**Unformatted text preview: **Chapter 5 Navier-Stokes Equations Problem 5.1. Stokes second problem
Consider an infinite flat plate y = 0 subject to oscillations with velocity Uw cos !t in the x-direction.
The fluid in the half-space y > 0 is Newtonian, homogeneous, and incompressible. Assume that body
forces are negligible, that the pressure is uniform and constant, and that the flow driven by the plate is
unidirectional along x.
(a) Write down the equation and boundary conditions satisfied by vx (y, t) (x-component of the velocity
field).
(b) Assume that the solution can be written
vx (y, t) = Re[U (y)ei!t ], (5.1) where Re denotes the real part. Find an equation and boundary conditions for U (y), and solve that equation. Infer the solution for vx (y, t).
(c) Sketch the velocity profile at t = 0. What is the characteristic thickness
moving next to the wall? of the layer of fluid that is Problem 5.2. Startup of shear flow
A Newtonian incompressible fluid with constant density ⇢ and kinematic viscosity ⌫ is placed between
two parallel infinite flat plates separated by a distance h. Initially, both plates are at rest. At t = 0, the
bottom plate (at y = 0) starts to translate with a constant velocity U = U ex . Assume that the flow is
unidirectional in the x-direction, that body forces can be neglected, and that the pressure is constant and
uniform everywhere in the fluid.
(a) Determine the velocity field u = u(y, t)ex between the two plates.
(b) Sketch the velocity profile for different values of the parameter ⇡ 2 ⌫t/h2 . What flow do you recover
when ⇡ 2 ⌫t/h2 ! 1?
Problem 5.3. Circular Couette flow
Consider the steady flow of a Newtonian fluid between two infinite concentric cylinders of radii Ri (inner)
and Ro (outer) that are rotating around their common axis at angular velocities !i and !o . The density
⇢ and viscosity µ of the fluid are both constant and uniform. In cylindrical coordinates (r, ✓, z), assume
that the flow is uniform in the z- and ✓-directions, and that its direction is azimuthal in the (r, ✓) plane:
u = u✓ (r)e✓ .
(a) Write down the Navier-Stokes equations and boundary conditions for this problem using cylindrical
coordinates. Simplify as much as you can using the stated assumptions.
(b) Solve for the velocity profile u✓ (r), and determine all the integration constants. Sketch the velocity
profile.
(c) Determine the pressure p(r) up to an additive integration constant.
Problem 5.4. Flow down an inclined plane
An incompressible liquid (constant density ⇢ and viscosity µ) is flowing under the influence of gravity g
17 (seeplane
figure).
Assume
the
flow
is the
steady,
unidirectional
along
x, and
uniform thickn
inalon
the
down a very long
at that
an angle
✓ to
horizontal,
forming
asteady,
film
of
constant
! inclined
(see
figure).
Assume
that
the
flow
unidirectional
atmosphere
atmosphere
downdown
a very
long
plane
inclined
at an
angle
✓angle
toishthe
forming
a
a very
long
plane
inclined
at
an
✓ horizontal,
tonthe
fo
y p
n horizontal,
= vthe
,isand
that
the
pressure
only
depends
on
y:
p
=
p(y).
v =!
vv
(y)e
and
the
pressure
only
depends
on
y:
=
p(y).
!
(see figure). Assume
flowxthat
steady,
unidirectional
along
x,
and
uniform
in
the
x
and
z
direc
y
xthat
x ,(y)e
v Assume
= vthat
and
that
theispressure
only
depends
on
y:
p unifo
=x,p(a
!
Assume
the
is
steady,
unidirectional
along
x,
and
! (see figure).
x (y)e
x ,flow
(seeliquid
figure).
that
the
flow
steady,
unidirectional
along
"
!y: p = p(y).
v = vx (y)ex , and that thev pressure
only
depends
onpressure
nonly depends
nthe pressure
= vx=
(y)e
, and
on yny:=p0on
= p(y).
vxx(y)e
and
that
only depends
y: p = p(y).
y the
y x ,that
v(y)
y
v(y)
a
!
n
y
x
n
!
n
atmosphere
y
y !
!
liquid
!
y =yh= h
liquid
!
atmosphere
atmosphere
g liquid
g
atmosp
!
!
!
v(y)v(y)
n
atmosphere
!
!
!
atmosphere
atmosphere
!liquid liquid!
x
!
!
xh h
liquid
!
