Solutions of a non-classical Stefan problem with nonlinear thermal coefficients and a Robin boundary condition

1.   Introduction

In ^[1], the authors introduced the following Stefan problem which is governed by a non-classical and nonlinear heat equation with heat source $F,$ thermal coefficients which depend on the temperature and a convective boundary condition at fixed face $x = 0$ (see also ^[2,3,4]):

where $\rho(T), c(T)$ and $k(T)$ are the density of the material, its specific heat, and its thermal conductivity, respectively; $T_{_{m}}$ is the phase-change temperature, $\rho_{0} > 0$ its the constant density of mass at the melting temperature; $l > 0$ is the latent heat of fusion by unity of mass and $s(t)$ is the position of phase change location. Also, $F$ depends on the evolution of the heat flux at the boundary $x = 0$ and assumed that

From the second equation in Pr.(1.1), we see that $T_{x}(0, t) < 0.$ This can be physically justified since $F$ in this particular study represents a heat sink due to "melting ice/water phase change".

To obtain a similarity solution, we introduce the new independent variable

and the dimension-free dependent variable, defined by

In terms of these new variables, the problem takes the following nonlinear boundary value problem ^[1]

where the nonlinear terms $L(f(\xi))$ and $N(f(\xi))$ are given by

and

with $Ste = \frac{c(T^{*}-T_{m})}{l} > 0$ (Stefan number) and the system parameters $k_{0},$ $\rho_{0},$ $c_{0}$ and $\alpha_{0} = \frac{k_{0}}{\rho_{0}c_{0}}$ are the reference thermal conductivity, density of mass, specific heat and thermal diffusivity, respectively.

Recently, the authors ^[1] proved the existence and uniqueness of the similarity solution of Pr.(1.1) under the restrictive Lipschitz conditions on the parameters and using techniques of functional analysis.

In this paper, explicit solutions of Pr.(1.1) are obtained when the thermal conductivity and specific heat are linear in temperature. Also, existence solutions of similarity-type are proved for this non-classical Stefan problem with nonlinear thermal conductivity and specific heat by using a technique based on lower and upper bounds of the solution. Our approach is simpler than the approach used in ^[1] and is more accessible to readers since it doesn't assume Lipschitz conditions on the parameters and doesn't involve tools of functional analysis.

2.   Linear conductivity

We consider the case where thermal conductivity and specific heat are linear in temperature ^[2,3,4]. This case has been largely discussed in the literature since it provides a good approximation of the actual values for some material such as water (see for example ^[5,6,7,8,9]). We also assume that $T$ is continuous differentiable, which is a natural assumption.

2.1.   Case 1

Let

Writing the nonlinear second-order ODE of Pr.(1.5) in a simple form

where $\alpha = \frac{kT^{*}}{k_{0}},$ $\beta = \frac{\rho cT^{*}}{\rho_{0} c_{0}},$ $\delta = \frac{k(T_{m}-T^{*})}{k_{0}}$ and $\gamma = \frac{\rho c(T_{m}-T^{*})}{\rho_{0} c_{0}}.$

Multiplying both sides of Eq (2.2) by $\delta \gamma,$ we obtain

or

Since $\alpha\gamma = \beta\delta = \frac{kT^{*}}{k_{0}}\frac{\rho c(T_{m}-T^{*})}{\rho_{0} c_{0}}.$ Then Eq (2.4) can be written as

where $a = \alpha\gamma$ and $b = \delta\gamma.$

By the change of variable

Eq (2.5) becomes

The solution of this linear first-order equation is given by

where $\varphi(\xi)$ is the imaginary error function, that is $\varphi(\xi) = \int_{0}^{\xi} \exp\left(\frac{\gamma}{\delta}t^{2}\right)dt$ and $z(0)$ can be determined by using $z(0) = (a+b f(0))f'(0) = \gamma (\alpha+\delta f(0))f'(0)$ and the first initial condition of Pr.(1.5) from which we can readily obtain

Hence

From Eq (2.6), we have

Substituting $z(\xi)$ into this equation, we obtain

Employing now the two boundary conditions of Pr.(1.5): $f(\xi_{0}) = 1$ and $f'(\xi_{0}) = M\xi_{0}$ to obtain

Thus, the condition on $\xi_{0}$ is given by

Thus

Theorem 2.1. The exact solution of Pr. (1.5) is given by

subject to the condition (2.14).

Remark 2.2. From $f(\xi) = \frac{T(x, t)-T^{*}}{T_{m}-T^{*}},$ it can be readily seen that $0 < f(0) < 1.$ Furthermore, for arbitrary $\epsilon > 0,$ it can also be shown using classical chain rule on this relation with $\xi = \frac{x}{2\sqrt{\alpha_{0} t}}$ that

This gives $0\leq f^{\prime}(0) = \frac{2\sqrt{\alpha_{0}t}}{T_{m}-T^{*}}T_{x}(0, t).$ Using the second equation in Pr.(1.1) we obtain $0\leq f^{\prime}(0) < \frac{2\sqrt{\alpha_{0}}h}{k(T(0, t))}.$ Therefore, from (2.14) we conclude that $\xi_{0} > 0.$

Consequently,

Theorem 2.3. The exact solution of the original Pr.(1.1) is given by

subject to the condition (2.14) and the free boundary is given by $s(t) = 2\xi_{0}\sqrt{\alpha_{0}t}.$

2.2.   Case 2

Another important case is $L(f(\xi)) = 1+ \delta f(\xi)$ and $N(f(\xi)) = 1$ ^[1]. For this case Pr.(1.5) becomes

A result on the existence and uniqueness of solution to the nonlinear boundary value problem Pr.(2.18) when $A = 0$ was proved in ^[8], where the solution was treated as a Generalized Modified Error.

