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Viscous fingering as a first step toward understanding dendrites
Superlattices
and Microstructures,
VISCOUS
FINGERING Il.
6’! 7
Vol. 3, No. 6, 1987
AS
A FIRST
w. DiFrancesco,
STEP S.
N.
TOWARD Rauseo,
UNDERSTANDING and
J.
V.
DENDRITES
Maher
Department of Physics and Astronomy University of Pittsburgh Pittsburgh, PA 15260 (Received
10
April
1987)
The development of vi8cous fingering pattern8 has been observed for Hele-Shav flows with both planar and circular formal have close interfaces. These flors initial Local curvature as a connection with dendritic growth. function of interfacial arclength ha8 been extracted for been performed on these and modal analyses have all cases A crude measure of the content of the curvature functions. analyses modal comes from the behavior of the average wavenumber, c, which at first increases during an initial ramifying flow period and then decrease8 with time as the pattern coarsens in the quiet regions behind the advancing fingers faster than it ramifies at the front.
In our this paper we summarize work recent on the ViSCOU8 fingering problemC1,21. In its initially planar viscous form, Seffman-Taylor flor[33, fingering represent8 the simplest of pattern formation problems. Thus the observation of its detail8 provides a valuable opportunity for direct testing which are of the computer calculations now just becoming feasible for pattern formationC41. Despite extreme its has simplicity, Saffman-Taylor flow much in common with the llullins-Sekerke which rnstabilityC51 gives rise to dendritic growth in alloys. Further the similarity to dendrite problem arises if the viscous fingering problem changed the flat is from initially interface of the Saffman-Taylor flow to an initially circular interfaceC61, in which case tip-splitting is forced on a growing interfacial the fingers as circumference can accommodate increasingly more the wavelengths of most likely disturbance. In this paper we first discus8 the formal similarity between the dispersion for relations the Saffman-Taylor and tlullins-Sekerke Instabilities, then we set out some of our recent results on the SeffmenTaylor problemtll and finally present Cl",- results for circular interface flowC21. Other papers in thrs conference discuss state-of-the-art calculations for these problems,
0749
-6036/87/060617+07
$02.00/O
sometimes with added complication8 even closer to realistic dendritic growth. instability~ll The Saffman-Taylor initially planar at the arises interface between two fluids flowing in a Hele-Shaw cell (a cell formed by parallel plates with a gap between them of thickness b where b is smaller than any other length scale in the problem). driven either by a pressure It ie the less viscous advancing gradient fluid against the more viscous or by density result of a gravity as a For the difference between the fluids. case of gravity driven flow in a closed cell where the average rectangular the velocity of the interface is zero, from linear diepersion relation, stability analysis, takes the form: lM'[?LlK] - (p2 - p,)gk
+ rr*k.3 ==0.
(1)
where K 1 b'/lZ is a mobility. The average shear viscosity, the effective interfacial surface tension, the density of fluid n, the and acceleration due to are gravity represented by ji, CT*. and g ?n* respectively. This dispersion relation predicts broad band instability for all wavenumbers. k, below a critical value set by cr*, Ap, ji, and b. When a binary liquid, in its two region, phase is quenched rapidly
01987Academic
Press Limited
618
Superlattices
and Microstructures,
Vol. 3, No
6, 7987
further into that region, a concentration gradient is imposed at the interface between the phases. It hae been proposedC73 that such a gradient leads to a Mullins-Sekerka type instability envisioned in ita simplest form, namely, the stationary symmetric model. In this picture, linear analysis reveals the dispersion relation:
iw-~kT?Dl,k3=0.
(2)
; where D is the diffusion coefficient of the liquid, 1. is a capillary length, and 1 characterizes the thickness of the region near the interface where the concentration gradient ie significant. The two diepersion relations deecribed above obviously ehare the cave generic form with a simple competition between a dertabilizing linear term and a etabilizing cubic term. This suggests that knowledge of one may well provide information about both, at least for the onset of the instability (where the linear stability analyeie should be valid) and possibly for stages of the nonlinear early pattern formation. Thie hope ie heightened by the formal similarity of calculatione of viscoue fingering and dendritic growth in the case where one of the fluids in the fingering flow ham negligible viscosity in comparison to With all these the other fluidC93. considerations in mind we preeent our reeulte for relatively early stages of viscoue fingering and diecuse enalyaee of these results for features which could be directly compared with feasible admittedly large scale but calculatione. Further work computer varying flow xi11 concentrate on parameter= to overlap with and then extend the etudy of equations 1 and 2 out of the ranges previously covered in solidification experiments.
