Abstract
A one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Required relationships between data are presented in order that these problems become equivalent to the problem where a Dirichlet condition at the fixed face is considered. Moreover, in the case where a Robin condition is prescribed, the limit behaviour is studied when the heat transfer coefficient at the fixed face goes to infinity.
1. Introduction
Stefan-like problems have attracted growing attention in the last decades due to the fact that they arise in many significant areas of engineering, geoscience, and industry [1–9]. The classical Stefan problem describes the process of a material undergoing a phase change. Finding a solution to this problem consists in solving the heat-conduction equation in an unknown region which has also to be determined, imposing an initial condition, boundary conditions, and the Stefan condition at the moving interface. For an account of the theory we refer the reader to [10].
In the classical Stefan problem the latent heat is assumed to be constant. In this paper, we are going to consider a variable one. This assumption is motivated by the fact that it becomes meaningful in the study of the shoreline movement in a sedimentary basis [11], in the one-dimensional consolidation with threshold gradient [12], in the artificial ground-freezing technique [13], and in nanoparticle melting [14], among others [15–21].
Many papers deal with a latent heat that depends on the position of the free boundary (size-dependent latent heat). In [18], a Stefan problem with a latent heat given as a function of the position of the interface has been considered. This hypothesis corresponds to the practical case when the influence of phenomena such as surface tension, pressure gradients, and nonhomogeneity of materials are taken into account. In [11], the shoreline movement in a sedimentary basin was studied, from where a one-phase Stefan problem with a latent heat arises that increases linearly with distance from the origin; i.e., (with a given constant). The generalization to the two-phase problem was done in [19]. Also, in [20] a latent heat defined as a power function of the position, i.e., (with a given constant and an arbitrary nonnegative integer), was considered. The extension to a noninteger exponent was done in [21] for flux and temperature boundary conditions, while the two-phase case was presented in [13]. In [22], a convective (Robin) condition was imposed for the one-phase case while for the two-phase case the analysis was done in [23].
In [12], a one-dimensional consolidation problem with a threshold gradient was studied. This problem can be reduced to a one-phase Stefan problem where the latent heat can be expressed as , that is to say, rate-dependent latent heat. It must be noticed that the case considered in [12] is not properly a Stefan problem because the velocity of the moving boundary disappears, and it has to be treated as a free boundary problem with implicit conditions [24, 25].
Recently, in [26] it was defined a generalized one-phase Stefan-like problem for a semi-infinite material with a latent heat given by (with a given constant and and arbitrary real constants), i.e., latent heat depending on the position and velocity of the moving boundary, taking a Dirichlet boundary condition. This paper intends to complete this model, by considering two new boundary conditions (Neumann and Robin conditions) at the fixed face .
In Section 2 we present a problem with a variable latent heat and a generalized boundary condition at the fixed face. We will obtain its exact solution, following the methodology given in [12, 21, 26], obtaining as immediate consequence the similarity solutions to two different problems: one with Neumann condition at and the other with a Robin one. Special cases will be treated in order to recover solutions recently reported in literature. Moreover, in Section 3, the equivalence between these problems and the problem with a Dirichlet condition considered in [26] will be proved under certain relationships between data. For the problem with a Robin boundary condition at the fixed face, the limit behaviour when the heat transfer coefficient goes to infinity will be also analysed in Section 4. This analysis will allow us to show that the Robin condition constitutes a generalization of the Dirichlet one, as happens in classical heat transfer problems [27]. Also, in Section 5, we will provide some plots and table of values in order to track the position of the free front and to show how the latent heat changes in time.
2. Formulation of the Problems and Exact Solution
2.1. Statement of the Problems
In this paper, the exact solution of two different free boundary problems is obtained. They will be defined as particular cases of the following problem that consists in finding the function and the moving boundary such thatwhere , , , (diffusivity), and (conductivity) are nonnegative constants.
The problem defined by specifying in will be referred to as problem (). In this case, condition (5) corresponds to the Neumann boundary condition:where a time dependent heat flux characterized by is applied at the fixed face . This flux is proportional to the power of time.
The problem defined by specifying in will be referred to as problem (). In this case, condition (5) corresponds to the Robin boundary condition:where characterizes the bulk temperature at a large distance from the fixed face and characterizes the heat transfer at the fixed face.
