# Explicit Solutions for the Solomon-Wilson-Alexiades’s Mushy Zone Model with Convective or Heat Flux Boundary Conditions

Authors: Domingo A. Tarzia
Journal: Journal of Applied Mathematics
 (2015)

---

## Abstract

We complete the Solomon-Wilson-Alexiades’s mushy zone model (Solomon, 1982) for the one-phase Lamé-Clapeyron-Stefan problem by obtaining explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get the explicit solutions for both cases which is new with respect to the original model. Moreover, when these conditions are satisfied, the two phase-change problems are equivalent to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces of the model is also obtained.


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## Body

## 1. Introduction

Heat transfer problems with a phase-change such as melting and freezing have been studied in the last century due to their wide scientific and technological applications [1–9]. A review of a long bibliography on moving and free boundary problems for phase-change materials (PCM) for the heat equation is shown in [10]. The importance of obtaining explicit solutions to some free boundary problems was given in the works [11–26].We consider a semi-infinite material, with constant thermal coefficients, that is initially assumed to be liquid at its melting temperature which is assumed to be equal to 0°C. At timet
=
0, a heat flux or a convective boundary condition is imposed at the fixed face x
=
0, and a solidification process begins where three regions can be distinguished [27, 28]:(H1)
liquid region at the temperature 0°C, inx
>
r
(
t
), t
>
0;
(H2)
solid region at the temperatureT
(
x
,
t
)
<
0, in 0
<
x
<
s
(
t
), t
>
0
(
w
i
t
h
s
(
t
)
<
r
(
t
)
);
(H3)
mushy region at the temperatureT
(
x
,
t
)
=
0, in s
(
t
)
≤
x
≤
r
(
t
), t
>
0. The mushy region is considered isothermal and we make the following assumptions on its structure:
(H3i)
the material contains a fixed portionε
l (with 0
<
ε
<
1) of the total latent heat l (see condition (3) in below);
(H3ii)
the width of the mushy region is inversely proportional to the gradient of temperature (see condition (4) below).Following the methodology given in [27–29] and the recent one in [30], we consider a convective boundary condition in Sections 2 to 4 and a heat flux condition in Sections 5 and 6 at the fixed face x
=
0, respectively. In both cases, we obtain explicit solutions for the temperature and the two free boundaries which define the mushy region. We also obtain, for both cases, the necessary and sufficient condition on data in order to get these explicit solutions given in Sections 2 and 5, respectively, which is new with respect to the original model when a temperature boundary condition at the face x
=
0 was imposed. Moreover, these two problems are equivalent to the same phase-change process with a temperature boundary condition on the fixed face x
=
0 studied in [27] and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained in Sections 4 and 6. Moreover, in Section 3, we obtain the convergence of the solution of the phase-change process with a convective boundary condition to the solution given in [27] for a temperature boundary condition at the fixed face x
=
0 when the heat transfer coefficient goes to infinity, and we also give the order of the corresponding convergence when the coefficient that characterized the transient heat transfer at x
=
0 goes to infinity.This paper completes the model given in [27] by considering two new boundary conditions (convective and heat flux) at the fixed face of the PCMs and obtaining explicit solutions for both cases when a restriction on data is satisfied.


