Page 52 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 37

CpdT = de + pd (*) =dh (as p = constant)

Cydt = de.

At the same time, from the transcription of Gibbs’ equation Tds =
de + pdv, and from C, — Cy = R, we have

d
Tds = C,dT — T(Cy — Cu)

or

Tds dp
G. (y-1) 0
=dT—-T(y-l—,

where y = a > 1.
C.
From C, — Cy = R we also get y = oe (Eucken’s formula), while
P
, ; _
the state equation, in y, becomes T= seabyrty 1)°
(y=
The relation ids = dT — T(7 — 1) together with the above expres-
sion for T, assuming that Cy and C. are constants, lead, by a direct
integration, to

Tp'~7 exp (=) = const
v
respectively

pp exp (S) = const.

If there is an adiabatic process (which means without any heat change
with the surrounding), from 6g = 0 we get ds = 0, ie., the entropy s
is constant along any trajectory and the respective fluid flow is called

isentropic (if the value of the entropy constant is the same in the whole
fluid, the flow will be called homentropic). In this case the perfect gas is
characterized by the equation of state T = Kop?—' and p = Kp’, where
Ko and K are constants while we also have

h=C,T, e=C,T.

Obviously in the case of an adiabatic process, the equation of state p =
Kp’, together with the Euler equations and the equation of continuity,

willbe sufficient for determining theunknowns (v;, p, p) (the temperature