Rcmulte
with
Rectangular
Cells
As wae mentioned above, a basic rectangular Hele-Shaw cell coneiste of two rectangular glass platea roperated by a uniform gap that ie very small with other linear compared the In euch a cell, fluid flow dimenrione. dimensional ie and effectively two described by Darcy's Law. Under the proper conditione, the interface between two immiscible liquids in a forms cell uneteble becomen and patterns like the onea ahown in Figure 1.
.F I”
,,;l--_J-I__J 1.13 0.76 0.36
w
0111II 1.51 I.13 0.76 0.38
w 0
0.36
0.73
1.09 I.46
0 0.36
Width
Figure
1:
0.73
1.09 1.46 0 0.36
0.73 1.09 1.46
(inches)
Time series of fingering patterns. Time progreeeion is down succeeding column8 starting with the pattern the upper left corner.
in
One goal of the study of patterns is to find ways to describe their rhapes and development that are universal and meaningful for comparison with the results of calculation. An obvious first approach is to observe some gross feature of the flow, such as the length of the fingers as a function of time. Figure 2 show6 the dimensionless length of the longest finger, as a function of 0, dimensionless time several for runs widely varying parameters with (i.e. viscosity contrast and surface tension). The lines labeled 5 and 6 correspond to flows driven by a pressure gradient the while others result from gravity driven flows. All show 6 increasing time with a with power law with an exponent of 2 1.5. There is a formal similarity between this and diffusion-limitedflow aggregation (DLA)C9,103 60 it may be no accident that the exponent observed here is came as that eeen in the computer eimuletions of the temporal growth of two-dimensional DLA patterns.
Superlattices
and Microstructures,
I
619
Vol. 3, NO. 6, 1987
I B 6.6
* lO-4
*
IO
100 t’
Figure
2:
Dimensionless length of the longeet finger vs dimensionless time as discussed in the text (see ref. 1).
k Figure
the A more detailed picture of gained through a interface has been Fourier analysis of its shape. Figure 3 shows an example of an interface four seconds a its into flov and corresponding Fourier transform exhibiting dominant ravenumbers which roughly with agree the number of fingers one would judge to be in the pattern. The Fourier transforms are useful in that they enable one to follow development of the individual wavenumbers. In this way, it is possible to compare the behavior of the various length scales imbedded in the patterns. For instance, the temporal developments of the dominant two wavenumbers in Figure 3 are depicted in Figure 4. One, hovever, soon encounters a difficulty in forming Fourier the transform of the flow patterns as they develop beyond the first few seconds. The pattern becomes a multivalued function of position across the cell as the fingers become more complicated, forming at their tips. We "balloons"
3:
Trace of a fingering pattern four seconds into a flow and its corresponding Fourier transform.
have thus decided to use, as an alternative, a plot of the local curvature arclength for each vs pattern's description. Curvature a* a has function of arclength clear two advantages: 1) it remains single-valued and thus allovs a modal analysis no matter how complicated the interface becomes and 2) no information about the local interface is discarded when this function replaces the spatial pattern. A concomitant disadvantage is that we have kept the considerable all physical detail and are now dealing with a function for which we have less intuition than ve had for the spatial pattern. first attempt at As a digesting the information contained in the curvatures, we have constructed "power spectra" - the square of the Fourier amplitudes resulting from a modal analysis of the curvature data. One of our current projects is to try
620
-a
Superlattices
1
and Microstructures.
Vol 3, No
6. 7.98
.
??