## 2. Explicit Solution with a Convective Boundary Condition

The phase-change process consists in finding the free boundariesx
=
s
(
t
) and x
=
r
(
t
) and the temperature T
=
T
(
x
,
t
) such that the following conditions must be verified (problem (P
1)):(1)
T
t
-
α
T
x
x
=
0
,
0
<
x
<
s
t
,
t
>
0
α
=
k
ρ
c
,
(2)
T
s
t
,
t
=
0
,
t
>
0
,
(3)
k
T
x
s
t
,
t
=
ρ
l
ε
s
˙
t
+
1
-
ε
r
˙
t
,
t
>
0
,
(4)
T
x
s
t
,
t
r
t
-
s
t
=
γ
>
0
,
t
>
0
with
γ
>
0
,
(5)
s
0
=
r
0
=
0
,
(6)
k
T
x
0
,
t
=
h
0
t
T
0
,
t
+
D
∞
,
t
>
0
h
0
>
0
,
D
∞
>
0
.Condition (6) represents a convective boundary condition (Robin condition) at the fixed face x
=
0 [31–33] with a heat transfer coefficient which is inversely proportional to the square root of the time [29, 30, 34, 35]. Now, we will obtain the solution of problem (1)–(6) when data satisfy the restriction (7).Theorem 1.
If the coefficienth
0 satisfies the inequality(7)
h
0
>
1
D
∞
γ
1
-
ε
ρ
l
k
2
=
h
0
∗
,then the solution of problem (1)–(6) is given by(8)
T
x
,
t
=
-
h
0
D
∞
π
α
/
k
erf
⁡
ξ
1
+
h
0
π
α
/
k
erf
⁡
ξ
1
-
erf
⁡
x
/
2
α
t
erf
⁡
ξ
,
0
<
x
<
s
t
,
t
>
0
,
(9)
s
t
=
2
ξ
α
t
,
t
>
0
,
(10)
r
t
=
2
μ
α
t
,
t
>
0
,with(11)
μ
=
ξ
+
γ
k
2
D
∞
h
0
α
e
ξ
2
1
+
h
0
π
α
k
erf
⁡
ξ
,and the coefficient ξ is given as the unique solution of the equation(12)
D
∞
c
l
π
F
x
=
G
x
,
x
>
0
,where the real functions G and F are defined by(13)
F
x
=
e
-
x
2
k
/
h
0
π
α
+
erf
⁡
x
,
G
x
=
x
+
γ
1
-
ε
π
2
D
∞
1
F
x
,
x
>
0
.Proof.
Taking into account thaterf
⁡
x
/
2
α
t is a solution of the heat equation (3) [3], we propose as a solution of problem (1)–(6) the following expression:(14)
T
x
,
t
=
C
1
+
C
2
erf
⁡
x
2
α
t
,
0
<
x
<
s
t
,
t
>
0
,where the two coefficients C
1 and C
2 must be determined.
From condition (4), we deduce the expression (9) for the free boundary s
(
t
), where the coefficient ξ must be determined. From conditions (6) and (2), we deduce the system of equations(15)
C
2
=
h
0
π
α
k
C
1
+
D
∞
,
C
1
+
C
2
erf
⁡
ξ
=
0
,whose solution is given by(16)
C
1
=
-
h
0
π
α
/
k
D
∞
erf
⁡
ξ
1
+
h
0
π
α
/
k
erf
⁡
ξ
,
C
2
=
h
0
D
∞
π
α
k
1
1
+
h
0
π
α
/
k
erf
⁡
ξ
,and then we get expression (8) for the temperature.
From condition (4), we deduce expression (10) for the interface r
(
t
) and expression (11) for μ. From condition (3), we deduce (12) for the coefficient ξ. Functions F
3 and G have the following properties:(17)
F
0
+
=
h
0
π
α
k
>
0
,
F
+
∞
=
0
+
,
F
′
x
<
0
,
∀
x
>
0
,
G
0
+
=
1
-
ε
γ
k
2
D
∞
h
0
α
>
0
,
G
+
∞
=
+
∞
,
G
′
x
>
0
,
∀
x
>
0
.
Therefore, we deduce that (12) has a unique solution when the coefficient h
0 satisfies the inequality(18)
D
∞
c
l
π
F
0
+
>
G
0
+
⟺
h
0
2
>
γ
1
-
ε
ρ
l
k
2
D
∞
2
;that is, inequality (7) holds.Corollary 2.
If the coefficienth
0 satisfies inequality (7), then the temperature, defined by (8), verifies the following inequalities:(19)
-
D
∞
<
T
0
,
t
≤
T
x
,
t
<
0
,
0
<
x
<
s
t
,
t
>
0
.Proof.
From (8), we obtain(20)
T
0
,
t
=
-
h
0
D
∞
π
α
/
k
erf
⁡
ξ
1
+
h
0
π
α
/
k
erf
⁡
ξ
=
-
D
∞
1
+
k
/
h
0
π
α
erf
⁡
ξ
>
-
D
∞
,
∀
t
>
0
.
Moreover, from (8) and (20), we also get(21)
T
x
,
t
+
D
∞
=
D
∞
1
+
h
0
π
α
/
k
erf
⁡
ξ
1
+
h
0
π
α
k
erf
⁡
x
2
α
t
≥
D
∞
1
+
h
0
π
α
/
k
erf
⁡
ξ
=
T
0
,
t
+
D
∞
>
0
,
0
<
x
<
s
t
,
t
>
0
;that is, (19) holds.


