1

Class #4

Retarding forces Stokes Law (viscous drag) 2-D motions with viscous drag Newton’s Law (inertial drag) Reynolds number Plausibility of Stokes law Plausibility of Newton’s Law Projectile motions with inertial drag

Worked Problems

:10

2

2-D Motion with viscosity

0

ˆ / :

1( ) ( )

:

(1 exp( )) exp( )

( )

z z z z Term

y Term y

xx x x

mmr mgz br Viscous drag w gravity

b

mv mg bv v v v z component

Solution from last timet tv v v

Now solve the x component which is even simpler than zv

mv bv v

0

'

'

0

0

ln( ) ln( )

exp( )

x

x

vx

vx

x x

x x

dv dt

vt

v v

tv v

dragF

gm

z

x :60

3

Velocity Dependent Force

Forces are generally dependent on velocity and time as well as position

Fluid drag force can be approximated with a linear and a quadratic term

= Linear drag factor(Stokes Law, Viscous or “skin” drag)

= Quadratic drag factor( Newton’s Law, Inertial or “form” drag)

2)( rcrbrFr

),,( trrFF

b

c

:15

quad

lin

fRatio

f

is important

4

The Reynolds Number

R < 10 – Linear drag1000< R < 300,000 –

Quadratic R > 300,000 – Turbulent

( )

( )

inertial quad dragR

viscous linear drag

density

viscosity

D

v

Dv

R

:20

5

Reynolds Number Regimes

R < 10 – Linear drag1000< R < 300,000 –

Quadratic R > 300,000 – Turbulent

6

The Reynolds Number II

DR OR

DR where

v

v

v

v

:20

2

2

163

163

inertial

viscous

inertial

viscous

F D

F D

DF DK K R

F D

2

2

v

v

v v

v

vD

density

viscosity

“D”= “characteristic” length

( )

( )

inertial quad dragR

viscous linear drag

7

The Reynolds Number III

R < 10 – Linear drag1000< R < 300,000 –

Quadratic R > 300,000 – Turbulent

1 22

1 222

1 22

1 22

1(1 / #)

R

dD

Linear Regime

D

Quadratic Regime

D

FC

v A

kD v DC

vDv A

Reynolds

kA vC k

v A

D

v

Dv

R

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Inertial Drag I

Plate with area “An” moves a distance through fluid with density

The mass of the fluid displaced is Mass “M” must acquire a velocity “v” to move

out of the way of the plate.The moving plate is causingRearranging we get

tv An

tv M Av t

( )p Mv Av t v 2p

Avt

2

2

vAkF

rcF

ndrag

drag

:35

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Inertial Drag II – A sphere

2vAkF ndrag

Previously demonstrated

“An” means “A normal to velocity”

Form factor for sphere

Plug ‘n’ play

2

4

1DAn

vvDFdrag ˆ16

22

4

1k

:40

10

Falling raindrops redux II

1) Newton

2) On z-axis

3) Rewrite in terms of v

4) Rearrange terms

5) Separate variables

2

2

2

2

2

2

2

ˆ

Assume vertical motion

(1 )

(1 )

zz

z z

z

z

mr mgz cr

mz mg cz

dv cg v

dt m

dv vmgDefine v g

c dt v

dvg dt

v

v

dragF

gm

z

x

:45

2

16c D

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Falling raindrops redux III

2

2(1 )

z

z

dvg dt

v

v

( )

20 0

2(1 )

v t tz

z

t

dvg dt gt

vv

( ) /

20arctanh( ( ) / )

(1 )

( ) tanh( / )

zz

v t v

vu dv v du

v

v duv v t v gt

u

v t v gt v

2arctanh( )

(1 )

dxx

x

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12

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x [degrees]

f(x)

sin, sinh and tanh

sin(x)sinh(x)tanh(x)

Tanh and sinh and cosh

cos2

sin2

tan

cosh2

tanh

ix ix

ix ix

ix ix

ix ix

x x

x x

x x

e ex

e ex

i

e ex

e e

e ex

e ex

e e

:55

sin( )x

sinh( )x

tanh( )x

13

Falling raindrops L4-1

A small raindrop falls through a cloud. It has a 1 mm radius. The density of water is 1 g/cc. The viscosity of air is 180 Poise. The density of air is 1.3 g/liter at STP.

a) What is the Reynolds number of this raindrop? (assume 10 m/s fall velocity)

b) Based on “a”, which type of drag should be more important?

c) What should be the terminal velocity of the raindrop, using quadratic drag?

d) What should be the terminal velocity of the raindrop, using linear drag?

e) Which of the previous of two answers should we use and why?

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Falling raindrops L4-2

A small raindrop is given an initial horizontal velocity of

and subsequently falls through a cloud. It has a 10 m radius. The density of water is 1 g/cc. The viscosity of air is 180 Poise.

a) Quantify the viscous force on the drop for a velocity of 10 mm/sec as well as the inertial force.

b) Should this drop be analyzed with linear or quadratic drag?

c) What is the Reynolds number of this raindrop?d) Write a formula for the position vector of the

raindrop as a function of time (set the origin to zero at point where it is released)

:50

20 /ˆ3 smxv

15

Pool Ball L4-3

A pool ball 6 cm in diameter falls through a graduated cylinder. The density of the pool ball is 1.57 g/cc. The viscosity of water is approximately 1 CentiPoise.

a) Quantify the force on the ball for a velocity of 100 mm/sec.

b) What should be the terminal velocity of the ball?

c) Quantify the force if we assume quadratic drag

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2 2

16dragF D v

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Lecture #4 Wind-up

.

:72

1gt

vzv v e

vvDFdrag ˆ

1622

Linear Drag (Sphere)

Quadratic Drag (Sphere)

xuDFdrag ˆ3

( ) tanh( / )v t v gt v

2

mg bv

mg cv