Aqueous Magnetic Fluids Instabilities in a Hele-Shaw Cell

Aqueous Magnetic Fluids Instabilities in a Hele-Shaw Cell M. RÄ‚CUCIU â€Å"Lucian Blaga† University, Faculty of Science, Dr.Ratiu Street, No.5-7, Sibiu, 550024, Romania In this study, it was investigated the interface patterns of two immiscible, viscous fluids into the geometry of a horizontal  Hele-Shaw cell, considering that one of the fluids is an aqueous magnetic fluid. In general, the total energy of a magnetic fluid  system consists of three components: gravitational, surface, and magnetic. For a magnetic fluid into a horizontal Hele-Shaw  cell, the gravitational energy is constant and can be inessential, thus leaving only the surface and magnetic energies. The  interface between two immiscible fluids, one of them with a low viscosity and another with a higher viscosity, becomes  unstable and starts to deform. Dynamic competition in such confined geometry leads finally to the formation of fingering  patterns of a magnetic fluid in a Hele-Shaw cell. The computational study upon the interface instabilities patterns in aqueous  magnetic fluids was accomplished evaluating fractal dimension in finger-type instabilities. Between the fr actal dimension and  experimental instability generation time a correlation was established. (Received January 30, 2009; accepted February 13, 2009) Keywords: Magnetic fluid, Hele-Shaw cell, Viscous finger instability 1. Introduction Magnetic fluids are composed of magnetic, around 10  nm, single-domain particles coated with a molecular  surfactant and suspended in a carrier liquid [1]. Magnetic  fluids confined within Hele-Shaw cell exhibit interesting  interfacial instabilities, like fingering instability. The patterns and shapes formation by diverse physical,  chemical and biological systems in the natural world has  long been a source of fascination for scientists [2]. Interface  dynamics plays a major role in pattern formation. Viscous  fingering occurs in the flow of two immiscible, viscous  fluids between the plates of a Hele-Shaw cell. A magnetic fluid is considered paramagnetic because  the individual nanoparticle magnetizations are randomly  oriented, even the solid magnetic nanoparticles are  ferromagnetic [3]. Due to Brownian motion, the thermal  agitation keeps the magnetic nanoparticles suspended and  the coating prevents the nanoparticles from adhering to  each other. In ionic magnetic fluids coating the magnetic  nanoparticles are replaced one an other by electrostatic  repulsion. Because the magnetic nanoparticles are much  smaller than the Hele-Shaw cell thickness, it may neglect  their particulate properties and it may consider a continuous  paramagnetic fluid. In this study, there was investigated the interface  patterns of two immiscible, viscous fluids into the geometry  of a horizontal Hele-Shaw cell, considering that one of the  fluids is an aqueous magnetic fluid. In general, the total energy of a magnetic fluid system  consists of three components: gravitational, surface, and  magnetic. For a magnetic fluid into a horizontal Hele-Shaw  cell, the gravitational energy is constant and can be  inessential, thus leaving only the surface and magnetic  energies. The interface between two immiscible fluids, one of  them with a low viscosity and another with a higher  viscosity, becomes unstable and starts to deform. Dynamic  competition in such confined geometry leads finally to the  formation of fingering patterns of a magnetic fluid in a  two-dimensional geometry (Hele-Shaw cell) [4-5]. Due to pressure gradients or gravity, the separation  interface of the two immiscible, viscous fluids undergoes  on to Saffman-Taylor instability [6] and develops  finger-like structures. The Saffman-Taylor instability is a  widely studied example of hydrodynamic pattern formation  where interfacial instabilities evolve. The computational study upon the interface  instabilities patterns in aqueous magnetic fluids was carried  out by evaluating the fractal dimension in finger-type  instabilities. 2. Experimental Aqueous magnetic fluids used in this study have been  prepared in our laboratory, by applying the chemical  precipitation method. In table I are presented the aqueous  magnetic fluid samples used in this experimental study. Table 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 cell In this paper, it studies the interface between two  immiscible fluids and with different viscosities, one of  them with a low viscosity and with magnetic properties  (aqueous magnetic fluid) and another with a higher  viscosity and non-magnetic (65% aqueous glycerin), when  the instability pattern diameter increased in time, instability  structure became more and more complex. The splitting of  the main finger streamer was occurred in time, determining  the irregularity increasing. For each instant image obtained  was computed the fractal dimension. a less viscous fluid is injected into a more viscous one in a  2D geometry (Hele Shaw cell). Fig. 1 shows a Hele-Shaw cell, used in this experiment,  with its two plates of plastic separated by cover microscope  slides placed in each corner of the cell, having the same  thickness about 300 micrometers. The used fluids are  injected through the center hole. Fig. 2. The finger instability dynamics