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PHYSICS OF FLUIDS

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- Includes liquids and gases
- Liquid has no fixed shape but nearly fixed volume
- Gas has neither fixed shape or volume
- Both can flow

Mass per unit volume

r = m/V (Greek letter “rho”)

m = rV

Example: Density of Mercury is 13.6 x 103 kg/m3 What is the mass of one liter?

M = rV = 13.6 x 103 kg/m3 x 10-3 m3 = 13.6 kg

Specific Gravity is ratio of its density to that of water

(1.00 x 103 kg/m3 =1.00 g/cm3)

- 1000 Kg per m3
- 1 Kg/liter
- 1 gram /cubic centimeter (cc)
- 1 cubic meter = 1000 liters
- 1 cubic centimeter = 1 milliliter
- 1 cubic meter = 1,000,000 cc

- Force per unit area
- Pressure = P = F/A
- Unit N/m2= Pascal(Pa)
- Exerted in all directions
- Force due to pressure is perpendicular to surface in a fluid at rest

rgh

- Let depth be h. Assume incompressible.
- Force acting on area is mg = rVg = rAhg
- P = F/A = rgh
- Pressure at equal depths is the same
- If external pressure is also present it must be added
DP = rg Dh

h

A

- What is thepressure at the bottom of a 20.0 meter deep lake?

P = rgh = 1.0 x 103 kg/m3 x 9.8 m/s2 x 20m =

1.96 x 105 N/m2 due to the water

What about the atmosphere pressing down on the lake?

- Average sea level pressure is 1 atm = 1.013 x 105 N/m2 (14.7 lbs/sq inch)
- Pressure gauges read pressure above atmospheric
- Absolute (total) pressure is gauge + atmospheric P = PA + PG

What is total pressure at bottom of lake?

Add 1.01 x 105 N/m3 to 1.96 x 105

- Finger holds water in straw
- How does pressure above water
compare with atmospheric? (hint:

atmospheric pushes up from below)

Pressure less because it plus weight of water must balance atmospheric

- rgH = 101,300 Pa; H = 101,300/(9.8 N/kg x 1000 Kg/m3) =10.3 m

- Probably not tested
- Pressure applied to a confined fluid increases pressure throughout by the same amount
- Pout = Pin
- Fout/Aout = Fin/Ain
- Fout/Fin = Aout/Ain
- Multiplies force by ratio of areas
- Principle of hydraulic jack and lift

Diagram courtesy Caduceus MCAT Review

- Open tube manometer simulation

The pressure difference is rgh

The (greater) pressure

P2 = P1 + rgh

How would this look if P1 was greater than P2 ?

Diagram courtesy Sensorsmag.com

- Submerged or partly submerged object experiences an upward force called buoyancy
- Pressure in fluid increases with depth
- Studied by Archimedes over 2000 years ago.

h = h2 – h1

- FB = F2 – F1 = P2A – P1A
= rFgA(h2 – h1)

= rFgAh

= rFgV = mFg

- Buoyant Force equals
weight of fluid displaced

h1

F1

A

h2

h

F2

- Buoyant force on a body immersed (or partly immersed) in fluid equals weight of fluid displaced.
- Argument in general: consider immersed body in equilibrium of any shape with same density as fluid. FBup must equal weight down. Replacing body by one with different density does not alter configuration of fluid so conclusion would not change.

Weighing Submerged object

Sfy =0

T +B - W = 0

T = W – B

T = W – rFVg

T is apparent weight W’

Diagram courtesy Caduceus MCAT Review

- Given crown mass 14.7 kg but weighed under water only 13.4 kg. Is it gold?
- W’ = W – FB W = rogV
- W-W’= FB = rF gV
- W/(W-W’) = rogV / rF gV = ro / rF
ro / rH2O= W/(W-W’) = 14.7/(14.7 – 13.4) = 14.7/1.3 = 11.3 LEAD

Another way to solve: isolatero = W/gV and then get V from FB /rfg

Objects float if density less than that of fluid.

FB = W at equilibrium

rFVdisp g = ro Vo g

Vdisp / Vo = ro /rF

Vo

Vdisp

Example: If an object’s density is 80% of the density of the surrounding fluid, 80% of it will be submerged

- 15 % of a log floats above the surface of the ocean. What is the density of the wood?

Vdisp / Vo = ro /rF

ro =Vdisp / Vo x rF = 0.85 x 1.025 x 103 kg/m3

= 0.87125=0.87 x 103 kg/m3

- What volume of helium is needed to lift a 60 Kg student?

FB= (mHe + 60 kg)g

rair Vg = (rHe V + 60 kg) g

V = 60 kg/(rair – rHe) = 60 kg/(1.29 – 0.18kg/m3)

= 54 m3

- Equation of Continuity
- Volume rate of flow is constant for incompressible fluids (not turbulent)
- A1v1 = A2 v2
- v is velocity

Erratic, contains eddies

Fluid follows smooth path

Courtesy MIT Media Laboratory

- A river narrows from 1000m wide to 100m wide with the depth staying constant. The river flows at 1.0 m/s when wide. How fast must it flow when narrow?

10 m/s

- What must be the cross sectional area of a heating duct carrying air at 3.0 m/s to change the air in a 300 m3 room every 15 minutes?
- A1v1 = A2v2 = A2l2/t = V2/t
- A1 = V2/ v1t = 300m3 /(3.0 m/s x 900s) = 0.11m2

A2

A1

l2

- Where velocity of fluid is high, pressure is low; where velocity is low, pressure is high
- Consequence of energy conservation
- P + ½ rv2 + rgy = constant for all points in the flow of a fluid
- P + ½ rv2 = constantifall on same level

If A1 is six times A2 how will the pressure in the narrow section compare with than in the wide section?

Hint: P + ½ rv2 = constant

P1 = P2 + ½ rv2 + rgz

P1 = P2 since both open to air

½ rv22 + rgz = 0

v2 =(2gz)1/2

Torricelli’s Theorem

If water pumped at 0.50 m/s through 4.0 cm diameter pipe in basement under 3.0 atm pressure, what will be flow speed and pressure in 2.6 cm diameter pipe 5.0m above?

1) find flow speed using continuity A1v1 = A2 v2

v2 = v1A1/A2 = v1pr12/pr22 =

1.2 m/s

2) Use Bernoulli’s Eq. to find pressure

P1 + ½ rv12 + rgy1 = P2 + ½ rv22 + rgy2

P2 = P1 +rg(y1 – y2) + 1/2r(v12 –v22)

=(3.0 x 105 N/m2) + (1.0 x 103 kg/m3)x

(9.8 m/s2)(-5.0m) + ½ (1.0 x 103 kg/m3)[(0.50 m/s)2 – (1.18m/s)2] =

= 2.5 x 105 N/m2

- P1 + ½ rv12 = P2 + ½ rv22
- Where speed is high, pressure is low
- Where speed is low, pressure is high

Courtesy Boston University Physics Dept. web site

Where is the pressure greater, less?

Courtesy The Aviation Group

Courtesy http://www.monmouth.com/~jsd/how/htm/airfoils.html

Courtesy NASA and TRW, Inc.

Sails are airfoils

Low pressure between sails helps drive boat forward

Courtesy Dave Culp Speed Sailing

Courtesy http://www.abdn.ac.uk/physics/streamb/fin13www/sld001.htm

- Helps explain why smoke rises up a chimney (air moving across top)
- Explains how air flows in underground burrows (speed of air flow across entrances is slightly different)
- Explains how perfume atomizer works
- Explains how carburetor works