Page 120 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 103

(b) Hence show that the Navier-Stokes equations for axisymmetric flow are given by
= pg, - - + p(F + -- - - + -)
ap
@u ldu u
@u
dr r2 r dr r2 dz2
=pgz--+p(-+--+-) law @W
ap
@W
dZ dr2 r dr dz2
6 Euler equations for two-dimensional flow in polar coordinates
(a) For the two-dimensional flow described in Ex. 2 show that the Euler equations
(inviscid momentum equations) take the form:

dr

[Hints: (i) The momentum components perpendicular to and entering and leaving
the side faces of the elemental control volume have small components in the radial
direction that must be taken into account; likewise (ii). the pressure forces acting on
these faces have small radial components.]
7 Show that the strain rates and vorticity for the flow and coordinate system of Ex. 6
are given by:
. du . ldv u
Err = -. QQ =--
rad+;

1
idu
?j =- av ---+--). idu c =---- av +-
r& 2 ( dr r ra+ ’ rw dr r
[Hint: (i) The angle of distortion (p) of the side face must be defined relative to the
line joining the origin 0 to the centre of the infinitesimal control volume.]
8 (a) The flow in the narrow gap (of width h) between two concentric cylinders of length
L with the inner one of radius R rotating at angular speed w can be approximated by the
Couette solution to the NavierStokes equations. Hence show that the torque T and
power P required to rotate the shaft at a rotational speed of w rad/s are given by

2rpwR3 L 2Tpw2~3~
T= P=
h ’ h
9 Axisymmetric stagnation-point flow
Carry out a similar analysis to that described in Section 2.10.3 using the axisymmetric
form of the NavierStokes equations given in Ex. 5 for axisymmetric stagnation-
point flow and show that the equivalent to Eqn (2.11 8) is
411’ + 2441 - 412 + 1 = 0

where 4’ denotes differentiation with respect to the independent variable c = m z
and 4 is defined in exactly the same way as for the two-dimensional case.