Page 56 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 4]

A second form is obtained by using the additional formulas

div (gradv) = grad (div v) — rotw = V*v, 2div[Q] = —rotw ;

more precisely, we have

v
div[o] = grad [(A + 241) (div v)| — rot uw+2grad (grad p)

+2 (grad p) x w — 2 (div v) grad p.

At last, by introducing some known vectorial-tensorial identities (see
Appendix A), one can get a third form,

div[o] = grad[(A + 2p) (div v)|] — rot pw

+ 2grad |(grad p) - v| — 2div [(grad 1) ® v]).

With regard to the energy equation, by using the nonconservative,
respectively the conservative form of this equation for an arbitrary de-
formable continuum, in the case of the viscous fluid we get

2
D v dq . .
PD; (« + =) = pz — div pv + div ([a]v) + pf-v

(the nonconservative form), respectively

o 2 2 6
a p (« + 5) +V: |p (« + -) y| = p= —divpv + div ({o]v)+pf-v

(the conservative form), where, obviously,

T
lalv = A (divv) v+y | (grad v) + (grad v)"| Vv.

If we are interested in the mathematical nature of these equations we
remark that, firstly, the equation of continuity is a partial differential
equation of first order which could be written, in Lagrangian coordi-
nates, Jo(R,t) = po, such that Jp = constant is a solution of this
equation which also defines the trajectories (obviously real). As these

trajectories are characteristic curves too, the equation of continuity is
then of hyperbolic type.