Comparing these problems with respect to the classical Stefan problem, the new feature to be observed is that condition (3) at the free interface can be thought of as a generalized Stefan condition where the latent heat term is not constant but rather a function of position and velocity of the moving boundary. Furthermore, in order to obtain a similarity type solution for problems , , and , will be specified aswith , , and being nonnegative given constants.
2.2. Similarity Type Solutions
Before finding the similarity type solution to problem , the subsequent analysis will be necessary.
Let us observe that if we use the similarity transformation presented in [20, 21]then it is obtained that Therefore, (1) is satisfied; i.e., if and only ifThis second-order ordinary differential equation, known in literature as Kummer’s differential equation (see [28]), has a general solution that is given bywith and being arbitrary constants. The function is called Kummer’s function or confluent hypergeometric function of the first kind and it is defined by the following series: where cannot be a nonpositive integer, and is the Pochhammer symbol defined by
The detailed proof of the fact that the general solution of Kummer’s equation (11) can be written as (12) may be found in [26].
The main properties of Kummer’s function to be used throughout this paper can be found in [28] and they are stated in the following way:where is the repeated integral of the complementary error function defined by
Now, we will look for the similarity solution to problem (). In order to make the notation clearer, we will refer to the solution of problem () as the pair that satisfies (1)-(5).
According to the previous analysis, will satisfy (1) if it is written aswith the similarity variable given by , where , are constants to be determined so that satisfies the rest of the conditions.
Observe that from (2) it should be noted that defined by the transformation (9) has to satisfy for all . Therefore the moving boundary must adopt the following form:where is a positive dimensionless coefficient to be determined.
Hence, bearing in mind that is written as (24) and as (25), finding the solution to problem () consists in determining the coefficients , , and .
The generalized boundary condition at the fixed face (5) and properties (15), (18)-(19) imply that
From condition (2), it can be deduced after some computations thatTherefore, replacing in (26), we getThen, we have obtained and as functions of .
Finally, the Stefan-type condition given by (3) will give us an equation for .
Applying the derivation formulas (18)-(19) we claim that Using relationships (16)-(17), the partial derivative of is reduced toReplacing (30) in (3) yields the following equality: which makes sense if and only if , due to the fact that neither , nor depends on time. Thus, the similarity solution for problem () will exist if and only ifand if is a positive solution of the following equation:with
The notation is adopted in order to emphasize the dependence of the solution to problem on and therefore to obtain easily the solutions to problems and .
The task is now to prove the existence and uniqueness of solution to (33). From the relationships (15), (18), (19), and (20), we obtain that satisfies
We can deduce that the l.h.s. of (33) is a strictly decreasing function that goes from to when increases from 0 to , while the r.h.s. of (33), if , is a strictly increasing function that goes from 0 to .
In conclusion we obtain that if , we can ensure that (33) has a unique positive solution.
It should be mentioned that due to restriction (32), i.e., and , we get that .
All the above analysis can be summarized in the following theorem.
Theorem 1. Let and be arbitrary real constants satisfying . Taking , there exists a unique solution of a similarity type for problem (), i.e., (1)-(5), which is given by (24) and (25), where and are given by formulas (27) and (28), respectively, and the dimensionless coefficient is defined as the unique positive solution of (33).
The solutions to problems and can be obtained as a consequence of Theorem 1, by fixing or , respectively. As an immediate consequence we have the following results.
Corollary 2 (case ). Let and be arbitrary real constants satisfying . Taking , there exists a unique solution of a similarity type for problem (), i.e., (1)-(4), and (6) which is given bywhere is the similarity variable. The coefficients , are defined by and is the unique positive solution of the equation that can be rewritten aswith
Corollary 3 (case ). Let and be arbitrary real constants satisfying . Taking , there exists a unique solution of a similarity type for problem (), i.e., (1)-(4) and (7), which is given bywhere is the similarity variable. The coefficients , are defined by and is the unique positive solution of the following equation:with defined by replacing in given in (34):
Specifying different values for and in the above results, several solutions reported in literature can be recovered as a corollary. For instance, consider the following.
Corollary 4. The solution to the classical Stefan problem with a Neumann boundary condition at the fixed face can be recovered from Theorem 1 by taking , .
Taking and thus , the latent heat is assumed to be constant like in the classical Stefan problem. In such case, fixing , the flux boundary condition is given by . Moreover, the property allows us to ensure that is the unique solution to the following equation: as was obtained in [29].
