U.S. patent application number 11/341096 was filed with the patent office on 2006-10-19 for system and method of obtaining entrained cylindrical fluid flow.
Invention is credited to Alfonso Ganan Calvo, Piotr Garstecki, Eric J. Ginsburg, Miguel-Angel Herrada-Gutierrez, Jose M. Lopez-Herrera, Pascual Riesco-Chueca, Wendy W. Zhang.
Application Number | 20060234051 11/341096 |
Document ID | / |
Family ID | 37108826 |
Filed Date | 2006-10-19 |
United States Patent
Application |
20060234051 |
Kind Code |
A1 |
Zhang; Wendy W. ; et
al. |
October 19, 2006 |
System and method of obtaining entrained cylindrical fluid flow
Abstract
A method and system for entraining fluids is provided. The
method and system may be used to create filaments. As one example,
a filament may be produced by entraining a first fluid within a
second fluid by flowing a third fluid, the flowing third fluid at
least partly constraining the second fluid in at least one
dimension. As another example, a filament may be produced by
entraining a first fluid within a second fluid based on a model of
a dynamic response of the first and second fluids as functions of
densities of the first and second fluids, viscosities of the first
and second fluids, Reynolds numbers of the first and second fluids,
and Weber numbers of the first and second fluids.
Inventors: |
Zhang; Wendy W.; (Chicago,
IL) ; Ginsburg; Eric J.; (Chicago, IL) ;
Calvo; Alfonso Ganan; (Seville, ES) ; Lopez-Herrera;
Jose M.; (Seville, ES) ; Riesco-Chueca; Pascual;
(Seville, ES) ; Herrada-Gutierrez; Miguel-Angel;
(Seville, ES) ; Garstecki; Piotr; (Warsaw,
PL) |
Correspondence
Address: |
BRINKS HOFER GILSON & LIONE
P.O. BOX 10395
CHICAGO
IL
60610
US
|
Family ID: |
37108826 |
Appl. No.: |
11/341096 |
Filed: |
January 27, 2006 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60648102 |
Jan 27, 2005 |
|
|
|
Current U.S.
Class: |
428/373 ;
428/374 |
Current CPC
Class: |
D01D 5/098 20130101;
Y10T 428/2929 20150115; Y10T 428/2931 20150115 |
Class at
Publication: |
428/373 ;
428/374 |
International
Class: |
D02G 3/00 20060101
D02G003/00 |
Claims
1. A filament being produced by: entraining a first fluid within a
second fluid by flowing a third fluid, the flowing third fluid at
least partly constraining the second fluid in at least one
dimension.
2. The filament of claim 1, further comprising solidifying at least
a part of the second fluid to form a fiber.
3. The filament of claim 2, further comprising solidifying at least
a part of the first fluid to form the fiber.
4. The filament of claim 1, wherein entraining a first fluid within
a second fluid is based only on fluid properties of the first fluid
and the second fluid
5. The filament of claim 4, wherein the fluid properties are
viscosity of the first fluid and the second fluid; and wherein the
second fluid is more viscous than the first fluid.
6. The filament of claim 1, wherein the first fluid comprises a
liquid or a gas that is immiscible with the second fluid.
7. The filament of claim 1, wherein a plurality of fluids are
entrained within the second fluid.
8. The filament of claim 7, wherein the third fluid flows at least
partly by co-electrospinning.
9. The filament of claim 7, wherein the third fluid flows at least
partly by pressure applied to the third fluid.
10. The filament of claim 7, wherein the second fluid is
axisymmetrically constrained by the flowing of the third fluid to
surround the second fluid.
11. The filament of claim 7, wherein the first fluid flows from a
first nozzle and the second fluid flows from a second nozzle; and
wherein the first nozzle is concentric with the second nozzle.
12. The filament of claim 7, wherein the second fluid flows to
entrain the first fluid; and wherein the flow of the second fluid
is due more to drawing of the second fluid by the flowing of the
third fluid than to pressure applied to the second fluid.
13. The filament of claim 12, wherein the entraining of the first
fluid within the second fluid is due more to drawing of the first
fluid by the flowing of the second fluid than to pressure applied
to the first fluid.
14. A filament being produced by: entraining a first fluid within a
second fluid based on a model of a dynamic response of the first
and second fluids as functions of densities of the first and second
fluids, viscosities of the first and second fluids, Reynolds
numbers of the first and second fluids, and Weber numbers of the
first and second fluids.
15. The filament of claim 14, wherein the model of the dynamic
response defines a phase space of a stable entraining of the first
fluid within the second fluid; and wherein values of the Reynolds
numbers of the first and second fluid and Weber numbers of the
first and second fluid are selected to be in the phase space of the
stable entraining of the first fluid within the second fluid.
16. The filament of claim 15, wherein the entraining of the first
fluid within the second fluid is performed by a system that
controls velocity of the first fluid and second fluid; and wherein
the velocity of the first fluid and the second fluid are selected
such that the model is in the phase space of the stable entraining
of the first fluid within the second fluid.
17. The filament of claim 15, wherein the model is derived
numerically.
18. The filament of claim 15, wherein the model is derived
analytically.
19. The filament of claim 18, wherein the model is a function of:
the ratio of the densities of the first and second fluid, the ratio
of the viscosities of the first and second fluid, the Reynolds
number of the first fluid Re.sub.1=.rho..sub.1VR.sub.1/.mu..sub.1
with .rho..sub.1 being the density of the first fluid, .mu..sub.1
being the viscosity of the first fluid, R.sub.1 being the radius of
a cylindrical jet through which the first fluid flows, and Vbeing
the uniform velocity of the first and second fluids relative to an
observer, the Reynolds number of the second fluid
Re.sub.2=.rho..sub.2VR.sub.2/.mu..sub.2 with .rho..sub.2 being the
density of the first fluid, .mu..sub.2 being the viscosity of the
first fluid, and R.sub.2 being the radius of a cylindrical jet
through which the second fluid flows, the Weber number of the first
fluid We.sub.1=.rho..sub.1V.sup.2R.sub.1/.sigma. where .sigma.
being the surface tension, and the Weber number of the second fluid
We.sub.2=.rho..sub.2V.sup.2R.sub.2/.sigma..
20 The filament of claim 14, further comprising solidifying at
least a part of the second fluid to form a fiber.
21. The filament of claim 14, wherein operating conditions are
selected in order to be in a specific phase space within the model
so that the flow of the first fluid is continuous for a
predetermined time.
22. The filament of claim 21, wherein the operating condition
comprises pressure; and wherein the pressure to at least one of the
first and second fluids is dynamically adjusted to maintain a shape
of an interface between the first fluid and the second fluid for
the predetermined time.
Description
REFERENCE TO RELATED APPLICATION
[0001] This application claims the benefit of U.S. Provisional
Application No. 60/648,102, filed Jan. 27, 2005. U.S. Provisional
Application No. 60/648,102, filed Jan. 27, 2005 is hereby
incorporated by reference herein in its entirety.
