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FLUID DYNAMICS
- Study of fluids(liquid and gas) in motion and its cause.
Kinds of Fluid Flow:
1. Steady Flow – if the overall flow pattern does not change in time.
2. Laminar Flow – adjacent layers of fluid slide smoothly past each other and the flow is steady.
3. Turbulent Flow – abrupt change in velocity, irregular and chaotic flow caused by high flow rates.
CONTINUITY PRINCIPLE
dm1
dm2
21 dmdm
from
;dV
dmρ dVρdm
2211 dsAρdsAρ
ds1 = v1dt
ds2 = v2dt
dtvAρdtvAρ 2211
If density does not change from one point to another;
2211 vAvA
Av – Rate of flow, Q (m3/s)
21 QQ
CONTINUITY PRINCIPLE
dm1
dm2
ds1 = v1dt
ds2 = v2dt
If density change from one point to another;
222111 vAρvAρ
Rate of flow, Q changes with factor of ρ1/ ρ2.
2211 QρQρ
12
12 Q
ρ
ρQ
BERNOULLI’S EQUATION
Conservation of Energy:
dKdUdW
ds1
ds2
F2 = P2A2
F1 = P1A1
222111 dsAPdsAP
1122 gymgym
211
222 2
12
1 vmvm
VAds
but
Vρm;V
mρ
From Continuity Principle:
21 mm 21 VV
BERNOULLI’S EQUATION
ds1
ds2
F2 = P2A2
F1 = P1A1
2211 VPVP
1122 gyVρgyVρ
211
222 2
12
1 vVρvVρ
21
221221 2
1 vvρyygρPP
Pressure Difference:
2222
2111 2
12
1 vρgyρPvρgyρP
Bernoulli’s Equation:
TORRICELLI’S THEOREM
y1
y2
h = y2 – y1
2222
2111 2
12
1 vρgyρPvρgyρP
Bernoulli’s Equation:
aPP 1 aPP 2
Since both the tank and the hole are exposed to air;
Continuity Principle:
2211 vAvA
Since A2 is too large compared to A1;
12
12 v
A
Av
02
1 AA
0
TORRICELLI’S THEOREM
y1
y2
h = y2 – y1 02
12
211 gyρPvρgyρP aa
Bernoulli’s Equation:
12212
1 yygρvρ
ghv 21
At one point in a pipeline the water’s speed is 3.00 m/s and the gauge pressure is 5 x 104 Pa. Find the gauge pressure at the second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.
Example 1
Water flows form an open tank as shown. The elevation of point 1 is 10.0 m, and the elevation of points 2 and 3 is 2.00 m. The cross-sectional area of point 2 is 0.0480 m2; at point 3 it is 0.0160 m2. The area of the tank is very large compared with the cross-sectional area of the pipe. Assuming Bernoulli’s equation applies, compute (a) the discharge rate in cubic meter per second; and (b) the gauge pressure at point 2.
Example 2
BERNOULLI’S EQUATION
The horizontal pipe shown has a cross-sectional area of 40.0 cm2 at the wider portions and 10.0 cm2 at the constriction. Water is flowing in the pipe, and the discharge from the pipe is 6.00 x 10-3 m3/s (6.00 L/s). Find (a) the flow speeds at the wide and the narrow portions; (b) the pressure difference between these portions; (c) the difference in height between the mercury columns in the U-shaped tube.
Example 3
BERNOULLI’S EQUATION
