Physics 1501: Lecture 33 Today ’ s Agenda

Physics 1501: Lecture 33, Pg 1

Physics 1501: Lecture 33Physics 1501: Lecture 33TodayToday’’s Agendas Agenda

Homework #11

(due Friday Dec. 2)

Midterm 2: graded by Dec. 2

Topics: Fluid dynamicsBernouilli’s equationExample of applications


Pascal and Archimedes’ PrinciplesPascal 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


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 FluidsIdeal 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


Recall the standard work-energy relationApply the principle to a section of flowing fluid with

volume V and mass m = V (here W is work done on fluid)

V

Conservation of Energy for Conservation of Energy for Ideal FluidIdeal Fluid

Bernoulli Equation


Lecture 33 Lecture 33 Act 1Act 1BernoulliBernoulli’’s Principles 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.

What is the pressure in the 1/2” pipe relative to the 1” pipe?

a) smaller b) same c) larger

v1 v1/2


Some applicationsSome applications Lift for airplane wing

Enhance sport performance

More complex phenomena: ex. turbulence


More applicationsMore applications

Vortices: ex. Hurricanes

And much more …


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

Ideal Fluid: Bernoulli ApplicationsIdeal 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



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



The tank is open to the atmosphere atthe top. Find and expression for the speed of the liquid leaving the pipe atthe bottom.

Example: Efflux SpeedExample: Efflux Speed


SolutionSolution


vh

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.

ExampleExampleFluid dynamicsFluid dynamics

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: SolutionExample: SolutionFluid dynamicsFluid 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 vh

y

A

B

C

O



What concepts and equations will you use to solve this problem? We have fluid in motion: fluid dynamicsFluid is water: incompressible fluidWe therefore use Bernouilli’s equation

Also continuity equation



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

vh

yA

B

C

O


Solve for v and ymax in term of symbols. Incompressible fluid: Av =constantA 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


vh

yA

B

C

O


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




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 vh

y

A

B

C

O


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)

Real Fluids: ViscosityReal Fluids: Viscosity

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 v when the fluid is in contact with a stationary surface a perpendicular distance y away.

area A


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

Real Fluids: ViscosityReal Fluids: Viscosity

Viscosity (Pa-s)

oilair glycerinH2O

area A

lower layers get pulled along (F = p/t)


p+p Qr

L

pR

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 ·p the velocity of the fluid remains constant power goes into heating the fluid: increasing its entropy

Real Fluids: Viscous FlowReal Fluids: Viscous Flow

How fast can viscous fluid flow through a pipe? Poiseuille’s Law


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?

Consider again the 1 inch diameter pipe and the 1/2 inch diameter pipe.

a) 3/2 b) 2 c) 4

L/2 L/2

Lecture 33 Lecture 33 Act 2Act 2Viscous flowViscous flow

Physics 1501: Lecture 33 Today ’ s Agenda

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