FIELD
[0002] The present application relates to a method and apparatus
for obtaining entrained fluid flow and more specifically, a method
and apparatus for obtaining a stable entrained cylindrical fluid
flow.
BACKGROUND
[0003] Formation of bubbles or filaments in a fluid, such as a
liquid or gas, may be of interest. For example, bubbles may be used
in a variety of scientific fields, such as in the fields of
biomedicine (e.g., in diagnosis, as potential gene therapy vectors,
to convey tiny amounts of therapeutic gases in the bloodstream
without the risk of embolism, etc.), advanced physics studies
(bubble sonoluminiscence, damping agents in neutron spallation
sources, etc.), chemical engineering, and as a strong allied in
environmental protection (e.g., microbubble drag reduction in
marine transport, dissolved air flotation water depuration
techniques, etc.). As another example, filaments may be used to
create fibers or the like.
[0004] Forming the bubbles or filaments in the fluid may be
difficult. The interface between the bubbles or filaments and the
surrounding fluid may be complicated, thereby making it difficult
to control a system to make stable and acceptable bubbles or
filaments. Therefore, there is need for a system that enables the
creation of stable and acceptable bubbles or filaments.
BRIEF SUMMARY
[0005] In one aspect of the invention, a filament is produced by
entraining a first fluid within a second fluid by flowing a third
fluid, the flowing third fluid at least partly constraining the
second fluid in at least one dimension. The third fluid may flow
through a variety of methods, such as by applying pressure to the
fluid flow. The third fluid flow may constrain the second fluid
axisymmetrically. Further, the third fluid may apply a force on the
second fluid to at least partly drag or withdraw the second fluid
from its nozzle and/or reservoir. In turn, the second fluid may
apply a force on the first fluid to at least partly drag or
withdraw the first fluid from its nozzle and/or reservoir. In this
manner, the force used to entrain the first fluid may be more due
to dragging or withdrawing rather than on a direct force on the
first fluid (such as increasing pressure applied to the reservoir
housing the first fluid). Specifically, if the force used to
entrain the first fluid may be due in part, or entirely, on the
fluid properties of the first and second liquid. For example, the
first fluid may be entrained within the second fluid based on the
viscosities of the first and second fluids. When a filament is
produced, a part of the second fluid and a part of the first fluid
(such as if the first fluid is a liquid) may be solidified in order
to form a fiber. When the first fluid is withdrawn, in order to
maintain a constant pressure, the first fluid's reservoir may need
to be filled with additional first fluid. Alternatively, instead of
applying no pressure on one or more of the fluids (such as moving
the fluid only by dragging or withdrawal), one or more of the
fluids may be "sucked upward," so that an additional amount of a
dragging force is required to move the fluid in the direction
opposite to the force sucking upward.
[0006] In another aspect of the invention, a filament being
produced by entraining a first fluid within a second fluid based on
a model of a dynamic response of the first and second fluids as
functions of densities of the first and second fluids, viscosities
of the first and second fluids, Reynolds numbers of the first and
second fluids, and Weber numbers of the first and second fluids.
The model of the dynamic response may define a phase space of a
stable entraining of the first fluid within the second fluid.
Further, the system may be controlled such that values of the
Reynolds numbers of the first and second fluid and Weber numbers of
the first and second fluid may be selected to be in the phase space
of the stable entraining of the first fluid within the second
fluid. The model may be derived numerically and/or analytically.
When a filament is produced, a part of the second fluid and a part
of the first fluid (such as if the first fluid is a liquid) may be
solidified in order to form a fiber.
[0007] The foregoing summary has been provided only by way of
introduction. Nothing in this section should be taken as a
limitation on the following claims, which define the scope of the
invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] FIG. 1 is an example of a block diagram of the system for
controlling fluid flow.
[0009] FIG. 2 is an example of a partial cross-section depicting
interaction of two flowing fluids.
[0010] FIG. 3 is an example of a depiction of three flowing
fluids.
[0011] FIG. 4 is another example of a cross-section depicting
interaction of two flowing fluids.
[0012] FIG. 5 is another example of a cross-section depicting
interaction of three flowing fluids.
[0013] FIG. 6 is example of a cross-section depicting interaction
of two flowing fluids, with the outer fluid driven by
co-electrospinning.
[0014] FIG. 7 is example of a cross-section depicting interaction
of three flowing fluids, with the outermost fluid driven by
co-electrospinning.
[0015] FIG. 8 is another example of a cross-section depicting
interaction of three flowing fluids, with the outermost fluid
driven by co-electrospinning.
[0016] FIG. 9a depicts a graph of the jetting-dripping transition
in the {.beta., We} plane (Re=11.2 and .alpha.=1).
[0017] FIG. 9b depicts a graph of several jetting-dripping
transition lines separating the {Re, We} plane into absolutely
unstable (dripping) and convectively unstable (jetting) regions,
for .alpha.=1, .alpha.=1.03 and diverse values of .beta..
[0018] FIG. 9c depicts a graph of absolute/convective instability
transition loci in the {Re, We} plane for different values of
{.alpha., .beta.}.
[0019] FIG. 10 depicts the theoretical jetting-dripping transition
for various (.alpha., .beta., Re, We).
[0020] FIG. 11 depicts a graph of a phase diagram with horizontal
axis z.sub.N, the nozzle location, and vertical axis
dR/dz(z.sub.N)=sqrt (.mu./4.mu..sub.0) cot .theta..sub.N.
[0021] FIGS. 12a-f summarize one set of experimental results.
[0022] FIG. 13 summarize another set of experimental results.
[0023] FIG. 14 depicts a graph ploting the experimental results of
FIG. 13 for three representative liquids.
[0024] FIG. 15 depicts the flow leading to (after solidification
downstream) creation of a single fiber containing multiple interior
cylinders which are either hollow, if a gas is used as the inner
fluid, or are composed of a second material.
[0025] FIG. 16 depicts nozzles aligned along a line, allowing the
fluid inside the nozzle to be entrained by a two-dimensional
viscous flow in a thin liquid film.
[0026] FIG. 17 illustrates a sheet containing parallel tubes
prepared by aligning nozzles along a line.
[0027] FIG. 18 depicts an arrangement of parallel ribbons
containing parallel long holes along them can be packed in the form
of a "long brick" with many parallel long holes along their
length.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0028] As discussed in the background, there are many applications
for bubbles or filaments. For example, in the context of filaments,
hollow cylinders or tubes and compound fibers with an inner
diameter of micron and sub-micron sizes have a variety of
applications. One of the most promising is in fiber optics. If
glass is used as the exterior fluid and the interior is either left
hollow or filled with a material with a different index of
refraction, one can create optical waveguides whose properties can
be varied in a flexible way by changing the composition of the two
fluids and by changing the dimensions of the fiber. In other
contexts, if a polymer solution or melt is used as the exterior
fluid and then solidified, the resultant tubes may be used in
membrane filters, capillaries for chromatography as well as
bioreactors and as cylindrical microfluidic channels. Compound
fibers comprised of an inner metal core and an outer insulator
sheath may be used as cables in microelectronic applications. The
outer sheath may also be a metal. Possible applications for a metal
structure comprised of one or more tubes include: (i) heat
exchangers for biology, biomedicine, and cardiovascular
applications; (ii) surgical devices; (iii) scientific &
laboratory optical or microfluidic instruments; and (iv) reactors
for chemical reactions. Other possible materials that may be used
include polymers such as polysaccharides and polypeptides.
