Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 1, pp. 301-319, Warsaw 2012

50th anniversary of JTAM

ON COMBINED SPATIAL AND TEMPORAL INSTABILITIES

OF ELECTRICALLY DRIVEN JETS WITH CONSTANT OR

VARIABLE APPLIED FIELD

Saulo Orizaga

Daniel N. Riahi

University of Texas-Pan American, Department of Mathematics, Edinburg, USA

e-mail: driahi@utpa.edu

We investigate the problem of combined spatial and temporal instabili-
ties of electrically driven viscous jets with finite electrical conductivity
in the presence of either constant or variable applied electric field.Ama-
thematical model leads to a lengthy equation for the unknown spatial
growth rate and temporal growth rate of the disturbances.This equation
is solved numerically usingNewton’smethod.We investigated two cases
of water jets and glycerol jets. For water jets and in the case of either
constant or variable applied field, we found two newmodes of instabili-
ties which grow simultaneously in time and space and lead to significant
reduction in the jet radius.However, in the case of glycerol jets, we found
two new modes of instabilities in the presence of constant applied field
but only onemode of instability in the presence of variable applied field.
For the glycerol jets, the combined temporal and spatial instabilities are
less stronger and lead to an increase in the jet radius. The instabilities
for both types of water and glycerol jets were found to be restricted
to particular domain in their wavelength and were enhanced with the
strength of the electric field.

Keywords: temporal instability, spatial instability, jet flow, electric field,
jet instability

1. Introduction

In this paper, we consider the problem of combined spatial and temporal in-
stabilities of electrically forced cylindrical jets of a viscous fluid with finite
electrical conductivity and in the presence of an externally imposed constant



302 S. Orizaga, D.N. Riahi

or variable electric field. The investigations of electrically forced jets are im-
portant particularly in applications such as those to electrospraying (Baily,
1988) and electrospinning (Hohman et al., 2001a, 2001b). Electrospinning is
a technology that uses electric fields to produce and control small fibers. The
aim is at producing non-woven materials that are unparalleled in their poro-
sity, high surface area, and the fineness and uniformity of their fibers. Such a
technology has been used in a number of areas including bioengineering, tis-
sue engineering, nano-electronics and in filtration media. Electrospraying is a
technology that uses electric field to produce and control sprays of very small
drops. The aim is at producing very small drops that are uniform in size and
are of chargedmacromolecules in the gas phase. This technology also has been
used in several areas including ink jet printing and the fuel injection process.

There have been a number of theoretical, computational and experimen-
tal studies of free shear flows and jets instabilities in the absence of electrical
effects (Michalke, 1965; Drazin and Reid, 1981; Tam and Thies, 1993; Soder-
berg, 2003;Healey, 2008) and in the presence of electrical effects (Taylor, 1969;
Reneker et al., 2000; Shkadov and Shutov, 2001; Hohman et al., 2001a, 2001b;
Li and Xia, 2004; Yu et al., 2004; Riahi, 2009; Orizaga and Riahi, 2009).

Riahi (2009) followed amodeling approach analog to that due to Hohman
et al. (2001a) and investigated analytically spatial instability of electrically
forced jets with an applied variable field but under a restricted condition of
neutral temporal stability and idealistic cases of a jet of zero electrical conduc-
tivity or a jet of infinite electrical conductivity andwith certain restrictions on
the frequency of the disturbances. He detected two spatial modes of instabili-
ty, each of which was enhanced with increasing the strength of the externally
imposed applied electric field. Thesemodeswere existed under certain restric-
ted ranges of the axial wave number of disturbances, and, in particular, did
not exist if the axial wave number was sufficiently small. Later, Orizaga and
Riahi (2009) followed amodeling approach similar to that carried out byRiahi
(2009) for the spatial instability of the electrically forced jet and again under a
neutral temporal stability condition, but they investigated numerically using
Newton’s method (Anderson et al., 1984) to determine results based on a set
of fluid parameters for the jet with finite conductivity and under a constant
or variable applied field. Orizaga and Riahi (2009) found two new modes of
spatial instabilities. One of these modes was enhanced by the strength of the
applied field, while the other mode decays with increasing the applied field.
The growth rates of both modes increased mostly with decreasing the axial
wavelength of the disturbances. For the case of a variable applied field, they
found the growth rates of the spatial instability modes to be higher than the



On combined spatial and temporal instabilities... 303

corresponding ones for the constant applied field, provided the strength of the
applied field was not too small.

