Page 71 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

56 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

the flow is homentropic then (as we have already seen) the fifth equation

will be p = Kop’.
Concerning the initial conditions for the Euler equations, they arise
from the evolution character of these equations. Such initial conditions
imply that we know gp, p, T and v at an “initial” moment so that these
conditions, together with the Euler equations, set up a Cauchy problem.
From the classical Cauchy—Kovalevski theorem we can conclude that this

Cauchy problem for the Euler system (with f = 0), the equation of con-
tinuity, the constancy of entropy on each path line ($ = 0, which means
in adiabatic evolution) and the state equation p = p(s, p), together with
the initial conditions v|,~9 = v° (r),pli-p = P° (r), Siig = 8° (x), TER,
where infrer p® (r) = p® > 0, is a well-posed problem and for any ini-
tial data and analytical state equations, i.e., there is a unique analytical
solution defined on the domain V = {r € R3,|t| < T (r)} Cc ‘+, where

T(r), for any r, is a function depending continuously on initial data in
the metrics of analytical spaces.
Of course the above mentioned result is a locally time existence and
uniqueness theorem which is valid only for continuous functions (data
and solution).
Generally, there are not global (for all time) existence and uniqueness

results, excepting the two-dimensional case due to the vorticity conser-
vation (4 = oy’. Nevertheless the practical applications require certain
sharp global uniqueness conditions for the Cauchy problem or more gen-
erally for the Cauchy mixed problem (with also boundary conditions, at
any time t) associated with the Euler system.
Before presenting such uniqueness results we remark that the “non-

uniqueness” of the Euler system solution would be linked to the “sud-
denness” of the approximation of a viscous and non-adiabatic fluid by
an ideal fluid in adiabatic evolution. R. Zeytonnian® has shown that the
loss of the boundary conditions associated with the mentioned approxi-
mation, in the circumstances of the presence of some bodies of “profile
type’, could be completed by the introduction of some Joukovski type

conditions (to which we will return) while in the case of some bodies
of “non-profile type”, the model should be corrected by introducing a
vortices separation (vortex sheets).
Let now U = (v1, 02,03, p,8)! be a solution of the Euler system for t >

0, a solution which is defined in a bounded domain 2 C IR*. We accept
that the boundary of this domain is composed of a three-dimensional
spatial domain wo, enclosed in the hyperplane ¢ = 0, and by a sectionally

*See R. Zeytonnian, Mécanique Fondamentale des Fluides, t.1, pp. 154 — 158 [160].
See R. Zeytonnian, Mécanique Fondamentale des Fluides, t.1, p. 126 [160].