Louisiana Tech University Ruston, LA 71272 Lubrication/Thin Film & Peristaltic Flows Juan M. Lopez Lecture 10 BIEN 501 Wednesday, March 28, 2007

Louisiana Tech UniversityRuston, LA 71272

Lubrication/Thin Film & Peristaltic Flows

Juan M. Lopez

Lecture 10

BIEN 501

Wednesday, March 28, 2007


Nondimensionalizing

• Momentum balance (and mass and energy balances) often written in nondimensional form– What is the advantage of nondimensionalizing our

equations in this way?• Solutions are more general

– Limits of integration are 0-1 regardless of what the characteristic length L0 is

• Dimensionless groups result– Reynolds number, Ruark number, Strouhal number, many

others for other balance equations

– Provide insight into physics of the problem – relative importance of different effects

0

*

L

zz i

i


Limiting Cases• Often can’t find an analytical solution to flow problems• Limiting cases – can provide significant insight• Dimensionless form of Navier-Stokes equations:

• What do we call the limiting case when NRe<<1?– Creeping flow

• How does this affect the Navier-Stokes equations?

• What are some examples of cases where such flows are important?– Microfluidics, flow in porous media, colloidal dispersions (small L0)– Polymer processing (large )

00

Re0

2000

*

Re

****

*

,,

111

vLN

vN

L

vtN

divNNtN

Ruo

St

RuSt

P

P vvvv

vvv

v

vvv

divt

ordivNNtN

divN

RuSt

PP *

Re

**

*

*

Re

**

111

1


Creeping Flow

• Creeping Flow approximation is an example of scaling– By comparing the order of magnitude of terms we can

make useful simplifications to complex equations– Resulting equations sometimes introduce some

inconsistencies• Creeping flow examples in text

– Cone and Plate Viscometer– Screw Extruder– Flow past a Sphere– Melt spinning

• Lubrication flows (Thin Draining Films)…


Reference Videos

modeltcompare.mov

Creeping Flow

eularian_Frame.mov

Boundary Layer

Lagrangian_Frame.mov

Boundary Layer


Lubrication Flows

• Liquid flows in long narrow channels and in thin films– Dominated by viscous stresses– Nearly unidirectional– Classic example: steady, 2D (x,y) flow in a thin

channel or narrow gap between solid objects• Pressure gradient much greater in x direction than y

direction, therefore treat P as a function of x only

x

y

dx

d

xdy

d

dx

d

P

PPPP

1

y

v

only

2x

2


Lubrication Flows

• Similar to plane Poiseuille flow except: is a function of x instead of constant and vx=vx(x,y) instead of vx=vx(y) only

• Lubrication approximation also assumes:

• Additional information from continuity equation

– Boundary conditions might be• Pressure at two points x0 and x1 or vx or vxy specified at two values of y

dx

dP

0dy

dv

dx

dv yx

2

2

2

2

2

2

2

2

y

v

x

vv

y

v

x

vv

y

v

x

v

xxy

xxx

xx


Limiting Cases

• Other important limiting case, NRe>>1

– What are these called?• Nonviscous or Stokes flows

• Reduces to

00

Re0

2000

*

Re

****

*

,,

111

vLN

vN

L

vtN

divNNtN

Ruo

St

RuSt

P

P vvvv

00

Re0

2000

****

*

,,

11

vLN

vN

L

vtN

NtN

Ruo

St

RuSt

P

Pvvv


Lubrication Flows

• This relates to your textbook derivation in the following way:

b1.7.4Eq.0

ly,Additional

ook.your textbin a1.7.4Eq.,y

v

yy

v

constant, is viscosityBecausey

v

Slide Previous From1

y

v

x2

x2

2x

2

2x

2

dy

d

dx

d

dx

d

dx

d

P

P

P

P


Lubrication Flows

• Following the derivation, our B.C.s:

4.7.2b Eq.v

4.7.2a Eq.00v

x

x

hyU

y

h

yUhyy

dx

d

dx

d

dx

d

P

PP

2

1v

4.5.3Section derivation theFrom

1v

yyy

v

y

x

xx


Lubrication Flows

• Now we find the flowrate:

212262

1

2

1

2

1

2

1

v

323

00

00

0

0 xx

Uh

dx

dhh

h

Uh

dx

d

dyyh

Udyhyy

dx

d

dyh

yUdyhyy

dx

d

dyh

yUhyy

dx

d

dyQ

xhxh

xhxh

xh

xh

PP

P

P

P


Introduction to Liebnitz

• From your Appendix A.1.H, we see how to differentiate an integral.