In the present study, we first use a method of approach similar to that
employed in Riahi (2009) and Orizaga and Riahi (2009) to arrive at a ma-
thematical model for the realistic electrically driven viscous jets with finite
conductivity, under externally imposed constant or variable applied field and
with no restriction on the neutral stability boundary for the time variation
of the superimposed perturbations. Here we consider more realistic jet flow
instabilities, where the perturbations grow naturally and simultaneously both
in time and space, and so no idealistic marginal stability conditions in time or
in space are imposed on themathematical modeling of the jet flow instability
system. We derive the dispersion relation which relates the growth rates of
the spatial and temporal instabilities of the growing disturbances to the wave
number in the axial direction, the frequency and the non-dimensional parame-
ters of the model.We solve numerically the dispersion relation for the spatial
and temporal growth rates of the growing disturbances using Newton’s me-
thod (Anderson et al., 1984). We found a number of interesting results which
were qualitatively different from the earlier studies (Riahi, 2009; Orizaga and
Riahi, 2009) where the jet perturbations were not allowed to grow in time. In
particular, we found new modes of instability which grow simultaneously in
time and space. For the two investigated cases of glycerol and water jets, the
growth rates of themodes of instabilities were increased as the strength of the
electric fieldwas increased.For the case ofwater jet, the temporal growth rates
were increasedmostly, while the spatial growth rates were increasedmostly in
the case of glycerol jet. The spatial and temporal growth rates for either water
jet or glycerol jet decrease mostly with decreasing the axial wave number of
the disturbances. For the case of variable applied field, the growth rates of
temporal and spatial growing disturbances were found to be mostly higher
than the corresponding ones for the case of constant applied field.

2. Formulation and analysis

The present mathematical modeling of the electrically driven jets is based
on the governing electrohydrodynamic equations (Melcher and Taylor, 1969),
whichwere already described in some details inRiahi (2009) andOrizaga and
Riahi (2009) and, thus, will not be repeated here. Here we briefly describe
the physical system and the correspondingmathematical model and refer the
reader to these two references for details. A cylindrical, axisymmetric, New-



304 S. Orizaga, D.N. Riahi

tonian and incompressible fluid jet is considered to be moving axially with
passive air as the external fluid. The governing equations are used in a cy-
lindrical coordinate system with the origin at the center of section where the
jet is emitted with the axial z-axis along the axis of the jet. The modeling
was based on the approximation that the jet is long and slender in the axial
direction with the large length scale in the axial direction in comparison to
that in the radial direction, so that a perturbation expansion is used in the
small jet aspect ratio. Expanding the dependent variables in the Taylor series
in the radial variable r, using such expansions in the full governing axisym-
metric system, keeping only the leading terms and non-dimensionalizing the
resulting equations, four relatively simple model equations are obtained for
the dependent variables h, v, σ and E as functions of t and z. Here h is the
radius of the jet cross section, v is the axial velocity, σ is the surface charge,
E is the induced electric field and t is the time variable. These equations con-
tain three non-dimensional parameters, which are the conductivity parameter
K∗ =K0{ρr30/[γβ(ε̃)2]}0.5, the charge induction parameter β = ε/ε̃−1 and
the viscosity parameter ν∗ = [ν2ρ/(γr0)]

0.5. Here r0 is radius of the cross-
sectional area of the nozzle exit at z =0, γ is the surface tension, K0 is the
reference conductivity, ε/(4π) is the permittivity constant in the jet, ϑε/(4π)
is the permittivity constant in the air, ρ is the fluid density and ν is the
kinematic viscosity.
Next, the electrostatic equilibrium solution for the four model equations,

referred to as the basic state solution, is considered as the one around which
perturbations can grow in time and space. The basic state solution for each
of the dependent variables, which is designated with a subscript b, is given
below

hb =1 vb =0 σb =σ0

Eb =
Ω

K̃(z)
=Ω
(
1− 8σ0π
Ω
√
β
z
) (2.1)

where both Ω and σ0 are constant quantities. Here σ0 is the background free
charge density.We set δ=8σ0π/(Ω

