Lecture 2Buoyancy.

Fluid dynamics.

Hot air balloon Buoyancy (in the Dead Sea)

Cohesion (water bubble in space)

Laminar flow

Vacuum gun

Sealed tube, air pumped out

Ping-pong ball

What happens if we punch a little hole on one side?

DEMO: Vacuum gun

Atmospheric pressure pushes ball through tube and accelerates to high speed. Realistic calculation of ballspeed is complicated and needs to take turbulent air and friction into account.

Buoyancy and the Archimedes’ principle

ybottom

ytop

hA

A box of base A and height h is submerged in a liquid of density ρ.

bottom topAp Ap

atm bottom atm topA p gy A p gy

A hg

Archimedes’s principle: The liquid exerts a net force upward called buoyant force whose magnitude is equal to the weight of the displaced liquid.

direction upVg

topbottomF F F

Ftop

Fbottom

Net force by liquid:

In-class example: Hollow sphere

A hollow sphere of iron (ρFe = 7800 kg/m3) has a mass of 5 kg. What is the maximum diameter for this sphere to be completely submerged in water? (ρwater = 1000 kg/m3)

A. It will always be submerged.

B. 0.11 m

C. 0.21 m

D. 0.42 m

E. It will always only float.

FB

mg

The sphere sinks if BF mg

3water

43

R g mg

3

water

30.106 m

4m

R maxMaximum diameter 2 0.21 mR

Density rule

A hollow sphere of iron (ρFe = 7800 kg/m3) has a mass of 5 kg. What is the maximum diameter necessary for this sphere to be fully submerged in water? (ρwater = 1000 kg/m3) Answer: R = 0.106 m.

And what is the average density of this sphere?

3

sphere water33

5 kg1000 kg/ m

4 40.106 m

3 3

m

R

An object of density ρobject placed in a fluid of density ρfluid

• sinks if ρobject > ρfluid

• is in equilibrium anywhere in the fluid if ρobject = ρfluid

• floats if ρobject ρfluid

This is why you cannot sink in the Dead Sea (ρDead Sea water = 1240 kg/m3 , ρhuman body = 1062 kg/m3 ) !

DEMO: Frozen helium

balloon

Attraction between molecules

Molecules in liquid attract each other (cohesive forces that keep liquid as such!)

In the bulk: Net force on a molecule is zero.

On the surface: Net force on a molecule is inward.

…And this force is compensated by the incompressibility of the liquid.

Wood floats on water because it is less dense than water. But a paper clip (metal, denser than water!) also floats in water… (?) .

Very small attraction by air molecules.

Surface tension

Overall, the liquid doesn’t “like” surface molecules because they try to compress it.

Liquid adopts the shape that minimizes the surface area.

Any attempt to increase this area is opposed by a restoring force.

The surface of a liquid behaves like an elastic membrane.

The weight of the paper clip is small enough to be balanced by the elastic forces due to surface tension.

Drops and bubbles

Water drops are spherical (shape with minimum area for a given volume)

Adding soap to water decreases surface tension. This is useful to:

• Force water through the small spaces between cloth fibers• Make bubbles! (Large area and small bulk)

How wet is water?

Molecules in a liquid are also attracted to the medium it is in contact with, like the walls of the container (adhesive forces).

Water in a glassWater in wax- or

teflon-coated glass

Fadhesive > Fcohesive

Fadhesive < Fcohesive

Or: surface tension in air-liquid interface is larger/smaller than surface tension in wall-liquid interface

Fluid flow

Laminar flow: no mixing between layers

Turbulent flow: a mess…

Dry water, wet water

Real (wet) fluid: friction with walls and between layers (viscosity)

Slower near the walls

Faster in the center

Ideal (dry) fluid: no friction (no viscosity)

Same speed everywhere

Within the case of laminar flow:

Flow rate

Consider a laminar, steady flow of an ideal, incompressible fluid at speed v though a tube of cross-sectional area A

dVAv

dtVolume flow

rate

A

dx = v

dt

dmAv

dtMass flow rate

Continuity equation

A1

A2

v1

v2

The mass flow rate must be the same at any point along the tube (otherwise, fluid would be accumulating or disappearing somewhere)

If fluid is incompressible (constant density):

ρ1 ρ2

Thin tube, large speed

Thick tube, small speedIncompressible fluid:

Example: Garden hose When you use your garden faucet to fill your 3 gallon watering can, it

takes 15 seconds. You then attach your 3 cm thick garden hose fitted with a nozzle with 40 holes at the end. You turn on the water, and 4 seconds later water spurts through the nozzle. When you hold the nozzle horizontally at waist level (1 m from the ground), you can water plants that are 5 m away.

a) How long is the hose?

b) How big are the openings in the nozzle?3

4 33 gallons 3.785 liters 1 m7.6 10 m / s

15 s 1 gallon 1000 literdVdt

Volume flow rate

hose hose

dVA v

dt

When you use your garden faucet to fill your 3 gallon watering can, it takes 15 seconds. You then attach your 3 cm thick garden hose fitted with a nozzle with 40 holes at the end. You turn on the water, and 4 seconds later water spurts through the nozzle. When you hold the nozzle horizontally at waist level (1 m from the ground), you can water plants that are 5 m away.

a) How long is the hose?

b) How big are the openings in the nozzle?

We use kinematics to determine vnozzle:

nozzlenozzle

0 x

x v t tv

x

h

202g

h t

hose hose nozzle nozzleA v A v

r

hose

2 vhose

40rnozzle

2 vnozzle

rnozzle

rhose

vhose

40vnozzle

1.5 cm 4.3 m / s

40 ×11 m / s1.4 m m