Page 17 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

2 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Axiomatically, the notion of mass is defined by the following axioms:
1) There is always an m: {M} —> IR,, i.e., an application which asso-

clates to a material system M, from the assembly of all material systems
{M}, a real positive number m(M) (which is also a state quantity joined
to M), called the mass of the system.
Physically, the association of this number m(M) to a material system
M is made by scaling the physical mass of M with the mass of another
material system considered as unit (i. e. by measurement);

2) For any “splitting” of the material system M in two disjoint subsys-
tems M, and My (M = M,UMg2 and M:;NMg2 = 9), the application
m satisfies the additivity property, i.e., m(M) = m(My1) + m(M2).
This additivity property attributes to the mass application the quality
of being a measure. Implicitly, the mass of a material system m(M) is
the sum of the masses dm of all the particles (molecules) which belong

to M, what could be written (by using the continuity hypothesis too)
as

m(M) = foam,
M

the integral being considered in the Lebesgue sense;
3) For any material system M, its mass m(M) does not change during
its evolution, 1e., it is constant and consequently m = 0 (the universal
principle of mass conservation).

Concerning the hypothesis of absolute continuity of the mass vis a vis
the volume of the region D occupied by the considered material system
M, this hypothesis obviously implies, besides the unity between the ma-
terial system and the region “filled” by it, that the mass of any material
subsystem P Cc M could become however small if the volume of the
region D Cc D, occupied by P, becomes, in its turn, sufficiently small
(but never zero, i.e., the principle of the indestructibility of matter is

observed). More, by accepting that the region D and all its subregions
D, are the closure of certain open sets which contain an infinity of fluid
particles occupying positions defined by the corresponding position vec-
tors r (vs. the inertial frame) and additionally the boundaries of these
sets are surfaces (in a finite number) with continuous normal, then ac-
cording to the Radon—Nycodim theorem, there is a positive numerical

function p(r,t), defined a.e. in D, such that the mass of a part PC M
can be expressed by

m(P) = few dv,

M