√
β) to be a small parameter (δ≪ 1) and

consider a series expansion in powers of δ for all the dependent variables for
the case of variable applied field.
In this paper, we investigate the cases where the applied electric field can

be either constant (δ = 0) or variable (δ 6= 0). We consider each dependent
variable as sum of its basic state solution plus a small perturbation, which is
assumed to be oscillatory in time and in the axial variable. Thus, we write

(h,v,σ,E) = (hb,vb,σb,Eb)+(h1,v1,σ1,E1) (2.2)



On combined spatial and temporal instabilities... 305

where the perturbation quantities, which are designated with subscript 1, are
given by

(h1,v1,σ1,E1)= (h
′,v′,σ′,E′)exp[(f+iω)t+(s+ik)z] (2.3)

Here (h′,v′,σ′,E′) are small constants, i is the pure imaginary number
√
−1,

f is the temporal growth rate of the growing disturbances, ω is the frequency,
s is the spatial growth rate of the growing disturbances and k is the axial
wave number. Using (2.1)-(2.3) in the fourmodel equations, we linearize with
respect to the amplitude of perturbation, consider the lowest order in δ (for
the variable applied field case) and divide each equation by the exponential
function exp[(f+iω)t+(s+ik)z].We then obtain 4 linear algebraic equations
for the unknown constants h′, v′, σ′ and E′. To obtain non-trivial (non-zero)
values of these constants, the 4× 4 determinant of the coefficients of these
unknownsmust be zero, which yields the following dispersion relation

1

2
(k2−1−s2−2iks)(k2−s2−2iks)

[
(iω+f)A+

4πK∗√
β

]

+(k2−s2−2iks)
[
(f+iω)

(12πν∗K∗√
β
+
ω2

4π

)
+
ω2K∗A√
β

]

− (ω− if)2
{4πK∗√
β
+[(f+iω)+3ν∗(k2−s2−2iks)]A

}

+(k− is)2
[
2(f+iω)Aπσ0

(
1+
ln0.89k

2+Λ

)

+
4πK∗√
β

(
2πσ20 −Ωσ0(s+ik)

4+(ln0.89k)−1

(k− is)2
√
β

)]
=0

(2.4)

where

A=1−
2

β(k− is)2 ln 1
χ

Λ=−β(k− is)2 ln0.89k χ=
1

0.89k

It can be seen from lengthy complex equation (2.4) that the spatial growth
rate s and temporal growth rate f can be found for given values of the para-
meters, k and ω. The effects of the unknowns s and f are highly nonlinear in
(2.4), and, in particular, it turns out that the solution for s alone with f =0
in (2.4) is quite different from the corresponding solution for swith f 6=0 as
our generated data for the two cases verified such outcome.



306 S. Orizaga, D.N. Riahi

3. Results and discussion

Dispersion relation (2.4) is investigated for both variable and constant applied
field cases. For the variable applied field, where δ 6= 0, we assume that δ is
small of order about 0.1, and here both Ω and β cannot take zero value,
so that we set 0 < σ0 ¬ 0.1, which turns out to keep the maximum value
of δ of order 0.1 in the range of values for the rest of the parameters that
are considered in the present study. For the constant applied field case, we set
σ0 =0. In this Section, we describe each of the cases that we investigated and
provide and discuss the corresponding results.

In the present computation, we consider 2 types of fluids for the jet, which
can be representatives for those used in the experimental investigation for
the problem such as water and a type of glycerol. For such fluids, we set
representative values of the parameters to be K∗ = 19.60, ν∗(glycerol) =
=9.05384, ν∗(water)= 0.00608, and β=77.00.

