Explicit solutions for the Solomon-Wilson-Alexiades's mushy zone model with convective or heat flux boundary conditions

We complete the Solomon-Wilson-Alexiades's mushy zone model (Letters Heat Mass Transfer, 9 (1982), 319-324) for the one-phase Lam\'e-Clapeyron-Stefan problem. We obtain 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 these explicit solutions. Moreover, when these conditions are satisfied the two problems are equivalents 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 is also obtained.

Moreover, when these conditions are satisfied the two problems are equivalents 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 is also obtained.

I. 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, 5-8, 11, 13, 18, 27]. A review of a long bibliography on moving and free boundary problems for phase-change materials (PCM) for the heat equation is shown in [24].
Explicit solutions to some free boundary problems was obtained in [3, 4, 9, 14 -17, 19, 21, 28, 29] We consider a semi-infinite material that is initially assumed to be liquid at its melting temperature which is assumed equals to 0 ºC. At time  0 t a heat flux or a convective boundary condition is imposed at the fixed face 0  x , and a solidification process begins where three regions can be distinguished [20,23]: H1) liquid region at the temperature 0 ºC, in  ( ), 0 x r t t ; H2) solid region at the temperature  0 (with ( ) ( )) x s t t s t r t ; H3) mushy region at the temperature  The mushy region is considered isothermal and we make the following assumptions on its structure: H3i) the material contains a fixed portion  (with   01 ) of the total latent heat (see condition (3) below); 3 H3ii) the width of the mushy region is inversely proportional to the gradient of temperature (see condition (4) below).
Following the methodology given in [20,23,25] and the recent one in [26] we consider a convective boundary condition in Sections II to IV, and a heat flux condition in Sections V and VI at the fixed face 0  x 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 II and V respectively.
Moreover, these two problems are equivalents to the same phase-change process with a temperature boundary condition on the fixed face 0  x and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained in Sections IV and VI. Moreover, in Section III we obtain the convergence of the phasechange process when the heat transfer coefficient goes to infinity.

II. EXPLICIT SOLUTION WITH A CONVECTIVE BOUNDARY CONDITION
The phase-change process consists in finding the free boundaries  () x s t and  () x r t , and the temperature  ( , ) T T x t such that the following conditions must be verified (Problem ( 1 P )): Condition (6) 4 then the solution of problem (1)-(6) is given by: and the coefficient  is given as the unique solution of the equation: where the real functions and GF are defined by: Proof. Taking into account that   is a solution of the heat equation (3) [6] we propose as a solution of problem (1)-(6) the following expression: where the two coefficients 12 and CC must to be determined. From condition (4) we deduce the expression (9) for the free boundary () st , where the coefficient  must be determined. From conditions (6) and (2) we deduce the system of equations: whose solution is given by: 5 and then we get expression (8) for the temperature.
From condition (4) we deduce expression (10) for the interface () rt and expression (11) for  . From condition (3) we deduce equation (12) for the coefficient FG have the following properties: Therefore, we deduce that equation (12) has a unique solution when the i.e. inequality (7) holds. □ (7) then the temperature, defined by (8), verifies the following inequalities:

Corollary 2. If the coefficient
Proof. From (8) we obtain: Moreover, from (8) and (22) we also get , G x h the functions defined in (13). We have the following result: Theorem 3. We obtain the following limits: where    ( , ), ( ) and ( ) T x t s t r t are the solutions of the following phase-change process with mushy region: (1)-(5) and instead of the boundary condition (6).
Proof. The solution of problem (1)- (5) and (25) is given by [20]: and the coefficient   given as the unique solution of the equation: where the real function 1 G is defined by:

CONVECTIVE AND TEMPERATURE BOUNDARY CONDITIONS
We consider the problem ( 2 P ) defined by the conditions (1) -(5) and temperature boundary condition at the fixed face 0  x , whose solution was given in [20]. We have the following property: where the coefficient  is given as the unique solution of equation (12) for Problem ( 1 P ) or as the unique solution of equation: for Problem ( 2 P ) where the real function 2 G is defined by: Proof. If the coefficient 0 h satisfies inequality (7) then the solution of the Problem ( 1 P ) is given by (8) - (12). Taking into account that: then we can define the Problem ( 2 P ) by imposing the temperature boundary condition (36) with data 0 D given in (37). By using this data 0 D in the Problem ( 2 P ) and the method developed in [26] we can prove that the solutions of both Problems ( 1 P ) and ( 2 P ) are the same and then the two problems are equivalents. □ (7) then the coefficient  of the solid-mushy zone interface of Problem ( 2 P ) verifies the following inequality:

Corollary 5. If the coefficient
Then,

V. EXPLICIT SOLUTION WITH A HEAT FLUX BOUNDARY CONDITION
Now, we will consider a phase-change process which consists in finding the free boundaries  () x s t and  () x r t , and the temperature  ( , ) T T x t such that the following conditions must be verified (Problem (43) 9 Condition (43) represents the heat flux at the fixed face 0  x characterized by a coefficient which is inversely proportional to the square root of the time [22].

Theorem 6.
If the coefficient 0 q satisfies the inequality then the solution of problem (1)-(5) and (43) is given by: and the coefficient   0 given as the unique solution of the equation: where the real function (50) Proof. Following the proof of the Theorem 1, we propose as a solution of problem (1)- (5) and (43) the following expression: where the two coefficients 12  G has the following properties: we can deduce that equation (49)  (1 ) i.e. inequality (44). □

Remark 3.
We have a relationship between * 0 q (the lower limit for the coefficient 0 q in order to have a phase-change process with a mushy region with a heat flux boundary condition at  0 x ) and * 0 h (the lower limit for the coefficient 0 h in order to have a phase-change process with a mushy region with a convective boundary condition at  0 x ) given by:

FLUX AND TEMPERATURE BOUNDARY CONDITIONS
Following Section IV, we will now study the relationship between the Problems ( 3 P ) and ( 2 P ). We have the following property: Proof. If the coefficient 0 q satisfies inequality (44) then the solution of Problem ( 3 P ) is given by (45) -(49). Taking into account that: we can define the Problem ( 2 P ) by imposing the temperature boundary condition (36) with the data 0 D given in (56). By using this data 0 D in Problem ( 2 P ) and the method developed in [26] we can prove that the solutions of both Problems ( 3 P ) and ( 2 P ) are the same and then the two problems are equivalents. □ Corollary 8. If the coefficient 0 q satisfies inequality (44) then the coefficient  of the solid-mushy zone interface of the Problem ( 2 P ) verifies inequality (42) which is the same that we have obtained through the equivalence between Problems ( 1 P ) and ( 2 P ).
By using the results of this work, we can now obtain new explicit expression for the determination of one or two unknown thermal coeffcient through a phase-change process with a mushy zone by imposing an overspecified convective boundary condition at the fixed face  0 x . This will complete and improve the results obtained previously in [23].

CONCLUSIONS
The goal of this paper is to complete the Solomon-Wilson-Alexiades's model for a mushy zone model for phase-change materials when a convective boundary or a heat flux condition at the fixed face 0  x is imposed. In both cases, explicit solutions for the temperature and the two free boundaries which define the mushy region was obtained; and, for both cases, the necessary and sufficient conditions on data in order to get these explicit solutions are also obtained. Moreover, the equivalence of these two phasechange process with the one with a temperature boundary condition on the fixed face 0  x was obtained. 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.