Page 31 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

16 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

derivative operator could then commute with that of integration. More
precisely we will perform the change of variables given by the equation

of motion expressed in Lagrangian coordinates, i.e, r = x(R,t), the
new integration domain becoming the fixed domain Dg from the initial
configuration and then we could come back to the current domain D(é).
More exactly, taking into account both Euler’s and Green’s theorems
we have

D D D - .
Bi gf POvae = Bef PORN) IAV = FF )
—
=
—
F(r,t)dv J F(X(R,t))JdV = — | (FJ+ FJ)dvV
= f (F + Fdiv v)dv = f (oF + vgradF + Fdivv)dv
Dit) D(t) a

pi | ot pit) 9 s(t)

where n is the unit external normal.
This transport formula will be useful in establishing the equations of

motion for continua (under the so-called conservation form).
Analogously, one establishes equivalent formulas for the total deriva-
tives of the curvilinear or surface integrals when the integration domains
depend upon time.
Thus

D OF ;
Di F-ndo = / Fe +rot(F xv)+v tiv® | -ndo,

S(t) C(t)

where C(t) is the contour enclosing the surface [52].
From this formula comes the necessary and sufficient condition for the
flux of a field F, through a material surface S(t), to be constant, which
condition is

OF
BE + rot(F x v) + vdivF = 0 (Zorawski condition).

In the formulation of the general principles of the motion equations

under a differential form (usually nonconservative), an important role is
taken by the following
LEMMA: Let g(r) be a scalar function defined and continuous in a
domain D and let D be an arbitrary subdomain of D. If f p(r)dv = 0,
D
for every subdomain D C D, then the function —(r) =0 in D.

The proof is immediate by using “reductio ad absurdum” and the
continuity of ~ [110].