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