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