In addition, σ0 = 0 for the constant applied field and σ0 = 0.10 for the
variable applied field.We then used (2.4) and carried outNewton’smethod to
generate data for spatial and temporal growth rates, for different values of Ω
and for both variable and constant applied field cases. The results are briefly
presented in the following two sub-sections.

3.1. The case of water jet

For Ω =1, we found two new modes of instabilities, each of which grows
in both time and space. Thesemodes of instability were present for both cases
of constant and variable applied field. The first mode of instability, referred
here to asmode I, favors relatively smaller values of the axial wave number k,
while the second mode of instability, referred to as mode II, favors relatively
larger values of k. For the case of variable applied field, the temporal growth
rate for both modes increases with the axial wave number k in the domain
0.04 < k < 0.38, while the spatial growth rate decreases with increasing k
for mode I in 0.36 < k < 0.46 and increases shapely with k for mode II in
0.70 < k < 0.74. For the case of constant applied field, the temporal growth
rate increases slowly with k for mode I in 0.10 < k < 0.28) and increases
sharply with k for mode II in 0.56 < k < 0.58, while the spatial growth
rate decreases sharply with increasing k for mode I in 0.32 < k < 0.34 and
increases with k for mode II in 0.48<k< 0.58.

For Ω = 2, we detected again the presence of two modes I and II of
instabilities which grow simultaneously in time and space and operated for



On combined spatial and temporal instabilities... 307

Ω = 1 case described in the previous paragraph. Figures 1 and 2 present
results for the growth rates s and f, respectively, versus k for 2 modes I
and II and for both constant (dashed lines) and variable (solid lines) applied
field cases. As can be seen fromFig.1, the growth rates s under the constant
applied field case decrease with increasing k for mode I in 0.30 < k < 0.38
and increase with k for mode II in 0.60<k < 0.64. For the case of variable
applied field, s decreases sharply with increasing k for mode I in 0.36<k<
< 0.38 and increase sharply with k for mode II in 0.58 < k < 0.60. It can
also be seen from Fig.1 that for the case of variable applied field, the spatial
growth rates formode I are larger than the correspondingones for the constant
applied field case, provided k is in the domain 0.36 < k < 0.38. The results
presented in Fig.2 for f versus k, indicate that for the variable applied field
case, f increases with k in 0.32<k< 0.38 formode I and in 0.50<k< 0.56
for mode II. For the constant applied field case, the temporal growth rates
increase with k in 0.42 < k < 0.46 for mode I and in 0.52 < k < 0.56
for mode II. For the constant applied field case, the temporal growth rates
f are greater than the corresponding ones for the variable applied field case,
provided the axial wave number is in the domain 0.58<k< 0.56.

Fig. 1. Spatial growth rate s versus the axial wave number k and for water jet
with Ω=2, constant applied field (dashed lines) and variable applied field

(solid lines)

For Ω =3, the two modes of combined temporal and spatial instabilities
continue to operate, where mode I favors relatively smaller values of k and
mode II favors relatively larger values of k. For the case of variable applied
field, s decreases with increasing k in 0.22<k < 0.24 and increases with k
in 0.54< k < 0.58. Similarly, for the case of constant applied field, s decre-
ases and increases, respectively, with increasing k in 0.28 < k < 0.30 and
0.60 < k < 0.68. For the case of constant applied field, s is larger than the
corresponding one for the variable applied field case if 0.28 < k < 0.30. For



308 S. Orizaga, D.N. Riahi

Fig. 2. The same as in Fig.1 but for the temporal growth rate f versus k

k in 0.54 < k < 0.58, s is larger for the case of variable applied field. The
temporal growth rates under the constant applied field case for both modes
increase with k in either 0.30 < k < 0.36 or 0.76 < k < 0.78. For the case
of variable applied field, f increases with k in either 0.12 < k < 0.24 or
0.38 < k < 0.46. For the case of constant applied field, f is larger than the
corresponding one for the variable applied field case, provided the axial wave
number is in 0.30<k< 0.36 or 0.76<k< 0.78.