!
v(y)
!
!
!
liquid
!
!
18
5 liquid
Navier-Stokes
Equations
liquid
v(y) v(y)
g
!
x
v(y)
x
"
!
v(y)
!! "
v(y)x v(y) y =y0= 0
h
y=h
down a very long plane inclined
at an angle ✓ to the horizontal,x forming a film of constant
thickness
h g ! !g
x
x !
!
= h isatmosphere
!
y=h
!
(see figure). Assume that they flow
steady,
unidirectional
along
x,
and
uniform
in
the
x
and
z
directions:
g
atmosphere
g
g
✓
!
v = vx (y)ex , and that the pressure
only
depends on y: p = p(y). !
h
liquid
!!
h!
!
liquid
!
!
!
x
! ! !
!h
!
nn
!
!
!
!
!
y
! "! "
atmosphere
v(y)
!
atmosphere
"
v(y)
" ! "y!=!0 " y
!
!
liquid
! ! x
y =!0
(a) Justifying
the momentum equation, projected
g every step in the derivation, show that
liquid
!along the
g
atmosphere
atmosphere
n
y directions,
leads
to
the
two
equations
!
h
!
!
n
atmosphere
atmosphere
h
atmosphereatmosphere
liquid liquid
!
!
v(y)
y=h
v(y)
2
!
liquidy = h
!!
n
liquid
!!
n liquidliquid
g d! vx
✓
!
!
n n ⇢g sin ✓n + µg 2 = 0,
n ✓
v(y)
v(y)
h h dy
x
v(y)
v(y)
v(y) x
g
v(y)
g
y = h dp
y
g g
g y
⇢g gcos ✓y+= h = 0.
h
h h
✓ ✓ dy
(a) Justifying yevery
in the
that the momentum equation, projected al
h derivation, show
h
= hystep
=
h
h
x show
(a)
Justifying
every
step
in
the
derivation,
that the momentum equation, projected along
y two
= hy equations
x
directions,
leads
to
the
=
h
y=
h
✓to the
y directions,
=y hy
(b) Knowing
that the ✓leads
pressure
the
interfacey between
the liquid and the atmosphere is given b
✓attwo
✓ equations
✓
x
y
=
0
x pxa , solve
✓
constantxatmospheric pressure
for
p(y).
d2 2 vx
x
⇢g
sin
✓
+
µ
0, in the derivation, show that
y
(a)
Justifying
every
d
vx2 =step
x⇢gviscous
dytensor)
y
(a)
Justifying
step
inisthe
derivation,
show that
the
sin ✓ +stress
µevery
=to0,the
(c) The viscous
traction t =yn·⌧y (where
⌧ is the
zero
at the interface
betwee
y
2
y
directions,
leads
(a) Justifying every step in the derivation, show
that the momentum
equation,
projected
xtwo
andequations
dyalong
(a) Justifying
every
step
in
the
derivation,
show
that
the
momentum
equation,
projec
dp
yyevery
directions,
leads
to
thethe
two
equations
(a)
Justifying
every
step
in
the
derivation,
show
that
the
momentum
equation,
liquid(a)and
the
atmosphere.
Using
this
fact,
derive
a
boundary
condition
for
the
velocity
v
(y)
at
y
(a)
Justifying
every
step
in
the
derivation,
show
that
the
momentum
equatio
x
(a)
Justifying
step
in
the
derivation,
show
that
the
momentum
Justifying every
step in the
derivation,
show
that
the ✓momentum
⇢g cos
+dp = 0.equation, projected along the
y directions, leads to theWhat
two equations
y directions,
leads
to
the
twothe
equations
is
the
boundary
condition
at
y
=
0?
y
directions,
leads
to
two
equations
y
directions,
leads
to
the
two
equations
dy
d
y
directions,
leads
to
the
two
equations
(a)
Justifying
every
step
in
the
derivation,
show
that
the
mom
⇢g
cos
✓
+
=
0.
y directions, leads to the two equations
⇢g sin ✓ +d2µvx
dy
2
2
y
directions,
leads
to
the
two
equations
✓ + µ 2d
(d) Solve for the velocity vxd(y).
vx Sketch the velocity profile.
