Sect. 10-8: Fluids in Motion (Hydrodynamics)

• Two types of fluid flow:

1. Laminar or Streamline: (We’ll assume!)

2. Turbulent: (We’ll not discuss!)

Streamline Motion

• Mass flow rate (mass of fluid passing a point per second): ρ1A1v1 = ρ2A2v2

Equation of Continuity

PHYSICS: Conservation of Mass!!• Assume incompressible fluid (ρ1 = ρ2 = ρ)

Then A1v1 = A2v2

Or: Av = constant– Where cross sectional area A is large, velocity v is

small, where A is small, v is large.

• Volume flow rate: (V/t) = A(/t) = Av

• PHYSICS: Conservation of Mass!!

A1v1 = A2v2 Or Av = constant

• Small pipe cross section larger v

• Large pipe cross section smaller v

Example 10-11: Estimate Blood Flowrcap = 4 10-4 cm, raorta = 1.2 cm

v1 = 40 cm/s, v2 = 5 10-4 cm/s

Number of capillaries N = ?

A2 = N(rcap)2, A1 = (raorta)2

A1v1 = A2v2

N = (v1/v2)[(raorta)2/(rcap)2]

N 7 109

Example 10-12: Heating DuctSpeed in duct:

v1 = 3 m/s

Room volume:

V2 = 300 m3

Fills room every

t =15 min = 900 s

A1 = ?

A1v1 = Volume flow rate = (V/t) = V2/t

A1 = 0.11 m2

Section 10-9: Bernoulli’s Equation • Bernoulli’s Principle (qualitative):

“Where the fluid velocity is high, the pressure is low, and where the velocity is low, the pressure is high.”– Higher pressure slows fluid down. Lower pressure

speeds it up!

• Bernoulli’s Equation (quantitative). – We will now derive it.– NOT a new law. Simply conservation of KE + PE (or

the Work-Energy Principle) rewritten in fluid language!

Work & energy in fluid

moving from Fig. a

to Fig. b :

a) Fluid to left of point 1

exerts pressure P1 on

fluid mass M = ρV,

V = A11. Moves it 1.

Work done:

W1 = F11= P1 A11.

Work & energy in fluid

moving from Fig. a

to Fig. b :

b) Fluid to right of point 2

exerts pressure P2 on

fluid mass M = ρV,

V = A22. Moves it 2.

Work done:

W2 = -F22 = -P2A22.

Work & energy in fluid

moving from Fig. a

to Fig. b :

a) b) Mass M moves

from height y1 to height

y2. Work done against

gravity:

W3 = -Mg(y1 - y2)

Sect. 10-9: Bernoulli’s Eqtn • Total work done from a) b):

Wnet = W1 + W2 + W3

Wnet = P1A11 - P2A22 - Mg(y1-y2) (1)

• Recall the Work-Energy Principle:

Wnet = KE = (½)M(v2)2 – (½)M(v1)2 (2)

• Combining (1) & (2):

(½)M(v2)2 – (½)M(v1)2

= P1A11 - P2A22 - Mg(y1-y2) (3)

• Note that M = ρV = ρA11 = ρA22 & divide (3) by V = A11 = A22

(½)ρ(v2)2 - (½)ρ(v1)2 = P1 - P2 - ρg(y1-y2) (4)

• Rewrite (4) as: P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2

Bernoulli’s Equation• Another form:

P + (½)ρ(v1)2 + ρgy1 = constant• Not a new law, just work & energy of system in fluid

language. (Note: P ρ g(y2 -y1) since fluid is NOT at rest!)

Work Done by Pressure = KE + PE

Sect. 10-10: Applications of Bernoulli’s Eqtn

P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2 Bernoulli’s Equation

Or:

P + (½)ρ(v1)2 + ρgy1 = constantNOTE! The fluid is NOT at rest, so ΔP ρgh !

• Example 10-13

Application #1: Water Storage Tank P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2 (1)

Fluid flowing out of spigot

at bottom. Point 1 spigot

Point 2 top of fluid

v2 0 (v2 << v1)

P2 P1

(1) becomes:

(½)ρ(v1)2 + ρgy1 = ρgy2

Or, speed coming out of

spigot: v1 = [2g(y2 - y1)]½ “Torricelli’s Theorem”

Application #2: Flow on the level

P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2 (1)

• Flow on the level y1 = y2 (1) becomes:

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

(2) Explains many fluid phenomena & is a quantitative statement of Bernoulli’s Principle:

“Where the fluid velocity is high, the pressure is low, and where the velocity is low, the pressure is high.”

Application #2 a) Perfume Atomizer

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

• High speed air (v) Low pressure (P)

Perfume is

“sucked” up!

Application #2 b) Ball on a jet of air(Demonstration!)

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

• High pressure (P) outside air jet Low speed

(v 0). Low pressure (P) inside air jet

High speed (v)

Application #2 c) Lift on airplane wing

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

PTOP < PBOT LIFT!

A1 Area of wing top, A2 Area of wing bottom

FTOP = PTOP A1 FBOT = PBOT A2

Plane will fly if ∑F = FBOT - FTOP - Mg > 0 !

Application #2 d) Sailboat sailing against the wind!

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2

(2)

“Where v is high, P is low, where v is low, P is high.”

Application #2 e) “Venturi” tubes

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

Auto carburetor

Application #2 e) “Venturi” tubes

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

Venturi meter: A1v1 = A2v2 (Continuity) With (2) this P2 < P1

Application #2 f) Ventilation in “Prairie Dog Town” & in chimneys etc.

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

Air is forced to

circulate!

Application #2 g) Blood flow in the body P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2)2 (2)

“Where v is high, P is low, where v is low, P is high.”

Blood flow is from right to left

instead of up (to the brain)

Problem 46: Pumping water up

Street level: y1 = 0v1 = 0.6 m/s, P1 = 3.8 atm Diameter d1 = 5.0 cm (r1 = 2.5 cm). A1 = π(r1)2

18 m up: y2 = 18 m, d2 = 2.6 cm (r2 = 1.3 cm). A2 = π(r2)2 v2 = ? P2 = ?Continuity: A1v1 = A2v2 v2 = (A1v1)/(A2) = 2.22 m/s Bernoulli:P1+ (½)ρ(v1)2 + ρgy1 = P2+ (½)ρ(v2)2 + ρgy2 P2 = 2.0 atm