3.1 Introduction

Consider a 1D domain (0 ? x ?L) through which a fluid with a velocity u is flowing. Then, the steady-state form of the first law of thermodynamics can be stated as

(3.1)

where

(3.2)

These equations are to be solved for two boundary conditions, T=T ₀ at x=0 and T=T _L at x=L. It is further assumed that ? u is a constant as are properties C _p and k.

Our interest in this chapter is to examine certain discretisational aspects associated with Equation 3.1. This is because in computational fluid dynamics (momentum transfer) and in convective heat and mass transfer, we shall recurringly encounter representation of the total flux in the manner of Equation 3.2. Note that if u=0, only conduction is present and the discretisations carried out in Chapter 2 readily apply. However, difficulty is encountered when convective flux is present. The objective here is to understand the difficulty and to learn about commonly adopted measures to overcome it. In the last section of this chapter, stability and convergence aspects of explicit and implicit procedures for an unsteady equation in the presence of conduction and convection are considered.

3.2 Exact Solution

Because our interest lies in examining the discretisational aspects associated with convective-conductive flux, we take the special case of S=0. For this case, an elegant closed-form solution is possible. Thus, we define

(3.3)

(3.4)

(3.5)

where P is called...