Page 49 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

34 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

A deformable continuum ts called isotropic, if there are not privileged

directions or, in other terms, the (“answering”) functional which defines
the stress tensor is isotropic or frame rotation invariant.
According to the Cauchy—Eriksen—Rivlin theorem [40], a tensor func-
tion f({A]), defined on a set of symmetric tensors of second order from
£3 and whose values are in the same set, is isotropic if and only if it
has the form f ({A]) = yo[I] + yi[A] + ve[A]?, where y,, are isotropic

scalar functions of the tensor [A] which could always be expressed as
functions of the principal invariants JI;, Jo, J3 of the tensors [A], Le.,
pr = vr (Ii, Io, Is).
As a corollary any linear isotropic tensor function [([A]) in E3 should
be under the form /({[A]) = cotr ({[A]) [I] + c:{[A], where co and c, are
constants.

3.3. Inviscid (Ideal) Fluids
The simplest of all the mathematical and physical models associated
to a deformable continuum is the model of the inviscid (ideal) fluid.

By an inviscid (ideal) fluid we understand that deformable continuum
which is characterized by the constitutive law [T] = —p[I] (or, on com-
ponents, 7;; = —pd;;) where p is a positive scalar depending only on r
and ¢ (and not on n), physically coinciding with the (thermodynamical)
pressure.

The “hydrostatic” form (characterizing the equilibrium) of the stress
tensor [T] = —p[I] shows that the stress vector T is collinear with the
outward normal n drawn to the surface element (and, obviously, of op-
posite sense) i.e., for an inviscid fluid the tangential stresses (which with-
stand the sliding of neighboring fluid layers) are negligible.
The same structure of the constitutive law for an inviscid fluid points

out that this fluid is always a homogeneous and isotropic medium.
In molecular terms, within an inviscid fluid, the only interactions
between molecules are the random collisions. Air, for instance, can be
treated as an inviscid fluid (gas).
With regard to the flow equations of an inviscid (ideal) fluid, known
as Euler equations, these could be got from the motion equations of a
deformable continuum (Cauchy equations), Le., from pa; = pf + Tij,j

where we use now the specific structure of the stress tensor 74; = —pd;;;
hence

pa; = pfi — Dy

or, in vector language