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Reminder: HW #10 due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade)

Reminder: HW #10 due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade) Recitation Quiz #11 tomorrow (last Recitation Quiz ) Formula Sheet for Final Exam posted on Bb Last Time : Pressure vs. Depth, Buoyant Forces and Archimedes’ Principle

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Reminder: HW #10 due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade)

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  1. Reminder: HW #10 due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade) • Recitation Quiz #11 tomorrow (last Recitation Quiz) • Formula Sheet for Final Exam posted on Bb • Last Time: Pressure vs. Depth, Buoyant Forces and Archimedes’ Principle • Today: Fluid Motion, Bernoulli’s Equation

  2. Fluids in Motion When a fluid is in motion, its flow can be either … Streamline (Laminar) : Every particle that passes a particular point moves along exactly the same smooth path Turbulent : In contrast, the flow of a fluid becomes irregular above a certain velocity or under conditions that cause abrupt changes in velocity Colorado River, Grand Canyon

  3. Viscosity In fluid flow, the term viscosity is used to describe the degree of internal friction in the fluid. This internal friction is associated with the resistance between two “adjacent layers” of the fluid moving relative to each other ! vs.

  4. Definition of an “Ideal Fluid” “Ideal fluids” satisfy the following conditions : (1) The fluid is non-viscous (no internal friction between adjacent layers) (2) The fluid is incompressible (density is constant) (3) The fluid motion is steady (velocity, density, and pressure each point in the fluid do not change with time) (4) The fluid moves without turbulence. Each element has no angular velocity about its center (no “eddy currents”)

  5. Continuity Equation Consider a fluid flowing through a pipe. Enters the pipe at (1) and exits at (2). Let’s apply conservation of mass to find a relation between the density, velocity, and cross-sectional area at (1) and (2) … Result is the “continuity equation” : ideal fluid

  6. Continuity Equation Implications of the continuity equation : (1) Product of the cross-sectional area and velocity is a constant (2) Speed is high when tube is constricted, low when larger (3) The continuity equation is equivalent to the statement that the volume of fluid entering the tube in some time interval Δt equals the volume of fluid exiting the tube in the same time interval

  7. Continuity Equation The continuity equation is also a measure of a “volume flow rate” Units of (A x v) are : (A x v) tells you how many m3 of the fluid flows past a particular point every second.

  8. Everyday Example: Garden Hose Q: What happens when you squeeze a garden hose ? A: Water sprays out with greater speed (and, thus, travels farther). This is just because of the continuity equation !! If A2 decreases, v2 must increase !!

  9. Example: 9.44 Water flowing through a garden hose of diameter 2.74 cm fills a 25-liter bucket in 1.5 minutes. (a) What is the speed of the water leaving the end of the hose ? (b) A nozzle is now attached to the end of the hose. If the nozzle diameter is 1/3 the diameter of the hose, what is the speed of the water leaving the hose ?

  10. Bernoulli’s Equation As a fluid flows through a pipe of varying cross-sectional area and elevation (height), the pressure changes along the pipe. Bernoulli’s Equation relates the pressure in the pipe to the kinetic and potential energy of the fluid along the pipe :

  11. Bernoulli’s Equation pressure kinetic energy per unit volume = KE/Volume potential energy per unit volume = PE/Volume The sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline.

  12. Venturi Tube Can be used to measure the speed of fluid flow. Horizontal  Heights are equal so y1 = y2 From continuity equation: v2 > v1 So this means: P1 > P2 Swiftly moving fluids exert less pressure than do more slowly moving fluids.

  13. Conceptual Question An ideal fluid flows through a horizontal pipe having a diameter that varies along its length. Does the sum of the pressure and kinetic energy per unit volume at different sections of the pipe … (a) decrease as the pipe diameter increases (b) increase as the pipe diameter increases (c) increase as the pipe diameter decreases (d) decrease as the pipe diameter decreases (e) remain the same as the pipe diameter changes

  14. Example: 9.52 Water flows through a constricted pipe. At the lower point, the pressure is 1.75 x 105 Pa, and the radius is 3.0 cm. At a point 2.5 m higher, the pressure is 1.20 x 105 Pa, and the radius is 1.5 cm. Find the speed of flow … (a) in the lower section, and (b) in the upper section. (c) What is the volume flow rate through the pipe ?

  15. Application: Airplane Wings higher air speed Lift on an airplane wing can be partly explained with the concepts of Bernoulli’s Equation. slight upward tilt lower air speed [Newton’s 3rd Law !!]

  16. Example An airplane has a mass M, and the two wings have a total area A. During level flight, the pressure on the lower wing surfaces is P1. Determine the pressure P2 on the upper wing surfaces.

  17. Next Class • 13.1 – 13.4 : Hooke’s Law, Simple Harmonic Motion

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