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

Pascal and Archimedes’ Principles Pascal’s Principle 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. Archimedes’ principle The buoyant force is equal to the weight of the liquid displaced. Object is in equilibrium

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 streamlines do not meet or cross velocity vector is tangent to streamline volume of fluid follows a tube of flow bounded by streamlines streamline 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

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) dV Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right. Bernoulli Equation

Lecture 33 Act 1 Bernoulli’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? a) smaller b) same c) larger

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

More applications Vortices: ex. Hurricanes And much more …

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 Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right.

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 Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right.

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 Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right.

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.

Solution

Example Fluid dynamics v h y A B C O 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 v of 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 ?

Example: Solution Fluid 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=Patm and v0=0 At A: PA and vA At B: PB=Patm and v0=v At C: PC and vC For ymax set PC=0 v h y A B C O 29

Example: Solution Fluid 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 29

Example: Solution Fluid dynamics v h y A B C O Solve for v and ymax in term of symbols. Let us first find v=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=v and yB=-h Solving for v 29

Example: Solution Fluid dynamics v h y A B C O Example: Solution Fluid 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=v and yB=-h set : PC=0 (cannot be negative) and vC=v and yC= ymax Solving for ymax 29

Example: Solution Fluid dynamics Solve for v and ymax in term of numbers. h = 1.00 m and 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 29

Example: Solution Fluid dynamics Verify the units, and verify if your values are plausible. [v] = L/T and [ymax] = L so units are OK v of a few m/s and ymax of a few meters seem OK Not too big, not too small Note on approximation Same as saying PA= PO =Patm or vA=0 i.e. neglecting the flow in the pipe at point A v h y A B C O 29

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: area A Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right. where F is the force required to move a fluid lamina (thin layer) of area A at the speed v when the fluid is in contact with a stationary surface a perpendicular distance y away.

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 area A lower layers get pulled along (F = Dp/Dt) Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right.    Viscosity (Pa-s) oil air glycerin H2O   

Real Fluids: Viscous Flow How fast can viscous fluid flow through a pipe? Poiseuille’s Law p+Dp Q r L p R 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

Lecture 33 Act 2 Viscous flow Consider again the 1 inch diameter pipe and the 1/2 inch diameter pipe. L/2 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? a) 3/2 b) 2 c) 4