For Ω = 4, we detected again two modes I and II of combined tempo-
ral and spatial instabilities which favor relatively smaller and larger values
of k, respectively. We generated data for either constant (σ0 =0) or variable
(σ0 6=0) applied field case. The growth rates s increase with k depending on
the range of value for the axial wave number. For σ0 = 0, s decreases and
increases with increasing k for k in 0.38 < k < 0.40 and 0.60 < k < 0.62,
respectively. For σ0 = 0.1, we found that s decreases and increases with in-
creasing k for k in 0.26 < k < 0.30 and 0.68 < k < 0.72, respectively. For
the case of constant applied field, the spatial growth rates are larger than
the corresponding ones for the variable applied field if k lies in the domain
0.38 < k < 0.40. The temporal growth rate was found to increase with k
for both modes I and II and under either constant or variable applied field
case.

Figures 3 and 4 present s and f versus k, respectively, for the case of
constant applied field but different values of Ω. We can see from Fig.3 that
larger values of the strength of the applied field can lead to higher values of the
spatial growth rate for particular values of the wave numbers for both modes
I and II. It can be seen fromFig.4 that for mode I, the temporal growth rate
increases with increasing Ω for k in the domain 0.16<k< 0.26, while f can
have higher value for larger value of Ω in the domain 0.68<k< 0.70 for k.



On combined spatial and temporal instabilities... 309

Fig. 3. Spatial growth rate s versus the axial wave number k for constant applied
field and for water jet with two values of Ω=1 (solid line) and 4 (dashed line)

Fig. 4. The same as in Fig.3 but for the temporal growth rate f versus k

Figures 5 and 6 present s and f versus k, respectively, for the case of
variable applied field and different values of Ω. It can be seen fromboth these
figures that the spatial or temporal growth rates for bothmodes I and II can
have higher values for larger values of Ω.

Fig. 5. Spatial growth rate s versus the axial wave number k for variable applied
field and for water jet with two values of Ω=1 (solid line), and 4 (dashed line)



310 S. Orizaga, D.N. Riahi

Fig. 6. Temporal growth rate f versus the axial wave number k for variable applied
field and for water jet with three values of Ω=1 (thick-solid line),

3 (thin-solid line) and 4 (dashed line)

Fig. 7. Perturbation quantities h1 (thin solid line), v1 (dashed line), σ1 (thick solid
line) and E1 (dotted line) versus the axial variable z for water jet with Ω=4,

σ0 =0.1, t=1, k=0.38, s=1.346087 and f =0.961981

Figure 7 presents variations of the perturbation quantities for mode I of
combined spatial and temporal instabilities versus the axial variable z and
for σ0 = 0.1, Ω = 4, k = 0.38, s = 1.3460 and f = 0.9619 but at one given
instant in time (t = 1). Due to the linear instability of the problem, we set
h′ =0.1 and determined the other perturbation constants v′, σ′ and E′ using
the procedure described in the previous Section. The real parts of the pertur-
bation quantities are used to collect the perturbation data for the instability
mode for different values of z. It can be seen from this figure thatmost of the
perturbation quantities begin to grow spatially after their generation at z=0
and their spatial growth is seen mainly as an exponential type of growth for
values of z beyond z=6 or so. For z > 6.4, the amplitudes of the perturba-
tions for v1 and h1 are sufficiently large that the present linear theory ceases
to be valid. In addition and most importantly, it is seen that the jet radius



On combined spatial and temporal instabilities... 311

perturbation can become negative for z > 5.6 and the magnitude increases
rapidly with increasing z. Using this result together with (2.1)1 and (2.2), it
implies significant reduction in the jet radius by the instabilities of mode I.
This result can have important implicationwith respect to the electrospinning
applications and making use of instabilities in a controlling manner for pro-
duction of very small- andnano-scale fibers.Figure 8 presents variations of the
perturbation quantities for mode I of combined spatial and temporal instabi-
lities versus time variable t for σ0 = 0.1, s = 1.3460, k = 0.38, f = 0.9619
and Ω=4 but at one given axial location (z=1). Again it can be seen that,
for example, for t > 5.4, the amplitude of v1 and E1 are sufficiently large
that the linear theory ceases to be valid. In addition, temporal instability of
mode I also exhibits significant reduction in the jet radius, which is enhanced
with increasing time. From the results presented in the Figs.7 and 8, which
were for the case of variable applied field, it can be concluded that combined
spatial and temporal instabilities exhibit both significant reduction in the jet
radius and a higher sensitivity of the jet flowwith respect to the disturbances
that can grow simultaneously in time and in space. We also generated simi-
lar data for the case of constant applied field and found that the spatial and
temporal instabilities are more effective and enhanced in the case of variable
applied field.

