fluid_mechanics

The net pressure
force on a particle
is determined by the
pressure gradient.
3.2 F ma along a Streamline 97
Thus, if dF ps is the net pressure force on the particle in the streamline direction, it follows that
dF ps 1p dp s 2 dn dy 1p dp s 2 dn dy 2 dp s dn dy
0p ds dn dy 0p
0s 0s dV
Note that the actual level of the pressure, p, is not important. What produces a net pressure
force is the fact that the pressure is not constant throughout the fluid. The nonzero pressure gradient,
§p 0p0s ŝ 0p0n nˆ , is what provides a net pressure force on the particle. Viscous forces,
represented by t ds dy, are zero, since the fluid is inviscid.
Thus, the net force acting in the streamline direction on the particle shown in Fig. 3.3 is given by
a dF s dw s dF ps ag sin u 0p
0s b dV
By combining Eqs. 3.2 and 3.3, we obtain the following equation of motion along the streamline
direction:
g sin u 0p 0V
rV (3.4)
0s 0s ra s
We have divided out the common particle volume factor, dV, that appears in both the force and
the acceleration portions of the equation. This is a representation of the fact that it is the fluid density
1mass per unit volume2, not the mass, per se, of the fluid particle that is important.
The physical interpretation of Eq. 3.4 is that a change in fluid particle speed is accomplished
by the appropriate combination of pressure gradient and particle weight along the streamline. For
fluid static situations this balance between pressure and gravity forces is such that no change in
particle speed is produced—the right-hand side of Eq. 3.4 is zero, and the particle remains stationary.
In a flowing fluid the pressure and weight forces do not necessarily balance—the force
unbalance provides the appropriate acceleration and, hence, particle motion.
(3.3)
E XAMPLE 3.1
Pressure Variation along a Streamline
GIVEN Consider the inviscid, incompressible, steady flow
along the horizontal streamline A–B in front of the sphere of radius
a, as shown in Fig. E3.1a. From a more advanced theory of
flow past a sphere, the fluid velocity along this streamline is
FIND Determine the pressure variation along the streamline
from point A far in front of the sphere 1x A and V A V 0 2 to
point B on the sphere 1x B a and V B 02.
as shown in Fig. E3.1b.
V V 0 a1 a3
x 3b
z
1 V o
0.75 V o
V
0.5 V o
V A = V ˆ O i
V = Vi ˆ V B = 0
A
B
a
x
0.25 V o
(a)
–3a –2a –1a 0
x
(b)
∂p __
∂x
0.610 ρV 2 0
/a
p
2
0.5 ρV 0
–3a –2a –a 0 x
(c)
–3a –2a –a 0
x
(d)
F I G U R E E3.1