[0029] A method and system are provided in order to create stable
and/or acceptable bubbles or filaments. One way to create filaments
is to produce an extensional flow, such as by entraining a fluid
within another fluid. For example, fluid that is more viscous than
the fluid being entrained may flow from a nozzle. The combination
of the entrained fluid and the outer fluid may be used to create
hollow cylinders (for example, by entraining a gas within a fluid)
or compound fibers on micron and sub-micron scales. As discussed
herein, fluid may comprise a liquid, a gas, or a combination of a
liquid and a gas.
[0030] As discussed in the background, the ability to create stable
and/or acceptable bubbles or filaments depends, at least in part,
on the forces at the interface between the multiple fluids. As
discussed below, a model is created to analyze the interface
between fluids. The model may be used in order to adjust the system
(including dynamically adjusting the system) in order to create
bubbles or filaments. For example, creating a filament by
entraining a fluid within another fluid may be achieved by applying
a force to the fluids (including the inner fluid or the outer
fluid). The model, discussed below, may be used to determine the
forces to entrain the fluid.
[0031] The force may be applied directly or indirectly to the
fluids in order to produce an extensional flow. There are several
examples of a direct force that may be applied to a fluid. Where
the fluid is in a tank or a reservoir, pressure may be applied so
that the fluid is forced from a nozzle connected to the tank or
reservoir. Where the fluid is conductive, a voltage may be applied
so that the electric field generated from the voltage applies a
force to the conductive fluid. Co-electrospinning is an example of
using a voltage to apply a force to the fluid. Further, a
combination of direct forces, such as applying pressure and
co-electrospinning, may be used to apply a direct force to a
fluid.
[0032] One or more of the fluids may also be subject to an indirect
force. As one example, the inner fluid may be entrained, not by any
force applied directly to it (such as applying pressure to the
inner fluid or co-electrospinning of the inner fluid), but by
proximity of the inner fluid to movement of another fluid (such as
movement of an outer fluid). The movement of the outer fluid may be
caused by a direct force (such as by applying pressure to the outer
fluid or co-electrospinning), or may be caused by an indirect force
applied to it as well. In effect, the inner fluid is dragged along
or withdrawn by the outer fluid, rather than pushed along.
[0033] Further, a fluid may be subject to multiple direct or
indirect forces. For example, a fluid may be subject to the
multiple direct forces of pressure and co-electrospinning. As
another example, the fluid may be subject to an indirect dragging
force (such as caused by an outer fluid dragging an inner fluid)
and may be subject to a direct force (such as pressure applied to
tank supplying the inner fluid). In the case of multiple direct and
indirect forces, one may compare the amount of each of the forces.
A fluid that has a greater indirect force may be considered to have
a greater dragging force than pushing force. Similarly, a fluid
that has a greater direct force may be considered to have a greater
pushing force than dragging force.
[0034] Referring to FIG. 1, there is shown an example of a block
diagram of the system 100 for controlling fluid flow. The system
100 may include a control system 102 that may select the parameters
so that the system performs acceptably (such as entrainment of an
inner fluid) and may send outputs so that the system operates under
the selected parameters. The control system 102 may include a
processor 104 that accesses a memory 106. The memory may include a
programs or models of dispersions, as discussed in more detail
below. The processor 104 may determine the parameters for the
system and may control the system, via I/O 108. The control of the
system may be performed in variety of ways, such as through voltage
control 110 (which may control co-electrospinning), through
pressure control 112 (which may control the pressure of one or more
of the fluids by applying pressure to a tank housing the fluid),
nozzle angle 114 (which may adjust the nozzle angle that ejects the
fluid), and radius control 116 (which may control the radius of one
or more of the nozzles).
[0035] A model may be derived for the interaction of the flowing
fluids. Referring to FIG. 2, there is shown an example of an
interaction of flowing fluids. Specifically, FIG. 2 depicts a
flow-focusing setup illustrating the spatio-temporal stability of a
core liquid jet produced by flow focusing (FF). The nozzle may emit
a liquid filament 1 which is surrounded by and pulled by the
indirect force of the co-flowing liquid 2. Co-flowing liquid 2 may
be subject to a direct force of being pressurized. Further,
co-flowing liquid 2 may be immiscible with liquid filament 1.
Though FIG. 2 depicts an inner liquid (liquid filament 1) and an
outer liquid (co-flowing liquid 2), any combination of liquids or
gases may be used for the inner and outer fluid.
[0036] The liquid filament 1 and co-flowing liquid 2 may have
liquid viscosities .mu..sub.1,2 and densities .rho..sub.1,2
cylindrical coordinates r, z. In analyzing the spatio-temporal
stability depicted in FIG. 2, one may make the following
assumptions: (i) the liquid filament 1 and co-flowing liquid 2 are
viscous and immiscible with an interfacial surface tension of
.sigma.; (ii) the Reynolds number of the co-flowing shell
Re.sub.2=.rho..sub.2VD.mu..sub.2.sup.-1 is large enough to ensure a
uniform stream-wise velocity Ve.sub.z around the liquid jet,
e.sub.z being the axial unit vector; (iii) the core liquid jet
diameter d.sub.j=2R.sub.j is roughly uniform and much smaller than
the orifice diameter D, d.sub.j/D<<1, so that the liquid jet
may be considered to flow in an unbounded co-flowing liquid domain
moving with uniform velocity V; this simplification (both in the
theory and in the setup) attempts to enforce cylindrical
homogeneity. This simplification circumvents the complexity of a
downstream variation of the flow regime (and hence, excludes the
possibility of heterogeneous stability behavior at different axial
locations). Thus purely cylindrical flow patterns are considered,
and disregarding the effects of local instability sources (e.g.,
oscillation of the meniscus); (iv) the axial length l required for
a full momentum diffusion from the shell to the core, l.about.92
.sub.1V.mu..sup.-1d.sup.2.sub.j, is small compared to the axial
length of the orifice L, 1/L<<1, and thus both liquid streams
may move with a similar uniform axial velocity V; (v) no azimuthal
modes are considered (i.e., pure axisymmetric motion), which is a
simplification supported by both theory and experiment, on the
jetting-dripping transition.
