 If q_{0} satisfies the inequality (1.11) then, an explicit solution to the problem (1.2)-(1.8) is given by

u_{1}(x, t) = T_{1} + (T_{2} − T_{1})μ_{1}^{−1} (b_{1} \frac{erf(\frac{χ_{1}}{2\sqrt{τ}} + λ_{1}) − erf(Λ̃_{1})}{g̃(λ_{1}) − erf(Λ̃_{1})}) ,

0 < χ_{1} < S(τ), τ > 0

u_{2}(x, t) = T_{1} + (T_{2} − T_{1})μ_{2}^{−1} (\frac{erf(\frac{χ_{2}}{2\sqrt{τ}} + λ̃_{2}) − erf(λ̃_{2})}{1 − erf(λ̃_{2})}) ,

χ_{2} > 0 , τ > 0

with

x = 2\frac{\sqrt{\frac{k_{0}t_{s}τ}{c_{0}}} a_{1}b_{1} (\frac{χ_{1}}{2\sqrt{τ}} + λ_{1})}{g̃(λ_{1}) − erf(Λ̃_{1})} [g̃(λ_{1}) − g̃(\frac{χ_{1}}{2\sqrt{τ}} + λ_{1})] , 0 < χ_{1} < S(τ) , τ > 0

x = [a_{2}b_{2} + θ_{2}a_{2} \frac{erf ˜(λ_{2})}{erf c(λ̃_{2})}] χ_{2}−

−\frac{2\sqrt{τ}a_{2}θ_{2}}{1 − erf(λ̃_{2})} [(\frac{χ_{2}}{2\sqrt{τ}} + λ̃_{2}) g̃(\frac{χ_{2}}{2\sqrt{τ}} + λ̃_{2}) − λ̃_{2}g̃(λ̃_{2})] + δ̃\sqrt{τ} , χ_{2} > 0 , τ > 0

t = t_{s}τ , τ > 0

where y(t) = \sqrt{\frac{k_{0}}{c_{0}}} δ̃\sqrt{t} is the free boundary, and the coefficients γ̃ and δ̃ are given by

γ̃ = \frac{Λ̃_{1}a_{1}b_{1} − a_{2}b_{2}λ̃_{2} [1 + a_{1}^{2}b_{1}L_{∗}]}{a_{1}^{2}b_{1} (1 − a_{2}^{2}b_{2}L_{∗}) − a_{2}^{2}b_{2}} , δ̃ = (λ̃_{2} − a_{2}γ̃) 2a_{2}b_{2},

where Λ̃_{1} and λ̃_{2} = λ_{2} \binom{}{Λ_{1}} are the solution of the system (3.29). Moreover this solution is unique when

\frac{L}{c_{0}(T_{2} − T_{1})} = L_{∗} ≥ \frac{1}{a_{2}^{2}b_{2}} or 0 < \frac{L}{c_{0}(T_{2} − T_{1})} = L_{∗} ≤ max \{0, \frac{1}{a_{2}^{2}b_{2}} − \frac{1}{a_{1}^{2}b_{1}} \} .

ACKNOWLEDGMENTS

This paper has been partially sponsored by CONICET - UA (Rosario-Argentina). This financial support was granted to the Project PIP No. 4798/96 ”Free Boundary Problems for the Heat Equation”.

References

[BrTa1] Briozzo, A. C. - Tarzia, D. A, Explicit solution of a free boundary problem for a non linear absorption model of mixed saturated-unsaturated flow, Adv. Water Resources, \underline{21}(1998) , 713-721.

[BrTa2] Briozzo, A. C. - Tarzia, D. A, An Explicit Solution for an Instantaneous Two-Phase Stefan Problem with Nonlinear Thermal Coefficients, Preprint (2000), Submitted to IMA J. Appl. Math.

[BrNaTa] Briozzo,A. C. - Natale, M. F. - Tarzia, D. A., Determination of unknown thermal coefficients for Storm’s type materials through a phase-change process, Int. J. of Nonlinear Mechanics, \underline{34} (1999) , 329-340.

[BrTrAv] Broadbridge, P. - Tritscher, P. - Avagliano, A., Free boundary problems with non-linear diffusion, Mathl. Comput. Modeling, \underline{18} (1993) , 15-34.

[HiHa] Hill, J. M. - Hart, V. G., The Stefan problem in nonlinear heat conduction, J. Appl. Math. Phys. (ZAMP), \underline{37} (1986), 206-229.

[KnPh] Knight, J. H. - Philip, J. R., Exact solution in nonlinear diffusion, J. Engng Math., \underline{8} (1974) , 219-227.

[NaTa] Natale, M. F. - Tarzia, D. A, Explicit solutions to the two-phase Stefan problem for Storm’s type materials, J. Phys. A:Math. Gen., \underline{37} (2000) , 395-404.

[Ro1] Rogers, C. - Broadbridge, P., On a class of moving boundary problems in nonlinear heat conduction: application of a Backlund transformation, Int. J. Nonlinear Mechanics, \underline{21} (1986), 249-256.

[Ro2] Rogers, C., Application of a reciprocal transformation to a two-phase Stefan problem, J. Phys. A: Math. Gen., \underline{18} (1985) , 105-109.

[SoWiAl] Solomon, A. D. - Wilson, D.G. - Alexiades, V., Explicit solutions to phase change problems, Quart. Appl.Math., \underline{51} (1983) , 237-243.

[St] Storm, M. L., Heat conduction in simple metals, J. Appl. Phys., \underline{22} (1951), 940-951.

[Ta1] Tarzia, D. A. , An inequality for the coefficient σ of the free boundary s(t) = 2σ\sqrt{t} of the Neumann solution for the two-phase Stefan problem, Quart. Appl.Math., \underline{39} (1981/82) , 491-497.

[Ta2] Tarzia, D. A., A bibiography on moving-free boundary problems for the heat diffusion equation. The Stefan and related problems, MAT - Serie A, # \underline{2} (2000) , (with 5869 titles on the subject, 300 pages). See www.austral.edu.ar/MAT-SerieA/2(2000)/

[TaTu] Tarzia D. A. - Turner, C. V., A note on the existence of a waiting time for a two-phase Stefan problem, Quart. Appl.Math; \underline{50} (1994) , 1-10.

[TrBr] Tritscher, P. - Broadbridge, P., A similarity solution of a multiphase Stefan problem incorporating general non-linear heat conduction, Int. J. Heat Mass Transfer; \underline{37} (1994), 2113-2121.