## 3. Asymptotic Behavior When the Coefficienth
0
→
+
∞

Now, we will obtain the asymptotic behaviour of solution (8)–(12) of problem (1)–(6) when the heat transfer coefficient is large, that is, when h
0
→
+
∞. From a physical point of view, it must be convergent to the solution of the same problem with a temperature boundary condition at the fixed face x
=
0 given by (23).For any coefficienth
0 satisfying inequality (7), we will denote the temperature T and the two free boundaries s and r (defined in (8), (9), and (10), resp.) by T
=
T
(
x
,
t
,
h
0
), x
=
s
(
t
,
h
0
), and x
=
r
(
t
,
h
0
), respectively, with coefficients ξ
=
ξ
(
h
0
) and μ
=
μ
(
h
0
). We will also denote by F
x
,
h
0 and G
x
,
h
0 the functions defined in (13). We have the following result.Theorem 3.
One obtains the following limits:(22)
lim
h
0
→
∞
⁡
T
x
,
t
,
h
0
=
T
∞
x
,
t
,
lim
h
0
→
∞
⁡
s
t
,
h
0
=
s
∞
t
,
lim
h
0
→
∞
⁡
r
t
,
h
0
=
r
∞
t
,where T
∞
(
x
,
t
), s
∞
(
t
), and r
∞
(
t
) are the solutions of the following phase-change process with mushy region: (1)–(5) and (23)
T
0
,
t
=
-
D
∞
,
t
>
0
,instead of the boundary condition (6).Proof.
The solution of problem (1)–(5) and (23) is given by [27](24)
T
∞
x
,
t
=
-
D
∞
1
-
erf
⁡
x
/
2
α
t
erf
⁡
ξ
∞
,
0
<
x
<
s
∞
t
,
t
>
0
,
(25)
s
∞
t
=
2
ξ
∞
α
t
,
t
>
0
,
(26)
r
∞
t
=
2
μ
∞
α
t
,
t
>
0
,with(27)
μ
∞
=
ξ
∞
+
γ
π
2
D
∞
e
ξ
∞
2
erf
⁡
ξ
∞
,and the coefficient ξ
∞ given as the unique solution of the equation(28)
G
1
x
=
D
∞
c
l
π
,
x
>
0
,where the real function G
1 is defined by (29)
G
1
x
=
G
∞
x
F
∞
x
,
x
>
0
,with(30)
G
∞
x
=
x
+
γ
1
-
ε
π
2
D
∞
1
F
∞
x
=
lim
h
0
→
∞
⁡
G
x
,
h
0
,
x
>
0
,
(31)
F
∞
x
=
e
-
x
2
erf
⁡
x
=
lim
h
0
→
∞
⁡
F
x
,
h
0
,
x
>
0
.Then,(32)
lim
h
0
→
∞
⁡
ξ
h
0
=
ξ
∞
,
lim
h
0
→
∞
⁡
μ
h
0
=
μ
∞
.And, therefore, the limits (22) hold.
Now, by studying the real functionsF
x
,
h
0 and G
x
,
h
0 as functions of the variable h
0, we can obtain the order of the convergence of solution (8)–(12) of problem (1)–(6) to solution (24)–(28) of problem (1)–(5) and (23) when h
0
→
∞.Theorem 4.
When the variableh
0
→
