Page 141 - Bird R.B. Transport phenomena

§4.2 Solving Flow Problems Using a Stream Function 125

To get the pressure distribution, we substitute these velocity components into the r- and
^-components of the Navier-Stokes equation (given in Table B.7). After some tedious manip-
ulations we get
fi7W

These equations may be integrated (cf. Eqs. 3.6-38 to 41), and, when use is made of the bound-
ary condition that as r —» °° the modified pressure 9? tends to p 0 (the pressure in the plane z =
0 far from the sphere), we get

p = p - pgz - I {~){jj cos в (4.2-17)
0

This is the same as the pressure distribution given in Eq. 2.6-4.
In §2.6 we showed how one can integrate the pressure and velocity distributions over
the sphere surface to get the drag force. That method for getting the force of the fluid on
the solid is general. Here we evaluate the "kinetic force" F by equating the rate of doing
k
work on the sphere (force X velocity) to the rate of viscous dissipation within the fluid,
thus

F k * = ~\ (TiVv)rrfr sin Ш&ф (4.2-18)
v
Jo Jo JR
Insertion of the function (—T:VV) in spherical coordinates from Table B.7 gives

Then the velocity profiles from Eqs. 4.2-13 and 14 are substituted into Eq. 4.2-19. When the in-
dicated differentiations and integrations (lengthy!) are performed, one finally gets

F = бтг/jiV^R (4.2-20)
k
which is Stokes' law.
As pointed out in §2.6, Stokes 7 law is restricted to Re < 0.1. The expression for the drag
force can be improved by going back and including the [v • Vv] term. Then use of the method
of matched asymptotic expansions leads to the following result 4

3
2
F = бтг/Ai^RIl + re Re + T^ Re (ln \ Re + у + | In 2 - fi)+ ^ Re In \ Re + О (Re )]
3
k
(4.2-21)
where у = 0.5772 is Euler's constant. This expression is good up to Re of about 1.

1. Proudman and J. R. A. Pearson, /. Fluid Mech. 2, 237-262 (1957); W. Chester and D. R. Breach,
4
/. Fluid. Mech. 37, 751-760 (1969).