Louisiana Tech University Ruston, LA 71272 Lubrication/Thin Film & Peristaltic Flows Juan M. Lopez...
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Transcript of Louisiana Tech University Ruston, LA 71272 Lubrication/Thin Film & Peristaltic Flows Juan M. Lopez...
Louisiana Tech UniversityRuston, LA 71272
Lubrication/Thin Film & Peristaltic Flows
Juan M. Lopez
Lecture 10
BIEN 501
Wednesday, March 28, 2007
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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)…
Louisiana Tech UniversityRuston, LA 71272
Reference Videos
modeltcompare.mov
Creeping Flow
eularian_Frame.mov
Boundary Layer
Lagrangian_Frame.mov
Boundary Layer
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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.
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
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
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
• 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
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
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:
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
• 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
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
• 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
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
• 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
Louisiana Tech UniversityRuston, LA 71272
Lubrication – Sliding Surface
• 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
Louisiana Tech UniversityRuston, LA 71272
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.
Louisiana Tech UniversityRuston, LA 71272
• QUESTIONS?