Page 39 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

24 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Then, by using the fundamental lemma, one gets the so-called con-

servative form of Cauchy’s equations

© (ov) + div (pv @ v ~ (T}) = pf,

which, on components, leads to

O ;
By (vi) + (ovivg — Ti3),5 = PFi(t = 1, 2,3).

Concerning the writing of Cauchy’s equations in Lagrangian coordi-
nates this requires the introduction of some new tensors as, for instance,
the Piola—Kirchoff tensor [33].
Concerning the objectivity (frame invariance) of the Cauchy equations

we remark that these equations are not frame invariants. Really while
the forces which correspond to the contact or distance direct actions are
essentially objective (frame invariants) as well as n = eee and div|T]
(these together with grad f and [T] respectively), the acceleration vector

which obviously depends on the frame of reference, is not objective.
An objective form of these equations obtained by the introduction of
some new vectors but without a physical meaning can be found in [33].
With respect to the mathematical “closure” of the Cauchy system of
equations (3 equations with 10 unknowns), this should be established
by bringing into consideration the specific behaviour, the connection be-

tween stresses and deformations, i.e., the “constitutive law” for the con-
tinuum together with a thermodynamic approach to the motion of this
medium.

2.6 Principle of Energy Variation.
Conservation of Energy

The fact that the energy of a material system does not change while
the system moves, 1.e., the so-called “energy conservation”, will lead
to another equation which characterizes the motion of the material

medium.
Obviously, by introduction of some thermodynamic considerations
later on, this energy equation will be rewritten in a more precise form.
Let us assess the elemental work done per unit time (the power) of the
forces exerted on a material subsystem P of the deformable continuum

M and whose configuration is D, 1.e.,

éL
Ta fv te Inda + [pt - var.
S D