1
time
(set) Figure
5:
The average dimensionless
wavenumber, time.
k,
ve
initial noisy period, that E 2 t-1.6. This sharp tendency toward longer wavelengths at late times may suggest a progressive smoothing of the patterns. Each value of k in plot is this itself an from an actually average ensemble of 14 runs under identical conditions. The approximately 5% standard deviations shown in the figure do not result from any - 5% difficultly in controlling the initial conditions rather intrinsic but represent the arising noise in the flow presumably from very small the fluctuations in initial conditions.
time Figure
4:
(set)
Results
Plots of the square Fourier amplitude vs time for the dominant wavenumbers, k = 6 and k = 7, shown in Figure 3.
to scale the power spectra over a wide For the conditions. variety of run present, however, our best insight is provided by a much more crude measure, the first moments of the power spectra they evolve in If we as time. P(k), power spectrum as represent a then this moment, k, is
k = Ikp(k)dk!jp(k)dk. Figure In dimensionless
5
a time
(3) plot shows,
of
L after
"S an
with
a
Circular
Cell
Circular cells, like Hele-Shaw their rectangular counterparts, consist of two plates separated by a gap that is small relative to all other lengths in the problem. Flow is initiated by one fluid (in this case injecting nitrogen gas) at the center of the cell to displace a more viscous fluid (oil) that initially fills the space between driving A dimensionless the plates. the force, C 3 Qu/bo, characterizes flow conditions. Here, Q is the area1 b IS the rate of nitrogen, injection u is the oil viscosity, gap thickness, the surface tension of the and o is SO far, we oil-nitrogen interface. have observed flows for which C had a 1 and 35. in the range between value In the future we intend to push both to much lower and to much higher values. circular VlSCOUS fingering in a represents a more complicated geometry
Superlattices
Figure
6:
and Microstructures,
62
Vol. 3, No. 6, 1987
Patterns formed by a growing oil-nitrogen interface (C = 17) at half-eecond intervals
(see
ref.
2).
I
.
.
. -
.
C= 16,t=2.23,
Figure
7:
:
’
Typical plot of the radius of gyration (R,) vs the area of the injected fluid, A (see ref 2), shoving the power-law relation R, - A".*
problem ir that tip-splitting is xorced on the fingers. As a demonstration of this, Figure 6 the time shows development of a flow in a typical circular cell tip-splitting with clearly evident. For these flows we have plotted the radius of gyration vs total area of the developing patterns, obtaining results typically like the one shown in Figure 7. For all of our
0
40
80
I20
160
L (cm) Figure
8:
A typical plot of curvature vs arclength along an interface (see ref. 2).
200
622
Superlattrces
and Mmostructures.
Voi 3. ho
t? i9H
60
I IO3
GO_ h 5
e
C= 16,1=0.739
Figure
:: :
??
/ :. ‘:.
60
C= 16,
0.6
1.2
t=2.23S
1.8
2.4
W/cm) Figure
9:
IO7
A&-:
ITi :: :i ;’
40 -yy.::. ::i:: : .a..: ” ,:.‘.:
IO5
The power spectrum of the curvature as a function of arclength depicted in Figure 8 (see ref. 2).
10:
k vs the area of mixing, Aa, for many values of C (see driven ref. 2). The harder runs show k decreasing with An.
flow conditions, such plots indicate where a = 1.79 f that R, 1 (Area)"' 0.07. This value of a agrees reasonably well with the value of 1.70 f 0.05 obtained from DLA. Despite the extreme simplicity of DLA, it seems to contain such of the basic physics of realistic pattern formation. We used plots of curvature "8 arclength our circular patterns for ones just as we did for the rectangular and in the same way derived values of wavenumber. Typical the average examples of curvature vs arclength are given in Figure 8 while Figure 9 shows their respective Fourier transforms. Plots of the average k VI dimensionless area (as a measure of mixing zone dimensionless time~83) for effective all of our ride variety of runs were fall on an combined and found to "universal" curve shown in apparently Figure 10. The driving forces in these runs differed by as such as a factor of The curve shows an increase of k 35. at early times as the patterns ramify. At times I; turns over and later eventually decreases as progressively more of the pattern, left behind by "anneals" in the low advancing fingers, pressure *fjords" thus gradient and introduces longer length into scales the curvature distributions. c is clearly a crude poorly quantity, designed to give a clear picture of this complicated pattern evolution, but
S~lperlattices
and Microstructures,
Vol. 3, No. 6, 1987
attractive to 1t is think of the temporal evolution of i; arising from a competition between rapid ramification in the steep pressure gradient region near the finger tips and surface tension driven annealing in the fjords. I" this picture the early stages of flow are dominated by the former effect while the latter effect governs later stages. This description is very reminiscent of the coarsening of dendrites. Whether the annealing in the fjords would eventually so fill in the fjords as to raise the fractal dimension away from the DLA value and Over toward a dense branching morphologyCll1 is not as yet clear and ~111 be addressed in our next series of experiments. I" summary, the tantalizing preliminary result presented above concerns the secular development of average wavenumbers E for power spectra of curvatures from both initially planar and circular viscous fingering patterns. These E, after a complicated initially ramifying flow period, decrease with time and represent a coursening of the pattern in the quiet regions behind the advancing fingers to a point where the interfacial capillary length is small in comparison to typical long smooth sections of the interface. Acknowledgement-This work was supported by the U. S. Department of Energy under grant #DE-FG02-84ER45131.
