Fluid Flows and Bernoulli s Principle. Streamlines demonstrating laminar (smooth) and turbulent flows of an ideal fluid

Transcription

1 Fluid Flows and Bernoulli s Principle Streamlines demonstrating laminar (smooth) and turbulent flows of an ideal fluid

2 CELEBRATING EINSTEIN March 31 - April 23, 2017 ART, DANCE AND SYMPHONY PERFORMANCES Danced Lecture and Interview Richard Price April 21, 7:30 p.m. Danced Lecture and Interview Shane Larson April 22, 2 p.m. April 22, 7:30 p.m. Danced Lecture and Interview Janna Levin April 23, 2 p.m. A Shout Across Time Community Symphony April 22, 4:30 p.m. April 23, 4:30 p.m. Einstein Artwork & Demonstrations April 21, 5 p.m. April 22, 4:30 p.m. April 23, 12 p.m. DOCUMENTARIES little green men Pulsar Search Collaboratory Documentary April 17, 7 p.m. LIGO, A Passion for Understanding April 19, 7 p.m. SPACE RACE FUN RUN April 22, 9 a.m. More events and information: einstein.wvu.edu Imagination is more important than knowledge. - Albert Einstein LECTURES Fast Radio Bursts: The Story So Far Duncan Lorimer, Center for Gravitational Waves & Cosmology March 31, 7:30 p.m. When Black Holes Collide! Gravitational Waves and Other Tales from the Horizon Zach Etienne, Center for Gravitational Waves & Cosmology, WVU Math Department April 7, 7:30 p.m. Einstein Unyielding: A Catalyst in a New Berlin Katherine Aaslestad, WVU History Department April 10, 7:30 p.m. Beginning the Exploration of the Universe with Gravitational Waves Over 100 years ago, Einstein predicted gravitational waves, and we are just beginning to detect them now. You are invited to join our interdisciplinary celebration to understand the beauty and significance of these transformative discoveries about our universe. Rainer Weiss, Massachusetts Institute of Technology April 13, 7 p.m. NANOGrav: Searching for Gravitational Waves with Pulsars Maura McLaughlin, Center for Gravitational Waves and Cosmology April 18, 7:30 p.m. Gravitational Wave Astronomy: Turning Imagination into Discovery Joan Centrella, Goddard Space Flight Center April 20, 3:30 p.m. PLANETARIUM SHOWS Einstein planetarium night and rooftop telescope observing March 31, 8:30 p.m. April 7, 8:30 p.m. April 10, 8:30 p.m. April 19, 8:30 p.m. April 20, 8:30 p.m. Extra credit opportunity (up to 10 points added to final exam grade) (this equates to 2.5 points added to your FINAL GRADE AVERAGE) Celebrating Einstein was originally produced by Montana WVU is an EEO/Affirmative Action Employer Minority/Female/Disability/Veteran State University and the extreme Gravity Institute. Must attend an event ON THIS FLIER. If you get a 100% on the curved grade this could become 110%. Means about 2.5 points added to your final grade.

3 Extra credit opportunity (up to 10 points added to final exam grade) (this equates to 2.5 points added to your FINAL GRADE AVERAGE) Attend an event.* I will NOT Write a report: ACCEPT any 1. What event did you attend? turned in after 2. What were the main (physics/ APRIL 28! astronomy) ideas discussed? 3. Relate them in some way to principles we learned in class. 4. What was the coolest thing you learned from the event? *space fun run excluded but will be really cool! Must attend an event ON FLIER ON PREVIOUS SLIDE. If you get a 100% on the curved grade this could become 110%. Means about 2.5 points added to your final grade. I ll put the schedule and a more detailed description online about how these will be assessed. STRICT DEADLINE OF APRIL 28.

4 Velocity v Δx/t Thinking about rates Acceleration a Δv/t Volume flow rate Volume through a surface per time I wanted to point out the concept of a rate: the change in something over time. We ve previously talked about velocity (change in distance over time), acceleration (change in velocity over time). Today we ll be talking about a VOLUME FLOW RATE, so volume per second (show different versions of this in terms of density and area and velocity).

5 Rates and fluid flow Cows sometimes eat small rocks and particulates! Water (an ideal fluid ) moves rapidly, and water/saliva help flush these materials from their stomachs. A drooly ruminant A cow swallows about 100 Liters (0.1 m 3 ) of saliva each day. Assuming cow swallows it all, what is the volume flow rate (volume per unit time) of saliva into the cow? Units of volume flow rate: m 3 /s A. 0.1 m 3 /s B. 1.2 x 10-3 m 3 /s C. 1.2 x 10-6 m 3 /s Q94 I know a lot of you are doing nutrition (either cows or humans). I encourage you to ask your profs about how fluid dynamics operates in cow nutrition, but here s what I came up with. Ruminants, or cows, produce tons of saliva. Published estimates for adult cows are in the range of 100 to 150 liters of saliva per day! Water/saliva flows through the rumen rapidly and appears to be critical in flushing particulate matter downstream. This is about 1 cubic centimeter per second! They d have to drool a lot or swallow a lot No wonder cows are so drooly. Anyways the point is: volume flow rate is the amount of volume that goes through some barrier per unit time. We will use this later.

6 Assumptions Today Non-viscous fluid (no internal friction.) Note: Honey is viscous. Mud is viscous. Water is not. Blood SHOULDN T be viscous! Density is constant. Fluid motion is steady. No turbulence in the fluid. There are a few limiting assumptions we ll use today. All of these make our analysis of fluid flows a lot more simplified in terms of mathematics. We ll be talking about something called Laminar or Streamline flows. The book (in unassigned reading the rest of this chapter) treats other more complex fluid flows, including viscosity.