[0037] With these assumptions, the system response may be analyzed
for small perturbations proportional to e.sup.i(kz-.OMEGA.t). In
analyzing the dispersion relation for a viscous liquid cylinder in
an immiscible viscous ambient liquid, as depicted in FIG. 2, wave
frequency .OMEGA., time t, wave number K, and coordinates {r, z}
may be scaled and made dimensionless with with V/R.sub.j,
R.sub.j/V, 1/R.sub.j and R.sub.j, respectively. Four dimensionless
parameters may arise: the Reynolds and Weber numbers, and the
liquid density and viscosity ratios, as shown in the following
equation: Re=.rho..sub.1VR.sub.j.mu..sub.1.sup.-1,
We=.rho..sub.1V.sup.2R.sub.j.sigma..sup.-1,
.alpha.=.rho..sub.2/.rho..sub.1, .beta.=.mu..sub.2/.mu..sub.1
(1)
[0038] The conservation equations of mass and momentum of the
liquid flow, together with the boundary conditions at the jet
surface (including normal and tangential stress balance) and at
infinity, lead to the dispersion relation between the perturbation
wave frequency .OMEGA. and its wavenumber k. Thus, a dimensionless
dispersion relation S({circumflex over (.omega.)}, x; Re, We,
.alpha.,.beta.)=0 (with {circumflex over
(.omega.)}=R.sub.j.OMEGA.V.sup.-1 and x=R.sub.jk) may be simplified
with the aid of a mathematical tool, such as Mathematica V5.01, by
introducing a speed V of the two fluids into: S .ident. ( x 2 - y 1
2 ) .function. [ N .function. ( x , y 1 , y 2 , .beta. ) M
.function. ( x , y 1 , y 2 , .beta. ) + 2 .times. ( 1 - .beta. ) ]
- Re 2 We .times. ( 1 - x 2 ) = 0 ( 2 ) ##EQU1##
[0039] where "viscous" wave numbers are defined for both liquids
as: y.sub.1.sup.2=x.sup.2-iRe{circumflex over (.omega.)},
y.sub.2.sup.2=x.sup.2-i.alpha..beta..sup.-1Re{circumflex over
(.omega.)}, (3)
[0040] and functions N and M are expressed as:
N.ident.2x.beta.y.sub.1y.sub.2[K.sub.0(y.sub.2)I.sub.1(y.sub.1)y.sub.1+I.-
sub.0(y.sub.1)K.sub.1(y.sub.2)y.sub.2]+x[x.sup.2(.beta.-1)-y.sub.1.sup.2+.-
beta.y.sub.2.sup.2].sup.2I.sub.0(x)I.sub.1(y.sub.1)K.sub.0(x)K.sub.1(y.sub-
.2)+4x.sup.3y.sub.1y.sub.2(.beta.-1).sup.2I.sub.0(y.sub.1)I.sub.1(x)K.sub.-
0(y.sub.2)K.sub.1(x)--y.sub.2I.sub.1(y.sub.1)K.sub.0(y.sub.2){[x.sup.4+y.s-
ub.1.sup.2y.sub.2.sup.2+x.sup.2(y.sub.1.sup.2-y.sub.2.sup.2)].beta.I.sub.1-
](x)+[y.sub.1.sup.4+x.sup.4(1-2.beta.).sup.2-2x.sup.2y.sub.1.sup.2(.beta.--
1)]I.sub.0(x)K.sub.1(x)}+y.sub.1I.sub.0(y.sub.1)K.sub.1(y.sub.2){[x.sup.4(-
.beta.-2).sup.2+2x.sup.2y.sub.2.sup.2.beta.(.beta.-1)+.beta..sup.2y.sub.2.-
sup.4]I.sub.1(x)K.sub.0(x)+[x.sup.2(x.sup.2-y.sub.1.sup.2)+y.sub.2.sup.2(x-
.sup.2+y.sub.1.sup.2)].beta.I.sub.0(x)K.sub.1(x)} (4)
M.ident.x{[y.sub.2K.sub.0(y.sub.2)K.sub.1(x)-xK.sub.0(x)K.sub.1(y.sub.2)]-
(y.sub.1.sup.2-x.sup.2)I.sub.1(x)I.sub.1(y.sub.1)+.beta.[y.sub.1I.sub.0(y.-
sub.1)I.sub.1(x)-xI.sub.0(x)I.sub.1(y.sub.1)](y.sub.2.sup.2-x.sup.2)K.sub.-
1(x)K.sub.1(y.sub.2)} (5)
[0041] Here, I.sub.0, I.sub.1, K.sub.0, and K.sub.1 stand for the
modified Bessel functions of order 0 and 1. Further, the limit
.alpha..fwdarw.0, .beta..fwdarw.0 yields: N M .fwdarw. ( x 2 + y 1
2 ) 2 x .function. ( x 2 - y 1 2 ) .times. I 0 .function. ( x ) I 1
.function. ( x ) - 4 .times. x 2 .times. y 1 ( X 2 - y 1 2 )
.times. I 0 .function. ( y 1 ) I 1 .function. ( y 1 ) ( 6 )
##EQU2##
[0042] This leads to the liquid-vacuum dispersion relation of: S
.ident. ( x 2 + y 1 2 ) 2 .times. I 0 .function. ( x ) I 1
.function. ( x ) - 4 .times. x 3 .times. y 1 .times. I 0 .function.
( y 1 ) I 1 .function. ( y 1 ) + 2 .times. x .function. ( x 2 - y 1
2 ) + Re 2 We .times. x .function. ( x 2 - 1 ) = 0 ( 7 )
##EQU3##
[0043] One may also use a spectral numerical code developed for the
stability analysis of swirling flows in pipes. Here, the linearized
equations may be discretized in the r-direction using Chebyshev
spectral collocation points (n.sub.i points for the inner fluid and
n.sub.e points for the outer one). For a given wave frequency
.OMEGA., one may linearize the non-linear (quadratic) eigenvalue
problem for the wave number x using a linear companion matrix
method. The resulting linear eigenvalue problem may be solved
numerically with the help of an eigenvalue solver subroutine
(DGVCCG from the IMSL library) which provides the entire spectrum
of eigenvalues and eigenfunctions. Spurious eigenvalues may be
excluded by comparing the computed spectra obtained for different
values of the number of collocation points. The use of the
numerical procedures allows: (i) to check that both techniques
render the same results, and (ii) to use the numerical spectral
technique in further studies to investigate the influence of other
parameters, such as the existence of other basic velocity profiles
or the case of a bounded liquid flow, which are not considered in
this work.
[0044] Based on the preceding, one may determine several things.
First, to change the reference system from a traveling observer to
a fixed observer anchored at the nozzle, one simply needs to
replace the wave frequency .OMEGA.=VR.sub.j.sup.-1{circumflex over
(.omega.)} by .OMEGA.'=VR.sub.j.sup.-1(.omega.-x) in the dispersion
relation. The new .omega. is the dimensionless wave frequency for
the static observer. This may be proved by retaining the linearized
convective terms Vu.sub.z and VR.sub.z in the momentum equation and
the kinematic condition at the interface, respectively. In a fixed
coordinate system, y.sub.1 and y.sub.2 are:
y.sub.1=.+-.[x.sup.2-iRe(.omega.-x)].sup.1/2,
y.sub.2=.+-.[x.sup.2-i.alpha..beta..sup.-1Re(.omega.-x)].sup.1/2
(8)
[0045] Second, both roots of the viscous wave numbers y.sub.1 and
y.sub.2 may be explored to encompass all potential sources of
spatio-temporal instability. The theoretical frame supporting the
results does not require such a precaution. The dispersion model
may be symmetric with respect to y.sub.1, so that identical
{(.omega., x} pairs may result, regardless of which root .+-.y is
chosen. This may not be the case with y.sub.2, whose positive and
negative roots may yield completely different results, implying
different wave solution pairs {.omega., x}.
