Page 33 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

18 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

of the point of application, ie., f = f(r,t). To avoid ambiguity we will

suppose, in this sequel, that all the external forces we work with are body
forces (gravity forces being the most important in our considerations).
To postulate the existence of the densities T and f (continuity hypothe-
ses) is synonymous with the acceptance of the absolute continuity of the
whole contact or external (body) actions with respect to the area or the
mass respectively. Then, by using the same Radon—Nycodim theorem,

the total resultant of the stresses and of the external body forces could
be written

Ro = [Baa R? = [tam = J etav,

S P D
representations which are important in the general principles formula-

tion.
In the sequel we will formulate the general principles for continua
by expressing successively, in mathematical language, the three basic
physical principles:
(1) mass is never created or destroyed (mass conservation);
(ii) the rate of change of the momentum torsor is equal to the torsor

of the direct exerted forces (Newton’s second law);
(111) energy is never created or destroyed (energy conservation).

2.2 Principle of Mass Conservation.
The Continuity Equation

Mass conservation, postulated by the third axiom of the definition of
the mass, requires that the mass of every subsystem P C M remains
constant during motion. Evaluating this mass when the subsystem is

located in both the reference configuration (i.e., for t = 0) Do and the
current configuration at the moment t, mass conservation implies that

m(P) f oo Ro)av f p(e,t)av f plx(R,O] say,
=
=
=
Do D Do

the last equality being obtained by reverting to the current reference
configuration.
In the continuity hypothesis of continuum motion ( p,v € C'), as the
above equalities hold for every subsystem P (and so for every domain
Do), the fundamental lemma, from the end of sub-section 1.1, leads to

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