Fig. 8. Perturbation quantities h1 (thin solid line), v1 (dashed line), σ1 (thick solid
line) and E1 (dotted line) versus the axial variable t for water jet with Ω=4,

σ0 =0.1, z=1, k=0.38, s=1.3460 and f =0.9619

3.2. The case of glycerol jet

For Ω = 1, we found two modes of combined temporal and spatial in-
stabilities similar to modes I and II described earlier in the case of water
(sub-section 3.1) that were present for the case of constant applied field. For



312 S. Orizaga, D.N. Riahi

this case, the growth rates s increase with the axial wave number in either
0.14 < k < 0.26 or 0.92 < k < 0.96, while the growth rates f remain quite
small and non-monotonic for k in either 0.24<k< 0.48 or 0.50<k< 0.70.
For the case of variable applied field, only one mode of combined temporal
and spatial instability was detected, which favored relatively intermediate or
large values of k. For this case, s increases with k in 0.84<k< 0.98, while
f decreases with increasing k in 0.30<k< 0.38.

For Ω=2, we found the twomodes of instabilities continue to be present
under the case of constant applied field. For this case, s increases with k
in either 0.02 < k < 0.16 or 0.58 < k < 0.98, while f increases with k
in either 0.06 < k < 0.16 or 0.22 < k < 0.28. For the case of variable
applied field,we found that sdecreases with increasing the axial wave number
in 0.16 < k < 0.32 and increases with k in 0.38 < k < 0.98, while the
temporal growth rate decreases with increasing k in 0.04 < k < 0.28. For
the case variable applied field, the spatial growth rates are larger than the
corresponding ones for the constant applied field case, provided k is in the
domain 0.66<k< 0.98. For the case of constant applied field, the temporal
growth rates are larger than those for the variable applied field case, provided
the axial wave number lies in 0.22<k< 0.26.

Figures 9 and 10 present the growth rates s and f, respectively, versus
the wave number k for both cases of constant and variable applied field. It
can be seen from Figs.9 and 10 that the spatial growth rates are enhanced
by the presence of variable applied field, while the temporal growth rates are
enhanced by the variable field only if the axial wave number is sufficiently
small.

Fig. 9. Spatial growth rate s versus the axial wave number k for glycerol jet
with Ω=2, constant applied field (dashed lines) and variable applied

field (solid lines)



On combined spatial and temporal instabilities... 313

Fig. 10. The same as in Fig.9 but for the temporal growth rate f versus k

For Ω = 3, again there are two modes of combined temporal and spatial
instabilities under the constant applied field case. In this case, s increases
with k in either 0.16<k< 0.44 or 0.50<k< 0.98. For the case of variable
applied field, s increases with k in 0.16<k < 0.98. For the case of variable
applied field, s is larger than the corresponding one for the constant applied
field case if k lies in 0.16<k< 0.44.Thegrowth rates of temporal instabilities
under either constant or variable applied fieldwere found to remain very close
to zero of order about 10exp(−17) or smaller for all the values of k in the
domain 0<k< 1.

