Page 19 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

4 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

of numbers representing, in fact, the coordinates of the point (particle)
within the chosen system. The synonymy between particle and material
point (geometrical point endowed with an infinitesimal mass) is often
used.
An important concept in the mechanics of continua will be that of
a “closed system” or a “material volume”. A material volume is an
arbitrary entity of the continuum of precise identity, “enclosed” by a

surface also formed of continuum particles. All points of such a material
volume, boundary points included, move with a respective local velocity,
the material volume deforming in shape as motion progresses, with an
assumption that there are no mass fluxes (transfers) in or out of the
considered volume, i.e., the volume and its boundary are composed by
the same particles all the time.

Finally, a continuum is said to be deformable if the distances between
its particles (1.e., the Euclidean metric between the positions occupied
by them) are changing during the motion as a reaction to the external
actions. The liquids and gases, the fluids in general, are such deformable
continua.

1.2 Motion of a Continuum.

Lagrangian and Eulerian Coordinates

To define and make precise the motion of a continuum we choose both
a rectangular Cartesian and a general curvilinear reference coordinate
systems, systems which can be supposed inertial.
Let R and r be, respectively, the position vectors of the contin-
uum particles, within the chosen reference frame, at the initial (refer-
ence) moment tp and at any (current) time ¢ respectively. We denote

by (X;) and (z;), respectively, the coordinates of the two vectors in
the rectangular Cartesian system while (X*) and (z*) will represent the
coordinates of the same vectors in the general curvilinear (nonrectan-
gular) system. Thus r referring to a rectangular Cartesian system 1s
r= 2,i, + oligo + 2313 = Teig , Where any two repeated indices imply
summation, and i, are the unit vectors along the x, axes respectively.

For a general system of coordinates (z!,x?, >), the same position vec-
tor r will be, in general, a nonlinear function r(x’) of these coordinates.
However its differential dr is expressible linearly in dz* for all coordi-
nates, precisely

_ Or
=a,,dz",
dr dz™ ™m — mm
~ Ox™
the vectors a,, being called the covariant base vectors. Obviously if 2™
are the Cartesian coordinates 2™ = z,, and, implicitly, am = im.