Page 21 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

6 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

“sweeping” the current configuration (at the time t) D = x(Do,t). In this

respect (1.1) can be also understood as a mapping of the tridimensional
Euclidean space onto itself, a mapping which depends continuously on
t € TF and the motion of the continuum in the whole time interval 7
will be defined by the vector function x (R,t) considered on Do x T.
Now, one imposes some additional hypotheses for the above mapping
joined to the equation of motion (1.1). These hypotheses are connected

with the acceptance of some wider classes of real motions which confer
their validity.
Suppose that r is a vectorial function of class C? (Do) with respect to
the R components. This means that the points which were neighbours
with very closed velocities and accelerations, at the initial moment, will

remain, at any time t, neighbours with velocities and accelerations very
closed too. Further, we presume that, at any moment ¢, there is a
bijection between Dg and DP except, possibly, of some singular points,
curves and surfaces. Mathematically this could be written through the
condition that, at any time t, the mapping Jacobian J = det(gradr)
#O ae. in D.
This last hypothesis linked to preserving the particles’ identity (they

neither merge nor break) is also known as the smoothness condition or
the continuity axiom. As from the known relation between the elemental
infinitesimal volumes of Dp and D, namely dv = JdV, one deduces,
through J # 0, that any finite part of our continuum cannot have the
volume (measure) of its support zero or infinite, the above hypothesis
also implies the indestructibility of matter principle.

In the previous hypotheses it is obvious that (1.1) has, at any moment
t, an inverse and consequently R = x71(r, t) € C?(D). Summarizing, in
our hypotheses, the mapping (1.1) is a diffeomorphism between Dp and
D.
The topological properties of the mapping (1.1) lead also to the fact
that, during the motion, the material varieties (1.e., the geometrical va-

rieties “filled” with material points) keep their order. In other words,
the material points, curves, surfaces and volumes don’t degenerate via
motion; they remain varieties of the same order. The same topological
properties imply that if Co(S9) is a material closed curve (surface) in
the reference configuration, then the image curve (surface) C(S), at any

current time t, will be also a closed curve (surface).
Further, if the material curves (surfaces) co) (50) and Cc?) (52)
are tangent at a point Po, then, at any posterior moment, their images
will be tangent at the corresponding image point P, etc.

The material description, the adoption of the Lagrangian coordinates,
is advisable for those motion studies when the displacements are small