etermination of Pi Terms
(Methods of Repeating Variables)
1. List all independent variables involved in the problem. Include
geometry (cg. diameter, length), uid properties (cg. density,
viscosity), and external effects driving the ow (e. g. velocity,
pre

N onuniform ow at the inlet and the outlet
Velocity in the (210% section, density is constant:
Pig + M + 9200? = Pf + I: m + 92m " 1053 + WSHAFTNETIN
where on is the kinetic energy coefcient (cc 2 I) and V is the average -
velocity. ' _ V1=30 mIS
31:3

Linear Momentum Equation
for Fixed Control Volume
Momentmn m mass times the velocity (of a particle)
Newton 3 second law. rate of change of momentum equals force:
d(mv)
dt =ma=F I '. -*. . . .
cambmed- With F # ma. for the system ylelds the linear momentu

Torque of Rotating Machinery
' Tshaft = milirinvei + moutiiroutveout]
'0 Torque is positive if it is oriented in the same direction. as w-
(pump)
«a Torque is negative if it is oriented in the opposite direction of w
(turbine)
o (irVQ) is positive (both f

Groundwater Flow in a Vertical Plane
Denition of head ¢[L]:
y is specic weight of water
2 is the elevation
Darcys Law:
(ML/T]- is the flow per unit area (specic: discharge) in the hor. dir.
qz[L/T] is the ow per unit area (specic discharge) in the ver. di

Conservation of .Mass
d fcv pdV
dt
+ J DVWdA = 0
(55
Net rate of mass outow in y direction
_ - a _
am") dy )dxdz -+ [pv - (W) 39-Mde 2 Ohm) dxdy dz
' 6y 2 6y
Net rate of mass outow in x direction
a
(9) dxdy dz
6):
Net rate of mass outow in z directi

Angular Momentum (Moment-ofMomentum)
Equation for Fixed Control Volume
A rigid body system (bad approximation for uids!):
at vi?
dt
a? m angular velocity *
T,
: mass moment of inertia
Ni :- moment that acts on the system
Fluids: for volume dV the angular

Euler Equations of Motion
Denition of p-ICSSUICS
1
p = 3(0xx + Uyy + Uzz)
For inviscid m nonviscous : frictionless ow (viscous stresses are
zero).
Since viscous normal stress is zero, the pressure is:
m 0') : guy : O'zz Bu 8v
Txy:Tyx:
Bu 6w
sz=sz=
Tyz=

Using the denition of the stream function:
1
dq 2 Lily + dx = (19
6y 6):
Using the denition of the stream function:
hence: ux between two streamlines m dierence in stream function
values.
Stream function in cylindrical coordinates Stream function example

Power of Rotating Machinery
Shaft power
'I
Wehaft shaftw
Wshaft I "min(irinwvein) + mendiroutwveout)
Wshaft = midiuinveix + mouiuoutveout)
a (iUVg) is positive (both for in and out) if V9 and. U are oriented
in the same direction '
I o (:tUVe) is negative