Page 18 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 3

The function p(r,t) is called the density or the specific mass accord-

ing to its physical meaning. By using the above representation for the
introduction of the density we overtake the shortcomings which could
arise by the definition of p (r,t) as a point function through

, m(P)

P (r, t) ~ val 60 vol(D)
vol(D)£0

a definition which, from the medium continuity point of view, specifies p
only at a discrete set of points.’ Obvious, the acceptance of the existence
of the density is a continuity hypothesis.

In the sequel, the region D occupied by the continuum M (and anal-
ogously D occupied by the part P) will be called either the volume
support of D, or the configuration at the respective moment in which
the considered continuum appears.
The regularity conditions imposed on DP and on its boundary will
support, in what follows, the use of the tools of the classical calculus (in
particular the Green formulas).

Obviously, the continuum will not be identified with its volume sup-
port or its configuration. We will take for the continuum systems the
topology of the corresponding volume supports (configurations), 1.e., the
topology which has been used in classical Newtonian mechanics. In par-
ticular, the distance between two particles of a continuum will be the

Euclidean distance between the corresponding positions of the involved
particles.
In the study of continua, in general, and of fluids, in particular, time
will be considered as an absolute entity, irrespective of the state of the
motion and of the fixed or mobile system of reference. At the same time
the velocities we will deal with are much less than the velocity of light
so that the relativistic effects can be neglected.

In the working space which is the tridimensional Euclidean space —
space without curvature —- one can always define a Cartesian inertial
system of coordinates. In this space we can also introduce another sys-
tem of coordinates without changing the basic nature of the space itself.
In the sequel, an infinitesimal volume of a continuum (i.e., with a
sufficiently large number of molecules but with a mass obviously in-
finitesimal) will be associated to a geometrical point making a so-called

continuum particle, a particle which is identified by an ordered triple

‘Since the function p defined by this limit cannot be zero or infinite (corresponding to the
outside or inside molecule location of the point where the density is calculated), Vol(D) can
never be zero.