[0046] Third, consistent with the spatial-temporal stability
analysis, both .omega. and x may be considered to be complex
variables. Therefore, .omega.=.omega..sub.r+i.omega..sub.i and
x=x.sub.r+i x.sub.i, {.omega..sub.r, .omega..sub.i,x.sub.r,x.sub.i}
being real numbers ("oscillation frequency", "local growth rate",
"wave number" and "spatial growth rate", respectively). Following
the well established spatial-temporal formalism to describe the
absolute/convective character of axisymmetric instabilities in the
{R.sub.e, W.sub.e, .alpha., .beta.} parametrical space of our
problem, solutions of d.omega./dx=0 in the dispersion relation (2)
with non-zero imaginary parts of .omega. and x. The system may be
defined to be absolutely unstable if d.omega./dx=0 for Im(x)<0,
and Im(.omega.)>0. One may specially care to choose all
solutions whose spatial branches departing from the saddle point
d.omega./dx=0 originate from distinct halves of the x-plane, i.e.,
the only ones providing the absolute instability growth rate.
[0047] A specific application of the above formulation may be
applied in the low Reynolds regime (discussed with regard to FIG.
11). Specifically, an analytical solution for the formation of gas
spouts in an extensional, high viscosity liquid flow that
presumably would allow the formation of extremely thin and long
holes in a viscous liquid bulk. The analytical solution may be
relevant to the formation of microbubbles from gas ligaments in
co-flowing liquids. Specifically, the solution may address whether
a long gas filament in a co-flowing liquid may be stable for any
combination of densities, viscosities, surface tension, and co-flow
velocity. Further, the solution may address a possible transition
from bubbling to jetting such as in liquid jets.
[0048] FIG. 3 is another example of a depiction of three concentric
fluids. As discussed below, the fluids selected may result in a low
Reynolds regime. As illustrated in FIG. 3, the entraining flow in
the exterior flow may be controlled by imposing a converging flow
in the fluid surrounding the two inner fluids. Analysis using the
long-wavelength model for entrainment, describes how, with a
suitably strong exterior entraining flow, such a compound fluid
cylinder may be stabilized against breakup into droplets due to
surface tension. This means that the liquid compound cylinder may
be easily solidified into solid fibers with an interior core of one
material and an exterior cylindrical sheath of a second material.
This setup also allows air to be entrained as the inner fluid, and
therefore can be used to manufacture hollow cylinders.
[0049] To create a stable entrained inner cylinder, the innermost
fluid may be less viscous than the entraining fluid, the two fluids
may be immiscible, and both may have finite surface tension. Given
any two such fluids, if the flow rates and reservoir pressures are
controlled to satisfy the continuous portion of the phase diagram
of FIG. 11, the diameter of the inner fluid may be tuned to a
desired value, and the inner fluid may remain a continuous cylinder
of, in theory, infinite length. Referring to FIG. 11, there is
shown a phase diagram with horizontal axis ZN, the nozzle location,
and vertical axis dR/dz(z.sub.N)=sqrt(.mu./4.mu..sub.0) cot
.theta..sub.N. FIG. 11, in the low Reynolds regime, depicts that
the entrainment dynamics at the threshold may be a function of
conditions at the nozzle, in particular, the dynamic contact angle
and the nozzle location relative to the imposed entraining flow.
Different nozzle conditions may result in different values for the
entrainment threshold, here expressed in terms of a dimensionless
quantity, Ca, the capillary number, which characterizes the ratio
of viscous stress in the entraining flow versus surface tension. As
shown in FIG. 11, values corresponding to entrainment transition at
one particular Ca.sub.c lie on a line segment, depicted as a dashed
line in the figure. Also, the minimum value of the inner fluid
diameter may be determined by the limits of the continuum
approximation.
[0050] Either the spout or the entraining fluid or both may be an
emulsion or a dispersion, if the concentration of the droplets or
particles is kept sufficiently dilute to maintain Newtonian
rheological properties. For example, carbon black may be
incorporated into the sheath to provide antistatic properties. In
contrast to existing techniques, e.g. chemical synthesis or
co-electrospinning, in one aspect of the invention, the process may
rely only on the fluid mechanics for the entrainment; therefore, it
may be robust and may function for a wide variety of materials. For
co-electrospinning, which also partially relies on a mechanical
process, to be successful the fluid pair used may satisfy certain
conditions on their viscosities, densities and electric
conductivities. In one aspect, the process may have the interior
fluid be less viscous than the exterior fluid. Alternatively, in
another aspect, the process may rely exclusively on chemical
synthesis or co-electrospinning, or may rely on a combination of
chemical synthesis, co-electrospinning and/or fluid mechanics.
[0051] With the same configuration but working in a different range
of flow rates, one can allow the compound liquid cylinder to
breakup into micron and submicron-sized compound liquid droplets or
bubbles. Again, because such droplets may be prepared of a size
comparable to the wavelength of light, they may have unique optical
properties and may have potential applications, such as in the
creation of an optical Bragg switch. Such compound drops may also
have applications in controlled drug delivery, catalysis, and in
various cosmetic and food applications which involve emulsions.
[0052] Referring to FIG. 4, there is shown another example of flow
of two fluids. Fluid 404 may be injected through tube 402. The
liquid may form a meniscus 410, which may be focused by an external
fluid, such as fluid 406. Fluid 406 may comprise an outermost
axisymetric flow, and may be composed of a polymer. Further, an
axisymetric entrance, such as an orifice or a channel 408, may be
used to focus the outermost fluid.
[0053] Referring to FIG. 5, there is shown yet another example of
flow of three fluids. Similar to FIG. 4, the innermost fluid 504,
for purposes of this figure termed the first fluid, may be injected
from tube 402 and form meniscus 410. Further, fluid 506, termed the
second fluid, is injected from tube 502, which is coaxial with tube
402. A third fluid, 510, may focus the second fluid. Further, an
axisymetric entrance, such as an orifice or a channel 508, may be
used to focus the third fluid 510. The first, second and third
fluids may comprise either a liquid or a gas. Further, fluids of
different viscosities may be selected. For example, the first,
second, and third fluids may be air, polymer, and air,
respectively; or may be liquid or gas, polymer, and polymer. Any
combination may be used.
[0054] Referring to FIGS. 6 and 7, there is shown another example
of flow of two and three fluids, respectively, using an outermost
fluid being focused by electrostatic forces. For example, in FIG.
