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Fluid Mechanics
Lecture #10
Review• Streamline
–A curve that is instantaneously tangent to the velocity vector of a flow.
• Streamwise acceleration21 ( )
2s s s s
ssdv dv dv d vdsa vdt ds dt ds ds
For constant density fluids
2
constant along the same streamline2
p Vgz
Bernoulli's Equation
2
constant across the streamlines p Vgz dnR
Stagnation PointThe point on the solid body at which the velocity is zero
Stagnation Pressure• A pressure is increased when the velocity becomes small.
• A stagnation point is where the velocity becomes zero and the pressure is maximized. 2
2p V z C
g
1 m/sV
20.01 mA
Force ?F
2 21 1
12 2
22 2p V p Vz z
g g
2 21 00 0 02(9.8) 9,800 2(9.8)
p
2 21 00 0 02(9.8) 2
pg
500 Pap
5 NF
Flow Rate (Discharge)• The measure of “how much a fluid flows”• Defined as the volume per time.• Unit: m3/s• Volume/Time = (Area) x (Length / Time)
= (Area) x (Velocity)• Q=VA
Example I• Determine the velocity at (2).
D
D 0 hV1
2
2g 00
V22
2g
11
2 21 2 2
22 2V Vz z
gp
gp
2 21 2 (1)
2 2V Vh
g g
One more equation required:2 2
1 2 (24 4
)V VDQ d
2Solve for V .
V2 2gh
1 (d / D)4
2If , 2VD d gh
DO IT YOUR SELF
Example II• Manometer is a device to measure pressure difference. Find the h.
Given Q(flowrate), A1, A2, SG.
Fluid is static within the tube (hydrostatic pressure)
There is a flow from left to right in a pipe (Bernoulli’s equation)
(1) (1') : apply Bernoulli equaion across streamlines
(1')
(2 ')
1 12 2
1 111
p V p Vz dn z dnR R
1 12
1
21
11p V p Vz dn z dn
11 11( )p p z z
2 2 2 2Similarly, ( )p p z z
Although there is a flow in between (1) and (1 ),the above equation tells the pressure distribution is hydrostatic.
Find the relation between p1 and p2.p1 (z2 z1) (l h) (SG)hl p2
1 2 1 2(1 )p z z SG h p
1 2 2 1 (1 )p p z z SG h
(1')
(2 ')
Apply Bernoulli Equation2 2
1 1 2 21 22 2
p V p Vz zg g
1 21
22
2 21( / ) ( / )
2 2p Q A p Q Az z
g g
1 22
2
2 2112
1 12
p p Qz zg A A
Solve equations
1 2 2 1 (1 )p p z z SG h 1 22
2
2 2112
1 12
p p Qz zg A A
2 1 2 1 22 1
2
21 1(1 )
21 Qz z SG h z z
g A A
1
2
2 22
1 12 (1 )
Qhg SG A A
DO IT YOUR SELF
Example III (Flow rate measurement)If one can measure p2‐p1, the flow rate Q can be calculated. Find Q in terms of p1, p2, A1, A2.
1 1 2 2Q V A V A
2 21 1 2 2
2 2p V p V
g g
2 2
221
1 222 2
p pQ QgA gA
2
2 21 2
21
1 12
p pQg A A
1 22 12 2
21
21 ( )A AQ pA A
p
Example IV (siphon)Calculate the flow rate through the siphon and the pressure at A.
2 21 1 2 2
1 2 between (1)----(2)2 2
p V p Vz zg g
(1)
(2)
220 3 0 0 0
2 9.8V
2 7.66 m/sV
2 2 32 (0.0(7.66 m/s) m4) 0.1 m /s
4Q V A
2 21 1
1 between (1)----(A)2 2
A AA
p V p Vz zg g
0 3 0 0 0Ap
29.4 kPaAp
Example V (Sluice Gate)
Calculate the flow rate through the sluice gate.
p1
z1
V12
2g
p2
z2
V22
2gV1z1B V2z2B
Q z2B2g(z1 z2 )1 (z1 / z2 )2
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