Page 35 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

20 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

remains constant as the particles are followed while they move (i.e., on

any pathline), but the value of this constant could be different from
trajectory to trajectory.
If the medium is homogeneous, 1.e., p is constant with respect to the
spatial variables, then it is incompressible if and only if p is constant vs.
the time too.
We note that if a continuum is homogeneous at the moment ¢ = 0,

it could become nonhomogeneous later on. In fact a continuum remains
homogeneous if and only if it is incompressible.
Within this book we will deal only with incompressible homogeneous
media (continua).

2.3. Principle of the Momentum Torsor Variation.

The Balance Equations
According to this principle of mechanics, applied within continua for

any material subsystem P C M, at any configuration of it D = x (P,t),
the time derivative of the momentum torsor equals the torsor of the
(direct) acting forces.
As the torsor is the pair of the resultant and the resultant moment,
while the (linear) momentum of the subsystem P is H (P) J vdm =
=

f pvdv and the angular (kinetic) momentum is Ko(P) = fr x vdm =
D P
fr x pvdv (O being an arbitrary point of £3), the stated principle can

D
be written as

P vdv= | Td a+ | pfdu,
pi | ever =
+/
fd
D S D
respectively

D
py | rxevdv= [xx tda+ fr x ptde,

D Ss D

the right members containing the direct acting forces resultant (1.e., the

sum of the stresses resultant and of the external body forces), respec-
tively the moment resultant of these direct forces (moment evaluated vs.
the same point OQ).
But, by using the continuity equation, we remark that 4), J pvdv =

f padv. In fact, on components, we have
D