6, there is shown a meniscus 602 which produces fluid flow 604. The
fluid flow 604 is focused by outer fluid 606, which may be a liquid
focused by electrostatic forces. If the fluid flow 604 is entirely
due to the dragging force applied by outer fluid 606, then the
pressure associated with the meniscus may be controlled to be
constant. Specifically, as the fluid flow 604 is drawn, the
meniscus may be replenished such that there is no or little
pressure change. This control of the pressure may be accomplished
by pressure sensors and valves associated with a tank housing the
innermost fluid. FIG. 7 depicts a three fluid system, with an
innermost fluid 708 flowing from meniscus 602. A second fluid 706
may also flow as depicted in FIG. 7. A third fluid 704 may comprise
an outermost fluid that is focused using electrostatic forces. The
third fluid 704 may thus draw the second fluid 706, which may in
turn draw the first fluid 708.
[0055] FIG. 8 depicts an expanded diagram of FIG. 7, showing tubes
806, 804, and 802 for the first, second and third fluids,
respectively. Further, the first, second and third fluids may enter
a liquid bath. Either before or in the liquid bath, one, some or
all of the fluids may be solidified. In the case of three liquids,
each of the liquids may be solidified. In the case where one or
more of the fluids is a liquid and the remaining fluids are a gas,
the liquids may be solidified. In order to collect the solidified
filaments, a take-up reel may be used, as is known in the art. The
take-up reel may thus spool the fibers produced.
[0056] Analyzing the dispersion model, one may investigate the
stability, and the type (absolute vs. convective) of the
instabilities that lead to break-up of hollow capillary jets in a
co-flowing unbounded liquid medium. This analysis may be over a
wide range (such as six orders of magnitude) of both Reynolds and
Weber (We) numbers. In contrast to what occurs with liquid jets,
hollow jets in unbounded co-flowing liquids may be absolutely
unstable, leading to local bubbling for all values of the Reynolds
and Weber numbers of the co-flowing liquid. The presence of a gas
(or any other fluid with finite viscosity) inside the jet may
elicit a transition from the absolute to convective character of
the instability for a finite {Re, We} pair. This transition may
correspond to the bubbling-jetting crisis similar to the
dripping-jetting transition for capillary liquid jets. Experiments
analyzing previously collected data and discussed below support
these conclusions, which delimit the parametrical realm of
micro-bubbling in unbounded co-flowing liquids.
[0057] The dispersion model may be analyzed in several ways. The
dispersion model may be easily reduced to a limit by setting
.beta.=0. At this limit, there may be a single wave solution,
absolutely unstable, for both positive and negative roots of
y.sub.1.sup.2 Traveling away from this limit solution, under fixed
Re and .alpha. (e.g. Re=11.2; .alpha.=1), .beta. may be changed to
evaluate the jetting-dripping transition loci. Referring to FIG.
9a, there is shown a graph of the jetting-dripping transition in
the {.beta., We} plane (Re=11.2 and .alpha.=1). Two theoretical
transition curves are depicted in FIG. 9a corresponding to positive
(WP) and negative (WN) y.sub.2.sup.2 roots are plotted. Both
solutions may collapse when .beta..fwdarw.0. Of note, the WN-branch
is the one to provide physically meaningful jetting-dripping loci
{.beta.,We}. In evolving on the {We,.beta.} space in FIG. 9a
(moving from jetting towards dripping), one first crosses the
negative branch WN. Clearly, as one moves deeper in the lower {Re,
We} region (smaller flow rates), one eventually hits the second
branch WP; however, this may be physically irrelevant since
absolute instability modes are already activated (leading to
dripping) once the transition through the WP-branch has taken
place. The formal procedure to determine the jetting-dripping loci
may be based on the simultaneous solution of the dispersion
relation along with the constraint that the group velocity be zero.
The resulting solutions {.omega.*,x*} may happen to be saddle
points; and some further checks may be required before identifying
them as jetting-dripping transition points. In short, one may look
for points {.omega.*,x*} where S=0 and
.differential.S/.differential.x=0, with x*.sub.i<0,
.omega.*.sub.i=0. In addition, the branches issuing from the saddle
point may cross the real x axis so as to define a range of unstable
wavelengths. These conditions may identify the associated
parametrical point {Re*,We*,.alpha.*,.beta.*} as being located at
the boundary (i.e., .omega.*.sub.i=0) of an absolutely unstable
behavior.
[0058] FIG. 9b depicts several jetting-dripping transition lines
separating the {Re,We} plane into absolutely unstable (dripping)
and convectively unstable (jetting) regions, for .alpha.=1,
.alpha.=1.03 and diverse values of .beta.. The limit of the
dispersion relation (by setting .beta.=0) is represented as a
dotted line. The jetting-dripping lines may exhibit an elbow or
fold-up which is not present in the limit (.beta.=0). Thus,
dripping may be obtained by sufficiently depressing either Re or
We. The plot also shows the limit to closely fit the dispersion
model when .alpha.=0.0012 and .beta.=0.001 (water jet in air). The
limit model may fail to describe low-Re situations, where the
influence of the outer fluid cannot be ignored.
[0059] Steady micro-jetting may give rise to greater productivity
(larger liquid flow rates), with a well controlled and small drop
size. On the contrary, dripping may give rise to much larger,
isolated droplets under similar Re and We. Dripping usually yields
highly monodisperse spray, but it may also exhibit bi-disperse or
polydisperse droplet distributions. Here, such non-linear complex
dynamics may arise whenever a given parametrical instance
{Re,We,.alpha.,.beta.} leads to more than one absolutely unstable
wave solutions and these solutions exhibit similar local growth
rate .omega..sub.i but different local oscillation frequency
.omega..sub.r.
[0060] Another analysis of the dispersion model comprises the
solution for each combination of {Re,We,.alpha.,.beta.} with the
largest temporal growth rate Im(.omega.)>0, i.e. the one that
dominates the break-up dynamics. FIG. 9c depicts a graph of
absolute/convective instability transition loci in the {Re, We}
plane for different values of {.alpha.,.beta.}. Regions below
(above) the curves correspond to absolutely (convectively) unstable
configurations. Thus, the graph depicts the loci in the {Re, We}
plane for which Im(.omega.) changes its sign. The line formed by
these points mark the transition from a convective to an absolute
(C/A) character of the instability for a given set of values of
{.alpha.,.beta.}.
[0061] Circles in FIG. 9 show the critical We numbers found with
the numerical code for .alpha.=.beta.=1000 (n.sub.i=31 and
n.sub.e=71 collocations points were used). There is a close
agreement with the results obtained by solving the analytical
dispersion relation (2). From the results above, one obtains that
in the limit Re <<1, for given {.alpha.,.crclbar.}, the C/A
transition depends on the capillary number Ca=We/Re only. For
example, for .alpha.=.beta., this results in an expression for the
critical capillary number as Ca*=0.139.alpha..sup.1/2 for .alpha.
above 100. Below this critical capillary number, the instability is
absolute (i.e., a gas spout breaks up--or bubbles--locally).
