Download presentation

Published byLeslie Maximilian Randall Modified over 4 years ago

2
**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!)

3
Streamline Motion

4
**PHYSICS: Conservation of Mass!!**

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

5
**PHYSICS: Conservation of Mass!!**

A1v1 = A2v2 Or Av = constant Small pipe cross section larger v Large pipe cross section smaller v

6
**Example 10-11: Estimate Blood Flow**

rcap = 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

7
**Example 10-12: Heating Duct**

Speed 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

8
**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!

10
**W1 = F11= P1 A11. 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.

11
**W2 = -F22 = -P2A22. 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.

12
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)

13
**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) 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

14
** Bernoulli’s Equation**

(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

15
**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 = constant NOTE! The fluid is NOT at rest, so ΔP ρgh ! Example 10-13

16
**Application #1: Water Storage Tank**

P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy (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”

17
**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) 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.”

18
**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!

19
**Application #2 b) Ball on a jet of air (Demonstration!)**

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2) (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)

20
**Application #2 c) Lift on airplane wing**

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2) (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 !

21
**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.”

22
**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

23
**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

24
**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!

25
**Application #2 g) Blood flow in the body**

P1 + (½)ρ(v1)2 = P2 + (½)ρ(v2) (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)

26
**Problem 46: Pumping water up**

Street level: y1 = 0 v1 = 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

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google