1. Fluid Mechanics
EIT Review

2. Shear Stress Tangential force per unit area rate of shear
change in velocity with respect to distance
rate of shear

3. Manometers for High Pressures
Find the gage pressure in the center of the sphere. The sphere contains fluid with g1 and the manometer contains fluid with g2.
What do you know? _____
Use statics to find other pressures. ? 1 g2
P1 = 0 h1 3 g1 h2 2 P1
+ h1g2
- h2g1
=P3
For small h1 use fluid with high density.
Mercury!

4. Differential Manometersp1
Water p2 h3
orifice h1 h2
Mercury
Find the drop in pressure between point 1 and point 2. p1
+ h1gw
- h2gHg
- h3gw
= p2
p1 - p2 = (h3-h1)gw + h2gHg
p1 - p2 = h2(gHg - gw)

5. Forces on Plane Areas: Inclined Surfaces
Free surface O q x A’
centroid B’ O
center of pressure y
The origin of the y axis is on the free surface

6. Statics Fundamental Equations Sum of the forces = 0
Sum of the moments = 0
pc is the pressure at the __________________
centroid of the area
Line of action is below the centroid

7. Properties of Areas yc b a
Ixc a
Ixc yc b d R yc
Ixc

8. Properties of Areas yc R
Ixc a yc b
Ixc R yc

9. Inclined Surface Summary
The horizontal center of pressure and the horizontal centroid ________ when the surface has either a horizontal or vertical axis of symmetry
The center of pressure is always _______ the centroid
The vertical distance between the centroid and the center of pressure _________ as the surface is lowered deeper into the liquid
What do you do if there isn’t a free surface?
coincide
below
decreases

10. Example using Moments Solution Scheme
An elliptical gate covers the end of a pipe 4 m in diameter. If the gate is hinged at the top, what normal force F applied at the bottom of the gate is required to open the gate when water is 8 m deep above the top of the pipe and the pipe is open to the atmosphere on the other side? Neglect the weight of the gate.
Solution Scheme
Magnitude of the force applied by the water
hinge
water F
8 m
4 m
Location of the resultant force
Find F using moments about hinge

11. Magnitude of the Force h = _____ 10 m Depth to the centroid pc = ___
hinge
water F
8 m
4 m Fr
h = _____
10 m
Depth to the centroid
pc = ___
b = 2 m
a = 2.5 m
Fr= ________
1.54 MN

12. Location of Resultant Force
hinge
water F
8 m
4 m Fr
Slant distance to surface
12.5 m
b = 2 m
a = 2.5 m cp
0.125 m

13. Force Required to Open Gate
hinge
water F
8 m
4 m Fr
How do we find the required force?
Moments about the hinge
=Fltot - Frlcp
lcp=2.625 m
2.5 m
ltot cp
F = ______
809 kN
b = 2 m

14. Example: Forces on Curved Surfaces
Find the resultant force (magnitude and location) on a 1 m wide section of the circular arc.
water
FV =
W1 + W2 W1
3 m
= (3 m)(2 m)(1 m)g + p/4(2 m)2(1 m)g
2 m
= 58.9 kN kN
= 89.7 kN W2
2 m
FH = x
= g(4 m)(2 m)(1 m)
= 78.5 kN y

15. Example: Forces on Curved Surfaces
The vertical component line of action goes through the centroid of the volume of water above the surface. A
water
Take moments about a vertical axis through A. W1
3 m
2 m W2
2 m
= m (measured from A) with magnitude of 89.7 kN

16. Example: Forces on Curved Surfaces
The location of the line of action of the horizontal component is given by A
water W1 b h
3 m
2 m W2
2 m
(1 m)(2 m)3/12 = m4
4 m x y

17. Example: Forces on Curved Surfaces
78.5 kN
horizontal
0.948 m
4.083 m
89.7 kN
vertical
119.2 kN
resultant

18. Cylindrical Surface Force Check
0.948 m
89.7kN
All pressure forces pass through point C.
The pressure force applies no moment about point C.
The resultant must pass through point C. C
1.083 m
78.5kN
(78.5kN)(1.083m) - (89.7kN)(0.948m) = ___

19. Curved Surface Trick Find force F required to open the gate.
The pressure forces and force F pass through O. Thus the hinge force must pass through O!
All the horizontal force is carried by the hinge
Hinge carries only horizontal forces! (F = ________) A
water W1
3 m
2 m O F W2
W1 + W2
11.23

20. Dimensionless parameters
Reynolds Number
Froude Number
Weber Number
Mach Number
Pressure Coefficient
(the dependent variable that we measure experimentally)

21. Model Studies and Similitude: Scaling Requirements
dynamic similitude
geometric similitude
all linear dimensions must be scaled identically
roughness must scale
kinematic similitude
constant ratio of dynamic pressures at corresponding points
streamlines must be geometrically similar
_______, __________, _________, and _________ numbers must be the same
Mach
Reynolds
Froude
Weber

22. Froude similarity Froude number the same in model and prototype
________________________
define length ratio (usually larger than 1)
velocity ratio
time ratio
discharge ratio
force ratio
difficult to change g
11.33

23. Control Volume Equations
Mass
Linear Momentum
Moment of Momentum
Energy

24. Conservation of Mass If mass in cv is constant2 1 v1 A1
Area vector is normal to surface and pointed out of cv
V = spatial average of v
[M/t]
If density is constant [L3/t]

25. Conservation of Momentum

26. Energy Equation
laminar
turbulent
Moody Diagram

27. Example HGL and EGL z velocity head pressure head elevation pump z = 0
energy grade line
hydraulic grade line z
elevation
pump
z = 0
datum

28. Smooth, Transition, Rough Turbulent Flow
Hydraulically smooth pipe law (von Karman, 1930)
Rough pipe law (von Karman, 1930)
Transition function for both smooth and rough pipe laws (Colebrook)
(used to draw the Moody diagram)

29. Moody Diagram 0.10 0.08 0.06 0.05 0.04 friction factor 0.03 0.02 0.01
0.015
0.04
0.01
0.008
friction factor
0.006
0.03
0.004
laminar
0.002
0.02
0.001
0.0008
0.0004
0.0002
0.0001
0.01
smooth
1E+03
1E+04
1E+05
1E+06
1E+07
1E+08 R

30. Solution Techniques find head loss given (D, type of pipe, Q)
find flow rate given (head, D, L, type of pipe)
find pipe size given (head, type of pipe,L, Q)

31. Power and Efficiencies
Electrical power
Shaft power
Impeller power
Fluid power
Motor losses IE
bearing losses Tw
pump losses Tw
gQHp

32. Manning Formula
The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way...
Hydraulic radius for wide channels

33. Drag Coefficient on a Sphere
1000
100
Stokes Law
Drag Coefficient 10 1
0.1
0.1 1 10
102
103
104
105
106
107
Reynolds Number