[0062] One result of the analysis is that in the limit of vanishing
viscosity and density of the inner fluid
(.alpha.,.beta..fwdarw..infin.), the jet may be absolutely unstable
for any combination of values of the Reynolds and Weber numbers.
When the viscosity and density of the inner fluid are small but
finite, for any value of the Reynolds number (either low or high)
there may be a finite value of the Weber number at which the
transition from an absolute to a convective character of the
instability takes place.
[0063] The dispersion model may be generated analytically, such as
by using the formulas shown above. Or, it may be generated
numerically, such as by analyzing experimental data. The dispersion
model developed analytically was compared with experimental
results. A stainless steel fluid focusing device with dimensions
D=D.sub.1=150 .mu.m, L=160 .mu.m, and H=125 .mu.m is used. The
inner edge of the orifice has been rounded to minimize vena
contracta effects. The ambient focusing fluid is distilled water at
T=23.degree. C. (.rho..sub.2=995 kgm.sup.-1, .mu..sub.2=0.001 Pas).
Three silicone oils with nominal Newtonian viscosities .mu..sub.1
equal to 0.005, 0.02 and 0.1 Pas and measured oil-water surface
tensions .rho.=33.2, 30.4, and 28 mN/m at T=23.degree. C. are used
as the jet-forming liquid. The oil densities are
.rho..sub.1=965.+-.0.2% kgm.sup.-1. Surfactants need not be used.
Thus, the study explores three values .beta.=0.2, 0.05, and 0.01,
with .alpha.=1.033. FIG. 10 depicts the theoretical
jetting-dripping transition for various (.alpha.,.beta.,Re,We). The
dotted lines show the experiments in FIGS. 12a-12f (small, joined
symbols: jetting; isolated large symbols: dripping); Q.sub.2=5.16
mL/min (empty symbols) and 1.82 mL/min (full symbols)
[0064] To compare the experiments with the theory, the diameter of
the unperturbed jet radius at the orifice exit is estimated by
assuming both liquids to issue at an approximately uniform velocity
V=4(Q.sub.1+Q.sub.2)/(.pi.D.sup.2), Q.sub.1 and Q.sub.2 being the
oil and water flow rates (Re.sub.2 ranges from 262 to 1362):
R.sub.j.about.(D/2)(Q.sub.1/Q.sub.1+Q.sub.2)).sup.1/2. Knowing
R.sub.j and Re, We can be calculated. There are dispersed five oil
flow rates Q.sub.1=2, 5, 10, 25, 50, and 100 ml/h, using a syringe
pump Harvard Apparatus mod. 4455 with B-D 1 cc plastic syringes
(Q.sub.1=1 and 2 ml/h), 5 cc (Q.sub.1=10 ml/h) and 20 cc
(Q.sub.1=25 and 50 ml/h). Two water flow rates Q.sub.2=1.82 and
5.16 ml/min (.+-.0.5%) were supplied, using a pressurized 300 cc
water reservoir.
[0065] FIGS. 12a-f summarize one set of experimental results.
Experiments are depicted for diverse values of Q.sub.1, Q.sub.2 and
.mu..sub.1 (.alpha.=1.033; .beta.=0.01 in FIG. 12a and FIG. 12b;
.beta.=0.05 in FIG. 12c and FIG. 12d; .beta.=0.2 in FIG. 12e and
FIG. 12f; exposure time: 100 ns. The scale is given by the dash,
measuring 200 .mu.m. FIG. 12e contains an insert showing the jet
breakup region. The jetting-dripping transition in flow focusing
exhibits a significant hysteresis effect: i.e., parametrical ranges
where (depending on the route followed) meta-stable dripping or
jetting may occur in regions corresponding to jetting or dripping,
respectively. Hysteresis may be caused by the global stability of
the system consisting of the capillary feeding tube, the
funnel-like meniscus attached to it, and the issuing jet. When the
steady jetting regime is abandoned to enter the dripping regime, a
considerable part of the apex region (at times, the whole meniscus)
oscillates, thus precluding the immediate recovery of steady jet
flow conditions, even after the operational parameters return to
their original setup in the former jetting regime. To enforce the
recovery of jetting conditions, the system may be turned back
sufficiently far into the convectively unstable (jetting)
parametrical subspace. The amplitude of the hysteretic effect is
associated to the meniscus size, i.e., to the relation between feed
tube size and orifice diameter: the smaller the feed tube, the
smaller the jetting-dripping hysteresis, and the reverse. This is
analogous to the jetting-dripping transition in electrospray.
Moreover, the higher the liquid viscosity, the larger the
associated hysteresis effect. In the experiments, a sharp-edged
feed tube (FIG. 2) was chosen whose inner diameter D.sub.1
coincides with the orifice diameter. The hysteresis effect was thus
minimized while maintaining a genuine fluid focusing geometry.
FIGS. 12a-f shows whenever it is feasible, the hysteresis range of
each experimental sequence.
[0066] Therefore, the experiments depicted in FIGS. 10 and 12a-f go
from jetting to dripping and not the other way around, because the
re-stabilization of a dripping mode requires driving the system
considerably deep inside the jetting mode. In FIGS. 12a-f, owing to
setup limitations, only four of the six experimental sequences were
brought to the dripping regime.
[0067] FIGS. 9b and 10 show a surprising jetting-dripping
transition topology: the upper branch is slanted so that a
potential for double transition exists; should one move along a
constant-Re line, one would first hit the jetting-dripping
transition and subsequently cross a jetting-dripping transition. Of
course, such result is to be taken cautiously, because of
hysteresis. The potential for double transition may be theoretical,
since experiments do not follow a constant-Reynolds route: when one
gradually increase the axial speed V, both the Weber and Reynolds
numbers grow (see Eq. 1), so that one would move over a parabolic
line crossing the jetting-dripping border just once. Another
intrinsic experimental difficulty is the water ambient, whose
divergent flow after the orifice exit tends to open up the core
stream leading to radial dispersal of the drops. The choice of a
sufficiently large L distance (L=160 .mu.m>D) helps to preserve
parallel flow conditions during a distance long enough to give rise
to a roughly cylindrical oil jet.
[0068] In spite of experimental difficulties, the model provides an
acceptable fit of the jetting-dripping transition data as
illustrated by the pictures in FIGS. 12a-f. Experimental dot curves
in FIG. 10 show a sequence of five measurements with different core
liquid flow rates Q.sub.1; the remaining physical parameters are
fixed. The jetting-dripping transition takes place at the lowest
Q.sub.1 flow rate compatible with a steady jet. Any flow rate below
such threshold produces the emission of large, isolated drops
(dripping): the drop-producing mechanism is the sequential filling
and detachment from a pulsating axial filament, an elongated
version of faucet-dripping (see FIG. 12d). Jetting drops may show a
diameter roughly similar to the jet diameter; while dripping drops
are substantially larger. The jetting-dripping transition is
remarkably predicted for the smallest viscosity oil (FIG. 12a) and
Q.sub.2=1.82 ml/min: it takes place at Q.sub.1=4.9 ml/h. The
transition is also very well predicted at .beta.=0.05 for both
Q.sub.2 flow rates. When dealing with the largest viscosity oil
(FIGS. 12e-f), the transition may be more difficult to assess,
although the experimental results fall very close to the predicted
transition within the experimental uncertainties associated to the
hysteresis effect. In FIGS. 12e-f, the top single pictures show
jetting behavior, while the remaining shots illustrate a typical
dripping process: a long pulsating tendril is filling a thick drop;
some recoil is observed, owing to braking effects from the ambient
liquid.
