Fluids Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 20

Floating and Buoyancy  Buoyant force F B = m displ fluid g  An object less dense than the fluid will float on top of the surface, then  object displaces fluid equal to its weight,  F B = m obj g  m displ fluid =  fluid V under = m obj.  An object denser than the fluid will sink. If submerged  object displaces fluid equal to its volume  m displ fluid =  fluid V object  F B =  fluid V object g

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 p 0 (e.g. the atmosphere above a column of water) then the equation should read: p=p 0 +  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 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, 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 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, W g = -  mg(y 2 -y 1 ) = -  g  V(y 2 -y 1 ) The work of the system due to pressure is, W p =Fd=pAd=  p  V=-(p 2 -p 1 )  V The change in kinetic energy is,  (1/2mv 2 )=1/2  V(v 2 2 -v 1 2 ) Equating work and  KE yields, p 1 +(1/2)  v  gy 1 =p 2 +(1/2)  v  gy 2

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  Use Bernoulli’s equation: p t +1/2  v t 2 =p b +1/2  v b 2 The difference in pressure is: p b -p t =1/2  v t 2 -1/2  v b 2 Pressure is F/A so: (F b /A)-(F t /A)=1/2  (v t 2 -v b 2 ) L=F b -F t and so: L= (½)  A(v t 2 -v b 2 ) If the lift is greater than the weight of the plane, you fly