Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 180764, 5 pages

http://dx.doi.org/10.1155/2014/180764

Research Article

On the Solutions of a Stefan Problem with Variable Latent Heat

Bougoffa Lazhar

Department of Mathematics, Faculty of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia

Received 17 April 2014; Accepted 3 July 2014; Published 13 July 2014

Academic Editor: Ashraf M. Zenkour

Copyright © 2014 Bougoffa Lazhar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with a reliable treatment of the Stephan problem with variable latent heat. The power series solutions have been presented. The method is examined for computational efficiency and accuracy. Also, an analytical solution based on a similarity variable is presented in the case when the Dirichlet boundary condition at the water-ice interface depends on time.

1. Introduction

The solution of the Stefan problem with variable latent heat consists of finding and the moving melt interface such that

which is the governing equation, subject to the following Neumann boundary condition at the left end of the domain describing the inlet heat flux:

the Dirichlet boundary condition at the water-ice interface:

the Stefan condition:

and the initial temperature distribution:

On comparison of this condition with the one-phase Stefan condition, we observe that the latent heat term is not a constant but, rather, a linear function of position [1]. Problems such as this have been treated under the name of Stefan problems with variable latent heat [1–3].

An analytical solution for this Stefan problem is presented in [1], by introducing the similarity variables and seeking the solution in the form with being an unknown function. Accordingly, it is natural that the interface location should be proportional to ; that is, , where is a constant.

The purpose of this paper is to apply the Adomian decomposition method [4–25] to find the solution of (1), (2), and (5), that is, the temperature distribution of the water , and then use a direct method to determine the position of the ice-water interface as a function of time. Also, an analytical solution based on a similarity variable is presented in the case when the Dirichlet boundary condition at the water-ice interface depends on time of the form , where is a constant.

2. The Adomian Decomposition Method

Based on the Adomian decomposition method, we write (1) in Adomian’s operator-theoretic notation as

where

We conveniently define the inverse linear operator as

Applying the inverse linear operator to (6) and taking into account that , we obtain

where the unknown boundary condition will be determined.

Define the solution by an infinite series of components in the form

Consequently, the components can be elegantly determined by setting the recursion scheme

for the complete determination of these components. In view of (11), the components are immediately determined as

Consequently, the solution is readily found to be

obtained by substituting (12) into (10).

We remark here that the unknown boundary condition can be easily determined by using the initial condition equation (5). Substituting into (13) and using the Taylor expansion of lead to

Equating the coefficients of like power of in both sides of (14) and taking into account that the compatibility conditions

yield

Thus

Accordingly, the solution equation (13) is completely determined by defining the function .

Once the function is obtained, we can rewrite the Stefan condition equation (4) in terms of the known function including the Dirichlet and Neumann boundary conditions. For this, integrating (1) with respect to from to and taking into account that , we obtain

Thus (4) can be replaced by

Using the following Leibniz rule for differentiation under the integral sign:

and taking into account that , we obtain

Substituting (21) into (19), we obtain

Since , we have

Applying the inverse linear operator to (23), we obtain

It follows that

Substituting the solution equation (13) of the heat equation into (25), we obtain

where

Thus

where

We now can determine the shoreline with time by solving the nonlinear equation (28). In order to demonstrate the feasibility and efficiency of this method, we consider the following case:

If we choose , then a simple calculation leads to , and (28) becomes

where .

Let us write and in series forms and . Thus

To compute we need the following theorem [4].

Theorem 1. If and are convergent, then is convergent, where

and are the Adomian polynomials.

Using this theorem with the given formula , we see that

Consequently, we obtain the recurrence relations for the coefficients

where and .

This solution, of course, is

3. Similarity Solution

We begin our approach by introducing the similarity variable

and look for solutions of (1) in the form

where the number and the function are to be determined. Substituting (39) into (1) we find that

and so

This is also difficult to solve for arbitrary values of but for special values we can do something. Define , and thus (41) can be transformed into

This has the general solution

where and are constants, provided . Hence

This gives a full solution for in the form

Taking the spatial derivative of the solution given by (45)

From the Neumann boundary condition at , we obtain

It follows that must be a constant. Hence

and from the Dirichlet condition we get

Let , and thus (49) becomes

The analytical solution based on a similarity variable [1] requires that . This means that does not depend on time. In this treatment we will assume that is a function of , that is, , where is a constant, which needs to be specified to realize an explicit solution. Thus

Since in (51) is a constant, it follows that must also be constant. Thus

The Stefan condition at the free boundary is

where

By putting (53) and (54) in the Stefan condition, we get the following transcendental equation:

Thus we have proved the following theorem.

Theorem 2. For the Stefan problem with variable latent heat

The solution is given by

where

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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