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6.642 Continuum Electromechanics
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Fall 2008
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.642 Continuum Electromechanics
Problem Set #8
Issued: 11/25/08
Fall Term 2008
Due: 12/09/08
Suggested Reading: Sections 8.17, 8.18
Final Exam: Dec. 9, 2004, 3-5PM. Open book, open notes. Focus on fluid interfacial stability.
Problem 1
Prob. 8.18.2 (Melcher, Continuum Electromechanics)
Corrections:
Part (b)
Part (c)

 c
2


hz  0
0 
Problem 2
Prob. 8.18.3 (Melcher)
k 2 kz2 I4 I1 I3

I1 I 2
2
Correction:
2
1
Prob. 8.18.1 (continued)
where, consistent with the usage in Section 8.14, E o is the equilibrium electric field evaluated
at the interface between layers.
(b) Show that the dispersion equation for the layer model, based on the results of Section 8.14,
takes the normalized form
1
8k
1
15
(c) Using k = 1, Dpm = 0, VoJVoJ = i1,Dq /jDqel = 1 and S = 1, compare the prediction of the first
eigenfrequency to the first resonance frequency predicted in the weak-gradient approximation and
to the "exact" result shown in Fig. 8.18.2a. Compare the analytical expression to that for the
weak-gradient imposed field approximation in the long-wave limit. Should it be expected that
the layer approximation would agree with numerical results for very short wavelengths?
(d) How should the model be refined to include the second mode in the prediction?
Prob. 8.18.2
A layer of magnetizable liquid is in static equilibrium, with mass density and permeability having vertical distributions ps(x) and Is(x) (Fig. P18.8.2).
The equilibrium magnetic field
Hs(x) is assumed to also have a weak gradient in the x direction, even though such a field is not irrotational. (For example, this gradient represents fields in the cylindrical annulus between concentric
pole faces, where the poles have radii large compared to the annulus depth k. The gradient in H is a
quasi-one-dimensional model for the circular geometry.) Assume that the fluid is perfectly insulating
and inviscid.
(a) Show that the perturbation equations can be reduced to
k2
D(PCDhR)
s )
-
s z
N
2
D(p sDv x) - k (
where k2 = 2
s
W
sx
k2H DM
)v + j
s2x
sH sD1
h = 0
z
4+
+ k2
z , H = Hsi z + h and
N
2
1
= -g Dp s + - DsDH
(b) As an example, assume that the profiles are Ps = Pm exp8x, ]s
that solutions are a linear combination of expyx, where
=
Pm exp~x, H s = constant.
k2 k2 H212
2
YT
-8
[
c+=
;
)2+k2+ai b1
b
2mak=
+
2
Show
1/2
PM
rlvfW,7
a = gSk2/2 2
(c) Assume that boundary conditions are v ( ) = 0, h ( ),
x 0z
2b
sinh cE sinh c
that eigenfrequencies
and
and show that the eigenvalue equation is
V-
i= 0
are
-
--- M
1L-~ioo
_
_
M--
and that eigenfrequencies are
2
Wn
=
k2k2H2
zasm
4
n m
2
gSk 2
K
2
n
2 ,-
2
2
n2
Kn = ( )
n
R
+(--8)
+ k
2
(d) Discuss the stabilizing effect of the magnetic field on
the bulk Rayleigh-Taylor instability.
Ps(x)
H,x)
2
-p
----
-----
----
----
----
Fig. P8.18.2
Fig. P8.18.2
8.77
Courtesy of MIT Press. Used
with permission.
Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. ISBN:
9780262131650.
Problems for Chap. 8
Fig. 8.18.2 (continued)
(e) Discuss the analogous electric coupling with
ps
s and H
-
s
configuration.
E
and describe the analogous physical
s
Prob. 8.18.3
As a continuation of Problem 8.18.2, prove that the principle of exchange of stabilities
holds, and specifically that the eigenfrequencies are given by
3
~"J~,~41
k 2 z 1141 2+
2
2
2
e
=
I1iZ
11i2
where
91
I1 =
(IDh
2
2
+ k
h
)dx ;
(ps ID
I2
+ k2Pv1 )dx
0
0
£
13
k 2Niv x 2 dx ;
=f
0
14=
H D
A*
^
vx h dx
0
Courtesy of MIT Press. Used with permission.
Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. ISBN: 9780262131650.
Problems for Chap. 8
8.78