Physics of fluids
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PHYSICS OF FLUIDS. Fluids. Includes liquids and gases Liquid has no fixed shape but nearly fixed volume Gas has neither fixed shape or volume Both can flow. Density. Mass per unit volume r = m/V (Greek letter “rho”) m = r V

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Fluids
Fluids

  • Includes liquids and gases

    • Liquid has no fixed shape but nearly fixed volume

    • Gas has neither fixed shape or volume

    • Both can flow


Density
Density

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)


Density of water
Density of water

  • 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



Pressure in fluids
Pressure in Fluids

  • 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


Pressure varies with depth as

rgh

Pressure Varies with Depth as

  • 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


Example pressure at bottom of a lake
Example: Pressure at Bottom of a Lake

  • 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?


Atmospheric gauge and absolute pressure
Atmospheric, Gauge and Absolute Pressure

  • 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


Example water in a straw
Example: Water in a Straw

  • 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


What is the tallest column of water that could be trapped like this
What is the tallest column of water that could be trapped like this?

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


Pascal s principle
Pascal’s Principle like this?

  • 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



Measuring pressure
Measuring Pressure like this?

  • 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


Buoyancy
Buoyancy like this?

  • Submerged or partly submerged object experiences an upward force called buoyancy

  • Pressure in fluid increases with depth

  • Studied by Archimedes over 2000 years ago.


Buoyant force on cylinder
Buoyant Force on Cylinder like this?

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


Archimedes principle
Archimedes Principle like this?

  • 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 like this?

Sfy =0

T +B - W = 0

T = W – B

T = W – rFVg

T is apparent weight W’

Diagram courtesy Caduceus MCAT Review


Example king s crown
Example: King’s Crown like this?

  • 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


Floating objects
Floating Objects like this?

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


Example floating log
Example: Floating Log like this?

  • 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


Example lifted by balloon
Example: Lifted by Balloon like this?

  • 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


Fluid flow
Fluid Flow like this?

  • Equation of Continuity

  • Volume rate of flow is constant for incompressible fluids (not turbulent)

  • A1v1 = A2 v2

  • v is velocity


Laminar vs turbulent flow
Laminar vs. Turbulent Flow like this?

Erratic, contains eddies

Fluid follows smooth path

Courtesy MIT Media Laboratory


Example narrows in a river
Example: Narrows in a River like this?

  • 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


Example heating duct
Example: Heating Duct like this?

  • 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


Bernoulli s equation
Bernoulli’s Equation like this?

  • 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


Question
Question like this?

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


Speed of water flowing through hole in bucket
Speed of Water Flowing Through Hole in Bucket like this?

P1 = P2 + ½ rv2 + rgz

P1 = P2 since both open to air

½ rv22 + rgz = 0

v2 =(2gz)1/2

Torricelli’s Theorem


Speed and pressure in hot water heating system
Speed and Pressure In Hot Water Heating System like this?

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 like this?

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


No change in height
No Change in Height like this?

  • P1 + ½ rv12 = P2 + ½ rv22

  • Where speed is high, pressure is low

  • Where speed is low, pressure is high


Why curveballs curve
Why Curveballs Curve like this?

Courtesy Boston University Physics Dept. web site


How an airfoil provides lift
How an Airfoil Provides Lift like this?

Where is the pressure greater, less?

Courtesy The Aviation Group


Crowding of streamlines indicates air speed is greater above wing than below
Crowding of streamlines indicates air speed is greater above wing than below

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


Lift illustrated
Lift Illustrated wing than below

Courtesy NASA and TRW, Inc.


Sailing against the wind
Sailing Against the Wind wing than below

Sails are airfoils

Low pressure between sails helps drive boat forward

Courtesy Dave Culp Speed Sailing


Venturi tube
Venturi Tube wing than below

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


Bernoulli s principle also
Bernoulli’s Principle also wing than below

  • 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


Giancoli website
Giancoli Website wing than below


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