Writing the nonlinear ODE of Pr.(2.18) as

The transformation $u(\xi) = (1+\delta f(\xi))^{2}$ leads to

subject to the initial-boundary conditions

2.2.1.   Solutions

In the special case $A = 0,$ Eq (2.20) is transformed to a homogeneous equation

Consider the following transformation ^[9]

Hence

The substitution of these into (2.22) gives

The substitution $\chi = w-v+1$ brings also Eq (2.25) to the Abel equation of the second kind

where

which is in general very difficult to handle. Since the explicit solution cannot be found, then we estimate the solution by finding upper and lower bounds of $u.$

2.2.2.   Upper and lower bounds of the solution

If we assume that $u'(\xi)\geq 0$ then

So that

From (2.22), we have

Thus

Integrating both sides of (2.31) from $\xi$ to $\xi_{0},$ we obtain

Integrating again (2.32) from $\xi$ to $\xi_{0},$ we obtain

where $u(\xi_{0}) = (1+\delta)^{2}$ and $u'(\xi_{0}) = \frac{4\xi_{0}\delta}{Ste}.$

The constants $u(0)$ and $u'(0)$ can be estimated from the mixed condition $u'(0) = 2p(\sqrt{u(0)}-1)$ and (2.32) (or (2.33)) to find

The case $u'(\xi)\leq 0$ follows in a similar fashion.

Based on this, and in view of the condition $u'(0) = 2p(\sqrt{u(0)}-1),$ we have

Lemma 2.4. 1). If $A = 0$ and $u'(\xi)\geq 0, \ \xi\in [0, \xi_{0}],$ then there is at least one solution $u(\xi)$ of Pr.(2.20) and (2.21) such that

subject to the following conditions

2). If $A = 0$ and $u'(\xi)\leq 0, \ \xi\in [0, \xi_{0}],$ then there is at least one solution $u(\xi)$ of Pr.(2.20) and (2.21) such that

subject to the following conditions

3.   Nonlinear conductivity

Let the thermal conductivity $k(\theta)$ given by $k(\theta) = \frac{\rho_{0}c_{0}}{(a+b\theta)^{2}}$ (see ^[10]) for a list of references that considered this case.

Thus

where $a, b, k^{*}$ are positive constants. Hence the nonlinear ODE of Pr.(1.5) can be written as

or in the equivalent form

where $k_{1} = \frac{\rho_{0}c_{0}}{k_{0}b}$ and $k_{2} = \frac{k^{*}}{b}.$

The change of variable $z(\xi) = a+b f(\xi)$ leads to an equation of the form

In turn, Eq (3.4) can be reduced, by the introduction of the new independent variable $u(\xi) = \frac{1}{z(\xi)},$ where $u'(\xi) = -\frac{z'(\xi)}{z^{2}(\xi)}$ and $z'(\xi) = -\frac{u'(\xi)}{u^{2}(\xi)}$ to the following form

The case $A = 0,$ brings Eq (3.5) to the simpler form

3.1.   Solutions

Consider now the following transformation

A simple computation leads to

The substitution of these into (3.6) gives

The following transformation $\chi = w-v+1$ reduces Eq (3.9) into the known Abel equation of the second kind (2.26), where $\Phi(v) = k_{1}-2(\frac{1}{v}-1)$ and $\Psi(v) = -2k^{2}_{1}\frac{(v-1)^{2}}{v},$ which is in general very difficult to handle. So that, we will find the lower and upper bounds of the solution $u(x).$

3.2.   Upper and lower bounds of the solution

Proceeding as before, we obtain

Lemma 3.1. 1). If $u'(\xi)\geq 0, \ \xi\in [0, \xi_{0}]$ then there is at least one solution $u(\xi))$ of Eq (3.6) subject to (2.21) such that

subject to the following conditions

2). If $u'(\xi)\leq 0, \ \xi\in [0, \xi_{0}]$ then there is at least one solution $u(\xi))$ of Eq (3.6) subject to (2.21) such that

subject to the following conditions

4.   The general case: $L(f(\xi))$ and $N(f(\xi))$ are nonlinear

Rewrite the general nonlinear equation

in the form

Let $F(\xi) = L(f(\xi))f'(\xi).$ Thus

Multiplying both sides of Eq (4.3) by $\exp\left(2 \int_{0}^{\xi}\frac{\eta N(f(\eta))}{L(f(\eta))}d\eta\right),$ we obtain

Thus

where $C$ is a constant of integration and can be found from the condition $F(0) = L(f(0))f'(0) = pf(0),$ that is $C = pf(0).$

Hence $f(\xi)$ can be immediately expressed by

If we assume that

Then a simple computation leads to

and

where $\int_{0}^{\xi} \exp\left(-\frac{ N_{m}}{L_{M}}\eta^{2} \right)d\eta = \frac{\sqrt{\pi L_{M} }}{2\sqrt{N_{m}}} \text{erf} \left(\frac{\sqrt{ N_{m} }}{L_{m}\sqrt{L_{M}}}\xi\right).$ Based on this and in view of the boundary conditions $f(\xi_{0}) = 1$ and $f'(\xi_{0}) = M \xi_{0},$ we have

Theorem 4.1. If the conditions (4.7) are satisfied, then there exists at least one solution $f(\xi)$ of Pr.(1.5) such that

where

and

subject to the following conditions

and

Acknowledgments

The authors would like to acknowledge the support of Prince Sultan University, Saudi Arabia for paying the Article Processing Charges (APC) of this publication.

Conflict of interest

The authors declare no conflict of interest.