Page 140 - Bird R.B. Transport phenomena

124 Chapter 4 Velocity Distributions with More Than One Independent Variable

SOLUTION For steady, creeping flow, the entire left side of Eq. D of Table 4.2-1 may be set equal to zero,
and the ф equation for axisymmetric flow becomes

EV = 0 (4.2-2)
or, in spherical coordinates
Г 2 sin в д ( 1 д W, v
п /лп Q

This is to be solved with the following boundary conditions:

B.C. 1: at r = R, v = — ^ — ^ = 0 (4.2-4)
r
r sin в °V
B.C. 2: at r = R, v 0 = +—^-- ^ = 0 (4.2-5)
Г Sin в дГ
2
B.C. 3: asr-^°o, ^-> -\v r 2 sin 0 (4.2-6)
x
The first two boundary conditions describe the no-slip condition at the sphere surface. The
third implies that v z —> v x far from the sphere (this can be seen by recognizing that v r = v x cos
в and v 0 = —v x sin в far from the sphere).
We now postulate a solution of the form

2
ф{г, в) =/(r) sin в (4.2-7)
since it will at least satisfy the third boundary condition in Eq. 4.2-6. When it is substituted
into Eq. 4.2-4, we get

d 2 _ 2_\l jF_ _ 2L
dr 2 r j\dr 2 r 2
2
The fact that the variable в does not appear in this equation suggests that the postulate in Eq.
4.2-7 is satisfactory. Equation 4.2-8 is an "equidimensional" fourth-order equation (see Eq.
C.l-14). When a trial solution of the form/(r) = Cr" is substituted into this equation, we find
that n may have the values -1,1,2, and 4. Therefore fix) has the form

2
fix) = Qr" 1 + C r + C r + C / (4.2-9)
2 3
To satisfy the third boundary condition, C must be zero, and C has to be -\v^ Hence the
4 3
stream function is
2
2
ф(г, в) = (С,г- ] + C r - {v^r ) sin в (4.2-10)
2
The velocity components are then obtained by using Table 4.2-1 as follows:
2
v = —r^— -^ = Uoc - 2 - - - 2 4 cos в (4.2-11)
r
6
r sin в <№ \ г r )

2
The first two boundary conditions now give Q = —\v R and C = \v R, so that
K 2 x
(4.2-13)

n0 (4.2-14)

These are the velocity components given in Eqs. 2.6-1 and 2 without proof.