Aqueous Magnetic Fluids Instabilities in a Hele-Shaw Cell

Aqueous Magnetic Fluids Instabilities in a Hele-Shaw CellM. RCUCIULucian Blaga University, Faculty of Science, Dr.Ratiu Street, No.5-7, Sibiu, 550024, RomaniaIn this subscribe to, it was investigated the user interface images of two non-miscible, viscous fluids into the geometry of a horizontalHele-Shaw booth, considering that one of the fluids is an aqueous charismatic fluid. In general, the total nada of a magnetised fluidsystem consists of tercet components gravitational, surface, and magnetized. For a magnetised fluid into a horizontal Hele-Shawcell, the gravitational energy is constant and can be inessential, thus leaving only the surface and charismatic energies. Theinterface between two immiscible fluids, one of them with a miserable viscosity and another with a higher viscosity, becomesunstable and starts to deform. Dynamic competition in such confined geometry leads finally to the formation of fingeringpatterns of a magnetized fluid in a Hele-Shaw cell. The comp utational get hold of upon the interface instabilities patterns in aqueousmagnetic fluids was accomplished evaluating fractal proportionality in finger-type instabilities. Between the fractal dimension and observational dissymmetry generation time a correlation was schematic.(Received January 30, 2009 accepted February 13, 2009)Keywords Magnetic fluid, Hele-Shaw cell, steamy finger instability1. IntroductionMagnetic fluids are composed of magnetic, around 10nm, single-domain particles coated with a molecularsurfactant and suspended in a carrier liquid 1. Magneticfluids confined within Hele-Shaw cell exhibit interestinginterfacial instabilities, like fingering instability.The patterns and shapes formation by diverse physical, chemical and biological systems in the natural world haslong been a source of fascination for scientists 2. Interfacedynamics plays a major role in pattern formation. Viscousfingering occurs in the flow of two immiscible, viscousfluids between the plates of a Hele-Shaw cell.A magnetic fluid is considered paramagnetic becausethe individual nanoparticle magnetizations are willy-nillyoriented, even the solid magnetic nanoparticles areferromagnetic 3. Due to Brownian motion, the thermalagitation keeps the magnetic nanoparticles suspended andthe coating prevents the nanoparticles from adhering toeach other. In ionic magnetic fluids coating the magneticnanoparticles are replaced one an other by electrostaticrepulsion. Because the magnetic nanoparticles are muchsmaller than the Hele-Shaw cell thickness, it whitethorn neglecttheir particulate matter properties and it may consider a continuousparamagnetic fluid.In this study, there was investigated the interfacepatterns of two immiscible, viscous fluids into the geometryof a horizontal Hele-Shaw cell, considering that one of thefluids is an aqueous magnetic fluid.In general, the total energy of a magnetic fluid systemconsists of three components gravitational, surface, andmagnetic. For a magn etic fluid into a horizontal Hele-Shawcell, the gravitational energy is constant and can beinessential, thus leaving only the surface and magneticenergies.The interface between two immiscible fluids, one ofthem with a low viscosity and another with a higherviscosity, becomes unstable and starts to deform. Dynamiccompetition in such confined geometry leads finally to theformation of fingering patterns of a magnetic fluid in atwo-dimensional geometry (Hele-Shaw cell) 4-5.Due to pressure gradients or gravity, the separationinterface of the two immiscible, viscous fluids undergoeson to Saffman-Taylor instability 6 and developsfinger-like structures. The Saffman-Taylor instability is awidely studied example of hydrodynamic pattern formationwhere interfacial instabilities evolve.The computational study upon the interfaceinstabilities patterns in aqueous magnetic fluids was carriedout by evaluating the fractal dimension in finger-typeinstabilities.2. ExperimentalAqueous magnetic fluids em ploy in this study have beenprepared in our laboratory, by applying the chemicalprecipitation method. In table I are presented the aqueousmagnetic fluid samples employ in this experimental study. prorogue 1. The magnetic fluid samples used in this study (d physical diameter of the magnetic nanoparticles,constituents of the magnetic fluid samples).Aqueous magnetic fluids instabilities in a Hele-Shaw cellIn this paper, it studies the interface between twoimmiscible fluids and with different viscosities, one ofthem with a low viscosity and with magnetic properties(aqueous magnetic fluid) and another with a higherviscosity and non-magnetic (65% aqueous glycerin), whenthe instability pattern diameter increased in time, instabilitystructure became more and more complex. The splitting ofthe main finger streamer was occurred in time, determiningthe irregularity increasing. For each fast image obtainedwas computed the fractal dimension.a less viscous fluid is injected into a more viscous o ne in a2D geometry (Hele Shaw cell).Fig. 1 shows a Hele-Shaw cell, used in this experiment,with its two plates of plastic separated by cover microscopeslides placed in each corner of the cell, having the samethickness about 300 micrometers. The used fluids areinjected through the center hole.Fig. 2. The finger instability dynamics for