Page 16 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Chapter 1

INTRODUCTION TO MECHANICS OF

CONTINUA

1. Kinematics of Continua

1.1 The Concept of a Deformable Continuum

The fluids belong to deformable continua. In what follows we will
point out the qualities of a material system to be defined as a deformable

continuum.
Physically, a material system forms a continuum or a continuum sys-
tem if it is “filled” with a continuous matter and every particle of it
(irrespective how small it is) is itself a continuum “filled” with matter.
As the matter is composed of molecules, the continuum hypothesis leads

to the fact that a very small volume will contain a very large number of
molecules. For instance, according to Avogadro’s hypothesis, lem? of air
contains 2,687 x 10!9 molecules (under normal conditions). Obviously,
in the study of continua (fluids, in particular) we will not be interested
in the properties of each molecule at a certain point (the location of the
molecule) but in the average of these properties over a large number

of molecules in the vicinity of the respective point (molecule). In fact
the association of these averaged properties at every point leads to the
concept of continuity, synthesized by the following postulate which is ac-
cepted by us: “Matter is continuously distributed throughout the whole
envisaged region with a large number of molecules even in the smallest
(microscopically) volumes”.

Mathematically, a material system filling a certain region DP of the
Euclidean tridimensional space is a continuum if it is a tridimensional
material variety (vs. an inertial frame of reference) endowed with a spe-
cific measure called mass, mass which will be presumed to be absolutely
continuous with regard to the volume of D.