Page 54 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 39

principles. At the same time, within the frame of Noll’s axiomatic sys-

tem, the postulate 3), which states the inertial frame invariance of [T],
is a consequence of the objectivity principle.
At last, the necessary and sufficient condition for the isotropy of a
tensorial dependence (the Cauchy—Eriksen—Rivlin theorem) shows that,
in our working space £3, the structure of this dependence should be of
the type

[T] — [—p(p, T) + yo(p, T, hh, qo, T3)](I]

+ PI (p, T, qh, Ip, T3)[D] + p2(p, T, Ii, Ie, I3){D]?,

where Yo, 91, y2 are isotropic scalar functions depending upon the princi-

—
pal invariants J), Iz, Iz of [D], where J; = tr{D] = div v, 212 = (tr[D])?
tr{D]? and I3 = det[D], and with the obvious restriction yo(p, T, 0,0,0) =
O (conditions required by the postulate 4)).
This general form for the constitutive law defines the so-called Reiner—
Rivlin fluids, after the names of the scientists who established it for the
first time.

Those real fluids characterized by a linear dependence between [T]
and [D] are called Newtonian or viscous. By using the corollary which
gives the general form of a linear isotropic tensorial function [([A]),
observing the hydrostatic form at rest, we necessarily have for these
fluids the constitutive law

[T}] = [—p (p, T) + A (e, T) tr[D]] [1] + 2p (p, T) [D],

where the scalars and A are called, respectively, the first and the
second viscosity coefficient. By accepting the Stokes hypothesis 3A +
2u = 0, which reduces to one the number of the independent viscosity
coefficients and which is rigorously fulfilled by the monoatomic gases
(helium, argon, neon, etc.) and approximately fulfilled by other gases

(provided that divv is not very large) we would have (from tr[T] =
—3p + (3A 2u) tr[D] ), that m1 + 722 + 733 = —3p, ie., the above
4+
mentioned result on the equality of pressure with the negative mean of
normal stresses.
Obviously, for a viscous fluid there are also tangential stresses and so
there is a resistance to the fluid layers sliding. The viscosity of fluids is
basically a molecular phenomenon.
For the incompressible viscous fluid from tr{[D] = divuv = 0 we get

[T] = —p{T] + 2u[D].