Page 29 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

14 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Following Kelvin, ifa material point (particle) belonging to S(t) moves
. d . . .
along the unit external normal n = ae with a velocity un, then its
infinitesimal displacement dr, in an infinitesimal interval of time (¢,t +
dt), will be ér = nu, dt. As this particle should remain on S(t) (to be a

material surface) we would obviously have f(r+ d6r,t+6t) = 0. Keeping
only the first two terms of the Taylor’s expansion which is backed by the
infinitesimal character of the displacement ér ( and correspondly of the
time dt ), we get

O
oi +un(n-gradf) = 0.

But, on the other side, any material point (particle) of the surface S(t)

should move with the continuum velocity at that point, 1.e., necessarily,
Un = V-n and thus we get the necessary condition

oly .gradf = PF =0.

To prove also the sufficiency of this condition we should point out
that (for instance) if this condition is fulfilled, then there will be at the
initial moment a material surface Sg, such that our surface S = y(So,t),

1.e., it is the image of Sg, through the motion mapping at the instant t.
But then, due to the conservation theorem of material surfaces, it comes
out immediately that S(t) should be a material surface.
Now let us attach to the first order partial differential equation of +

ui Zt = 0 its characteristic system, i.e., let us consider the differential
system

dex _ dep _ des _

VI V2 U3
It is known that if ya (r,t) = Xq, Xo, being constants (a = 1,2,3 ),

is afundamental system of first integrals of our characteristic differential
system, the general solution of the above partial differential equation is
f = ®(¢~1, v2, 93) = B(X1, Xo, X3),where © is an arbitrary function of
class C!. But, then, the particles ofcoordinates Xq (a = 1,2,3 )which
fulfil the equation ®(X X3) = 0 will also fulfil f = 0, ie., at the
1,
Xo,
time t, they will be on the material surface of equation f = 0 (in other
words, the surface S(t) is the image, at the moment ¢, of the material

surface ®(X,, Xo, X3) = 0 from a reference configuration).
This result, which gives the necessary and sufficient condition for an
(abstract) surface to be material is known as the Euler—Lagrange crite-
rion.