fluid_mechanics

102 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation
pressure increases without bound as r → q, whereas for case (b)
the pressure approaches a finite value as r → q. The streamline
patterns are the same for each case, however.
Physically, case (a) represents rigid body rotation (as obtained
in a can of water on a turntable after it has been “spun up”) and
case (b) represents a free vortex (an approximation to a tornado, a
hurricane, or the swirl of water in a drain, the “bathtub vortex”).
See Fig. E6.6 for an approximation of this type of flow.
The sum of pressure,
elevation, and
velocity effects is
constant across
streamlines.
If we multiply Eq. 3.10 by dn, use the fact that 0p0n dpdn if s is constant, and integrate
across the streamline 1in the n direction2 we obtain
dp
r V 2
dn gz constant across the streamline
r
(3.11)
To complete the indicated integrations, we must know how the density varies with pressure and
how the fluid speed and radius of curvature vary with n. For incompressible flow the density is
constant and the integration involving the pressure term gives simply pr. We are still left, however,
with the integration of the second term in Eq. 3.11. Without knowing the n dependence in
V V1s, n2 and r r1s, n2 this integration cannot be completed.
Thus, the final form of Newton’s second law applied across the streamlines for steady, inviscid,
incompressible flow is
p r V 2
dn gz constant across the streamline
r
(3.12)
As with the Bernoulli equation, we must be careful that the assumptions involved in the derivation
of this equation are not violated when it is used.
3.4 Physical Interpretation
In the previous two sections, we developed the basic equations governing fluid motion under a
fairly stringent set of restrictions. In spite of the numerous assumptions imposed on these flows,
a variety of flows can be readily analyzed with them. A physical interpretation of the equations
will be of help in understanding the processes involved. To this end, we rewrite Eqs. 3.7 and 3.12
here and interpret them physically. Application of F ma along and normal to the streamline results
in
p 1 2rV 2 gz constant along the streamline
(3.13)
z
p +
1 rV 2 + gz
2
= constant
p + r V
2
dn + gz
= constant
and
p r V 2
dn gz constant across the streamline
r
(3.14)
as indicated by the figure in the margin.
The following basic assumptions were made to obtain these equations: The flow is steady
and the fluid is inviscid and incompressible. In practice none of these assumptions is exactly
true.
A violation of one or more of the above assumptions is a common cause for obtaining an
incorrect match between the “real world” and solutions obtained by use of the Bernoulli equation.
Fortunately, many “real-world” situations are adequately modeled by the use of Eqs. 3.13
and 3.14 because the flow is nearly steady and incompressible and the fluid behaves as if it were
nearly inviscid.
The Bernoulli equation was obtained by integration of the equation of motion along the “natural”
coordinate direction of the streamline. To produce an acceleration, there must be an unbalance
of the resultant forces, of which only pressure and gravity were considered to be important. Thus,