PHY 2311

PHYSICS 231Lecture 22: fluids and viscous flow

Remco ZegersWalk-in hour: Tue 4-5 pm

Helproom

PHY 2312

P = P0+ fluidghh: distance between liquid surface and the point where you measure P

P0

P

h

B = fluidVobjectg = Mfluidg = wfluid

The buoyant force equals the weight of the amount of water that can be put in the volume taken by the object.If object is not moving: B=wobject object= fluid

Pressure at depth h

Buoyant force for submerged object

Buoyant force for floating objecthB

w

The buoyant force equals the weight of the amount of water that can be put in the part of the volume of the object that is under water.objectVobject= waterVdisplaced h= objectVobject/(waterA)

PHY 2313

quiz (extra credit)

A block of weight w is placed in waterand found to stay submerged as shownin the picture. The liquid is thenreplaced by another liquid of lowerdensity. What will happen if the block is placed in the liquid of lower density?

a) the block will float on the surface of the liquidb) the block will be partially submerged and partially above the liquidc) the block will again be submerged as shown in the pictured) the block will sink to the bottom

initially B = fluidVobjectg w=Mblockg B=wlower density liquid: w remains the same, B becomes smallerthe block will sink to the bottom

PHY 2314

Bernoulli’s equation

P1+½v12+gy1=

P2+½v22+gy2

P+½v2+gy=constant

The sum of the pressure (P), the kinetic energy per unit volume (½v2) and the potential energy per unit volume (gy)is constant at all points along a path of flow.

Note that for an incompressible fluid:A1v1=A2v2

This is called the equation ofcontinuity.

PHY 2315

hole in a tank

Pdepth=h =Pdepth=0+ ghh

x

y If h=1m & y=3m what is x?Assume that the holes are smalland the water level doesn’t dropnoticeably.

P0

PHY 2316

h

x1

y

If h=1 m and y=3 m what is X?

Use Bernoulli’s law

PA+½vA2+gyA= PB+½vB

2+gyB

At A: PA=P0 vA=? yA=y=3At B: Pb=P0 vB=0 yB=y+h=4P0+½vA

2+g3=P0+g4

vA=(g/2)=2.2 m/s

A

P0 B

PHY 2317

Each water element of mass m has the samevelocity vA. Let’s look at one element m.vA=(g/2)=2.2 m/s

x1

3m

vA

In the horizontal direction:x(t)=x0+v0xt+½at2=2.2t

In the vertical direction:y(t)=y0+v0yt+½at2=3-0.5gt2

= 0 when the water hits the ground, so t=0.78 s

so x(0.78)=2.2*0.78=1.72 m

0

PHY 2318

Viscosity

Viscosity: stickiness of a fluidOne layer of fluid feels a largeresistive force when slidingalong another one or along asurface of for example a tube.

PHY 2319

ViscosityContact surface A

fixed

movingF=Av/d

=coefficient of viscosityunit: Ns/m2

or poise=0.1 Ns/m2

PHY 23110

Poiseuille’s Law

How fast does a fluid flowthrough a tube?

Rate of flow Q= v/t=R4(P1-P2)

8L(unit: m3/s)

PHY 23111

Example

PP=106 Pa P=105 Pa

Flow rate Q=0.5 m3/sTube length: 3 m=1500E-03 Ns/m2

What should the radius of the tube be?

Rate of flow Q=R4(P1-P2)

8L

R=[8QL/((P1-P2))]1/4=0.05 m

PHY 23112

If time permits, I will do additional problems here.

PHY 23113

Buoyant forces

When submerged in water an object weighs 1.6N. At thesame time, the water level in the water container (withA=0.01 m2) rises 0.01 m. What is the specific gravity (sg) of the object? (water=1.0x103 kg/m3)

A=0.01 m2

Use the fact that the Buoyant forceon a submerged object equals the weight of the displaced water.W =Fg-B

=Mobjectg-Mwater,displacedg =objectVobjectg-waterVobjectg

=Vobjectg (object-water)1.6N =0.01*0.01*g(object-water)=1.0x10-4*9.8*water(sg-1)sg=2.63

PHY 23114

Keep it coming.A plastic bag contains a glucosesolution. The part of the bag thatis not filled is under vacuum. If thepressure in a blood vein is 1.33x104 Pa,how high must one hang the bag tomake sure the solution (specific gravity1.02) enters the body? (w=1.0x103kg/m3)

P=P0+gh1.33x104=0+1.02*1.0x103*9.8*hh=1.33 m

PHY 23115

Titanic: After the Titanic sunk, 10 peoplemanage to seek refuge on a 2x4m wooden raft. It is still 1.0 cm above water. A heavy debate follows when another person (60 kg) wants to board as well. Fortunately, a PHY231 student is among the 10. Can she convince the others that it is safe to pull the person on board without the whole raft sinking? (w=1.0x103 kg/m3)

With 10 people: Fg=B (Mraft+M10)g=Vdisplaced,before wg

With 11 people: Fg=(Mraft+M10+M1)g B=(Vdisplaced,before+Vextra) wg

stationary if Fg=B (Mraft+M10)g+M1g=(Vdisplaced,before+Vextra)wg M1g=Vextra wg so Vextra=(M1/w)

Vextra=60/1.0x103=0.06m3

Vextra=LxWxH=8H=0.06 so H=0.75cm so there is still 0.25cm to spare!

PHY 23116

BernoulliA=5cm2

A=2cm2, P=1 atm

2m

Water flows over a height of 2m through an oddly shaped pipe.A) If the fluid velocity is 1 m/s at the bottom, what is it at the top? B) What is the water pressure at the top?A) Use the equation of continuity: A1v1=A2v2

5*vtop=2*1 vtop=0.4 m/sB) Use Bernoulli. Ptop+½vtop

2+ghtop= Pbot+½vbot2+ghbot

Ptop+0.5*(1E+03)*0.42+(1E+3)*9.8*2=(1E+05)+0.5*(1E+03)12

Ptop=80820 Pa.

=1.0x103 kg/m3

PHY 23117

question

Imagine holding two bricks under water. (brick> water)Brick A is just beneath the surface of the water, BrickB is at a greater depth. The force needed to hold brickB in place is:a) largerb) the same asc) smallerthan the force required to hold brick A in place.

Grav. force is the same on both.Buoyant force B: weight of water displaced by brick.Also same for both!

PHY 23118

question

Two cups are filled to the same level with water.One of the two cups has ice cubes floating in it.When the ice cubes melt, in which cup is the level of water higher?

a) The cup without ice cubesb) The glass with ice cubesc) It is the same in both

Before the ice melts: B=Fg so wgVdisp= icegVice

Vdisp= iceVice/water

After it melts: Mmelted water=Mice

waterVmelted= iceVice

Vmelted=iceVice/water

the same!

global warming?