Page 75 - Bird R.B. Transport phenomena

60 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

is — (p + r )| r=R (cos в). We now multiply this by a differential element of surface
rr
R 2 sin в dO йф to get the force on the surface element (see Fig. A.8-2). Then we inte-
grate over the surface of the sphere to get the resultant normal force in the z direction:
77
n) f- Г _ ^|
= ( ( + T r=R e)R2 s i n
Jo Jo
5
According to Eq. 2.6-5, the normal stress r is zero at r = R and can be omitted in the in-
rr
tegral in Eq. 2.6-7. The pressure distribution at the surface of the sphere is, according to
Eq. 2.6-4,
V \г-к = Po ~ PgR cos в - | ^ cos в (2.6-8)

When this is substituted into Eq. 2.6-7 and the integration performed, the term contain-
ing p gives zero, the term containing the gravitational acceleration g gives the buoyant
0
force, and the term containing the approach velocity v gives the "form drag" as shown
x
below:
3
F {n) = $irR pg + 2TTHRV (2.6-9)
X
3
The buoyant force is the mass of displaced fluid (f тгК р) times the gravitational accelera-
tion (g).
Integration of the Tangential Force

At each point on the solid surface there is also a shear stress acting tangentially. The
force per unit area exerted in the —0 direction by the fluid (region of greater r) on the
solid (region of lesser r) is +т \ . The z-component of this force per unit area is (r \ )
re r=R
гв г=к
sin 0. We now multiply this by the surface element R 2 sin в dвdф and integrate over the
entire spherical surface. This gives the resultant force in the z direction:
2
7
f(» = Г " Г ( Tr0 \ r=R sin 0)R sin в dd dф (2.6-10)
Jo Jo
The shear stress distribution on the sphere surface, from Eq. 2.6-6, is
T e\r=R = \^Y~ S i n в (2.6-11)
r
Substitution of this expression into the integral in Eq. 2.6-10 gives the "friction drag"

F U) = ATTLJLRV,» (2.6-12)
Hence the total force F of the fluid on the sphere is given by the sum of Eqs. 2.6-9 and
2.6-12:
3
F = \TrR pg + infiRVoo + A7rfiRv x (2.6-13)
buoyant form friction
force drag drag
or
3
i<rrR pg + бтг/LtRi;» (2.6-14)
buoyant kinetic
force force

5
In Example 3.1-1 we show that, for incompressible, Newtonian fluids, all three of the normal
stresses are zero at fixed solid surfaces in all flows.