∞, one obtains the following estimations:(33)
0
<
ξ
∞
-
ξ
h
0
=
O
1
h
0
w
h
e
n
h
0
⟶
∞
,
(34)
0
<
s
∞
t
-
s
t
,
h
0
=
O
1
h
0
,
∀
t
≥
0
w
h
e
n
h
0
⟶
∞
,
(35)
μ
h
0
-
μ
∞
=
O
1
h
0
w
h
e
n
h
0
⟶
∞
,
(36)
r
t
,
h
0
-
r
∞
t
=
O
1
h
0
,
∀
t
≥
0
w
h
e
n
h
0
⟶
∞
,
(37)
T
x
,
t
,
h
0
-
T
∞
x
,
t
=
O
1
h
0
,
∀
x
≥
0
,
∀
t
>
0
w
h
e
n
h
0
⟶
∞
.Proof.
As the variableh
0
→
∞, we can consider that h
0
>
h
0
∗ and then solution (8)–(12) of problem (1)–(6) is well defined.
FunctionF
(
x
,
h
0
) is an increasing function in variable h
0; therefore, function G
(
x
,
h
0
) is a decreasing function in variable h
0. Then, function G
(
x
,
h
0
)
/
F
(
x
,
h
0
) is a decreasing function in variable h
0, ∀
x
>
0, which is convergent to G
∞
(
x
)
/
F
∞
(
x
) as h
0
→
∞ because (30) and (31) hold.
By using (13) and (31), we have(38)
0
<
F
∞
x
-
F
x
,
h
0
=
F
∞
x
k
/
h
0
π
α
erf
⁡
x
+
k
/
h
0
π
α
<
e
-
x
2
erf
2
⁡
x
k
h
0
π
α
=
O
1
h
0
,
∀
x
>
0
when
h
0
⟶
∞
.
By using (13) and (30), we have(39)
0
<
G
x
,
h
0
-
G
∞
x
=
γ
1
-
ε
π
2
D
∞
F
∞
x
-
F
x
,
h
0
F
∞
x
F
x
,
h
0
<
γ
1
-
ε
π
2
D
∞
F
∞
x
-
F
x
,
h
0
F
∞
x
F
x
,
h
0
∗
=
O
1
h
0
,
∀
x
>
0
when
h
0
⟶
∞
.
Therefore, we have(40)
0
<
G
x
,
h
0
F
x
,
h
0
-
G
∞
x
F
∞
x
<
F
∞
x
-
F
x
,
h
0
x
+
γ
1
-
ε
π
/
D
∞
1
/
F
x
,
h
0
∗
F
∞
x
F
x
,
h
0
∗
=
O
1
h
0
,
∀
x
>
0
when
h
0
⟶
∞
.
Then, the estimation (33) holds and(41)
0
<
s
∞
t
-
s
t
,
h
0
=
2
α
t
ξ
∞
-
ξ
h
0
=
O
1
h
0
,
∀
t
≥
0
when
h
0
⟶
∞
.By using (11) and (27), we get(42)
μ
∞
-
μ
h
0
=
ξ
∞
-
ξ
h
0
-
γ
π
2
D
∞
α
e
ξ
2
h
0
h
0
+
γ
π
2
D
∞
e
ξ
∞
2
erf
⁡
ξ
∞
-
e
ξ
2
h
0
erf
⁡
ξ
h
0
=
O
1
h
0
when
h
0
⟶
∞
,
r
t
,
h
0
-
r
∞
t
=
2
α
t
μ
h
0
-
μ
∞
=
O
1
h
0
,
∀
t
≥
0
when
h
0
⟶
∞
.
Finally, by using (8) and (24), we get(43)
T
x
,
t
,
h
0
-
T
∞
x
,
t
=
D
∞
1
+
h
0
π
α
/
k
erf
⁡
ξ
h
0
1
+
erf
⁡
x
/
2
α
t
erf
⁡
ξ
0
h
0
π
α
k
erf
⁡
ξ
∞
-
erf
⁡
ξ
h
0
=
O
1
h
0
,
∀
x
>
0
,
∀
t
>
0
when
h
0
⟶
∞
,and the thesis holds. In the particular case, when x
=
0, we have(44)
0
<
T
0
,
t
,
h
0
-
T
∞
0
,
t
=
D
∞
1
+
h
0
π
α
/
k
erf
⁡
ξ
h
0
<
k
D
∞
h
0
π
α
erf
⁡
ξ
h
0
∗
=
O
1
h
0
,
∀
t
>
0
when
h
0
⟶
∞
.