References [llJ. V. Maher, Phys. Rev. Lett. 54. in, The Physics of_ 1498 (1385); Finely Divided Matter, ed. by N. Boccare and H. Daoud (SpringerVerlag, Berlin, 1985) p. 252 - 257. [2lS. N. Rauseo, P. D. Barnes, Jr., and J. V. Haher, Phys. Rev. A 35, 324', (1987).
623
and G. I. Taylor, C31P. G. Saffman Ser. A 245, Proc. R. Sot. London 312 (1958). C4lD. Kessler and H. Levine, Phys. Rev. A 33, 2632 (1986); 33, 2639 (1986); L. n. Sander, P. Ramanlal, and E. Ben-Jacob, ibid. 32, 3160 (1985); E. Ben-Jacob, N. D. Goldenfeld, J. Kopllk, H. Levine, T. Hueller, and L. M. Sander, Phys. Rev. Lett. 55, 1315 (1985); G. Daccord, J. Nittmen, and H. E. Stanley, Ibid. 56, 1. Shraimen, Ibid.
336 56,
(1986,; B. 2028 (1986); Ibid. Langer,
D.
C. Hong and J. S. 2032 (1986); R. Combescot, T. Y. Pomeau. and A. Dombre, V. Hekim, Pumir, Ibid. 56, 2036 (1986); 5 Bensimon, L. P. Kadanuff, S. Llang, 56.
El. I. Shralman, (unpublxshed);
and C. Tang A. J. DlGregorla
and
L. W. Schwartz, J. FluId Mech. lh4, 383 (1986); J. Nittman, H. E. Stanley, Nature 321, 663 11986). [SIW. W. Mull~ns and R. F. Sekerka, J. Appl. Phys. 34, 323 (1963;; 35, 444 (1964). 16lL. Paterson. 513 t1981);
(1585,. I713. S. Langer Metall. 25, hydrodynamzc added
J.
Fluld
Phys. and 1113
degrees
D.
"8,
;!h
L. A. Turekl, (1977);
Jasnow, Ohta, Phys.
by
Mect~,. llJ*
Flulds
of
Acta
treedom
D. A. Nlcole, Hrv. A _!:*. 31'7
and T. (1381). CalSee discussion of failure ~31 t ~rne scaling In refererlce 2 arid alsu !li, C. W. Perk and G. M. Horns?, 1. Fluid Nech. 139, 291 11%+4~. [YIL. P. Kadanoff, J. Z:tat. P'hva. 14'3, 267 (1985). [lQlT. A. ?hys. Phys.
Wltter, and Rev. Lett. Rev. H 27,
L. M. !z:::andcr, 47, 1449 tl’:iHl,: 56tii I~Y&+I.
[lllY. Sawada. A. Dougherty. dnd J. P. Gollub, Phys. Rev. Lrtt. 5h, l,'t.ti E. Berl-Jacob, G. Iwutschrr-, (1986); P. Garlk, N. D. Goldrnfeld, ar,d Y. Lareah, IbId. 57, 1303 3148t ,.
and Microstructures,
VISCOUS
FINGERING Il.
6’! 7
Vol. 3, No. 6, 1987
AS
A FIRST
w. DiFrancesco,
STEP S.
N.
TOWARD Rauseo,
UNDERSTANDING and
J.
V.
