fluid_mechanics

100 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation
The difference in fluid velocity between two points in a flow field, V 1 and V 2 , can often be
controlled by appropriate geometric constraints of the fluid. For example, a garden hose nozzle
is designed to give a much higher velocity at the exit of the nozzle than at its entrance where it
is attached to the hose. As is shown by the Bernoulli equation, the pressure within the hose must
be larger than that at the exit 1for constant elevation, an increase in velocity requires a decrease
in pressure if Eq. 3.7 is valid2. It is this pressure drop that accelerates the water through the nozzle.
Similarly, an airfoil is designed so that the fluid velocity over its upper surface is greater 1on
the average2 than that along its lower surface. From the Bernoulli equation, therefore, the average
pressure on the lower surface is greater than that on the upper surface. A net upward force,
the lift, results.
3.3 F ma Normal to a Streamline
V3.4 Hydrocyclone
separator
In this section we will consider application of Newton’s second law in a direction normal to
the streamline. In many flows the streamlines are relatively straight, the flow is essentially
one-dimensional, and variations in parameters across streamlines 1in the normal direction2 can
often be neglected when compared to the variations along the streamline. However, in numerous
other situations valuable information can be obtained from considering F ma normal
to the streamlines. For example, the devastating low-pressure region at the center of a tornado
can be explained by applying Newton’s second law across the nearly circular streamlines of
the tornado.
We again consider the force balance on the fluid particle shown in Fig. 3.3 and the figure in
the margin. This time, however, we consider components in the normal direction, nˆ , and write Newton’s
second law in this direction as
n
δm
V
a dF n dm V 2
(3.8)
r r dV V 2
r
where g dF n represents the sum of n components of all the forces acting on the particle and dm
is particle mass. We assume the flow is steady with a normal acceleration a n V 2 r, where r is
the local radius of curvature of the streamlines. This acceleration is produced by the change in direction
of the particle’s velocity as it moves along a curved path.
We again assume that the only forces of importance are pressure and gravity. The component
of the weight 1gravity force2 in the normal direction is
dw n dw cos u g dV cos u
To apply F ma
normal to streamlines,
the normal
components of
force are needed.
V3.5 Aircraft wing
tip vortex
If the streamline is vertical at the point of interest, u 90°, and there is no component of the particle
weight normal to the direction of flow to contribute to its acceleration in that direction.
If the pressure at the center of the particle is p, then its values on the top and bottom of the
particle are p dp n and p dp n , where dp n 10p0n21dn22. Thus, if dF pn is the net pressure
force on the particle in the normal direction, it follows that
dF pn 1p dp n 2 ds dy 1p dp n 2 ds dy 2 dp n ds dy
0p ds dn dy 0p
0n 0n dV
Hence, the net force acting in the normal direction on the particle shown in Fig 3.3 is given by
a dF n dw n dF pn ag cos u 0p
0n b dV
By combining Eqs. 3.8 and 3.9 and using the fact that along a line normal to the streamline
cos u dzdn 1see Fig. 3.32, we obtain the following equation of motion along the normal direction
g dz
dn 0p
0n rV 2
r
(3.9)
(3.10a)