fluids physics 202 professor vogel (professor carkner’s notes, ed) lecture 20
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Fluids
Physics 202Professor Vogel (Professor Carkner’s
notes, ed)Lecture 20
Floating and Buoyancy Buoyant force FB= mdispl fluidg An object less dense than the fluid will float
on top of the surface, then object displaces fluid equal to its weight, FB= mobjg
mdispl fluid=fluidVunder = mobj.
An object denser than the fluid will sink. If submerged object displaces fluid equal to its volume mdispl fluid=fluidVobject
FB= fluidVobjectg
Fluids at REST We will normally deal with fluids in a
gravitational field Fluids in the absence of an external gravitational field
will form a sphere Fluids on a planet will exert a pressure which
increases with depth For a fluid that exerts a pressure due to gravity:
p=gh Where h is the height of the fluid in question,
and g is the acceleration of gravity and is the density
Gauge Pressure If the fluid has additional material pressing
down on top of it with pressure p0 (e.g. the atmosphere above a column of water) then the equation should read:
p=p0+gh Pressure usually depends only on the height
of the fluid column The gh part of the equation is called the
gauge pressure A tire gauge that shows a pressure of “0” is really
measuring a pressure of one atmosphere
Measuring Pressure If you have a U-shaped tube with some liquid in it and
apply a pressure to one end, the height of the fluid in the other arm will increase
Since the pressure of a fluid depends only on its height, this set-up can be used to measure pressure This describes an open tube manometer
Since air is pressing down on the open end, the manometer actually measures gauge pressure above air pressure or overpressure
If you close off one end of the tube and keep it in vacuum, the air pressure on the open end will cause the fluid to rise This is called a barometer
Measures atmospheric pressure
Barometers
MOVING Fluids
We will assume: Steady -- velocity does not change with
time (not turbulent) Incompressible -- density is constant Nonviscous -- no friction Irrotational -- constant velocity through
a cross section Real fluids are much more complicated
The ideal fluid approximation is usually not very good
Moving FluidsConsider a pipe of cross sectional area A with a fluid
moving through it with velocity vWhat happens if the pipe narrows?
Mass must be conserved so,Av = constant
If the density is constant then, Av= constant = [dV/dt] = volume flow rate
Since rate is a constant, if A decreases then v must increaseConstricting a flow increases its velocityBecause the amount of fluid going in must equal the
amount of fluid going outOr, a big slow flow moves as much mass as a small fast flow
Continuity
[dV/dt]=Av=constant is called the equation of continuityYou must have a continuous flow of material
You can use it to determine the flow rates of a system of pipesFlow rates in and out must always balance
outCan’t lose or gain any material
Continuity
The Prancing Fluids
As a fluid flows through a pipe it can have different pressures, velocities and potential energies
How can we keep track of it all?The laws of physics must be obeyed
Namely conservation of energy and continuity
Neither energy nor matter can be created or destroyed
Bernoulli’s EquationConsider a pipe that bends up and gets wider at
the far end with fluid being forced through itThe work of the system due to lifting the fluid is,
Wg = -mg(y2-y1) = -gV(y2-y1) The work of the system due to pressure is,
Wp=Fd=pAd=pV=-(p2-p1)VThe change in kinetic energy is,
(1/2mv2)=1/2V(v22-v1
2)Equating work and KE yields,
p1+(1/2)v12+gy1=p2+(1/2)v2
2+gy2
Fluid Flow
Consequences of Bernoulli’s Equation
If the speed of a fluid increases the pressure of the fluid must decrease
Fast moving fluids exert less pressure than slow moving fluids
This is known as Bernoulli’s principleBased on conservation of energy
Energy that goes into velocity cannot go into pressure
Note that Bernoulli holds for moving fluids
Constricted Flow
Bernoulli in Action
Blowing between two pieces of paper
Getting sucked under a trainConvertible top bulging outAirplanes taking off into the windBut NOT Shower curtains getting
sucked into the shower – ask me why!
Lift
Consider a thin surface with air flowing above and below itIf the velocity of the flow is less on the
bottom than on top there is a net pressure on the bottom and thus a net force pushing up
This force is called liftIf you can somehow get air to flow
over an object to produce lift, what happens?
December 17, 1903
Deriving LiftConsider a wing of area A, in air of density Use Bernoulli’s equation:
pt+1/2vt2=pb+1/2vb
2
The difference in pressure is: pb-pt=1/2vt
2-1/2vb2
Pressure is F/A so: (Fb/A)-(Ft/A)=1/2(vt
2-vb2)
L=Fb-Ft and so: L= (½)A(vt
2-vb2)
If the lift is greater than the weight of the plane, you fly