For Ω=4,we found that the twomodes of combined temporal and spatial
instabilities continue to exist for the case of constant applied field.The growth
rates s increase with k in either 0.24 < k < 0.44 or 0.52 < k < 0.98, while
the growth rates f increase and decrease, respectively with increasing k in
0.04 < k < 0.12 and 0.48 < k < 0.70. For the case of variable applied
field, we found that s increases with k in the domain 0.26 < k < 0.98,
but f decreases with increasing k in the domain 0.16 < k < 0.30. For the
case of variable applied field, the spatial growth rates are larger than the
corresponding ones for the temporal growth rates if k lies in 0.24<k< 0.44.
For the case of constant applied field, the temporal growth rates were found
to be larger than the corresponding ones for the variable applied field if k is
in the domain 0.06 < k < 0.12. Figures 11 and 12 present the growth rates
s and f, respectively, versus k for the case of constant applied field and for
different values of Ω. It can be seen from these figures that the spatial or
temporal growth rates can be enhanced by the strength of the applied field.
For f, we found increasing and decreasingmodes of instability along the axial
wave number.Figures 13 and14present sand f, respectively, versus k for the
case of variable applied field and for different values of Ω. As can be seen from
these figures, the spatial growth rates can be enhanced by the strength of the
applied field, while the temporal growth rates can be decreased by increasing
the strength of the field.



314 S. Orizaga, D.N. Riahi

Fig. 11. Spatial growth rate s versus the axial wave number k for glycerol jet with
constant applied field and for three values of Ω=1 (solid line), 3 (thin line) and

4 (dashed line)

Fig. 12. Temporal growth rate f versus the axial wave number k for glycerol jet
with constant applied field and for two values of Ω=1 (thick line) and

4 (dashed line)

Fig. 13. Spatial growth rate s versus the axial wave number k for glycerol jet with
variable applied field and for three values of Ω=1 (solid line), 3 (thin line) and

4 (dashed line)



On combined spatial and temporal instabilities... 315

Fig. 14. Temporal growth rate f versus the axial wave number k for glycerol jet
with variable applied field and for two values of Ω=1 (thick line) and

4 (dashed line)

Fig. 15. Perturbation quantities h1 (thin solid line), v1 (dashed line), σ1 (thick
solid line) and E1 (dotted line) versus the axial variable z for glycerol jet with

Ω=4, σ0 =0.1, t=1, k=0.16, s=1.6251 and f =0.0280

Figure 15presents variations of theperturbationquantities versus theaxial
variable z for σ0 = 0.1, Ω = 4, k = 0.16, s= 1.6251 and f = 0.0280 but at
one given instant in time (t = 1). It can be seen from this figure that most
of the perturbation quantities begin to grow spatially after their generation
at z = 0 and their spatial growth is seen mainly as an exponential type of
growth for values of z beyond z = 4.5 or so. For z > 5.5, the amplitudes
of perturbations for σ1 and h1 are sufficiently large that the present linear
theory ceases to be valid. In addition, it can be seen that in contrast to the
case ofwater jet, the glycerol jet radius is enhanced by themode of instability.
Figure 16 presents variations of the perturbations versus time t for Ω = 4,
k=0.16, σ0 =0.1, s=1.6251 and f =0.0280 but for a given location in the
axial direction (z=1). Again it can be seen that, for example, for t> 55, the



316 S. Orizaga, D.N. Riahi

amplitude of E1 is sufficiently large that the linear theory ceases to be valid.
It is also seen that the jet radius is increasedwith time. Comparing the results
presented in Figs.7, 8, 15 and 16, it can be concluded that in the presence
of variable applied field, combined spatial and temporal instabilities exhibit
higher strength and sensitivity and can lead to significant reduction in the
jet radius, while such instabilities exhibit lower strength and sensitivity and
lead to an increase in the jet radius in the case of glycerol jets. Our additional
collected data for the constant applied field and for both water and glycerol
jets lead to similar conclusions stated above.

