Page 44 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 29

By introducing now the heat flux principle (Fourier—Stokes) which

states that there is a vector q(r,t), called heat density vector, so that

q(r,n, t) =n: q(r,t)’,

the Gauss divergence theorem leads to

[ e+ diva - 2w — pra) = 0,
D

that is, using the fundamental lemma too,

pé = 2w — divq+ pra.

Obviously if we did not “split” dQ into the conduction heat and the
radiation heat, the last two terms of the above relation would be repre-
sented by the unique term p24, é6q being the total heat density per unit

of mass.
To conclude, the energy equation together with the first law of ther-
modynamics could be written both in a nonconservative form

D v dq
—
—\)=p—4+di
PD (c+ 5) py, + divlT]v + p f. Vv;
. . . 10
and in a conservative form or of divergence type

O v v2 6g
Bi e (c+ 5) +V: \¢ (c+ 5) v| = pz, + div[T]v + pf -v,

this last form playing a separate role in CFD.

The second law of thermodynamics, known also as the Kelvin—Planck
or Clausius principle, is a criterion which points out in what sense a
thermodynamic process is irreversible.
It is well known that all the real processes are irreversible, the re-
versibility being an attribute of only ideal media. While the first law of
thermodynamics does not say anything on the reversibility of the pos-
tulated transformations, the second law tries to fill up this gap. More

*For sake of simplicity we consider only the case of the heat added to P and corresponding
“yn” will represent the unit inward normal drawn to S and this is the right unit normal
vector we deal with in our case.
The heat flux principle could be got by applying the above form of the first law of ther-
modynamics to a tetrahedron of Cauchy type (that is a similar tetrahedron with that used
in the proof of the Cauchy theorem)
'°The transformation of the left side could be done by using the derivative of a product and
the equation of continuity.