aqueousmagnetic liquid stabilized with citric acid (A2 sample).Between the fractal dimension and experimentalinstability generation time a additive correlations wereestablished for all magnetic fluid samples used in this study(correlation coefficient, R2, 0.962). In Fig. 3 is presentedthe dynamics of the fractal dimension during the surfaceimage recording, for A1 magnetic fluid sample.Fig. 1. Hele- Shaw cell used in this experimental study.In the experiment firstly was injected aqueous glycerinthrough the central hole to fill the cell. After the cell wasfilled with glycerin, the magnetic fluid was injected. Thesurface image recordings were made with a digital camera.The computational study upon the interfaceinstabilities patterns between aqueous magnetic fluids andglycerin was accomplished evaluating fractal dimension infinger-type instabilities. The fractal analysis was carried outusing the box-counting method 7 as a computationalalgorithm. In order to apply box-counting method thesurface images were analyzed following the HarFA 5.0software steps, computing the fractal dimension of the eachimage.Fig. 3. The dynamics of the fractal dimension in the A1magnetic fluid samplecase, stabilized with3. Results and discussionThe computational study was accomplished on threesets of images, for aqueous magnetic fluids analyzed in thisstudy, representing finger instabilities real betweentwo immiscible fluids and with different viscosities.In Fig. 2 are presented the finger instability dynamicsfor aqueous magnetic liquid stabilized with citric acid (A2sample), having a radial symmetry between the fluidsinjected in the Hele-Shaw cell (glyce rin and magneticfluid).From images shown in Fig. 2 it may be observed thattetrametilamoniu hydroxide.Also, between the fractal dimension and physicaldiameter of the magnetic nanoparticles, constituents of themagnetic fluid sample, a correlation was established (alinear regression with a correlation coefficient, R2 = 0.926).In Fig. 4 it may be observed that for increasing physicaldiameter of the magnetic nanoparticles was obtained adecreased fractal dimension, after a flow time of magneticfluids about 9 seconds.134Fig. 4. Graphical dependence of fractal dimension infunction of physical diameter of the magneticnanoparticles, constituents of the magnetic fluid samplesanalyzed in this study.M. Rcuciuwithin the Hele-Shaw cell about 7 seconds.In the Saffman-Taylor instability, if a forward bump isformed on the interface between the fluids, it enhances thepressure gradient and the local interface f number. Becausethe velocity of a point on the interface is proportional to thelocal pressur e gradient, the bump grows faster than otherparts on the interface. On the other hand, the set up ofsurface tension competes with this diffusive instability.Surface tension operates to reduce the pressure at highlycurved parts of an interface, and sharp bumps are forcedback. Thus, as a result we have the formation of the viscousfinger instabilities.The fractal dimension based analysis proposed in thispaper of the aqueous magnetic fluid instability is intendedto lead to further mathematical modeling of fingerinstabilities patterns focused on non-linearity of themagnetic fluid-non-magnetic fluid interface stability.A linear correlation (correlation coefficient, R2, 0.988)was established between the fractal dimension andviscosity appreciate of the magnetic fluid samples used In Fig. 5it may be observed that for increased viscosity value of themagnetic fluid was observed increasing the fractaldimension of the interface fluids instability structure.Fig. 5. Linear correlation between fr actal dimension andmagnetic fluid viscosity value.Table 2. Correlation coefficient and standard deviation tothe fractal dimension calculation after a flow time ofmagnetic fluids within the Hele-Shaw cell about 7seconds.4. ConclusionsIn this paper it was investigated the flow of twoimmiscible, viscous fluids in the confined geometry of aHele-Shaw cell.It may conclude that the fractal dimension values of thefinger instabilities pattern images are cypher proportionalwith the magnetic fluid viscosity value and instabilitygeneration time, while with the physical diameter of themagnetic nanoparticles constituents of the magnetic fluid alinear negative dependence was evidenced.The next theoretical analysis step will follow thedeveloping of a convenient model to describe thenon-magnetic fluid settle on magnetic fluid surfacestability.References1 P. S. Stevens, Patterns in Nature, Little Brown, Boston(1974).2 S.S. Papel, Low viscosity magnetic fluid obtained bythe colloidal suspension of m agnetic particles. USPatent 3, 215, 572 (1965).3 R. E. Rosensweig, Ferrohydrodynamics, CambridgeUniversity Press, Cambridge (1985).4 C. Tang, Rev. Mod. Phys., 58, 977 (1986).5 D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman,C. Tang, Rev.Mod. Phys., 58, 977 (1986).6 P. G. Saffman, G. I. Taylor, Proc. R. Soc. London, Ser.A, 245, 312 (1958).7 T. Tel, A. Fulop, T. Vicsek, Determination of FractalDimension for Geometrical Multifractals, Physica A,159, 155-166 (1989).________________________*In Table 2 the fractal dimension value, correlationcoefficient and standard deviation to the fractal dimensioncalculation, for all magnetic fluid samples used in this experimental study after a flow time of magnetic fluids