The steady heat transfer through a rectangular longitudinal fin is studied. The thermal conductivity and heat transfer coefficient are assumed to be temperature dependent making the resulting ordinary differential equation (ODE) highly nonlinear. An asymptotic solution is used as a means of understanding the relationship between key parameters. A dynamical analysis is also employed for the same purpose.

In this paper we consider the temperature profile in a longitudinal rectangular fin attached to a stationary base surface. Fins are defined as extended surfaces used to enhance the heat dissipation from a hot surface [

As far as the author knows, there is no or very little work which has been done on obtaining asymptotic solutions to a problem of the form presented here. An investigation of such solutions is of value given the prevalence of many parameters whose impact and relationship with each other has yet to be fully understood. It is the purpose of asymptotic solutions to reveal the dominant physical mechanisms of the model. In Moitsheki and Harley [

We consider a rectangular longitudinal one-dimensional fin with a cross-sectional area

Schematic representation of a longitudinal fin with a rectangular profile.

The heat transfer coefficient may be given as the power law [

In the above equations,

Introducing the following dimensionless variables:

In (

The dimensionless boundary conditions become

These conditions ensure an insulated fin tip and a constant base temperature.

In this section we will assume that

Plot of the asymptotic solution (

It needs to be noted that the solution does not allow for the case

In this section we will consider the case where

The method of asymptotics is effectively used here to inspect the behaviour of the solution for small

For the case where

In order to investigate the dynamics of a system in which

As a means of further investigating the behaviour of the system we conduct a dynamical system analysis. We do this by rewriting (

The Jacobian matrix

To be able to do a phase plane analysis of the relevant equation we need to linearise the system. This is done via the calculation of the Jacobian (

This in turn produces the eigenvalues

The phase diagrams produced for this equilibrium point—to be seen in Figure

Plot of the phase trajectories for system (

Takens [

Division of the vector field by

We now turn to our other two equilibrium points. Evaluating

When we consider

For these two equilibrium points we find that we have two cases to consider:

eigenvalues with no zero or purely imaginary eigenvalues,

eigenvalues which are purely imaginary.

In the first instance we turn to the Hartman-Grobman theorem which states that if

For case (a) where

Plot of the phase trajectories for system (

If however, as in the case of (b), any one of the eigenvalues has zero real part, then stability cannot be determined by linearisation. Thus, given that the linearised system does not describe the nonlinear system, we consider phase diagrams for specific values of our parameters as a means of understanding the dynamics of the system. If we choose

Plot of the phase trajectories for system (

For

When

As such, for these values of

In this paper, we were able to obtain an asymptotic solution which clarifies the behaviour of the system when

As a means of further investigating the effects of

C. Harley, thanks H. Ockendon, C. Hall, and C. Please for useful discussion and acknowledges support from the National Research Foundation, South Africa, under Grant no. 79184. Furthermore, this paper was made possible (in part) by a grant from the Carnegie Corporation of New York. The statements made and views expressed are, however, solely the responsibility of the author.