Page 34 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 19

which represents the equation of continuity in Lagrangian coordinates.

In spatial (Eulerian) coordinates, by making explicit the third axiom
from the mass definition, 1.e., m = 0, we get

0=1n(P) = = [ p(r,t)av = f (p+ paivv) dv,

D D
where the Reynolds transport theorem has been used. Backed by the
same fundamental lemma, the following forms of the continuity equation
can also be obtained:

p+ pdivv =0 (the nonconservative form)

or

0
+ div (pv) =0 (the conservative form).

We remark that if in the theoretical dynamics of fluids, the use of
nonconservative or conservative form does not make a point, in the ap-
plications of computational fluid dynamics it is crucial which form is
considered and that is why we insist on the difference between them.

2.2.1 Incompressible Continua
A continuum system is said to be incompressible if the volume (mea-

sure) of the support of any subsystem of it remains constant as the
continuum moves.
By expressing the volume (measure) of the arbitrary system P at both
the initial and the current moment, we have

[a= [av f say,

Do D Do
i.e., the incompressibility, in Lagrangian coordinates, implies that J = 1

and consequently the equation of continuity becomes

po (R) = p(x (R,t)).

We can arrive at the same result, in Eulerian coordinates, if we write

D io
0= = [w= | (i+aivv) do,

D D

which leads to dzuv = 0 and, from the continuity equation, to FP =
O. We conclude that for incompressible continua, the (mass) density