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dimensional analysis

y 0

x v

v

x y

   

 

2 2

2 2

v

x y

t x y x y

^ ^  ^  ^ ^      ^ ^  ^ ^

v v v v

g

* *

/ _{x x} /

x  x L   v_

* *

/ _{y y} /

y  y L   v_

* t * *

t v v / L

L

^ _

    

* * * * 2 * 2 *

* *

x y

* * * 2 2 *2 *2

v v v L P v v

g L

t x y x y

  

_ _  _

 

             

2 /

_



Fr gL

/ 2

Eu  P v_ Re  Lv_ /

2

/

D / 2 C F A

v_

 (inertial to gravitational forces)

(pressure to inertial forces) (inertial to viscous force)

 0



 u ^{ u u u}_^{   ug}



 



  



 ₂

t p

/ 2

, / ,

/ ,

/L U t Ut L p p U

x

x^  u^ u ^  ^   ^{ } _^_x ^ _L¹ _^_x^

0



^ u^

p g t

u g u

u u

Fr 1 Re

1  ₂ 













 

 _{      }







UL

 Re

gL U² Fr  Boundary conditions

0 on

0 

 y

u_x ^{ux 0 on y 0}

B y U

u_x  on 

L y B u_x^ 1 on ^ 

1 0 1

2 1





 



 





 R R

p  _nn  _Re¹ _We^{1 1 1 0}

2 1





 



 





 ^ _{ }



R p _nn R



U²L We 

dynamic similarity

If two flows occur in geometrically similar systems in which viscous effects and interfacial phenomena occur, and if Re, We, Fr are the same in both systems, the two systems are dynamically similar

Inkjet printing

Goal: to know how much a drop spreads on impact with a surface

s m U

m N s

Pa m

nanoliters

V 8 810^{12 3},  0.005  ,   0.04 / , 1 / We want to design a scaled-up version of this process to facilitate observation and measurement of the drop dynamics

(designing a dynamically similar system) Length scale

Velocity scale

m 10 2 m ) 10 8

( ^{12 1/3 4}

3 /

1      

V L

m/s

1 U

Assume gravity is not a significant factor (this may not be the case) -> ignore Froude number

005 40 . 0

0002 .

0 ) 1 (

Re   1000 



UL

04 5 . 0

) 0002 . 0 ( ) 1 ( We 1000

2

2  

 

U L

exp

40

Re 

 



 







UL

exp 2

5

We 



 



 

 

U L We must design an experiment such that

5 parameters, 2 constrains -> 3 free parameters m

L 0.002 ,

kg/m 1000

N/m, 05

.

0  ³ 







s Pa 018 . 0 m/s,

354 .

0  





U

Removing oil from water surface

Moving belt with hydrophobic surface

Goal: design an experimental model to learn more about how the rate of oil entrainment is related to operating parameters such as belt speed and physical properties Dynamics are dependent on the balance between inertial, viscous, surface, gravitational forces

syst em model

Re 

 



 



 











UL UL

syst em 2

model 2

We 



 







 











U L U L

syst em 2

model 2

Fr 



 







 





gL U gL

U

5 parameters, 3 constraints -> 2 free parameters

system model kL

L 



^model 



^system

system 2

/ 1

model k U

U 

system 2

/ 3

model 

 k

system 2

model





 k

1.Large k does not meet surface tension

2.Cannot take all three dimensionless group together 3.Need more physics to make knowledgeable

evaluations of the relative importance of the three dynamic groups, Re, We, Fr

Experimental design

At some critical height, the free surface forms a vortex that is sucked into the tube,

entraining air in the liquid; we wish to avoid

s m L

L D

D_r 1m _t 0.03m ₁  0.5m ₂  0.3m Q 1.210^{2 3}/ N/m

0.25 kg/m

2400 s

Pa

2.4   ³ 

  



Molten ceramic at 1000K in large reservoir -> need scale-down

Dynamic similarity

Re

model

 Re

real

Fr

_model

 Fr

_real

We

_model

 We

_real

7 . 25 84

. 0

03 . 0 8 . 9 2400 Fr

Bo We

2

2    



 

gL

If the length scale is large, the radius of curvature of the vortex may be so large that surface tension effect will be negligible

Surface tension effect is unimportant -> neglect We

4 509 Re

t

t  

  







D Q

UD 16 980

Fr ₅

t 2

2

t

2  

 gD

Q gD

U



3 parameters (tube diameter, flow rate, kinematic viscosity), 2 constraints -> select tube diameter 0.3cm (scale down by one order of magnitude)

t real model

t



 



 



 





D Q D

Q



 _t⁵ _real

2

model 5

t 2



 



 



 





D Q D

Q

/s cm 38 /s

m 10 8 .

3 ^{5 3 3}

model   ^ 

Q

/s m 10 16 .