d vx d2 vd2 vx ⇢gd2sin
vx
dy
⇢g sin ✓that
+ µ the 2pressure
= 0, at the interfaced2⇢g
(5.2)
x the atmosphere
(b) Knowing
between
the
liquid
and
✓ +sin
µ sin
=sin
0,
vxsin ⇢g
⇢g
✓
+
µ
=
0,
2
⇢g
✓
+
µ
= 0, is
✓
+
µ
=
0,
dy
2
dy
(e) Calculate
the
volumetric
flow
rate
Q
(per
unit
length
in
the
z
direction),
defined
as
⇢g
sin
✓
+
µ
=
0,
2
2
2
dy
(b)
Knowing
that
the
pressure
at
the
interface
between
the
liquid
and
the
atmosphere
is
⇢g
cos
+
dp
dy
dy
constant atmospheric pressure pa , solve for p(y).
d v✓x+✓gi
dy 2
⇢g
cos
dp
dp
⇢g
sin
✓
+
µ
=
0
dp
constant atmospheric
pressure
p
,
solve
for
p(y).
Z
dp
a
2
dy
dp
⇢g costraction
✓ + t==0.n·⌧ (where ⌧ his the viscous
(5.3)
✓ +cos
=
0.
dp⇢g cosstress
(c) The viscous
is=
zero
at the=dy
interfac
⇢gtensor)
cos
✓⇢g
+cos
=+0.
✓0.
0.
⇢g
✓
+
dy
dy
⇢g
cos
✓
+
=
0.
dy
Q =fact,
vthe
dy.
dyvelocity
x (y)
dyat
(c) The
traction t =Using
n·⌧ (where
⌧ isderive
viscous
stress
is
zero
the
interface
liquid
andviscous
the atmosphere.
this
a boundary
condition
for
vbx
dy
dpbetw
(b)
Knowing
that
the tensor)
pressure
theatthe
interface
0
⇢g cos
✓+
=
0
Knowing
that
the pressure
at the
interface
between
liquid
atmosphere.
Using
fact,
derive
a
boundary
condition
for
the
velocity
v
(y)
What
is and
the the
boundary
condition
at this
y =(b)
0?
x
constant
atmospheric
pressure
pa , solve
for
p(y).
dy
(b) Knowing that the pressure at the interface
between
the
liquid
and constant
the
atmosphere
is at
given
by
the
(b) Knowing
that
the
pressure
atpressure
theatmospheric
interface
between
the
liquid
and
the
atmosph
(b)
Knowing
that
the
at
the
interface
between
the
liquid
and
the
(b)
Knowing
that
the
pressure
the
interface
between
the
liquid
pressure
pa , solve
for p(y).
What
is the
condition
y =pressure
0?thebetween
Knowing
that
the
at thethe
interface
between
the liquid
and
the at
Problem
6.5.
Wind-driven
flow
inside
aSketch
lake
Knowing
thatboundary
the(b)
pressure
theat
interface
liquid
and
the
atmosphere
is given
Solve
the
velocity
vxat(y).
velocity
profile.
constant
atmospheric
pressure
pa ,p(y).
solve
p(y).
constant atmospheric pressure(b)
pa(d)
, solve
forfor
p(y).
constant
atmospheric
pressure
p(c)
solve
for
viscous
traction
t for
= n·⌧
(where
⌧ is
thex-dire
visco
constant
atmospheric
pressure
p
solve
p(y).
a , The
a , for
Consider
a
large
lake,
over
which
wind
is
blowing
and
exerts
a
constant
shear
stress
S
in
the
constant
atmospheric
pressure
p
,
solve
for
p(y).
constant
atmospheric
pressure
p
,
solve
for
p(y).
a
(c)
The
viscous
traction
t = at
n·⌧
(where
⌧ isbetween
the viscous
(d)Calculate
Solve forthe
thevolumetric
velocity vxa(y).
Sketch
the
velocity
profile.
(b)
Knowing
that
the
pressure
the
interface
the
liquid
and
the
atmosphere.
Using
this
fact,
derive
azer
bl
(e)
flow
rate
Q
(per
unit
length
in
the
z
direction),
defined
as
(c)
The
viscous
traction
tand
=
(where
⌧ is the
viscous
stress
tensor)
is
viscous
traction
t=
n·⌧
(where
⌧=is
the steady
viscous
stress
tensor)
isderive
zero
atain
the
in
(c) The viscous traction t(see
= n·⌧
(where
⌧ is(c)
theThe
viscous
tensor)
isis
zero
at
then·⌧
between
figure
below).