7 Volume 1 Volume2 time time Rate of mass in Rate of mass out If the flow rate is constant, the mass going in for each time interval has to equal the mass coming out. This leads to the realization that the VOLUME FLOW RATE at two different points in the pipe should be the same. So WHAT DOES THIS MEAN? This means as much volume as you push in should come out in the same amount of time.

8 AΔx t Av Δx1 A1 Volume 1 Volume2 time time Δx2 A2 AΔx t Av Rate of mass in Δx1 Δx2 Rate of mass out AΔx t Av A1 A2 AΔx t Av Therefore, the amount of volume going passing through one end of the pipe at a given time will be the same amount of volume coming out. You can see that we can write the volume passing through the tube at a given time as A delta x. [LIGHT BOARD DERIVATION] THIS PRODUCT A v IS CALLED THE VOLUME FLOW RATE or the VOLUME FLUX, JUST LIKE WE DID EARLIER. Basically, it tells you that if the cross-sectional area of a channel or pipe is larger, you get slower flow. Smaller channels/pipes get faster flow.

9 Continuity Equation Flow rate is FASTER if pushed through a smaller cross-sectional area. A1v1 A2v2 This is called the continuity equation, and it s really cool! If you know the volume of fluid flowing into or out of a channel, you can determine the velocity of that fluid at any point along the channel. Big tubes have low velocity, small tubes have high velocity. [See light board notes for proportionality] There are a lot of applications where this applies! Plumbing, watering your garden, circulation, GI track.

10 What do you do if your garden hose does not reach all of your plants? Physics! A1v1 A2v2 Covering part of the hose opening makes the water flow through a smaller cross-sectional area, so the water must flow faster. PHYSICS! It s all around us if you take the time to think about it!

11 Aneurysms Is the blood flow faster in a normal blood vessel or in a blood vessel with aneurysm? saccular aneurysm fusiform aneurysm Q95 A. Normal blood vessel B. Aneurysm C. Same in healthy and aneurysed vessel D. Not enough information to determine It s also inside of us. In an aneurysm, your blood vessel gets bulged out. I d like you to tell me, where is the velocity faster? ANSWER: A. Again I encourage you to ask your nutrition profs about fluid dynamics in the circulatory and GI system.

12 Bernoulli s Principle P1, v1, y1 If volume flow rate is constant, and conservation of energy applies to fluids, then P1 + 1/2 ρv1 2 + ρgy1 P2 + 1/2 ρv2 2 + ρgy2 P2, v2, y2 We re not going to go through the derivation but you can see it in the book. If you take the continuity equation, and apply conservation of energy principles to the fluid, then you can show that for two different points in a fluid flow, the PRESSURE plus the KINETIC ENERGY PER VOLUME plus the POTENTIAL ENERGY PER VOLUME will be the same. So if you have a pressure, velocity, and height for one section of the pipe, this will relate with this equation to the change in those values at another point in the pipe.

13 An important consequence As a fluid goes through a region where it changes speed or height, the pressure of the fluid will change. lower P P1 + 1/2 ρv1 2 + ρgy1 P2 + 1/2 ρv2 2 + ρgy2 higher P As height increases, pressure decreases. higher P lower P CAREFUL!!! As speed increases, pressure decreases. One very important consequence of this, and the one that is most used in typical analyses, is understanding that as a fluid goes through a region where it changes speed OR it changes height, the pressure of the fluid will change.

14 Careful!.Counter-intuitive! P1 + 1/2 ρv1 2 + ρgy1 P2 + 1/2 ρv2 2 + ρgy2 This is something people often get confused so I ll show you one more time ***in a continuous flow***, LOW VELOCITY (larger channel) regions have HIGHER PRESSURE. Fast-moving regions (smaller tubes) have LOWER PRESSURE ***in a continuous flow***.

15 Aneurysms Is the PRESSURE higher in a normal blood vessel or in region with aneurysm? saccular aneurysm fusiform aneurysm Q96 A. Normal blood vessel B. Aneurysm C. Same in healthy and aneurysed vessel D. Not enough information to determine Let s think back to aneurysms. ANSWER: B Velocity is LOWER, pressure is HIGHER in an aneurysm! Increased pressure can cause the artery to rupture. This is really dangerous.

16 Bernoulli s Most Important Implication. A slow-moving fluid exerts more pressure than a fast-moving fluid (depends also on elevation of volume flow). If you run the numbers on the Bernoulli equation, you ll find this is true.

17 Consider a house with a very thin (Δy ~ 0), flat roof of area square meters. During a hurricane with winds of 140 mph (62.6 m/s), what is the net force on the roof? Outside house v air 2 1 v air ~ 0 Inside house P1 + 1/2 ρv1 2 + ρgy1 P2 + 1/2 ρv2 2 + ρgy2 Density of air: kg/m 3 Let s see an example of this. [See light board notes for problem solution] This pressure difference exerts a net force on the roof that s about the equivalent of a motorcycle slamming into the roof at about 20mph! Build a stronger roof if you live in florida.

18 Consider a house with a very thin (Δy ~ 0), flat roof of area square meters. During a hurricane with winds of 140 mph (62.2 m/s), what is the net force on the roof? Outside house v air 2 1 v air ~ 0 Inside house P1 + 1/2 ρv1 2 + ρgy1 P2 + 1/2 ρv2 2 + ρgy2 Just to reinforce a concept ANSWER: D higher pressure where there s SLOWER VELOCITY AIR. If the wind is blowing very hard outside, What direction does the net force point? A B C D Q97

19 Bernoulli s Most Important Implication. A slow-moving fluid exerts more pressure than a fast-moving fluid (depends also on elevation of volume flow). [Showed demo]

20 Airplane wings! Faster air over Slower air under Higher pressure under the wing creates LIFT! Wings are designed so there s faster air over and slower air under Produce LIFT!