hw_5 - ME152A
Homework
#5
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Unformatted text preview: ME152A
Homework
#5
 Due
FRIDAY,
NOVEMBER
14th,
2008
 
 Question
#1:
 In
this
problem,
assume
gravity
acts
downwards
in
the
y‐direction
and
x
is
the
 horizontal
direction.
List
all
other
assumptions
and
SHOW
ALL
WORK.

 You
are
on
the
planet
Zoezeb,
similar
to
Zumeratron
from
the
midterm
exam,
but
 oceans
are
filled
with
zoe‐water
and
the
atmosphere
is
zoe‐air.
The
surface
of
the
 planet
(h
=
0)
is
known
as
zoe‐level
(as
opposed
to
sea‐level)
Properties
of
these
 fluids
are
below:
 zoe‐water:
 
 
 
 zoe‐air:
 ρ
=
100
kg/m3

 ν
=
0.01
m2/s
 p0
=
100003
N/m2


 ρ
=
p/N0

 where:
p
=
zoe‐
atmospheric
pressure
(N/m2)
 
 
 p0
=
zoe‐atmospheric
pressure
at
zoe‐level
 N0
=
12.5
(N.m/kg)
(constant)
 
 The
acceleration
of
gravity
on
this
planet
is
the
same
as
earth
(and
acts
in
the
same
 direction,
g0
=
‐
9.8
m/s2.
However
additionally
there
is
a
linear
electric
field
on
the
 planet
that
acts
IN
THE
SAME
DIRECTION
AS
GRAVITY:
 
 E
=
‐Ky+Eo
 
 
 where:
K
=
0.23
V/m2
 
 
 
 
 
 
 y
=
height
above
zoe‐level
 
 
 
 
 
 
 Eo
=
electric
field
at
zoe‐level
=
‐1
V/m
 Note
that
the
body
force
per
unit
volume
that
attributed
to
an
electric
field
is
ρeE,
 where
ρe
is
the
volumetric
charge
density,
C/m3.
In
zoe‐water,
ρe,
is
10
C/m3,
and
in
 zoe‐air,
ρe
is
0.05
C/m3.

 a)
Develop
an
expression
for
the
pressure
variation
as
a
function
of
y,
p(y),
in
zoe‐ air.
Express
the
answer
in
terms
of
N0,
g0,
K,
ν, ρe,
E0
and
p0.
Do
not
substitute
 numerical
values
for
constants.
 b)
Develop
an
expression
for
the
hydrostatic
pressure
variation
in
zoe‐water
as
a
 function
of
y,
p(y)
in
terms
of
N0,
g0,
K,
ρ, ρe,
E0, ν
and
p0.

What
is
the
pressure
at
h
=
 5
km
below
zoe‐level?
 c)
Solve
for
the
pressure
distribution,
p(y)
in
terms
of
N0,
g0,
K, ρe,
E0,
ρ, ν
and
p0,
in
 zoe‐water
if
the
velocity
field
is
 
 
 
 
 
 









 d)
A
tank
filled
with
zoe‐water
at
zoe‐level
is
subject
to
constant
linear
acceleration
 ax
=
5
m/s
in
zoe‐air.
Calculate
the
height
of
zoe‐water
in
the
tank
as
a
function
of
x,
 when
the
15
m
radius
tank
filled
with
zoe‐water
of
H
=
2
m,
is
subject
to
the
 constant
acceleration
ax.
Assume
that
the
electric
field
does
not
affect
the
velocity
 distribution
in
the
tank.
 
 
 Question
#2:
 Derive
the
Navier‐Stokes
equations
using
the
differential
control
volume
approach
 that
we
used
in
class
to
derive
linear
motion,
deformation,
angular
deformation,
and
 conservation
of
mass.
Assume
the
flow
is
incompressible
and
that
the
forces
that
are
 acting
on
the
element
are
pressure,
shear
stress,
and
gravity.
 
 Question
#3:
 A
viscous,
incompressible
fluid
flows
in
the
horizontal
direction
in
a
circular
tube
 radius
R
due
to
a
pressure
gradient,
dp/dx.
Determine,
by
the
use
of
the
Navier‐ Stokes
equations,
an
expression
from
the
velocity
distribution
in
the
x‐direction,
u,
 as
a
function
of
r,
u(r).
(HINT:
Use
cylindrical
coordinates).

Next,
determine
the
 average
velocity
and
the
corresponding
average
flow
rate.
Express
you
answer
in
 terms
of
dp/dx,
the
coefficient
of
dynamic
viscosity
µ,
and
the
channel
radius,
R.
 
 Question
4:
 Parallel
flow
in
a
two‐dimensional
microfluidic
channel
shown
below
is
induced
by
 pulling
two
thin
plates
through
a
liquid
shown.
The
plates
are
each
pulled
with
a
 constant
force
F
and
eventually
attain
a
velocity
V0.
The
liquid
has
a
dynamic
 viscosity
µ
and
the
area
of
plates
is
A.
Apply
the
Navier‐Stokes
equations
to
find
a
 relation
between
the
plate
velocity
V
and
the
channel
flow
rate
Q
attained
for
given
 values
of
a,
b,
µ,
F,
and
A.
 
 
 
 Question
5:
 A
long,
thin,
cylindrical
tube
connects
two
large
reservoirs
as
shown
below.
The
 walls
of
the
tube
are
negatively
charged
and
the
liquid
in
the
tube
has
a
uniform
net
 positive
charge
density
,
ρe,
(C/m3).
A
potential
difference
V
is
applied
along
the
 length
L
of
the
tube
using
two
electrodes.
 
 
 a.

Express
the
Force
per
unit
volume
(body
force)
in
terms
of
ρe,
L
and
V
 b.

Assuming
zero
pressure
gradient
in
the
channel,
find
the
velocity
profile
in
 the
tube
as
a
function
of
the
parameters
shown
(assume
the
no‐slip
 conditions
holds)
 c.
After
some
time,
the
electric‐field‐induced
flow
generates
a
pressure
 difference
along
the
channel
as
the
liquid
levels
in
the
two
tanks
change,
 thereby
causing
backflow
due
to
pressure.
Find
the
maximum
pressure
that
 can
exist.
(Hint:
How
much
backflow
can
there
possibly
be?
This
would
be
 the
point
of
maximum
pressure).
 
 ...
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