The
goal
ofstress
this
problem
totraction
determine
the
flow
established
the
(c)
The
viscous
tinterface
n·⌧
(where
⌧the
is field
the
viscous
stress
tenso
liquid
the
atmosphere.
Using
this
fact,
boun
(c)
The
viscous
traction
t
=
n·⌧
(where
⌧
is
the
viscous
stress
tensor)
is
zero
a
(c) The
viscous
traction
t
=
n·⌧
(where
⌧
is
the
viscous
stress
tensor)
is
zero
at
the
interface
betwe
constant
atmospheric
pressure
pdirection),
for
p(y).
What
is
boundary
condition
ydefined
=boundary
0?condition
a , solve
(e) Calculate
theaand
volumetric
flow
rate
Q
(per
unit
the
zyfact,
liquid
and
atmosphere.
Using
this
fact,
derive
aatboundary
for
liquid
the
atmosphere.
Using
this
fact,
athis
boundary
condition
forasThe
the condi
veloc
liquid
and
the
atmosphere.
Using
derive
a0?
liquid and the atmosphere.
this
fact, derive
boundary
condition
for
the
vxin
(y)
at
=
h.
by Using
the wind.
Assume
that
the
lake
has
athe
constant
depth
hthe
before
the
wind
starts
wind
Zlength
0
What
isavelocity
the
boundary
condition
y =blowing.
hderive
liquid
and
the
atmosphere.
Using
this
fact,
derive
a
boundary
condition
for
th
liquid and the atmosphere.
Using
this
fact,
derive
boundary
condition
for
the
velocity
v
(y)
at
y
x
is theiscondition
boundary
condition
at vythe=
What
the(c)
boundary
at
y=
0?v(where
boundary
at
ySolve
=
The
viscous
traction
t0?=
n·⌧
⌧ is thethe
viscous
stress
What is the boundary condition at y = 0? What is theWhat
(d)
velocity
velocity
p
Z0?yfor
Q
=condition
x (y). Sketch
hthe
is the at
boundary
at
= xvelocity
0?(y) dy.vx (y).
What is the boundaryWhat
condition
y = 0? condition
(d) Solve
for
Sketch the velocity profi
0
liquid
and
the
atmosphere.
Using
this
fact,
derive
a
boundary
(d)
Solve
for
the
velocity
v
(y).
Sketch
the
velocity
profile.
x
Q
=
v
(y)
dy.
(d)
Solve
for
the
velocity
v
(y).
Sketch
the
velocity
profile.
(d) Solveprofile.
for the velocity vx (y).(e)Sketch
thexvelocity
profile. flow rate Q (per unit leng
xthe volumetric
(d) Solve for the velocity vx (y). Sketch the velocity
Calculate
(d) Solve for the velocity
vx (y).
the
velocity
profile.
(d) Solve
forSketch
the velocity
vCalculate
(y).boundary
Sketch
the
velocity
profile.
(e)
flow
(per unit length
0 the volumetric
What
isxlake
the
condition
at
ylength
=rate
0? Q
Problem
6.5.
Wind-driven
flow
inside
a
(e)
Calculate
the
volumetric
flow
rate
Q
(per
unit
in
the zin
direction),
(e)
Calculate
the
volumetric
flow
rate
Q
(per
unit
length
the Zz dir
(e)
Calculate
thethe
volumetric
flowdefined
rate Q as
(per unit length
in
the
z direction),
defined
as
(e) Calculate the volumetric flow
rate
Q
(per
unit
length
in
z
direction),
(e) Consider
Calculate
volumetric
flowwhich
rate
Q
(per
unit
length
in
z vdirection),
defined
Z hS inhdet
(e)
Calculate
the volumetric
flow
ratevelocity
Qthe
(per
unit
inthe
the
zasdirection),
large
lake,
over
wind
is
blowing
and
exerts
a length
constant
shear
stress
(d) Solve
for the
Sketch
velocity
profile.
Problemathe
6.5.
Wind-driven
flow
inside
a lake
x (y).
Z
Z
Q
=
v
h
Z
h
Z The goal of this problem
h the steady flow
(see
figure abelow).
determine
field
establish
Q =S
vthe
Z his to and
Consider
largehlake,
over which wind
is blowing
exerts
aQflow
constant
shear
stress
inin
Z=hrate
0x (yx
v
(y)
dy.
x
Q
=
v
(y)
dy.
