Page 53 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

38 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

T being determined at the same time with p). For the same perfect

gas, under the circumstances of the constancy of the specific principal
heats but in the nonadiabatic case, the first law of thermodynamics (by
neglecting the radiation heat) leads to

ph =p—divq

or, by using the Fourier law, we arrive at

pCyT = p + div (xgradT),

an equation which allows the determination of the temperature not sep-

arately, but together with Euler’s equations, 1.e., using the whole system
of six equations with six unknowns (1, p, p,T).
Generally, the fluid characterized by the equations of state under the
form f(p,p) = Owith f satisfying the requirements of the implicit func-
tions theorem, are called barotropic. For these fluids, the determination
of the flow comes always to a system offiveequations with five unknowns,
with given initial and boundary conditions.

3.4 Real Fluids

By definition a deformable continuum is said to be a real fluid if it
satisfies the following postulates (Stokes):
1) The stress tensor [T] is a continuous function of the rate-of-strain
tensor [D], while it is independent of all other kinematic parameters (but
it may depend on thermodynamical parameters such as p and 7);
2) The function [T] of [D] does not depend on either a space position
(point) or a privileged direction (1.e., the medium is homogeneous and

isotropic);
3) [T] is a Galilean invariant;
4) At rest ({D] = 0), [T] = —p[I],p > 0.
The scalar p > 0 designates the pressure of the fluid or the static
pressure. A fundamental postulate states that p is identical with the
thermodynamic pressure. We will see later in what circumstances this

pressure is an average of three normal stresses.
Generally the structure of the stress tensor should be [T] = —p[I]+[o],
where the part “at rest” —p[I] is isotropic while the remaining [a] is an
anisotropic part. For the so-called Stokes (“without memory”’) fluids,
[o] = ®(v,gradv,(D]), with restriction {[o] = 0 for the fluid flows of
“rigid type” (without deformations), while for the fluids “with memory”,
[a] depends upon the time derivatives of [D] too.
The postulate 2) implies, through the medium isotropy, that the func-

tion [T] is also an isotropic function in the sense of the constitutive laws