3 ^{5 2}

model

 

  ^{3.16 10 Pa s}

2

model   ^ 



Test with an aqueous solution of corn syrup or glycerol at 1/10 of full scale (geometrically and dynamically similar) 5

. 06 1

. 0

) 003 . 0 )(

8 . 9 ( Bo 1000

2 



Surface tension may be important in this small length scale

Need more experiments with

liquids of several surface tensions and look for any influence

Dimensional analysis

The problem of a drop of liquid formed at the lower end of a vertical capillary

Step 1: Make a list of the relevant parameters

requires a good sense of the physics of the process Step 2: List the fundamental dimensions of each parameter

t L U

Lt m t

L g

t m L

m L

D L

V

/ ] [ /

] [ /

] [

/ ] [ /

] [ ]

[ ]

[

2

2 3

c 3

drop





















The effect of temperature arises only through its effect on the physical properties

Step 3: Determine, from the Buckingham pi theorem, the number of dimensionless groups that characterize this problem

# of independent dimensionless groups = # of parameters – # of dimensions (4 = 7 – 3)

rank of dimensional matrix

Step 4: From the list of the independent parameters (all but V_drop) select a

number equal to D (=3, in this case) that will be used as ‘recurring parameters’.

It is wise to pick a set of parameters that include all the dimensions



 ,

c

, D

Step 5: Form, in turn, dimensionless groups that are proportional to each of the remaining nonrecurring parameters

parameter ng

nonrecurri ] [

parameters recurring

c drop drop







 a b c

D V

V  

c b

D

a

g

g

^



_c

 

c b

Da

U

U^  _c

 

c b

Da 



^  _c

a,b,c are different for each of these four equations

Step 6: For each of the four equations above, solve for the set of exponents

c b

Da

V

V_drop^  _{drop c} 

c b

a m t m L

L L t

L

m^{0 0 0}  ³ ( / ²) ( / ³) c

b m: 0  

c a L: 0  3 3

b t : 0  2

3 ,

0 ,

0   

 c a

b

3 c drop

drop D

V^  V

c b

a m t m L

L t L t

L

m^{0 0 0} ( / ²) ( / ²) ( / ³) c

b

 0

c a 3 1

0   b 2 2 0 

2 1

1  



 c a

b



D_c² g^  g

c b

a m t m L

L t L t

L

m^{0 0 0} ( / ) ( / ²) ( / ³) c

b

. 0

a a 3 1

0    b 2 1 0   

2 1 2

1 2

1  



 c a

b

2 / 1

c 



 



 





 U D

U

c b

a m t m L

L Lt m t

L

m^{0 0 0} ( / ) ( / ²) ( / ³) c

b



1 0

c a 3 1

0    b 2 1 0  

2 1 2

1 2

1  



 c a

b



_c



^1/2

   

 D

 





 



 



 



 





2 / 1 c 2

c 2

/ 1 c 3

c drop

drop

, , ( )

D D

gD U D

V f

V 











Dispersion of an oil stream in an aqueous pipe flow

Predict the mean droplet diameter as a function of the parameters that characterize this flow

Step 1: Make a list of parameters.

Step 2: List the fundamental dimensions.

Lt m L

m t

L q

t L Q

Lt m

t m L

m L

d L L

L D

L D

/ ] [ /

] [ /

] [ /

] [ /

] [

/ ] [ /

] [ ]

[ ]

[ ]

[ ]

[

3 3

3

2 3

p p

 

 





























Step 3: Use the Buckingham pi theorem.

The number of fundamental dimensionless groups = 11 – 3 = 8

Step 4: Select the recurring parameters.

D

_p

,  , 

Step 5: Form, in turn, dimensionless groups.

c b

Da

D

D^  _p  L^_p  L_pD_p^a^b^c ^  D_p^a ^b^c ^  ^D_p^a^b^c

c b

Da 



^  ^ _p d^  dD_p^a^b^c q^  qD_p^a^b^c Q^ QD_p^a ^b^c

Step 6: Solve for the coefficients for each of the equations.

c b

Da

D

D^  _p 

c b

a m t m L

L L t

L

m^{0 0 0}  ( / ²) ( / ³) c

b m: 0  

b a L: 01 3

b t : 0 2

1 0

0   

 c a

b

Dp

D^  D

c b

a m t m L

L Lt m t

L

m^{0 0 0} ( / ) ( / ²) ( / ³) c

b



1 0

c a 3 1

0   b 2 1 0   

2 1 2

1 2

1  



 c a

b



p



^1/2

   

 D



































 

 

 ^{ }



2 / 1 3 p 2

/ 1 3 p p

p p 2

/ 1 p

2 / 1 p

p

, ,

, , , ) (

, )

( q D

Q D D

L D D d

D D F

D D











 





























 

 



 ^



Q q Q D

D L D D d

D F

D D ( ) , , , , , ,

2 / 1 3 p p

p p 2

/ 1 p

p 









 



Speculation about the physics of the process

- viscosity of oil is not significant if it is comparable to that of water

- inlet tube diameter is of no significance if it is large compared to the drop size - most liquid densities lie in a narrow range -> no effect of density ratio

- as long as pipe length is large, drop size reaches equilibrium and does not change - if q/Q is small, it does not affect the drop breakup























 





2 / 1 3 p

p F Q D

D D D