(e)
Calculate
the
volumetric
Q
(per
unit
length
the
x
0
Q = h0 before
vx (y)
dy.
Q=
vxThe
(y)
dy.
(5.4)
by(see
thefigure
wind.
Assume
thatgoal
theof
lake
constant
depth
the
wind
starts
blowing.
below).
thishas
problem
is vtox (y)
determine
the
steady
flow
Qa =
dy. Q
0 field established
v0flow
dy.
x (y)
0 =
0
Problem
6.5. Wind-driven
inside
a
lake
Z
0
0the
h
by the wind. AssumeProblem
that the 6.5.
lake has Problem
a constant6.5.
depth
h0 before
starts
blowing.
The
Wind-driven
flowwind
inside
a lake
flow
inside
a lake
ProblemWind-driven
6.5.flow
Wind-driven
flow
inside
a lake
Consider
a lake
large
lake,
over
which wind
is blowing
an
Problem
6.5.
Wind-driven
inside
a
Q
=
v
(y)
dy.
Problem 5.5. HydrodynamicProblem
slip
Consider
a
large
lake,
over
which
wind
is
blowing
and
ex
x
6.5. Wind-driven
flow
inside
lake
Consider
aWind-driven
large
lake,
over
which
wind
iswind
blowing
andthis
exerts
constant
she
Problem
6.5.
flow
inside
a lake
Consider
aalarge
lake,
over
which
is blowing
andaexerts
atocons
(see
figure
below).
The
goal
of
problem
isstress
det
0
Consider
a
large
lake,
over
which
wind
is
blowing
and
exerts
a
constant
shear
S
Experiments in microfluidic devices
have
shown
that
the
no-slip
boundary
condition
can
sometimes
be
in(see
figure
below).
The
goal
of
this
problem
is
to
determ
Consider a large lake,
over
which
wind
isbelow).
blowing
and
exerts
athis
constant
shear
stress
Sconstant
insteady
thethe
x-dir
(see
figure
below).
The
goal
ofgoal
thisAssume
is to
determine
the
flow
(see
figure
The
ofproblem
problem
is to
determine
ste
Consider
a large
lake,
over
which
wind
isto
blowing
and
exerts
a aconstant
shear
by
the
wind.
that
the
lake
has
dep
(see
figure
below).
The
goal
of
this
problem
is
determine
the
steady
flow
field
est
accurate, especially when the (see
channel
walls
are made
hydrophobic
surfaces.
more
accurate
boundary
by
the
Assume
the
lake
has
a constant
depth
before
wind
figure
below).
The of
goal
of
this
isthat
toA
determine
the
steady
flow
field
established
in th
th
Problem
6.5.
Wind-driven
inside
ahdepth
lake
0the
bywind.
the problem
wind.
Assume
that
the
lake
has
constant
h0 the
before
figure
below).
goal
of
this
problem
isflow
toahdetermine
steady
flow
fis
by the(see
wind.
Assume
thata The
the lake
has
a constant
depth
the
wind
starts
blow
0 before
condition in this case is the following:
by the wind. Assume
that
the
lake
has
constant
depth
h
before
the
wind
starts
blowing.
The
win
0 a constant
largehas
lake,
over which
wind
blowing
exerts
by the wind. AssumeConsider
that thealake
depth
h0isbefore
theand
wind
sta vt = b n · rvt at the wall, (see figure below). The goal of this problem is to determine t
(5.5)
by the wind. Assume that the lake has a constant depth h0 bef where vt is the tangential component of the velocity vector, n is a unit normal vector pointing into the
fluid, and b is a given constant.
(a) What are the dimensions of b?
(b) Consider the pressure-driven flow of an incompressible homogeneous Newtonian fluid in a cylindrical microchannel of radius a (cylindrical Poiseuille flow). Solve for the velocity vx (r) in cylindrical
coordinates, using the boundary condition (5.5). < 0 a uniform3µ
and removing it6 through
the other,
cr
when ↵the 1plates
<0
Navier-Stokes
Equations
of the plates, and y-direction
aligned
⇢gH 3 to the 2p
Q = normal
⇣
⌘
3
(x, y, z) with originwhere
on the
centerplane
of
↵
: the entrance
Q = ⇢gH
(6.27)
(1 ↵) t
8
=
⌧0 /⇢gH. say
component
u↵
constant,
U3µ
. Show
that
y is
:
(1 <↵)02 1 +
when ↵of
< the
1velocity
when
↵
1y-direction
plates,
and
aligned
normal
to
the
plate
3µ
2
⇣
⌘
3
⇢gH
↵
Q=
Problem
6.11. Flow in a (6.27)
channe where
2 the
component
u⌧y0 /⇢gH.
is constant,
say(pU0 . porous
Show
that
:
(1 ↵)2 1 +velocity
when
↵ < ↵1 =An
pL )H
incompressible Newtonian
fluid of1 de
3µ
2
ux (y) =
where ↵ = ⌧0 /⇢gH.
