Page 41 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

26 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

t2
[ Aa | dvs ff B. ndadt = [ [cava

D(t2) D(t1) ti S(t) ti D(t)

Obviously if all considered variables (1.e., the motion) are assumed
continuous in time, the general conservation principle becomes

Di m,| aes | B nda = [ cae,

S(t) D(t)

where n is the unit outward vector drawn normal to the surface S.
The above relation states that for a volume support D, the rate of
change of what is contained in D, at moment ¢#, plus the rate of flux
out of S, is equal to what is furnished to D. The quantities A,B,C are
tensorial quantities, A and C having the same tensorial order. If B 4 0,
then it is a tensor whose order is one unity higher than A.

If we use the Reynolds transport theorem for the first integral and the
Gauss divergence theorem for the second integral, we have

/ [Se + divt - c| dv =0
0
D(t)

where f = Av + B.
Since the above result is valid for any material subsystem P of the
deformable continuum (1.e., for any D) the fundamental lemma and the
same hypothesis on the motion continuity allows us to write

“- + divf = C,

which is the unique general differential equation, in conservative form,
associated to the studied principles.

3. Constitutive Laws. Inviscid and real fluids

3.1 Introductory Notions of Thermodynamics.
First and Second Law of Thermodynamics

Thermodynamics is concerned with the behaviour of different mate-
rial systems from the point of view of certain state or thermodynamic
variables parameters. The considered thermodynamic (state) variables
will be the absolute temperature (the fundamental quantity for thermo-
dynamics), the pressure p, the mass density p, the specific (per mass

unity) internal energy e and the specific entropy s. The last two state
variables will be defined in what follows.