Fluid dynamics is a branch of mechanics, or physics, that seeks to describe or explain the nature of physical phenomena that involve the flow of liquids and/or gases.

Chapter 1

What is fluid dynamics?

Definition & Goal

„

Fluid dynamics is a branch of mechanics, or physics, that seeks to describe or explain the nature of physical phenomena that involve the flow of liquids and/or gases.

„

One of our primary goals will be to produce mathematical models that permit us to understand, describe, and design engineering

systems and processes that involve the mechanics and dynamics of fluids.

„

Thinking about fluid dynamics with some typical problems

Drop breakup in a stirred tank

Suppose we place a liquid in a tank and agitate it with a high speed rotating impeller

Flow field in a CVD reactor

In semiconductor device manufacturing, thin films are grown on silicon wafer Uniformity of film growth is essential to the success of this process

Control of the flow field is very important

Predict the response of the flow field to changes in reactor geometry, gas flow rate, the positions of gas inlet and outlets, and operating conditions

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

=

=

=

=

=

=

=

μ

σ ρ

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

No. of independent dimensionless groups = No. of parameters – No. of dimensions (4 = 7 – 3)

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

σ ρ ,

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

σ ρ

c b

D

g

g

=

σ ρ

c b

Da

U

U^∗

=

σ ρ

c b

Daσ ρ μ

μ^∗ = _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}σ ρ

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

σ ρD_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 ⎟

⎠

⎜ ⎞

⎝

= ⎛

∗

σ ρ 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

(

)

∗ = μ ρσ

μ D

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

= ⎛

∗

≡

2 / 1 c 2

c 2

/ 1 c 3

c drop

drop

, , ( )

D D

gD U D

V f

V ρσ

μ σ

ρ σ

ρ

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

′ =

′ =

=

=

=

=

=

=

=

=

=

μ ρ

μ

σ ρ

Step 3: Use the Buckingham pi theorem.

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

Step 4: Select the recurring parameters.

D

, ρ , σ

Step 5: Form, in turn, dimensionless groups.

c b

D a

D

D ^∗ = _pσ ρ L^∗_p = L_pD_p^aσ ^bρ ^c μ ^∗ = μ D_p^aσ ^bρ ^c μ ′^∗ = μ^′D_p^aσ ^bρ ^c

c b

D aσ ρ ρ

ρ′^∗ = ^′ _p d ^∗ = dD _p^aσ ^bρ ^c q^∗ = qD _p^aσ ^bρ ^c Q^∗ = QD _p^aσ ^bρ ^c

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

c b

Da

D

D^∗ = _pσ ρ

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

(

)

∗ =μ ρσ −

μ 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

σ ρ σ

ρ ρ

ρσ ρ μ

ρσ μ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

′ ⎛

′ ′

=

= ⁻

∗

Q q Q D

D L D D d

D F

D D ( ) , , , , , ,

2 / 1

3 p p

p p 2

/ 1 p

p σ

ρ ρ

ρ μ ρσ μ

μ

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

σ ρ