Page 50 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 35

pa = pf — gradp,

a system which should be completed by the equation of continuity.
Of course the Euler equations could be rewritten in a “conservative”
form (by using the continuity equation and the differentiating rule of a

product), namely

dO (pv)
+ V-(pv v) = pf — gradp.
@

at
If the fluid is incompressible, the Euler equations and the equation
of continuity, together with the necessary boundary (slip) conditions
(characterizing the ideal media) which now become sufficient conditions
too, ensure the coherence of the respective mathematical model, Le.,
they will allow the determination of all the unknowns of the problem

(the velocity and pressure field). If the fluid is compressible one adds the
unknown p(r,¢), which leads to a compulsory thermodynamical study of
the fluid in order to establish the so-called equation of state which closes
the associated mathematical model.
The thermodynamical approach to the inviscid fluid means the use

of the energy equation (together with the first law of thermodynam-
ics) and of Gibbs’ equation which, being valid for any ideal continuum,
synthesizes both laws of thermodynamics.
The energy equation, either under nonconservative form or under con-
servative form, comes directly from the corresponding forms of an arbi-
trary deformable continuum, namely from

2
= (e+ 5) = po! — div (pv) + pf-v

respectively

O v vy" dq
mye (e+ 5) +¥: lo(e+ 5) v]} - q 7 div (pv) + pf v.

Concerning the Gibbs’ equation, pé = [T] - [D] + p7’s, in the case of
an inviscid fluid it becomes ([T] = —p[I] so that [T] -[D] = —p[I] -{D] =
—ptr ({(D]) = —pdivv)

pe = —pdivv + pl's

or, by eliminating divv from the equation of continuity (divv = —8),
we get