Page 73 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

58 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

In the case of the incompressible inviscid isochrone ( dp = 0) or baro-

tropic compressible fluid flows, Dario Graffi has given a uniqueness result
which requires [57]:
(i) the functions v(r,t), p(r,t) and p(r,t) are continuously differen-
tiable with bounded first derivative on [0, 7] x D,
(ii) the initial conditions v(r, 0), p(r,0), the boundary conditions v-n

and the external mass forces f are given, respectively, on } and [0,7] x,
(ili) the state equation (in the barotropic evolution) is of the C? class.
We remark that these results keep their validity if D becomes un-
bounded — the most frequent case of fluid mechanics —- under the re-
striction of a certain asymptotic behaviour at far distances (infinity) for
the magnitude of velocity, pressure and mass density, namely of the type

V = Vo + O (r-G+*)) »P =Po tO (r-G+)) P= Po tO (r-G+)) ;

where € is a positive small parameter.
We conclude this section with a particular existence and uniqueness
result which implies an important consequence about the nonexistence of

the Euler system solution for the incompressible, irrotational and steady
flows.
More precisely, if D is a simply connected and bounded region, whose
boundary 0D moves with the velocity V, it can easily shown that [19]:
(1) there is a unique incompressible, potential, steady flow in D, if and
onlyif { V-nds = 0,
aD
(ii) this flow minimizes the kinetic energy Egin = 4 f pv*dv over all
D
the vectors u with zero divergence and satisfying u-nlgpn = V- nlgp-
We remark that this simple result, through (11), associates to the prob-
lem of solution determining a minimum problem for a functional, that
is a variation principle. Such principles will be very useful in numerical
approaches to the fluid dynamics equations and we will return to them
them later in this book.

At the same time if our domain D is bounded and with fixed bound-
ary 0D (V = 0), only the trivial solution v = 0 (the rest) corresponds to
a potential incompressible steady flow. Obviously in the case of the un-
bounded domains this result will be not true provided that the boundary
conditions on 0D should be completed with the behaviour at infinity.
The same result (the impossibility of an effective flow) happens even
if the domain is the outside of a fixed body or a bodies system, the

fluid flow being supposed incompressible with uniform potential (without
circulation) and at rest at infinity.