DENDRITES
Maher
Department of Physics and Astronomy University of Pittsburgh Pittsburgh, PA 15260 (Received
10
April
1987)
The development of vi8cous fingering pattern8 has been observed for Hele-Shav flows with both planar and circular formal have close interfaces. These flors initial Local curvature as a connection with dendritic growth. function of interfacial arclength ha8 been extracted for been performed on these and modal analyses have all cases A crude measure of the content of the curvature functions. analyses modal comes from the behavior of the average wavenumber, c, which at first increases during an initial ramifying flow period and then decrease8 with time as the pattern coarsens in the quiet regions behind the advancing fingers faster than it ramifies at the front.
In our this paper we summarize work recent on the ViSCOU8 fingering problemC1,21. In its initially planar viscous form, Seffman-Taylor flor[33, fingering represent8 the simplest of pattern formation problems. Thus the observation of its detail8 provides a valuable opportunity for direct testing which are of the computer calculations now just becoming feasible for pattern formationC41. Despite extreme its has simplicity, Saffman-Taylor flow much in common with the llullins-Sekerke which rnstabilityC51 gives rise to dendritic growth in alloys. Further the similarity to dendrite problem arises if the viscous fingering problem changed the flat is from initially interface of the Saffman-Taylor flow to an initially circular interfaceC61, in which case tip-splitting is forced on a growing interfacial the fingers as circumference can accommodate increasingly more the wavelengths of most likely disturbance. In this paper we first discus8 the formal similarity between the dispersion for relations the Saffman-Taylor and tlullins-Sekerke Instabilities, then we set out some of our recent results on the SeffmenTaylor problemtll and finally present Cl",- results for circular interface flowC21. Other papers in thrs conference discuss state-of-the-art calculations for these problems,
0749
-6036/87/060617+07
$02.00/O
sometimes with added complication8 even closer to realistic dendritic growth. instability~ll The Saffman-Taylor initially planar at the arises interface between two fluids flowing in a Hele-Shaw cell (a cell formed by parallel plates with a gap between them of thickness b where b is smaller than any other length scale in the problem). driven either by a pressure It ie the less viscous advancing gradient fluid against the more viscous or by density result of a gravity as a For the difference between the fluids. case of gravity driven flow in a closed cell where the average rectangular the velocity of the interface is zero, from linear diepersion relation, stability analysis, takes the form: lM'[?LlK] - (p2 - p,)gk
+ rr*k.3 ==0.
(1)
where K 1 b'/lZ is a mobility. The average shear viscosity, the effective interfacial surface tension, the density of fluid n, the and acceleration due to are gravity represented by ji, CT*. and g ?n* respectively. This dispersion relation predicts broad band instability for all wavenumbers. k, below a critical value set by cr*, Ap, ji, and b. When a binary liquid, in its two region, phase is quenched rapidly
01987Academic
Press Limited
618
Superlattices
and Microstructures,
Vol. 3, No
6, 7987
further into that region, a concentration gradient is imposed at the interface between the phases. It hae been proposedC73 that such a gradient leads to a Mullins-Sekerka type instability envisioned in ita simplest form, namely, the stationary symmetric model. In this picture, linear analysis reveals the dispersion relation:
iw-~kT?Dl,k3=0.
(2)
; where D is the diffusion coefficient of the liquid, 1. is a capillary length, and 1 characterizes the thickness of the region near the interface where the concentration gradient ie significant. The two diepersion relations deecribed above obviously ehare the cave generic form with a simple competition between a dertabilizing linear term and a etabilizing cubic term. This suggests that knowledge of one may well provide information about both, at least for the onset of the instability (where the linear stability analyeie should be valid) and possibly for stages of the nonlinear early pattern formation. Thie hope ie heightened by the formal similarity of calculatione of viscoue fingering and dendritic growth in the case where one of the fluids in the fingering flow ham negligible viscosity in comparison to With all these the other fluidC93. considerations in mind we preeent our reeulte for relatively early stages of viscoue fingering and diecuse enalyaee of these results for features which could be directly compared with feasible admittedly large scale but calculatione. Further work computer varying flow xi11 concentrate on parameter= to overlap with and then extend the etudy of equations 1 and 2 out of the ranges previously covered in solidification experiments.
Rcmulte
with
Rectangular
Cells
As wae mentioned above, a basic rectangular Hele-Shaw cell coneiste of two rectangular glass platea roperated by a uniform gap that ie very small with other linear compared the In euch a cell, fluid flow dimenrione. dimensional ie and effectively two described by Darcy's Law. Under the proper conditione, the interface between two immiscible liquids in a forms cell uneteble becomen and patterns like the onea ahown in Figure 1.