for aqueous  magnetic liquid stabilized with citric acid (A2 sample). Between the fractal dimension and experimental  instability generation time a linear correlations were  established for all magnetic fluid samples used in this study  (correlation coefficient, R2, 0.962). In Fig. 3 is presented  the dynamics of the fractal dimension during the surface  image recording, for A1 magnetic fluid sample. Fig. 1. Hele- Shaw cell used in this experimental study. In the experiment firstly was injected aqueous glycerin  through the central hole to fill the cell. After the cell was  filled with glycerin, the magnetic fluid was injected. The  surface image recordings were made with a digital camera. The computational study upon the interface  instabilities patterns between aqueous magnetic fluids and  glycerin was accomplished evaluating fractal dimension in  finger-type instabilities. The fractal analysis was carried out  using the box-counting method [7] as a computational  algorithm. In order to apply box-counting method the  surface images were analyzed following the HarFA 5.0  software steps, computing the fractal dimension of the each  image. Fig. 3. The dynamics of the fractal dimension in the A1  magnetic fluid sample  case, stabilized with 3. Results and discussion The computational study was accomplished on three  sets of images, for aqueous magnetic fluids analyzed in this  study, representing finger instabilities developed between  two immiscible fluids and with different viscosities. In Fig. 2 are presented the finger instability dynamics  for aqueous magnetic liquid stabilized with citric acid (A2  sample), having a radial symmetry between the fluids  injected in the Hele-Shaw cell (glycerin and magnetic  fluid).From images shown in Fig. 2 it may be observed that  tetrametilamoniu hydroxide. Also, between the fractal dimension and physical  diameter of the magnetic nanoparticles, constituents of the  magnetic fluid sample, a correlation was established (a  linear regression with a correlation coefficient, R2 = 0.926). In Fig. 4 it may be observed that for increasing physical  diameter of the magnetic nanoparticles was obtained a  decreased fractal dimension, after a flow time of magnetic fluids about 9 seconds. 134 Fig. 4. Graphical dependence of fractal dimension in  function of physical diameter of the magnetic  nanoparticles, constituents of the magnetic fluid samples  analyzed in this study. M. Răcuciu within the Hele-Shaw cell about 7 seconds. In the Saffman-Taylor instability, if a forward bump is  formed on the interface between the fluids, it enhances the  pressure gradient and the local interface velocity. Because  the velocity of a point on the interface is proportional to the  local pressure gradient, the bump grows faster than other  parts on the interface. On the other hand, the effect of  surface tension competes with this diffusive instability. Surface tension operates to reduce the pressure at highly  curved parts of an interface, and sharp bumps are forced  back. Thus, as a result we have the formation of the viscous  finger instabilities. The fractal dimension based analysis proposed in this  paper of the aqueous magnetic fluid instability is intended  to lead to further mathematical modeling of finger  instabilities patterns focused on non-linearity of the  magnetic fluid-non-magnetic fluid interface stability. A linear correlation (correlation coefficient, R2, 0.988)  was established between the fractal dimension and  viscosity value of the magnetic fluid samples used In Fig. 5  it may be observed that for increased viscosity value of the  magnetic fluid was observed increasing the fractal  dimension of the interface fluids instability structure. Fig. 5. Linear correlation between fractal dimension and  magnetic fluid viscosity value. Table 2. Correlation coefficient and standard deviation to  the fractal dimension calculation after a flow time of  magnetic fluids within the Hele-Shaw cell about 7  seconds. 4. Conclusions In this paper it was investigated the flow of two  immiscible, viscous fluids in the confined geometry of a  Hele-Shaw cell. It may conclude that the fractal dimension values of the  finger instabilities pattern images are direct proportional  with the magnetic fluid viscosity value and instability  generation time, while with the physical diameter of the  magnetic nanoparticles constituents of the magnetic fluid a  linear negative dependence was evidenced. The next theoretical analysis step will follow the  developing of a convenient model to describe the  non-magnetic fluid influence on magnetic fluid surface  stability. References [1] P. S. Stevens, Patterns in Nature, Little Brown, Boston  (1974). [2] S.S. Papel, Low viscosity magnetic fluid obtained by  the colloidal suspension of magnetic particles. US  Patent 3, 215, 572 (1965). [3] R. E. Rosensweig, Ferrohydrodynamics, Cambridge  University 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 Fractal  Dimension for Geometrical Multifractals, Physica A,  159, 155-166 (1989). ________________________ * In Table 2 the fractal dimension value, correlation  coefficient and standard deviation to the fractal dimension  calculation, for all magnetic fluid samples used in this experimental study after a flow time of magnetic fluids