Fig. 16. Perturbation quantities h1 (thin solid line), v1 (dashed line), σ1 (thick
solid line) and E1 (dotted line) versus the axial variable t for glycerol jet with

Ω=4, σ0 =0.1, z=1, k=0.16, s=1.6251 and f =0.0280

4. Concluding remarks

Mathematical modeling and numerical investigation of combined linear spa-
tial and temporal instabilities of electro-hydrodynamic system for electrically
forced slender viscous and finite conducting jet flows with externally imposed
either constant or variable applied field were carried out. We were able to
uncover two new modes of instabilities in the case of water jet in presence of
either constant or variable applied field. For the case of glycerol jet, we found
two modes of combined temporal and spatial instabilities under the imposed
constant applied field and only one such a mode in the case of variable ap-
plied field. For the case of variable applied field, only one mode of instability
was found for the glycerol jet. The detected temporal-spatial instabilities were
found to reduce significantly the jet radius and exhibit higher strength and



On combined spatial and temporal instabilities... 317

sensitivity in the water jet cases, in which the jet radius is enhanced and in-
stabilities are less strong with lower sensitivity in the case of glycerol jets. In
addition, it was found that the instability modes are more effective mostly in
the case of variable applied field.

In regard to the relevance of the present detectedmodes of combined tem-
poral and spatial to the available experimental results (Taylor, 1969; Hoh-
man et al., 2001a,b) and in applications, we note that for electrically for-
ced jets, Taylor (1969) documented the presence of axisymmetric instabilities
at lower values of the strength of the applied field and non-axisymmentric
whipping modes of instabilities at higher values of the field strength. Hoh-
man et al. (2001a,b) observed experimentally axisymmetric excitations and
growth in time as well as thickening blobs along the axial direction, and
instabilities appeared to grow in time as well as they move downstream.
These observations indicate the presence of axisymmetric instabilities and
also temporally and spatially growing disturbances in the jet flow. Since
non-axisymmetric whipping modes of instabilities have also been observed
to be strong leading to significant jet radius reduction and ultimately bre-
ak up the jets (Hohman et al., 2001b), it was thought that the whipping
mode is the important instability mechanism for jet radius reduction and
the corresponding advantage for controlling such instability in order to pro-
duce higher quality fibers of very small- and nano-scale size. However, the
present results for the axisymmetric modes of combined temporal and spa-
tial instabilities in water jets, that can lead to significant reduction in the
jet radius, could stimulate future investigations for ways to make use of
such results in an optimized manner to production of higher quality fiber
materials.

Further extensions of the present study that are planned by the second
author to be investigated in future, are the cases of combined spatial and
temporal instabilities due to the non-axisymmetric disturbances.As is evident
from the well known experiments by Taylor (1969) that the electrically forced
jets can be non-axisymmetric and whip for sufficiently large values of the
strengthof the electric field,we can expect that combined spatial and temporal
instabilities due to the non-axisymmetric disturbances can dominate over the
corresponding axisymmetric ones if Ω is sufficiently large.

Acknowledgement

This researchwas supported by a research grant fromUTPA-FRC.



318 S. Orizaga, D.N. Riahi

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O złożonych, przestrzenno-czasowych niestabilnościach elektrycznie

indukowanych strumieni w stałym i zmiennym polu

Streszczenie

Autorzy prezentują problem złożonych przestrzenno-czasowych niestabilności
elektrycznie indukowanych strumieni wiskotycznych o skończonej przewodności elek-
trycznej w obecności stałego albo zmiennego pola. Model matematyczny sprowadzo-
no do skomplikowanego równania, w którym niewiadomą jest czasowe i przestrzenne
tempo wzrostu zakłóceń w strumieniu. Równanie to rozwiązano numerycznie, uży-
wając metody Newtona. Zbadano dwa przypadki – dla strumienia wody i gliceryny.
Dla wody, niezależnie od tego, czy pole elektryczne jest stałe, czy zmienne, wykryto
dwie nowepostacie niestabilności rosnących równocześniew czasie i przestrzeni, które
znacznie ograniczająpromień strumienia.Wprzypadku gliceryny, dwie nowepostacie
niestabilności znaleziono tylko przy stałympolu elektrycznym.Wpolu zmiennym za-
obserwowano jedną postać.Dla gliceryny, złożoneniestabilności przestrzenno-czasowe
wykazują słabszą intensywność i zwiększają promień strumienia. Niezależnie od ro-
dzaju ośrodka, niestabilności te ograniczają się do pewnego zakresu długości falowej
i wzrastają z natężeniem pola elektrycznego.

Manuscript received August 9, 2010; accepted for print June 20, 2011