## 4. Equivalence between the Mushy Zone Models with Convective and Temperature Boundary Conditions

We consider problem (P
2) defined by conditions (1)–(5) and temperature boundary condition(45)
T
0
,
t
=
-
D
0
<
0
,
t
>
0
,at the fixed face x
=
0, whose solution was given in [27]. We have the following property.Theorem 5.
If the coefficienth
0 satisfies inequality (7), then problem (P
1), defined by conditions (1)–(6), is equivalent to problem (P
2), defined by conditions (1)–(5) and (45), when the parameter D
0 in problem (P
2) is related to parameters h
0 and D
0 in problem (P
1) by the following expression:(46)
D
0
=
D
∞
erf
⁡
ξ
k
/
h
0
π
α
+
erf
⁡
ξ
>
0
,where the coefficient ξ is given as the unique solution of (12) for problem (P
1) or as the unique solution of equation(47)
G
2
x
=
D
0
c
l
π
,
x
>
0
,for problem (P
2), where the real function G
2 is defined by (48)
G
2
x
=
G
0
x
F
∞
x
,
G
0
x
=
x
+
γ
1
-
ε
π
2
D
0
1
F
∞
x
,
x
>
0
.Proof.
If the coefficienth
0 satisfies inequality (7), then the solution of problem (P
1) is given by (8)–(12). Taking into account that(49)
T
0
,
t
=
-
h
0
π
α
/
k
D
∞
erf
⁡
ξ
1
+
h
0
π
α
/
k
erf
⁡
ξ
=
-
D
∞
erf
⁡
ξ
k
/
h
0
π
α
+
erf
⁡
ξ
<
0
,
t
>
0
,we can define problem (P
2) by imposing the temperature boundary condition (45) with data D
0 given in (46). By using this data D
0 in problem (P
2) and the method developed in [30], we can prove that the solutions of both problems (P
1) and (P
2) are the same and then the two problems are equivalent.Corollary 6.
If the coefficienth
0 satisfies inequality (7), then the coefficient ξ of the solid-mushy zone interface of problem (P
2) verifies the following inequality:(50)
erf
⁡
ξ
<
D
∞
D
0
D
∞
-
D
0
2
c
π
γ
1
-
ε
l
,
∀
D
∞
>
D
0
.Then,(51)
erf
⁡
ξ
<
D
0
2
c
π
γ
1
-
ε
l
.Remark 7.
The real functionsG
∞, defined in (30), and G
0, defined in (48), are similar; the difference between them is the parameters D
∞ or D
0 used in each definition.


## 5. Explicit Solution with a Heat Flux Boundary Condition

Now, we will consider a phase-change process which consists in finding the free boundariesx
=
s
(
t
) and x
=
r
(
t
) and the temperature T
=
T
(
x
,
t
) such that the following conditions must be verified (problem (P
3)): conditions (1)–(5) and(52)
k
T
x
0
,
t
=
q
0
t
,
t
>
0
q
0
>
0
.Condition (52) represents the heat flux at the fixed face x
=
0 characterized by a coefficient which is inversely proportional to the square root of the time [34].Theorem 8.
If the coefficientq
0 satisfies the inequality(53)
q
0
>
γ
1
-
ε
ρ
l
k
2
=
q
0
∗
,then the solution of problem (1)–(5) and (52) is given by(54)
T
x
,
t
=
-
q
0
π
α
erf
⁡
ω
k
1
-
erf
⁡
x
/
2
α
t
erf
⁡
ω
<
0
,
0
<
x
<
s
t
,
t
>
0
,
(55)
s
t
=
2
ω
α
t
,
t
>
0
,
(56)
r
t
=
2
ν
α
t
,
t
>
0
,with(57)
ν
=
ω
+
γ
k
2
q
0
α
e
ω
2
,and the coefficient ω
>
0 given as the unique solution of the equation(58)
G
3
x
=
q
0
ρ
l
α
,
x
>
0
,where the real function G
3 is defined by(59)
G
3
x
=
x
+
γ
1
-
ε
k
2
q
0
α
e
x
2
e
x
2
,
x
>
0
.Proof.
Following the proof of Theorem1, we propose as a solution of problem (1)–(5) and (52) the following expression:(60)
T
x
,
t
=
A
1
+
A
2
erf
⁡
x
2
α
t
,
0
<
x
<
s
t
,
t
>
0
,where the two coefficients A
1 and A
2 must be determined.
From condition (2), we deduce expression (55) for the free boundary s
(
t
), with the coefficient ω to be determined. From conditions (2) and (52), we deduce(61)
A
1
=
-
q
0
π
α
k
erf
⁡
ω
,
A
2
=
q
0
π
α
k
,and then we get expression (54) for the temperature.
From condition (4), we deduce expression (56) for the interface r
(
t
) and expression (57) for ν. From condition (3), we deduce (58) for the coefficient ω. Since function G
3 has the following properties:(62)
G
3
0
+
=
γ
1
-
ε
k
2
q
0
α
>
0
,
G
3
+
∞
=
+
∞
,
G
3
′
x
>
0
,
∀
x
>
0
,we can deduce that (58) has a unique solution when the coefficient q
0 satisfies the inequality(63)
q
0
ρ
l
α
>
G
3
0
+
⟺
q
0
2
>
γ
1
-
ε
ρ
l
k
2
,which is inequality (53).Remark 9.
We have a relationship betweenq
0
∗ (the lower limit for coefficient q
0 in order to have a phase-change process with a mushy region with a heat flux boundary condition at x
=
0) and h
0
∗ (the lower limit for the coefficient h
0 in order to have a phase-change process with a mushy region with a convective boundary condition at x
=
0) given by(64)
q
0
∗
=
D
∞
h
0
∗
.


