Page 22 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 7

and we may watch the whole motion of the individualized (by their
positions in the reference configuration) particles.
In the case of fluids, in general, and of gases, in particular, the

molecules are far enough apart that the cohesive forces are not suffi-
ciently strong (in gases, for instance, an average separation distance
between the molecules is of the order 3,5 x 10~’cm). As a consequence
to follow up such particles during their motion becomes a difficult task,
the corresponding displacements being very large (a gas sprayed inside
“fills” immediately the respective room).

That is why for fluids, in general, and for gases, in particular, another
way to express the parameters of the motion, to choose the independent
variable, should be considered. This new type of motion description is

known as the spatial or Eulerian description, the corresponding variables
being the spatial or Eulerian coordinates.

Precisely, as Eulerian coordinates (variables) the components ofr ( 2;
or x’) and tare to be considered. In other words, in this description, we
focus not on the continuum particles themselves but on their position
in the current configuration and we determine the motion parameters
of those particles (not the same !) which are locating at the respective
positions at that time. Thus to know v = v(r,t), for a fixed r at t € T,
means to know the velocities of all the particles which, in the consid-

ered interval of time, pass through the position defined by r. On the
other hand, if we know the velocity field v(r,¢) on Dx T, by integrating
the differential equation o = v(r,t), with initial conditions (assuming
that the involved velocity field is sufficiently smooth to ensure the exis-
tence and uniqueness of the solution of this Cauchy problem) one gets r
= x(R,7¢), which is just the equation of motion (1.1) from the material

(Lagrangian) description. Conversely, starting with (1.1) one could im-
mediately set up v(r,t), etc., which establishes the complete equivalence
of the two descriptions.

In what follows we calculate the time derivatives of some (vectorial or
scalar) fields f expressed either in Lagrangian variables ( f(R,¢) ) or in
Eulerian variables ( f(r, #) ).

In the first case f = of and this derivative is called a local or material
. . . . . 2
derivative. Obviously, in this case, v = ce (R,t) anda= oF (R, t).

But, in the second case, we have f = ef +(v-V)f, where V is, in
Cartesian coordinates, the differential operator V = grad = se ii. This
derivative is designed to be the total or spatial or substantive derivative

or the derivative following the motion. In particular a(r,t) = a +
(v-V)v.