[0069] Additional comparison of results is shown in FIG. 13 which
shows bubbling to jetting. Liquid: ethanol-water (50/50% vv),
T=28.degree. C. Orifice diameter D=200 .mu.m. (a) Q.sub.1=13
mL/min, Q.sub.g=0.65 mL/min; (b) Q.sub.1=23 mL/min, Q.sub.g=0.25
m/min; (c) Q.sub.1=23 mL/min, Q.sub.g=0.5 mL/min. The arrow
indicates the flow direction. The results are of experiments where
a gaseous jet was pulled by a stream of liquid flowing through a
co-axial orifice of diameter D (Flow Focusing technique). Both the
liquid and the gas issued from a stagnant region upstream of the
orifice, where pressure is kept constant and equal for both fluids.
In order to assess the diameter of the undisturbed gaseous jet, the
assumption was made that the gas is withdrawn by tangential viscous
stresses by the liquid stream, and therefore that the gas velocity
inside the undisturbed gas jet diameter does not depart
significantly from that of the co-flowing liquid. Under this
assumption, mass conservation for an incompressible fluid yields:
d.sub.j=[4Q.sub.g/(.pi.U)].sup.0.5. This approximation may be more
accurate for smaller radii of the gaseous jets (owing to the
growing importance of viscous action inside them) and for smaller
driving pressure gradients. An analysis of the equations of the
steady jet (in the high Re regime) may lead to the conclusion that
a plug-flow velocity profile in the jet at the orifice exit is a
sufficiently accurate solution, and the only one consistent with
the use of the dispersion relation. In the experiments, it has been
observed what seems a clear transition to jetting, as shown in FIG.
13, although the onset of convective instability is often blurred
by unsteadiness and chaotic bubbling. FIG. 14 plots the
experimental data for three representative liquids (corresponding
to the non dimensional parameters {.alpha.=760,.beta.=68},
{.alpha.=800,.beta.=324}, and {.alpha.=850,.beta.=544}), as well as
the experiments of FIG. 13 (under the assumption that
R.sub.o<<D). All experimental data points, which correspond
to monodisperse microbubbling, are located below the calculated
Weber numbers corresponding to the transition from bubbling to
jetting. The observation that all experimental data for bubbling
fall into the region of absolute instability established by the
model supports it.
[0070] One conclusion from this analysis is that a cylindrical gas
spout in a high viscosity liquid moving with speed Vmay be
absolutely unstable, and may prevent the formation of long gas
spouts, below a finite value of the Weber number or, alternatively,
below a finite value of the Capillary number Ca=We/Re. Thus, the
solution for gas spouts entrained in extensional viscous flows
discussed in regard to FIG. 11, may be convectively unstable as
long as the outer liquid is sufficiently viscous, provided that
there is minimum finite Capillary number for which the steady
solution can exist. For Re<<1, below the critical value of
the Capillary number, the spouts formed by the extensional flow in
the solution may form a train of microbubbles, a fact observed in
related experiments.
[0071] In those experiments, to provide a flow as close as possible
to the true extensional one, the liquid stream is surrounded by
another gas stream in order to avoid the liquid contact with the
exit orifice wall. This is not a true unbounded liquid flow, but
the diameter of the spout is small in comparison to the outer
diameter of the liquid-jet, and the results illustrate well our
findings. The influence on the ratio d.sub.l/d.sub.g of the outer
diameter of the liquid-jet to the diameter of the gaseous jet is
verified using an alternative spectral collocation code. For
d.sub.l/d.sub.g=100, the differences in the critical Weber numbers
with those of unbounded liquid were less than 3% for large Reynolds
numbers, while the differences were less than 1% for small Reynolds
numbers.
[0072] The model may be applied in a variety of instances. For
example, a hollow optical fiber with a cylindrical hole may only be
drawn if the Weber number is above the critical one, or if there is
an alternative mechanism providing local stability. This mechanism
may be a longitudinal positive gradient of viscosity (e.g., the one
due to "fiber quenching", when the glass solidifies) or an external
negative pressure gradient (a potential alternative in ultra-high
speed fiber drawing).
[0073] The model may also be applied in other configurations, such
as that shown in FIGS. 15-17. FIG. 15 depicts the flow leading to
(after solidification downstream) creation of a single fiber
containing multiple interior cylinders which are either hollow, if
a gas is used as the inner fluid, or are composed of a second
material. As shown in FIG. 15, there are three nozzles that extend
a certain finite distance, and each of which axisymmetrically may
constrain a fluid of one composition ("Composition B") around a
second entrained fluid of a different composition ("Composition
A"), and which themselves may be surrounded by a larger diameter
flow of more fluid of Composition B. This may be an efficient way
to manufacture, for example, special optical fibers. Second,
nozzles may be aligned along a line, allowing the fluid inside the
nozzle to be entrained by a two-dimensional viscous flow in a thin
liquid film. For example, FIG. 16 depicts nozzles aligned along a
line, allowing the fluid inside the nozzle to be entrained by a
two-dimensional viscous flow in a thin liquid film. After
solidification this creates a thin sheet of material with thin
ribbons which are either hollow or comprised of a second material.
An example of this is depicted in FIG. 17, which illustrates a
sheet 1700 containing parallel tubes 1702 prepared by aligning
nozzles along a line. The tubes 1702 may be hollow or filled with a
material different from that of the bulk. The figures, including
FIG. 17, are not necessarily to scale. Such composite thin sheets
with precisely patterned texture may have interesting material
properties. For example, they may be very strong in one direction
(perpendicular to the ribbons) and more fragile in the direction
along the ribbons. An arrangement of parallel ribbons containing
parallel long holes 1802 along them can be packed in the form of a
"long brick" 1800 with many parallel long holes along their length
distributed as shown in FIG. 18.
[0074] This methodology enables creating compound fibers wherein
the inner diameter can be made far smaller than the outer diameter
of the fiber. The disparity in the interior and exterior diameters
has at least two useful consequences. First, by placing a group of
nozzles in close proximity, the entrainment process can be used to
create a single fiber containing multiple interior cylinders which
are either hollow or made of a second material (FIG. 16).
[0075] It is therefore intended that the foregoing detailed
description be regarded as illustrative rather than limiting, and
that it be understood that it is the following claims, including
all equivalents, that are intended to define the spirit and scope
of this invention. Other variations may be readily substituted and
combined to achieve particular design goals or accommodate
particular materials or manufacturing processes.
* * * * *