• Our continuity equation is:

dt

tdataf

dt

tdbtbfdx

dt

txdfdxtxf

dt

d tb

ta

tb

ta,,

,,

integrate now We0

,v

flow, film thin a is thisBecause0vv

x

yx

dx

yxd

dy

d

dx

d


Introduction to Liebnitz

dx

dh

dx

d

dx

dhU

dx

dhU

dx

dh

dx

d

dx

dhU

dx

dh

dx

d

dx

dhU

dx

dhU

dx

dh

dx

d

dx

dhU

Uh

dx

dh

dx

d

dx

dhU

dx

ddx

dhhxdy

dx

ddx

dhhx

dx

dxdy

dx

ddy

dx

d

xh

xhxh

P

PP

PP

3

33

33

x

x0 x

xx0 x0

x

16

2

12

12

12

212

10

2122120

Qfor result previousour in put weNow,Q

0

,vv0

,v0

0,vvv

0

There is a problem in the textbook derivation, that becomes apparent if we are to apply Liebnitz accurately. Because the terms on the RHS are dx, the LHS must be dy. This also is required in order to make the Q substitution.


Example 4.6 and Problem 4.11

• We analyze our lubrication flow result on a simple geometry. This is a simplification of synovial fluid lubrication between joints.

L x to0 xfrom Valid,

:as described becan profileOur

211

x

L

hhhxh


Lubrication – Sliding Surface

• Previously established,

L

hh

h

hhhh

hh

U

xL

hhhxh

dx

dh

dx

d

dx

dhU

21

212

21

211

3

where

6

:obtain weg,Integratin

h(x),for in ngSubstituti1

6

aPxP

P



• We solve for the forces in the vertical and horizontal directions. We change the integration variable to h instead of x.

dhh

hhhh

hh

UW

dhW

dxW

h

h

h

h

L

2

1

2

1

212

212

12

N

2

12

02

12

N

6

1F

slide previous thefrom resultsour ngSubstituti

1

1F

aPxP

aPxP



1

2

22

12

212

212

12

N

where

1

12ln

6

1

1

6

1F

2

1

h

h

UW

dhh

hhhh

hh

UW h

h

• With W being the depth into the page for the lubrication flow, the normal force is found by:



• Now, we can plug in some numbers:

• How large must sliding velocities be to support a normal component of weight of 1000 N for a fluid with a viscocity μ = 1 centipoise = 10-3 Pa-s (water) vs. a fluid with viscosity μ = 10 poise = 1 Pa-s (viscous oil).

5 cm 0.004 0.8333L W L



• From the results above:

• First, the low viscosity fluid

cEq

UWFN 10.7.4.

1

12ln

6

1

12

2

12

s

mU

m

sNN

mUm

sN

N

831,105000504.075.189999.01000

18333.0

18333.028333.0ln

004.0

05.106

004.01

11000

2

23

2

12



• Second, the high viscosity fluid:

• It becomes readily apparent that the change in viscosity has a great effect upon the feasibility of our system.

s

mU

m

sNN

mUm

sN

N

8.105000504.0187509999.01000

18333.0

18333.028333.0ln

004.0

05.16

004.01

11000

2

2

2

12



• Discussion:– Do you expect this to be a different reaction if

a different fluid is used?– Let’s perform an experiment.

• Corn Starch• Water• Shear-Thickening Fluid


Announcements/Reminders

• HW 4 has been posted on blackboard.– It is for 2 weeks, so don’t panic!– Extra office hours have been worked in for the

homework AND the exam preparation.

• HW 3 due this week (Friday)

• Office hours 1 hours shorter today (I have a meeting after lunchtime).

• We DO have tutorial lab tonight.


• QUESTIONS?

Documents

Louisiana Tech University Ruston, LA 71272 Lubrication/Thin Film & Peristaltic Flows Juan M. Lopez Lecture 10 BIEN 501 Wednesday, March 28, 2007