Corollary 5. The solutions provided in [20, 21] can be recovered from Theorem 1 by taking , , and .
Taking , we get that , i.e., a power function of the position. For such a case, by taking into account the fact that we automatically obtain the solutions already presented in literature. It must be pointed out that if is an integer, properties (21)-(22) should be applied.
Remark 6. It must be noticed that in case we want to recover a latent heat defined as , we have to set , , and thus . However we cannot recover the solution given in [12], due to the fact that the boundary condition imposed at the fixed face (Dirichlet) does not agree with the boundary condition considered in problem .
Corollary 7. The solution to the classical Stefan problem with a Robin boundary condition at the fixed face can be recovered from Theorem 1 by taking , (See [27]).
Corollary 8. The solution to the Stefan problem studied in [22] can be recovered from Theorem 1 by taking , , and .
3. Equivalence to the Problem with Dirichlet Condition
In [26], the unique similarity solution of a problem defined by (1)-(4) with a Dirichlet boundary condition at the fixed face characterized by was obtained; i.e.,The problem defined by conditions (1)-(4) and (49) will be referred to as problem () and its solution will be referred to as the pair .
According to [26], if and are arbitrary real constants satisfying , taking , the unique solution to problem () is given bywhere and is the unique positive solution of the following equation:with
In this section we will study conditions on the data of the problem () that guarantee its equivalence with the problem . For equivalence it will be understood that both problems have the same solution.
Consider the problem () with given data , , whose solution is given by formulas (24) and (25) under the hypothesis that and are arbitrary real constants with , and . Computing it is obtained that with defined as the unique positive solution to (33). Suppose now that we fix and we solve the problem () obtaining . Notice that the moving boundary is characterized by a dimensionless coefficient that will be the unique solution to (53); i.e.,
Notice that if we put , the prior equation reduces to (33), meaning that constitutes a solution to (57). Therefore, as the unique solution to (57) is given by , results. Then, it follows easily that , obtaining as a consequence that the solution with the data given in function of coincides with the solution of the problem ().
Conversely, consider the problem () with a given data whose solution is given by formulas (50) and (51) under the assumption that and are arbitrary real constants, , and . Computing , the following is obtained: Let us consider () with the data given by fixing and such that . The solution of this problem can be obtained by (24) and (25). The free boundary is characterized by a dimensionless coefficient that is the unique solution of (33), i.e., satisfies The prior equation has as a solution due to the fact that if we replace by , it is obtained that (60) is equivalent to (53). As (60) has a unique solution given by , we claim that . In addition, by some computations, it becomes , and so the solution to problem () given by a data in function of is equal to the solution to problem (). Therefore the following theorem holds.
Theorem 9. If and are arbitrary real constants satisfying and , then the problem () defined by condition (1)-(5) is equivalent to problem () defined by (1)-(4) and (49), when the parameters , and in the problem () are related to the parameter in problem () by the following expression:The coefficient makes reference to the unique solution of (33) for problem () which will coincide with the unique solution of (53) for problem ().
As a consequence of the above result, by fixing and , respectively, we can obtain the following corollaries.
Corollary 10 (case ). If and are arbitrary real constants satisfying and , then the problem () defined by conditions (1)-(4) and (6) is equivalent to problem () defined by (1)-(4) and (49), when the parameter in the problem () is related to the parameter in problem () by the following expression:The coefficient makes reference to the unique solution of (41) for problem () which will coincide with the unique solution of (53) for problem ().
Corollary 11 (case ). If and are arbitrary real constants satisfying and , then the problem () defined by conditions (1)-(4) and (7) is equivalent to problem () defined by (1)-(4) and (49), when the parameters , in the problem () are related to the parameter in problem () by the following expression:The coefficient makes reference to the unique solution of (46) for problem () which will coincide with the unique solution of (53) for problem ().
4. Asymptotic Behaviour When the Coefficient
In this subsection we are going to analyse the behaviour of the problem () when the coefficient which characterizes the heat transfer coefficient at the fixed face tends to infinity. Due to the fact that the solution of this problem depends on , we will rename it. Thus, we will consider and defined by (43)-(44).
Let us define the problem () defined by conditions (1)-(4) and the following condition of Dirichlet type at the fixed face given bywhere corresponds to the data of the problem (). Notice that the solution to problem () can be obtained from (50) and (51) replacing by .
