Page 55 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

40 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Sutherland, in the hypothesis that the colliding molecules of a perfect

or quasiperfect gas are rigid interacting spheres, got for the viscosity
1
coefficient yz the evaluation p = aT'!/? (1 + h) , where a and # are

constants [153].
Fluids that do not observe a linear dependence between [T] and [D]
are called non-Newtonian. Many of the non-Newtonian fluids are “with

memory”, blood being such an example.
In the sequel we will establish the equation for viscous fluid flows
without taking into account the possible transport phenomenon with
mass diffusion or chemical reactions within the fluid.
Writing the stress tensor under the form [T] = —p{I]+[o], the Cauchy
equations for a deformable continuum lead to

= = pf gradp + divia|
—

or, in conservative form,

= (pv) + V- (pv @v) = pf — gradp + div[G].

We remark that all the left sides of these equations could be writ-
ten in one of the below forms, each of them being important from a

mathematical or physical point of view:

pa= oor + (v -grad)v v] = p E + (gradv)v v|
=

14,2
Bt
_— =p [Se + aiv(v @v) - vaive| = |% + grad ($v ) +0 xv].
Ot
Concerning div [ao] = div [A (div v) [I] + 24 [D]], which is a vector,

by using the formulas 2[D] = (gradv) + (gradv)", div(gradv)? =
grad (div v), 2[Q]a =wxa ({Q] being the rotation tensor — the skew-
symmetric part of gradv and aan arbitrary vector), we get for divia]
a first form

div{o] = (A + pn) grad (div v) + pV7v+ (div v) grad »

+ 2gradv (grad p) + (grad) x w,

where

V2v = div (gradv), grad v(gradu) = (gradu - grad) v