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EML3701 Outline
Final exam outline
Fluid properties
• Choice, true/false problems (n = 10)
• Calculation problems (n = 3)
Hydrostatic pressure and forces
• Hydrostatic forces, e.g., place surfaces, curved surfaces
• Control volume analysis, e.g., mass conservation, reaction forces on a plate
from a jet
• Viscous flow in a pipe, e.g., laminar flow – pressure drop, shear stress, flow
rate measurement/manometry
Steady flow of an inviscid, incompressible fluid
Motion and visualization
Conservation of mass and energy
Dimensionless quantities and groups
1
2
What is a Fluid? ...
What is a Fluid? ...
F
b
B
B’
F
U
b
Fluid:deforms continuously when a shear force of any magnitude is
applied
B’
U
• No-Slip Condition
• fluid and solid interface have
same velocity
• fluid “sticks” to the boundary
Viscosity: measure of the resistance of a fluid to flow
µ: dynamics viscosity
: rate of shearing strain
3
B
4
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Viscosity
Surface Tension
Newtonian, linear relationship
Newtonian vs NonNewtonian
Shear thickening,
viscosity increases as τ increases
3
2
μ: slope
1
low viscosity
Shear stress, τ
Shear stress, τ
high viscosity
du
Rate of shearing strain
dy
Surface tension: intensity of molecular
attraction per unit length.
Shear thinning,
viscosity decreases as τ increase
= force/length [FL-1]
Rate of shearing strain
Viscosity: fluid resistance to deform
τ = μ du
dy
5
6
Fluid Statics
Some Important Equations
Specific weight
Specific gravity
Shear stress for
Newtonian fluids
Ideal gas law
7
Force surface
tension
F3= Ps dsdx
Fst =σl
F2= Py dzdx
θ
θ
Capillary rise in
tube
mg
Speed of sound
ideal gas
F1= Pz dydx
Py = Ps = Pz
k: specific heat ratio
R: specific gas constant
Bulk modulus
Pascal’s Principle: pressure is transmitted undiminished,
equal in all direction (neglecting gravity)
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Pressure in a Column of Fluid (considering gravity)
p2
Manometers Pressure
patm : atmospheric pressure at sea level
dA
h
A
dz
p1
mg
x
h2
(2)
gage pressure
p1 = ρgh + patm
fluid, ρ1
(3)
gage fluid, ρ2
pA = ρ2gh2 - ρ1gh1
pA = γ2h2 - γ1h1
dA
p1
z
(1)
h1
p1 - p2 = ρgdz
fluid 1 gas & fluid 2 liquid (example air)
y
9
pA ≈ γ2h2
10
Hydrostatic Force on an Arbitrary Plane Surface
Resultant Force inArbitrary Plane Surface
Step1: Find the centroid of the surface (gate)
θ
hc
- if the surface is completely vertical (θ=90°)
O
hc = yc
l1
- if the surface is inclined vertical
hc = yc sin(θ)
𝑦
l2
Magnitude of the resultant force
FR = γhc A
Step2 :Find the location of Resultant Force
yR = Ixc/(yc A) + yc
usually, just yR
11
12
xR = Ixyc/(yc A) + xc
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Hydrostatic Force on an Curved Surface
y
h1
Fy
A
Hydrostatic Force on an Curved Surface
h2
W
CG
FH
Fy = ρgh1 Ay
x
Fx
FV
W
B
Ay
FH
Summation of forces
FH = Fx
FV = Fy + W
FR = √(FH)2 +(FV)2
Fx = ρghc Ax
= ρg(h1 +h2)/2 Ax
W
Ax
O
O
Fx = ρghcAx
= ρg(h1 +h2)/2 Ax
FV
Summation of forces
FH = Fx
FV = Fy - W
FR = √(FH)2 +(FV)2
13
Fy = ρgh2 Ay
14
Review Force: Curved Surface
Buoyancy
Floating body
ρf
FB
c
CG
mg
FB =ρf gV
Free body diagram water
V
displaced volume
Buoyancy force = weight of the displaced volume of fluid
acts in the centroid of the displaced volume
15
16
FB =ρf gV = mbodyg
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Stagnation point
The Bernoulli Equation
Continuity Equation (conservation of mass)
17
18
Flowrate Measurements
19
Flow from large tank (no losses or viscous effects)
20
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Eulerian description of velocity field - represented in
Energy line & hydraulic grade line
21
different coordinates
22
Flow visualization
• Terms
• Streaklines
• Streamlines
• Pathlines
23
24
Equation for Streamlines
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Reynolds Transport Theorem (RTT)
25
26
Pipe
flow
27
28
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Buckingham Pi theorem
There are k-r independent dimensionless parameters to describe the
problem r basic dimensions that describe the problem; Usually r = 3
Mass (M), Length (L) and Time (T)
Reynolds
number
29
30
Viscous pipe flow
Dimensionless numbers, groups
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Extended Bernoulli Equation
(with no shaft
work/pump )
Poiseuille’s law!
33
34