Page 128 - Bird R.B. Transport phenomena

Problems 113

rectangular fluid element. The Gauss theorem for a tensor in which e is a third-order tensor whose components are
is needed to complete the derivation. the permutation symbol e^ (see §A.2) and v = /x/p is the
k
This problem shows that applying Newton's second law kinematic viscosity.
of motion to an arbitrary moving 'Ъ1оЬ" of fluid is equivalent (b) How do the equations in (a) simplify for two-dimen-
to setting up a momentum balance over an arbitrary fixed re- sional flows?
gion of space through which the fluid is moving. Both (a) and
(b) give the same result as that obtained in §3.2. 3D.3 Alternate form of the equation of motion. 8 Show
(c) Derive the equation of continuity using a volume ele- that, for an incompressible Newtonian fluid with con-
ment of arbitrary shape, both moving and fixed, by the stant viscosity, the equation of motion may be put into
methods outlined in (a) and (b). the form
+
7
3D.2 The equation of change for vorticity. :<x> — :7) (3D.3-2)
(a) By taking the curl of the Navier-Stokes equation of where
motion (in either the D/Dt form or the d/ dt form), obtain = Vv 4- (Vv) and a) = Vv - (Vv) + (3D.3-2)
+
an equation for the vorticity, w = [V X v] of the fluid; this
equation may be written in two ways:
+ [w • Vv] (3D.2-1)
_D_ i
Dt"
8 R G S a f f m a n
[ [(Vv) • (Vv)ll (3D.2-2) ' ' Vortex Dynamics, Cambridge University
+ e:
Dt w = Press, corrected edition (1995).