Then we are going to state the following result.
Theorem 12. If and are arbitrary real constants satisfying and the problem () converges to problem () when tends to infinity; i.e., In this context the term “convergence” means that
Proof. On the one hand, the free boundary solution to problem () is characterized by a dimensionless parameter that is the unique solution to (46); i.e., where and given by (47). On the other hand, the moving boundary is characterized by a dimensionless parameter which will be defined as the unique solution of (53) replacing by ; i.e., where is defined by (54).
We are going to prove that when , the coefficient converges to the coefficient . We know that is a strictly increasing function that goes from 0 to when increases from 0 to ; is a strictly decreasing function that goes from to 0 and is a strictly decreasing function in as well but decreases from to 0 when goes from 0 to . After some computations, it can be seen that Therefore it can be concluded that , for all . In addition, when it can be easily seen that and so . Once this equality has been proved, by taking the limit in the definitions of and one can obtain the required convergence for and .
5. Computational Examples
In this section, we present and discuss some computational examples.
From Theorem 1, the solution to problem is characterized by a dimensionless parameter defined as the unique positive solution to (33). This equation can be rewritten as withTo solve the nonlinear equation we apply the following Newton iteration formula:where is the value of at the th iteration step and
We have implemented a MATLAB program to compute the dimensionless coefficient for different values of the parameters. The stopping criterion used is the boundedness of the absolute error .
In addition, given that the latent heat behaves as a function of the free front, we will plot in order to show how it changes in time. Observe thatTherefore it is deduced that the latent heat behaves as a power of time, i.e., with if , for and in case . It must be pointed out that in all cases should be in order to meet the hypothesis of Theorem 1.
Let us first analyse the problem with a Neumann boundary condition at the fixed face. From Corollary 2, the solution of the problem () is characterized by a dimensionless parameter defined as the unique solution of (41). This equation can be rewritten as (70) specifying :where is given by (42) and the dimensionless parameter is defined byIn Table 1 we present the computational results of for different values of .
In Figure 1, we plot the coefficient that characterizes the free front , for different values of the parameters , , and .
In Figure 2, we can observe graphically what can be analytically deduced, in the sense that when , , we obtain that the latent heat behaves as power of time, i.e., with . In the case that , , we obtain that and for , , the power becomes .
Now, we turn to the problem with a Robin boundary condition at the fixed face. From Corollary 3, the solution of the problem () is characterized by a dimensionless parameter defined as the unique solution of (46). This equation can be rewritten as (70) fixing :where is given by (47). Introducing the dimensionless parameter Ste, which constitutes a generalization of the Stefan number, and the generalized Biot numberwe get that can be rewritten as
In Table 2 we present the computational results of for different values of Bi, , and , fixing Ste=0.5. Observe that the last column of the table intends to show that when Bi increases, becomes closer to which is the dimensionless parameter of the free front to the problem with Dirichlet condition. This convergence is in agreement with the prior section, taking into account the fact that analysing is equivalent to analysing .
In Figure 3, we plot against Bi for different values of and , fixing Ste=0.5.
6. Conclusions
In this paper two different one-phase Stefan-like problems were studied for a semi-infinite material. The main feature of both problems resides in the fact that a variable latent heat depending on the power of the position and the rate of change of the moving boundary is considered (). Using Kummer functions, exact solutions of similarity type were obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face.
In addition, the necessary and sufficient relationships between the data of the two problems in order to obtain an equivalence with the problem with a Dirichlet condition are obtained.
For the problem with Robin boundary condition, the limit behaviour of the solution when the heat transfer coefficient at the fixed face goes to infinity was analysed, obtaining as a result the convergence to the solution of a Stefan problem with a Dirichlet boundary condition.
This paper constitutes a mathematical generalization of the classical one because it can be obtained by fixing the parameters , . Also, the results obtained when a latent heat is considered as a linear or power function of the position of the free boundary can be recovered.
We have provided tables and plots in order to show how the free front evolves in each case for specific values of the parameters.
It is worth mentioning that finding exact solutions is meaningful not only to understand better the physical processes involved but also to verify the accuracy of numerical methods that solve Stefan problems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The present work has been partially sponsored by CONICET-UA Project PIP no. 0534 and ANPCyT PICTO Austral 2016, no. 0090, Rosario, Argentina.