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Physics 1501: Lecture 33 Today ’ s Agenda

Physics 1501: Lecture 33 Today ’ s Agenda. Homework #11 (due Friday Dec. 2) Midterm 2: graded by Dec. 2 Topics: Fluid dynamics Bernouilli ’ s equation Example of applications.

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Physics 1501: Lecture 33 Today ’ s Agenda

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  1. Physics 1501: Lecture 33Today’s Agenda • Homework #11 (due Friday Dec. 2) • Midterm 2: graded by Dec. 2 • Topics: • Fluid dynamics • Bernouilli’s equation • Example of applications

  2. Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel. The buoyant force is equal to the weight of the liquid displaced. Pascal and Archimedes’ Principles • Pascal’s Principle • Archimedes’ principle • Object is in equilibrium

  3. streamlines do not meet or cross • velocity vector is tangent to streamline • volume of fluid follows a tube of flow bounded by streamlines streamline Ideal Fluids • Fluid dynamics is very complicated in general (turbulence, vortices, etc.) • Consider the simplest case first: the Ideal Fluid • no “viscosity” - no flow resistance (no internal friction) • incompressible - density constant in space and time • Flow obeys continuity equation • volume flow rate Q = A·v • is constant along flow tube: • follows from mass conservation if flow is incompressible. A1v1 = A2v2

  4. dV Bernoulli Equation Conservation of Energy for Ideal Fluid • Recall the standard work-energy relation • Apply the principle to a section of flowing fluid with volume dV and mass dm = r dV (here W is work done on fluid)

  5. a) smaller b) same c) larger Lecture 33 Act 1Bernoulli’s Principle • A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 What is the pressure in the 1/2” pipe relative to the 1” pipe?

  6. Some applications • Lift for airplane wing • Enhance sport performance • More complex phenomena: ex. turbulence

  7. More applications • Vortices: ex. Hurricanes • And much more …

  8. Ideal Fluid: Bernoulli Applications • Bernoulli says: high velocities go with low pressure • Airplane wing • shape leads to lower pressure on top of wing • faster flow  lower pressure  lift • air moves downward at downstream edge  wing moves up

  9. Ideal Fluid: Bernoulli Applications • Warning: the explanations in text books are generally over-simplified! • Curve ball (baseball), slice or topspin (golf) • ball drags air around (viscosity) • air speed near ball fast at “top” (left side) • lower pressure force  sideways acceleration or lift

  10. Ideal Fluid: Bernoulli Applications • Bernoulli says: high velocities go with low pressure • “Atomizer” • moving air ‘sweeps’ air away from top of tube • pressure is lowered inside the tube • air pressure inside the jar drives liquid up into tube

  11. Example: Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom.

  12. Solution

  13. C y O A v h B ExampleFluid dynamics • A siphon is used to drain water from a tank (beside). The siphon has a uniform diameter. Assume steady flow without friction, and h=1.00 m. You want to find the speed vof the outflow at the end of the siphon, and the maximum possible height y above the water surface. • Use the 5 step method • Draw a diagram that includes all the relevant quantities for this problem. What quantities do you need to find v and ymax ?

  14. C y O A v h B Example: SolutionFluid dynamics • Draw a diagram that includes all the relevant quantities for this problem. What quantities do you need to find v and ymax ? • Need P and v values at points O, A, B, C to find v and ymax • At O: P0=Patmand v0=0 • At A: PAand vA • At B: PB=Patmand v0=v • At C: PCand vC • For ymax set PC=0

  15. Example: SolutionFluid dynamics • What concepts and equations will you use to solve this problem? • We have fluid in motion: fluid dynamics • Fluid is water: incompressible fluid • We therefore use Bernouilli’s equation • Also continuity equation

  16. C y O A v h B Example: SolutionFluid dynamics • Solve for v and ymax in term of symbols. • Let us first findv=vB • We use the points O and B where : P0=Patm=1 atm and v0=0 and y0=0 where: PB=Patm=1 atm and vB=vand yB=-h • Solving forv

  17. C y O A v h B Example: SolutionFluid dynamics • Solve for v and ymax in term of symbols. • Incompressible fluid: Av =constant • A is the same throughout the pipe vA= vB= vC = v • To get ymax , use the points C and B (could also use A) where: PB=Patm=1 atm and vB=vand yB=-h set : PC=0(cannot be negative) and vC=vand yC= ymax • Solving for ymax

  18. Example: SolutionFluid dynamics • Solve for v and ymax in term of numbers. • h = 1.00 mand use g=10 m/s2 • Patm=1 atm = 1.013  105 Pa (1 Pa = 1 N/m2 ) • density of waterwater = 1.00 g/cm3 = 1000 kg/m2

  19. C y O A v h B Example: SolutionFluid dynamics • Verify the units, and verify if your values are plausible. • [v] = L/T and [ymax] = L so units are OK • vof a few m/s andymaxof a few meters seem OK • Not too big, not too small • Note on approximation • Same as saying PA= PO =Patmor vA=0 • i.e. neglecting the flow in the pipe at point A

  20. area A Real Fluids: Viscosity • In ideal fluids mechanical energy is conserved (Bernoulli) • In real fluids, there is dissipation (or conversion to heat) of mechanical energy due to viscosity (internal friction of fluid) Viscosity measures the force required to shear the fluid: where F is the force required to move a fluid lamina (thin layer)of area A at the speed vwhen the fluid is in contact with a stationary surface a perpendicular distance y away.

  21. area A air H2O oil glycerin       Viscosity (Pa-s) Real Fluids: Viscosity • Viscosity arises from particle collisions in the fluid • as particles in the top layer diffuse downward they transfer some of their momentum to lower layers • lower layers get pulled along (F = Dp/Dt)

  22. L r p+Dp Q R p Real Fluids: Viscous Flow • How fast can viscous fluid flow through a pipe? • Poiseuille’s Law • Because friction is involved, we know that mechanical energy is not being conserved - work is being done by the fluid. • Power is dissipated when viscous fluid flows: P = v·F = Q ·Dp the velocity of the fluid remains constant power goes into heating the fluid: increasing its entropy

  23. L/2 L/2 a) 3/2 b) 2 c) 4 Lecture 33 Act 2Viscous flow • Consider again the 1 inch diameter pipe and the 1/2 inch diameter pipe. 1) Given that water is viscous, what is the ratio of the flow rates, Q1/Q1/2, in pipes of these sizes if the pressure drop per meter of pipe is the same in the two cases?

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