.F I”
,,;l--_J-I__J 1.13 0.76 0.36
w
0111II 1.51 I.13 0.76 0.38
w 0
0.36
0.73
1.09 I.46
0 0.36
Width
Figure
1:
0.73
1.09 1.46 0 0.36
0.73 1.09 1.46
(inches)
Time series of fingering patterns. Time progreeeion is down succeeding column8 starting with the pattern the upper left corner.
in
One goal of the study of patterns is to find ways to describe their rhapes and development that are universal and meaningful for comparison with the results of calculation. An obvious first approach is to observe some gross feature of the flow, such as the length of the fingers as a function of time. Figure 2 show6 the dimensionless length of the longest finger, as a function of 0, dimensionless time several for runs widely varying parameters with (i.e. viscosity contrast and surface tension). The lines labeled 5 and 6 correspond to flows driven by a pressure gradient the while others result from gravity driven flows. All show 6 increasing time with a with power law with an exponent of 2 1.5. There is a formal similarity between this and diffusion-limitedflow aggregation (DLA)C9,103 60 it may be no accident that the exponent observed here is came as that eeen in the computer eimuletions of the temporal growth of two-dimensional DLA patterns.
Superlattices
and Microstructures,
I
619
Vol. 3, NO. 6, 1987
I B 6.6
* lO-4
*
IO
100 t’
Figure
2:
Dimensionless length of the longeet finger vs dimensionless time as discussed in the text (see ref. 1).
k Figure
the A more detailed picture of gained through a interface has been Fourier analysis of its shape. Figure 3 shows an example of an interface four seconds a its into flov and corresponding Fourier transform exhibiting dominant ravenumbers which roughly with agree the number of fingers one would judge to be in the pattern. The Fourier transforms are useful in that they enable one to follow development of the individual wavenumbers. In this way, it is possible to compare the behavior of the various length scales imbedded in the patterns. For instance, the temporal developments of the dominant two wavenumbers in Figure 3 are depicted in Figure 4. One, hovever, soon encounters a difficulty in forming Fourier the transform of the flow patterns as they develop beyond the first few seconds. The pattern becomes a multivalued function of position across the cell as the fingers become more complicated, forming at their tips. We "balloons"
3:
Trace of a fingering pattern four seconds into a flow and its corresponding Fourier transform.
have thus decided to use, as an alternative, a plot of the local curvature arclength for each vs pattern's description. Curvature a* a has function of arclength clear two advantages: 1) it remains single-valued and thus allovs a modal analysis no matter how complicated the interface becomes and 2) no information about the local interface is discarded when this function replaces the spatial pattern. A concomitant disadvantage is that we have kept the considerable all physical detail and are now dealing with a function for which we have less intuition than ve had for the spatial pattern. first attempt at As a digesting the information contained in the curvatures, we have constructed "power spectra" - the square of the Fourier amplitudes resulting from a modal analysis of the curvature data. One of our current projects is to try
620
-a
Superlattices
1
and Microstructures.
Vol 3, No
6. 7.98
.
??
1
time
(set) Figure
5:
The average dimensionless
wavenumber, time.
k,
ve
initial noisy period, that E 2 t-1.6. This sharp tendency toward longer wavelengths at late times may suggest a progressive smoothing of the patterns. Each value of k in plot is this itself an from an actually average ensemble of 14 runs under identical conditions. The approximately 5% standard deviations shown in the figure do not result from any - 5% difficultly in controlling the initial conditions rather intrinsic but represent the arising noise in the flow presumably from very small the fluctuations in initial conditions.
time Figure
4:
(set)
Results
Plots of the square Fourier amplitude vs time for the dominant wavenumbers, k = 6 and k = 7, shown in Figure 3.
