Page 72 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 57

smooth hypersurface I (for ¢ > 0) which has a common border with the

domain wo. Let also € = (£0,£1, €2,€3) be the outward unit normal to Lr.
It is proved that the uniqueness of U in 2Q is intimately linked with the
hyperbolicity of the Euler equations which requires the fulfilment of the
following complimentary hypothesis: at each point of the hypersurface
I the inequality

Eo + u1€1 + eke + vsé3 > a (Ef + 5 + 63)? (2.1)

should be satisfied.
More precisely, one states that ({160]) if the solution U of the Euler
system exists in the class C! (Q) and this solution satisfies the condition
(2.1), while inf,,, p°(r) > 0, then for any other solution U‘ € C1 (Q) of
the Euler system, one could find a constant kg > 0 such that 6U = U'—U
fulfils ||}6U; || < ko ]}6U;t = Ol|* for t > 0. Consequently if the equality
U' = U holds on wo (that means in the hyperplane t = 0), then it will be
satisfied at any point-moment (r,t) € Q (wo). Obviously 2 (wo), called

the determination domain for the solution of the Cauchy problem with
the initial data on wo, is the union of all the domains which back on wo
and on whose boundary the inequality (2.1) is satisfied.
It has been also proved that if [ (wg) is a smooth boundary (of C?
class) of the determination domain 2Q (wo), then this hypersurface will

be a characteristic surface of the Euler system, the inequality sign of
(2.1) being replaced by that of equality.
We now remark that in the conditions of an Euler system in adiabatic
evolution with a state equation p = p(p, s) of C? class, assuming that the
domain D(é) of the fluid flow has the boundary & (t), which is composed
of both rigid and “‘free” parts, and vy is the propagation velocity of the
surface % [33] then, if
(i) v(r, t), p(r, t), (r,t) are functions of class C! on [0, 7] x D,

(ii) the initial conditions v(r,0), p(r,0), s(r,0) are given together with
(iii) the boundary conditions v,-n = 0 on [0,7] x © and, similarly,
Vv, p,s in the regions where v, < 0,
then the Euler system (even with f # Q), in adiabatic evolution, with
the state equation p = p(p,s), has a unique solution’.
The uniqueness is still kept even in the case when there are not bound-

ary conditions at the points of &: where vy > c, ¢ being the speed of
sound.

‘For the definition of the norm we deal with, we should first consider all the cuts w(t) of Q
by the hyperplane t = constant > 0. Then by introducing the vectorial function v = {v;}
on &, its norm corresponding to the cut w(t) will be defined by |[v;¢{| = fff (95 v2) dw.
w(t)
‘J.Serrin [135].