Dynamic similarity

Chapter 5

Dimensonal analysis

Dynamic similarity

= 0

⋅

∇ u

⎠

⎜⎜ ⎞

⎝

⎛ + ⋅∇

∂

∂ 2

t p

/ 2

, / ,

/ ,

/L U t Ut L p p U

x

x^∗ = u^∗ = u ^∗ = ^∗ = ρ ∂ ∗

= ∂

∂

≡ ∂

∇ x L x

1

=0

⋅

∇^∗ u^∗

p g t

u g u

u u

Fr 1 Re

1 ∇ 2 + +

∇

−

=

∇

⋅

∂ +

∂ _{∗ ∗ ∗ ∗ ∗ ∗ ∗}

∗

∗

μ ρUL 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

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ +

− Δ

− R R

p μ _nn σ _Re¹ _We^{1 1 1 0}

2 1

⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛ +

− Δ

− ^∗ _{∗ ∗}

∗

R p _nn R

σ ρU²L We=

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}, μ = 0.005 ⋅ , σ = 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 =

μ ρUL

04 5 . 0

) 0002 . 0 ( ) 1 ( We 1000

2

2 = =

= σ

ρU L

exp

40

Re ⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

=

μ

ρ

exp 2

5

We ⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

= σ

ρU 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 =

= ρ

σ

s Pa 018 . 0 m/s,

354 .

0 = ⋅

= μ

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

system model

Re ⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

μ ρ μ

ρUL UL

system 2

model 2

We ⎟⎟

⎠

⎜⎜ ⎞

⎝

=⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

σ ρ σ

ρU L U L

system 2

model 2

Fr ⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

gL U gL

U

5 parameters, 3 constraints -> 2 free parameters

system model kL

L =

ρ

= ρ

system 2

/ 1

model k U

U =

system 2

/ 3

model μ

μ = k

system 2

model σ

σ = 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

A roll coating system

H/L ~ 0.01; roll appears as a plane Critical roll speed U* for air

entrainment

No characteristic length scale Re Ca

We = =

σ

μU Ca*; dimensionless critical entrainment velocity

Predict Ca* from a knowledge of physical properties

c b

N = Re

Fr We

c a b

σ α ρ μ

ρ ⎟⎟⎠ =

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛

4 / 1

4 3 3/4

1/4

We Fr

Re ⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

= g

N μ

ρ

σ _γ

μ

σ ρ ⎟⎟ ≡

⎠

⎜⎜ ⎞

⎝

= ⎛

3 / 1 4 1/3

3 / 4

We Fr Re

g

)

(

Ca

= ℜ γ

Inspectional analysis

] factors shape

Fr,

2 = Ψ[Re, Δ

U P ρ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ + ∂

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

∂

−∂

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

∂ ^∗

∗

∗

∗

∗

∗ ∗

∗

∗

∗

∗

∗

∗ ∗

∗

∗

∗

∗

∗ ∗

2 2 2 2

2 1

1 Re

1

θ θ

θ z z z z

z z z

r

u r

z u r

r u r r z

p z

u u u r u r

u u

Fully developed laminar flow

(

,

,

) = ( 0 , 0 ,

(

, θ ))

⎥⎦

⎢ ⎤

⎣

⎡

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

∂

−∂

= _∗^∗ 1_{∗ ∗} ^∗ _∗^∗ 1_∗2 ² ₂^∗ Re

0 1

θ ^z

z u

r r

r u r r z

p

∗

∗

∂

−∂

= r

0 p

) factors shape

Re, , (

) factors shape

Re, , , (

∗

∗

∗

∗

∗

∗

=

=

z p p

r u

u_{z z} θ

Shape factor from boundaries

) , ( )

( ^∗ = − _∗^∗ = F r^∗ θ dz

z dp C

Constant; dimensionless pressure gradient

L U

D z P

C( ) ₂ ρ

− Δ

∗ =

) factors shape

Re, (

constants that appear in the differential equations

and the boundary conditions that define the flow

= ′

Re

∗ ∗

∗ = z

z _⎥

⎦

⎢ ⎤

⎣

⎡

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

∂

− ∂

= _∗^∗_∗ 1_{∗ ∗} ^∗ _∗^∗ 1_∗2 ² ₂^∗

0 ^z θ ^z

u r

r r u r r z

p

) factors shape

, (

) factors shape

, , (

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

=

=

z p p

r u

u_{z z} θ

⎥⎦

⎢ ⎤

⎣

⎡

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

∂

− ∂

= _∗^∗ 1_{∗ ∗} ^∗ _∗^∗ 1_∗2 ² ₂^∗ Re

0 1

θ ^z

z u

r r

r u r r z

p

) factors shape

Re, , (

) factors shape

Re, , , (

∗

∗

∗

∗

∗

∗

=

=

z p p

r u

u_{z z} θ

Force balance; the pressure drop arises from the wall shear stress

[ ]

∫ ∫ ⁻

=

Δ

s znds dz PAc 0

τ

A contour integral along the cross-sectional perimeter

[

]

∫ ∫

^Δ

Δ =

dz U ds

P

s zn /

2 0

Re 1

ρ

K U

P

/ Re

) factors shape

Re, (

2

= ′ Δ ρ

) Re,

,

( shape factor

zn

= f θ Δ

[

]

∫ ∫

^Δ

Δ =

dz U ds

P ^{L D}

s zn Re)

/ ( 2 0

ρ

P

DΔ =

ρ

Dimensional analysis; 6-3=3 2 fn^(Re, L^/ D^{,shape factors)} UP =

Δ

[

] [

]

PA s zn

L

s−τzn = −τ =τ

=

Δ c

∫ ∫

∫

Fully developed Wall shear stress averaged along the perimenter; S is the wetted perimeter Friction factor

2 2

1

h 2

2 2 1

2 1

) / ( /

U r L P U

SL PA

f U ^c

ρ ρ

ρ

τ = Δ = Δ

= D^h = 4r^h

μ ρ U r_h Re = 4

) factors shape

( Re F

f =

Entry region flow

[ ] ^{[ ]}

2 ) Re

2 ( Re 2

Re Re ^{( / Re)}

0 Re)

/ (

0 G L D

L dz D

z L h

dz D L ds

f D ^{L D L D}

s Δzn = =

=

∫ ∫

∫

) factors shape

, / Re (

Re

f

=

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 =

= ρ σ

μ

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

Dynamic similarity

Re

= Re

Fr

= Fr

We

= We

7 . 25 84

. 0

03 . 0 8 . 9 2400 Fr

Bo We

2

2 = × × =

=

= σ

ρgL

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 = =

= π μ

ρ μ

ρ

D Q

UD 16 980

Fr ₅

t 2

2

t

2 = =

= gD

Q gD

U

π

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

ν

ν _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

× −

ν = ^{3.16 10 Pa s}

2

model = × ⁻ ⋅

μ

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

Priciple of dynamic similarity

„

For isothermal, incompressible, low Mach number flow, the fluid dynamics are completely controlled by the values of no more than three dynamic dimensionless groups

„

Reynolds number; relative importance of inertial forces to viscous force

„

Froude number; of inertial force to gravitational force

„

Weber number; of inertial force to interfacial force

„

All the dimensionless groups that might come out from the

Buckingham pi theorem can be expressible in terms of these 3

groups, hence are not independent of them