Problem length
6.11. Flow
in a porous
with
inje
L, separation
2Hchannel
⌧µL
L, 2and
infinite
Re
(pthe
pLof)H
1 ⇢plate
y
0 fluid
An incompressible
Newtonian
density
over
the
length
of
plates.
The
two
where
↵
=
⌧
/⇢gH.
0
Problem 6.11. Flow in a porous channel with injection/suction
ux (y)
=⌧ L, and infinite width.and
length
L,
separation
2H
Th
the
plates
and
removing
it
through
the
oth
µL
Re
H
wheretwo
Reparallel
= ⇢U H/µ
is the19
crossflow Reynolds nu
5 An
Navier-Stokes
Equations
incompressible
Newtonian
fluid
density channel
⇢ and viscosity
µ flows beween
plates
of
over the length
the
plates.
The
plates
(x,flat
y,of
z)
with
origin
ontwo
thefor
centerplane
Problem
6.11. Flow
in of
a porous
with injection/suction
plates.
Sketch
the
axial
velocity
profile
Reare⌧porou
1,ofR
length L, separation
2H
⌧
L,
and
infinite
width.
The
flow
is
induced
by
a
pressure
difference
(p
p
)
the
plates
and
removing
it
through
the
other,
unifor
0
L
of
the
plates,
and
y-direction
aligned
nor
An incompressible Newtonian fluid of density ⇢ and viscosity
µRe
flows
beween
two
parallel
flat plates
of anumb
where
=
⇢U
H/µ
is
the
crossflow
Reynolds
(c)over
What
the value
v (a) ofThe
the velocity
at theporous:
wall? Sketch
the velocity
profile,
andy,
give
an origin
interpreta(x,
z)
with
on of
the centerplane
of)the say
entran
theislength
of the
injecting
of
the
same
fluid
through
one
velocity
component
u
is
constant,
U
y
length
L,xplates.
separationtwo
2H plates
⌧ L,are
and infiniteby
width.
Themore
flow
is
induced
by
a
pressure
difference
(p
p
Problem
6.12.
Pipevelocity
flow of two
immiscible
0 ReL⌧liquids
plates.
Sketch
theplates,
axial
profile
for
ofCartesian
the
and y-direction
aligned
normal1,toRe
the ⇠
tion
the and
constant
b. it through the other, a uniform crossflow is generated.
thefor
plates
removing
Use
coordinates
over
the length
of the plates. The two plates are porous: by injecting
more
of
the
same
fluid
through
one
of
Consider
the incompressible
of two
velocity
component uy isflow
constant,
sayimmiscible
U . Show
(p0 thatp
(x, y, z) with origin
on theremoving
centerplane
of the entrance
to athe
plates, crossflow
x-direction
aligned
with
the
length
ux (y)
= pressu
plates
it through
the other,
uniform
generated.
Use
Cartesian
coordinates
Problem
Pipe
flow
of
two immiscible
liquids
dricalis6.12.
pipe
of
radius
R,
driven
by
a constant
Problem 5.6. the
Flow
in aand
porous channel
with injection/suction
µL
2
of the plates,(x,
and
aligned
normal
to the plates.
crossflow
means
that the transverse
y, y-direction
z) with origin
centerplane
of the Uniform
entrance
the(see
plates,
aligned
the
length
(p
pL )H
1
thex-direction
incompressible
flow
of two
immiscible
0
figure
below).
1ofuwith
(with
viscosity
µ1 ) hom
occ
An incompressible
Newtonian
fluidonofthe
density
⇢ and viscosity
µ flowstoConsider
beween
two
parallel
flatLiquid
plates
x (y) =
of the plates,
and y-direction
aligned
normal
the
plates.
cr...

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- Spring '16
- rubenanaya
- Fluid Dynamics, Velocity, crossflow Reynolds