Page 70 - Mechanical Engineers' Handbook (Volume 4)

5 Fluid Momentum 59

udA
2
A
1
2
VA

u 2
n
1
2
Vn i 1 i
For laminar flow in a circular tube, ⁄3; for laminar flow between parallel plates,
4
1.20; and for turbulent flow in a circular tube, is about 1.02–1.03.
5.2 Equations of Motion
For steady irrotational flow of an incompressible nonviscous fluid, Newton’s second law
gives the Euler equation of motion. Along a streamline it is
V 1 p z
V g 0
s s s

and normal to a streamline it is
V 2 1 p g z 0
r n n
When integrated, these show that the sum of the kinetic, displacement, and potential energies
is a constant along streamlines as well as across streamlines. The result is known as the
Bernoulli equation:

V 2 p
gz constant energy per unit mass
2
V 2 V 2
1 p gz 2 p gz constant total pressure
2 1 1 2 2 2
and

V 2 1 p 1 V 2 2 p 2
z z constant total head
2g g 1 2g g 2
For a reversible adiabatic compressible gas flow with no external work, the Euler equation
integrates to

p
p
V 2 k V 2 k
1 1 gz 2 2 gz
2 k 1 1 1 2 k 1 2 2
which is valid whether the flow is reversible or not, and corresponds to the steady-flow
energy equation for adiabatic no-work gas flow.
Newton’s second law written normal to streamlines shows that in horizontal planes
2
dp/dr V /r, and thus dp/dr is positive for both rotational and irrotational flow. The
pressure increases away from the center of curvature and decreases toward the center of
curvature of curvilinear streamlines. The radius of curvature r of straight lines is infinite,
and thus no pressure gradient occurs across these.
For a liquid rotating as a solid body