## 6. Equivalence between the Mushy Zone Models with Heat Flux and Temperature Boundary Conditions

Following Section4, we will now study the relationship between problems (P
3) and (P
2). We have the following property.Theorem 10.
If the coefficientq
0 satisfies inequality (53), then problem (P
3), defined by conditions (1)–(5) and (52), is equivalent to problem (P
2), defined by conditions (1)–(5) and (45), when the parameter D
0 in problem (P
2) is related to the parameter q
0 in problem (P
3) by the following expression:(65)
D
0
=
q
0
π
α
k
erf
⁡
ω
>
0
,where the coefficient ω is given as the unique solution of (58) for problem (P
3) or as the unique solution of (47) for problem (P
2).Proof.
If the coefficientq
0 satisfies inequality (53), then the solution of problem (P
3) is given by (54)–(58). Taking into account that(66)
T
0
,
t
=
-
q
0
π
α
k
erf
⁡
ω
<
0
,
t
>
0
,we can define problem (P
2) by imposing the temperature boundary condition (45) with the data D
0 given in (65). By using this data D
0 in problem (P
2) and the method developed in [30], we can prove that the solutions of both problems (P
3) and (P
2) are the same and then the two problems are equivalent.Corollary 11.
If the coefficientq
0 satisfies inequality (53), then the coefficient ξ of the solid-mushy zone interface of problem (P
2) verifies inequality (51) which is the same as that we have obtained through the equivalence between problems (P
1) and (P
2).Remark 12.
At last, for a suggestion of an anonymous referee, we will transform problem (P
1), given by the equations and conditions (1)–(6), and inequality (7) in a dimensionless form. We define the following dimensionless change of variables:(67)
η
=
x
L
,
τ
=
α
t
L
2
,
S
τ
=
s
t
L
,
R
τ
=
r
t
L
,
θ
η
,
τ
=
T
x
,
t
D
∞
,where L is a characteristic length. Therefore, the equations and conditions (1)–(6) are transformed as(68)
θ
τ
-
θ
η
η
=
0
,
0
<
η
<
S
τ
,
τ
>
0
,
θ
S
τ
,
τ
=
0
,
τ
>
0
,
θ
η
r
τ
,
τ
=
1
Ste
ε
S
′
τ
+
1
-
ε
R
′
τ
,
τ
>
0
,
θ
η
r
τ
,
τ
R
τ
-
S
τ
=
γ
D
∞
,
τ
>
0
,
S
0
=
R
0
=
0
,
θ
η
0
,
τ
=
B
τ
θ
0
,
τ
+
1
,
τ
>
0
,where S
t
e is the Stefan number and B
/
τ is the Biot number defined by the following expressions:(69)
Ste
=
c
D
∞
l
>
0
,
(70)
B
=
h
0
α
k
=
h
0
ρ
k
c
>
0
.
Moreover, inequality (7) for the physical coefficient h
0, which characterized the heat transfer coefficient in the boundary condition (6), is transformed in the following way: (71)
B
>
1
D
∞
γ
1
-
ε
l
2
c
=
B
∗
.
Therefore, limith
0
→
∞ in problem (1)–(6) in physical variables is equivalent to limit B
→
∞ in problem (68) in dimensionless variables.
By using the results of this work, we can now obtain new explicit expression for the determination of one or two unknown thermal coefficients through a phase-change process with a mushy zone by imposing an overspecified convective boundary condition at the fixed facex
=
0. This will complete and improve the results obtained previously in [28].


## 7. Conclusions

The goal of this paper is to complete the solution of Solomon-Wilson-Alexiades’s model for a mushy zone model for phase-change materials when a convective or a heat flux boundary condition at the fixed facex
=
0 is imposed. In both cases, explicit solutions for the temperature and the two free boundaries which define the mushy region were obtained and, for both cases, the necessary and sufficient conditions on data in order to get these explicit solutions are also obtained which is new with respect to the original model when a temperature boundary condition at the face x
=
0 was imposed. Moreover, the equivalence of this two phase-change process with the one with a temperature boundary condition on the fixed face x
=
0 was obtained and an inequality for the dimensionless coefficient that characterizes the first free boundary is also given. On the other hand, the convergence of the phase-change process with mushy zone when the heat transfer coefficient goes to infinity was also obtained and the order of the convergence is also shown.



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*Source: 375930-2015-11-23.xml*