Page 45 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

30 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

precisely, in a simplified form, one postulates that a transformation, a

thermodynamical process, takes place in such a way that the entropy
does not decrease or remain the same.
What is the entropy ? In the case of reversible processes, the spe-
cific entropy (per mass unit) s is defined by the differential relation
ds = ‘4 where 6g is the total heat per mass unit while 7 is the abso-

lute temperature — an objective and intensive quantity (.e., it is not an
absolute continuous function of volume) — whose values are strictly pos-
itive and which is the fundamental quantity of thermodynamics. But,
generally, the entropy S for the material subsystem P will also be a
state quantity which is an absolute continuous function of mass (exten-
sive quantity) and it can be expressed, via Radon—Nycodim’s theorem

as S = fs(r,t)dm, s being the specific entropy. In the case of an
P
irreversible process this entropy changes as a result of both interaction
with surroundings (external action) and inside transformations (internal
actions) such that we have ds = ds, + ds;.

Since ds; > 0(a result coming from kinetics) and ds, = 4 we have
that ds > 4 which is the local form of the second law, also known as
the Clausius—Duhem inequality. We remark that the “equality symbols”
would belong to the case ds = ds, and, implicitly, to the reversible

(ideal) processes. Obviously for these reversible processes, using also
the first law of thermodynamics under the form pé = [T] - [D] +p%4, one
obtains the so-called Gibbs equation

pé |T] -[D] + pT,
=

which is fundamental in the study of ideal continua.
Concerning the general (unlocal) formulation for the second law of
thermodynamics, the condition of some real (irreversible) processes, this

could be the following:
For any material subsystem P of the deformable continuum M, which
is seen in the configuration D of boundary OD, there is a state quantity
S, called entropy, whose rate of change, when the subsystem is passing
from a state to another (neighboring) one, satisfies

» 6Q q:n Tq
> =~
gaat | pifdv >0
S> / —_——— — > Q.
aD D