ABSTRACT
The laws describing the nonisothermal ßow of an incompressible Newtonian ßuid were presented in Chapter 1 and are summarized here for ready reference. The equations are written for a ßuid region Ωf using a Cartesian coordinate system xi in a Eulerian reference frame, with the index i taking the values i = 1, 2, 3 for the three-dimensional case and i = 1, 2 for the two-dimensional case; the usual summation convention on repeated indices is used [see Eqs. (1.5.7)—(1.5.9)]; time is denoted by t. An extended form of the Boussinesq approximation is used, which allows the ßuid properties to be functions of the thermodynamic state (e.g., pressure P and temperature T ) and the density ρ to vary with temperature T according to the relation
ρ = ρ0[1− β(T − T0)] (5.2.1) where β is the coefficient of thermal expansion and the subscript zero indicates a reference condition. The variation of density as given in (5.2.1) is permitted only in the description of the body force; the density in all other situations is assumed to be that of the reference state, ρ0. The governing equations of convective heat transfer are summarized below.
