Chapter 14

Fluids

Fluids at Rest

(Fluid = liquid or gas)

Density ()

• unit: kg/m3

• we will consider only = m/V = constant

• (incompressible fluid)

• we will also assume g = constant

dV

dm

Pressure (p)

• A fluid exerts a force dF normal to any area dA you consider in it.

• For a fluid at rest, the force is equal and opposite on each side.

Pressure (p)

• Why is there a force?

• Microscopically: the fluid particles are in motion and collide with dA

• Macroscopically: the fluid is at rest

Pressure (p)

• pressure p is a scalar (no intrinsic direction)

• reason: it acts normal to any surface dA

dA

dFp

Pressure (p)

• units used:pascal, millibar, atm

• 1 Pa = 1 N/m2

1 millibar = 100 Pa 1 atm = 1.013×105 Pa

dA

dFp

Terminology

• ‘gauge pressure’ = p – patm

• (can be > 0 or < 0)

• (e.g., read on a car tire pressure gauge)

• p = absolute pressure (> 0) = atmospheric pressure + gauge pressure = patm + gauge pressure

Pressure and depth

gdy

dp

• pressure: p coordinate: y

• pressure decreases with ‘elevation’ y:

Derive this result and integrate itDerive this result and integrate it

Pressure and depth

ghpp

yygpp

21

1212 )(

• coordinate: y distance: h > 0

• p increases with depthfor any shape of vessel

Demonstration: depth and shape of containerDemonstration: depth and shape of container

Exercise 14-10

• The dangers of a long snorkel tube:

• Find the gauge pressure at the depth shown. Will this cause the snorkeler’s lungs to collapse?

Demonstration: atmospheric pressureDemonstration: atmospheric pressure

Pascal’s Law

ghpp

yygpp

21

1212 )(

• If any change in pressure p is applied at one point, it is transmitted to all points in the fluid and to walls enclosing it.

Example: Hydraulic Lift At equilibrium, p = F1/A1 = F2/A2

DemonstrationDemonstration

Two Pressure Gauges

Notes on (a) and Exercise 14-9Notes on (a) and Exercise 14-9Notes on (b) firstNotes on (b) first

Homework Hints: Exercise 14-55

Buoyancy and Buoyant Force

A (fully or partially) submerged object feels an upward force equal to the weight of fluid it displaces

(a) fluid element with weight wfluid

(b) body of same shape feels buoyant force B = wfluid

DemonstrationDemonstration

Surface Tension• Molecules of liquid

attract each other (else no definite volume)

• center: net force = 0

• surface: net force is directed inward

Surface Tension• So the surface acts like a

membrane under tension (like a stretched drumhead)

• The surface resists any change in surface area

• Strength characterized by ‘surface tension’

DemonstrationDemonstration

Surface Tension • = F/d

= cohesive force per unit length

• surface tension force F = d

• We can measure by just balancing F

Do Example 14-23 Do Example 14-23 Notes on measuring Notes on measuring

cohesion: attraction of like molecules example: liquid-liquid forces (surface tension)

adhesion: attraction between unlike moleculesexample: liquid-glass forces

(a) adhesion > cohesion: water wets glass(b) adhesion < cohesion: mercury beads up

Capillarity

• For these two cases, the surface tension force F pushes the column of liquid either up or down:

• (a) up for water• (b) down for mercury

Notes on capillary tubesNotes on capillary tubes

Homework Announcements

• Homework Set 5: Correction to hints for 14-55 (handout at front and on webpage)

• Recent changes to classweb access (see HW 5 sheet at front and webpage)

• Homework Sets 1, 2, 3: returned at front(scores to be entered on classweb soon)

Midterm Announcements

• Friday:

• review required topics

• practice problems (from class, HW, new?)

• Monday: (midterm)

• you can bring a sheet of notes (both sides)

• you will be given a list of equations

Fluid Flow

(Fluid Dynamics)

Flow Fluid

• Flow line =path of fluidelement

• Flow tube =bundle of flow lines passingthrough area A (just a useful construct)

Simplifying Assumptions

• Steady flow:

• At any given point in the fluid, its properties (v, , p) don’t change in time

Simplifying Assumptions

• Steady flow:

• different flow lines never cross each other

• fluid entering a flow tube never leaves it

Simplifying Assumptions

• Incompressible fluid:

• = constant

• No friciton:• no ‘viscosity’

Continuity Equation

A1v1 = A2v2 • the same volume dV of

fluid enters and exits tube:

• dV = volume passing through A in dt = Av dt

Avdt

dV

NotesNotes

Continuity Equation

A1v1 = A2v2

• along the flow: • A = area of flow tube

v = speed of fluid

• if one increases, theother must decrease

Notes and Demonstration: water flowNotes and Demonstration: water flow

Continuity Equation

A1v1 = A2v2

• Where the flow lines are crowding together, the fluid speed is increasing

Bernoulli’s Equation

• only valid for: steady flow, incompressible fluid,no viscosity!

2222

2111

2

1

2

1

vgyp

vgyp

NotesNotes

Bernoulli’s Equation

• if v1= v2= 0:

• reduces to previous result for fluid at rest

2222

2111

2

1

2

1

vgyp

vgyp

Bernoulli’s Equation

• if y1= y2 then for p and v:

• if one increases, theother must decrease

2222

2111

2

1

2

1

vgyp

vgyp

DemonstrationDemonstration

Applications of Bernoulli’s Equation

Venturi Meter (Example 14-10) horizontal flow tube

NotesNotes

Note: if viscosity is present, then v decreases with distance from tube center

Venturi Meter:Homework Problem 14-90 (c)

NotesNotes

DemonstrationDemonstration

Wing Lift

Can’t predict flow lines but they indicate low pressure above wing, so net force up

Demonstration: propellorDemonstration: propellor

Efflux Speed: vertical flow tube

NotesNotes

Siphon: flow tube points up, then down

• First: you must fill the tube

• There is a limit:

H + h < 10 m

Curve Ball: viscosity makes it possible

Warm-up demonstrationsWarm-up demonstrations

Viscosity drags air with spinning ball: low pressure=net force so the ball curves

DemonstrationDemonstration

Homework Announcements

• Homework Set 5: Correction to hints for 14-55 (handout at front and on webpage)

• Recent changes to classweb access (see HW 5 sheet at front and webpage)

• Homework Sets 1, 2, 3: returned at front(scores to be entered on classweb soon)