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ON TEMPORAL INSTABILITY
OF ELECTRICALLY FORCED JETS WITH NONZERO
BASIC STATE VELOCITY
Sayantan Das(SD)
Masters Student @ UT Pan Am
Mentors :Dr . D.N. Riahi & Dr. D. Bhatta
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IN OTHER WORDS…
Modeling instabilities of the Electro spinning process
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WHAT IS ELECTRO -SPINNING?
Process of producing nano-fibers
http://nano.mtu.edu/Elect
rospinning_start.html
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QUALITY NANOFIBERS
unparalleled in their porosity, high surface area
fineness and uniformity
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STABILITY?
. Here stability in terms of perturbation is considered
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IN DETAIL
Schematic Representation
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To detect and understand temporal
instabilities
Parameter regime under which
Instabilities are strong
Subsequent ways to control and eliminate such instabilities
WHY
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WE USE,ELECTRO-HYDRODYNAMIC
EQUATION mass conservation D/Dt+.u=0
(1a) momentum Du/Dt =P+.( u)+qE
(1b) charge conservation Dq/Dt+.(KE)=0
(1c) electric potential E=
(1d) D/Dt =/t+ u. - total derivative t-time variable
u-velocity vector P-pressure E -
electric field vector -electric potential q- charge
-fluid density -dynamic viscosity K-conductivity
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HOW WE MODEL?
We non- dimensionaliz
e 1(a-d)
We get four non
dimensional equation 2(a-
d)
Using perturbation technique we linearize the
PDE,s
Forming 4x4
determinant , we get the Dispersi
on relation
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THE NON DIMENSIONAL EQN
022
vhz
ht
(2a)
2
2
3
2
2
2
24
811
E
z
h
z
h
z
h
h
zz
vv
t
v
z
vh
zhh
E 22
32
(2c)
hz
Ehz
EzEb 42
)ln()( 22
2
(2d)
02
1 2
KEhz
hvz
ht
(2b)
All the constant
parameters are from
Hohman et al 2001
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PERTURBATION TECHNIQUE
We consider (h,v, , E)=
Perturbation quantities , by subscript ‘1’
Where, =()exp(
)are assumed to be small in magnitude
Basic state solution , by subscript ‘b’
Linearized w.r.t. amplitude
The complex growth rate ()
k is the axial wave number
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MATHEMATICALLY
We plug in (h,v, , E) in the non dimensional equation
We then get the coefficient of each dependent
variable for each equations
Then we form a 4X4 Determinant of the coefficients .
Then by finding a nontrivial solution , we find the
DISPERSION RELATION
DISPERSION RELATION tells us about the growth
rate &frequency of the perturbations
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OUR WORK
Hohman et.al ,2001, considered the basic state
velocity to be zero
We considered basic state velocity to be a non
zero and a constant quantity
Considering this case we derived the
DISPERSION RELATION
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DISPERSION RELATION
Where,
0322
13 TTT
K
ivkkT b
431
41
44
12
2
1 22
22
2
b
b
EKkkT
k
Kkiviv bb
863
212
442
14 2222
3
k
EiEkKkT bb
bb
KEk
vik bb
b
121
44
42
1 222
3
bb ikvK
kvk
43 222
2,),(ln,89.0
1 2 kk
We get ,
with;
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COMPUTATIONAL
We use Matlab to produce the zeroes of the
dispersion relation
In Matlab we used the inbuilt function Fzero
Fzero finds the root of a function
For growth rate we considered the real part of
For frequency we considered the imaginary part
of
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RESULTS
Growth rate v/s Wave number for K*=inf ,vb=1, and variable applied field
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Growth rate v/s Wave number for K*=0,vb=1,and variable applied field
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MORE…
Growth rate v/s Wave number for K*=19.3 ,vb=1, for variable applied field
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Growth rate v/s Wave number for K*=19.3,v*=0.3,sigmab=0.1vb=1,Eb varied
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Contd….
Primary and Secondary modes with K*=19.3,sigmab=0.1,vb=1,Eb=2.9,&v*=0
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SO…
• The variable applied field is stabilizing
• The finite values of either viscosity or conductivity are stabilizing
• There are two modes of instability for small values of the wavenumber
• All above results comply with Hohman et al with zero basic state velocity
• Hence,the growth rate in temporal instabilty is unaffected by the value of the basic
state velocity, but significant changes are already seen in spatial instability cases.
•So is our work is of no importance ? with vb being nonzero
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NO
The non zero basic state velocity significantly
affects the frequency of the perturbed state
Hence also affects the period
Which is significant for producing quality fibers
LETS SEE HOW
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FIGURE
Frequency v/s k, with K*=0,v*=0,sigmab=0.1,Eb=2.9
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Frequency v/s k, with K*=19.3,v*=0,sigmab=0.1,Eb=2.9
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HENCE
More the vb less is the frequency , hence more is
the period
Presence of conductivity increases the period
As velocity of the wave is proportional to the
negative frequency
As vb increases the velocity of the wave increases
(Obvious)
Hence production of nanofibers will be affected
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FUTURE STUDIES…
• Investigate the case for spatial instability with
non zero basic state velocity
• Investigate combined spatial and temporal
instability with non zero basic state velocity
• Investigate non-linear model
• Investigate non axisymmetric case
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THANK YOU ALL…
My special thanks to Dr Bhatta, & Dr Riahi for the support
and enthusiasm…..
Any questions or comments are gladly welcomed