Transcript fluids, Bernoulli

Fluids
Physics 202
Professor 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=rfluidVunder = mobj.
An object denser than the fluid will sink. If
submerged
object displaces fluid equal to its volume
mdispl fluid=rfluidVobject
FB= rfluidVobjectg
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=rgh
Where h is the height of the fluid in question,
and g is the acceleration of gravity and r 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+rgh
Pressure usually depends only on the height
of the fluid column
The rgh 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 Fluids
 Consider a pipe of cross sectional area A with
a fluid moving through it with velocity v
 What happens if the pipe narrows?
 Mass must be conserved so,
Avr = 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
increase
 Constricting a flow increases its velocity
 Because the amount of fluid going in must
equal the amount of fluid going out
 Or, a big slow flow moves as much mass as a
small fast flow
Continuity
 [dV/dt]=Av=constant is called the equation
of continuity
 You must have a continuous flow of material
 You can use it to determine the flow rates of a
system of pipes
 Flow rates in and out must always balance out
 Can’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 Equation
 Consider a pipe that bends up and gets wider at
the far end with fluid being forced through it
 The work of the system due to lifting the fluid is,
Wg = -Dmg(y2-y1) = -rgDV(y2-y1)
 The work of the system due to pressure is,
Wp=Fd=pAd=DpDV=-(p2-p1)DV
 The change in kinetic energy is,
D(1/2mv2)=1/2rDV(v22-v12)
 Equating work and DKE yields,
p1+(1/2)rv12+rgy1=p2+(1/2)rv22+rgy2
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 principle
 Based 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 train
Convertible top bulging out
Airplanes taking off into the wind
But NOT Shower curtains getting
sucked into the shower – ask me why!
Lift
Consider a thin surface with air flowing
above and below it
If 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 lift
If you can somehow get air to flow over
an object to produce lift, what happens?
December 17, 1903
Deriving Lift
 Consider a wing of area A, in air of density r
 Use Bernoulli’s equation:
 pt+1/2rvt2=pb+1/2rvb2
 The difference in pressure is:
 pb-pt=1/2rvt2-1/2rvb2
 Pressure is F/A so:
 (Fb/A)-(Ft/A)=1/2r(vt2-vb2)
 L=Fb-Ft and so:
 L= (½)rA(vt2-vb2)
 If the lift is greater than the weight of the
plane, you fly