Page 51 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

36 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

de = Tds — pd (*) ;
p

This last differential relation could be the departure point in the ther-
modynamical study of the ideal fluids. If the internal energy e is given

as a function of the independent parameters s and ; = v, Le., if we
know e = €(s,v), then we will immediately have the equations of state
p= — 9 (s,v) and T = ae (s,v) or, in other words, the function é(s, v) ,
determining the thermodynamical state of the fluid, is a thermodynami-

cal potential for this fluid. Obviously, this does not occur if e is given as a
function of other parameters when we should consider other appropriate
thermodynamical potentials.
If the inviscid fluid is incompressible, from d (4) = dy = 0 we have

that T = T'(s) or s = 8(T) and hence e = €(T). More, if in the energy

equation, written under the form

pé = [T] -(D] — divg + pra,

we accept the use of the Fourier law q = — xgradT, where x is the
thermal conduction coefficient which is supposed to be positive (which

expresses that the heat flux is opposite to the temperature gradient), we
get finally

pe = div (ygradT) + pra.

As e = €(T) and rg (the radiation heat) is given together with

the external mass forces, the above equation with appropriate initial
and boundary conditions, allows us to determine the temperature T
separately from the fluid flow which could be made precise by considering
only the Euler equations and the equation of continuity.
This dissociation will not be possible, in general, within the compress-
ible case. Even the simplest statics (equilibrium) problems for the fluids
testify that.

An important situation for the compressible fluids is that of the perfect
fluids (gases), the air being one of them.
By aperfect gas, we understand an ideal gas which is characterized by
the equation of state (Clapeyron) p = pRT (where R is a characteristic

constant). For such a perfect gas the relation Cp, — Cy = T (2). (3t),
becomes Cy — Cy, = R,even if C, and C, are functions of temperature
(Joule). Since (6g), = CpdT, (6q),, = CydT, the first law of thermody-

namics under the form dq = de + pdv, leads to