physics 231 lecture 22: fluids and viscous flow
DESCRIPTION
PHYSICS 231 Lecture 22: fluids and viscous flow. Remco Zegers Walk-in hour: Tue 4-5 pm Helproom. P 0. h. B. w. Pressure at depth h. P =P 0 + fluid gh h: distance between liquid surface and the point where you measure P. h. P. Buoyant force for submerged object. - PowerPoint PPT PresentationTRANSCRIPT
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 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?