to scale the power spectra over a wide For the conditions. variety of run present, however, our best insight is provided by a much more crude measure, the first moments of the power spectra they evolve in If we as time. P(k), power spectrum as represent a then this moment, k, is
k = Ikp(k)dk!jp(k)dk. Figure In dimensionless
5
a time
(3) plot shows,
of
L after
"S an
with
a
Circular
Cell
Circular cells, like Hele-Shaw their rectangular counterparts, consist of two plates separated by a gap that is small relative to all other lengths in the problem. Flow is initiated by one fluid (in this case injecting nitrogen gas) at the center of the cell to displace a more viscous fluid (oil) that initially fills the space between driving A dimensionless the plates. the force, C 3 Qu/bo, characterizes flow conditions. Here, Q is the area1 b IS the rate of nitrogen, injection u is the oil viscosity, gap thickness, the surface tension of the and o is SO far, we oil-nitrogen interface. have observed flows for which C had a 1 and 35. in the range between value In the future we intend to push both to much lower and to much higher values. circular VlSCOUS fingering in a represents a more complicated geometry
Superlattices
Figure
6:
and Microstructures,
62
Vol. 3, No. 6, 1987
Patterns formed by a growing oil-nitrogen interface (C = 17) at half-eecond intervals
(see
ref.
2).
I
.
.
. -
.
C= 16,t=2.23,
Figure
7:
:
’
Typical plot of the radius of gyration (R,) vs the area of the injected fluid, A (see ref 2), shoving the power-law relation R, - A".*
problem ir that tip-splitting is xorced on the fingers. As a demonstration of this, Figure 6 the time shows development of a flow in a typical circular cell tip-splitting with clearly evident. For these flows we have plotted the radius of gyration vs total area of the developing patterns, obtaining results typically like the one shown in Figure 7. For all of our
0
40
80
I20
160
L (cm) Figure
8:
A typical plot of curvature vs arclength along an interface (see ref. 2).
200
622
Superlattrces
and Mmostructures.
Voi 3. ho
t? i9H
60
I IO3
GO_ h 5
e
C= 16,1=0.739
Figure
:: :
??
/ :. ‘:.
60
C= 16,
0.6
1.2
t=2.23S
1.8
2.4
W/cm) Figure
9:
IO7
A&-:
ITi :: :i ;’
40 -yy.::. ::i:: : .a..: ” ,:.‘.:
IO5
The power spectrum of the curvature as a function of arclength depicted in Figure 8 (see ref. 2).
10:
k vs the area of mixing, Aa, for many values of C (see driven ref. 2). The harder runs show k decreasing with An.
flow conditions, such plots indicate where a = 1.79 f that R, 1 (Area)"' 0.07. This value of a agrees reasonably well with the value of 1.70 f 0.05 obtained from DLA. Despite the extreme simplicity of DLA, it seems to contain such of the basic physics of realistic pattern formation. We used plots of curvature "8 arclength our circular patterns for ones just as we did for the rectangular and in the same way derived values of wavenumber. Typical the average examples of curvature vs arclength are given in Figure 8 while Figure 9 shows their respective Fourier transforms. Plots of the average k VI dimensionless area (as a measure of mixing zone dimensionless time~83) for effective all of our ride variety of runs were fall on an combined and found to "universal" curve shown in apparently Figure 10. The driving forces in these runs differed by as such as a factor of The curve shows an increase of k 35. at early times as the patterns ramify. At times I; turns over and later eventually decreases as progressively more of the pattern, left behind by "anneals" in the low advancing fingers, pressure *fjords" thus gradient and introduces longer length into scales the curvature distributions. c is clearly a crude poorly quantity, designed to give a clear picture of this complicated pattern evolution, but
S~lperlattices
and Microstructures,
Vol. 3, No. 6, 1987
attractive to 1t is think of the temporal evolution of i; arising from a competition between rapid ramification in the steep pressure gradient region near the finger tips and surface tension driven annealing in the fjords. I" this picture the early stages of flow are dominated by the former effect while the latter effect governs later stages. This description is very reminiscent of the coarsening of dendrites. Whether the annealing in the fjords would eventually so fill in the fjords as to raise the fractal dimension away from the DLA value and Over toward a dense branching morphologyCll1 is not as yet clear and ~111 be addressed in our next series of experiments. I" summary, the tantalizing preliminary result presented above concerns the secular development of average wavenumbers E for power spectra of curvatures from both initially planar and circular viscous fingering patterns. These E, after a complicated initially ramifying flow period, decrease with time and represent a coursening of the pattern in the quiet regions behind the advancing fingers to a point where the interfacial capillary length is small in comparison to typical long smooth sections of the interface. Acknowledgement-This work was supported by the U. S. Department of Energy under grant #DE-FG02-84ER45131.
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