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Ferrofluid Capillary Rise in Porous Medium under the Action of Non-Uniform Magnetic Field

The capillary flow of a ferrofluid in a single cylindrical capillary tube and through a sandy porous medium under the action of a non-uniform magnetic field is studied experimentally. The dynamics of the capillary rise and the static case have been considered. It has been shown that the non-uniform magnetic field with upward directed gradient accelerates the capillary rise; contrary, the non-uniform magnetic field with downward directed gradient decelerates the capillary rise. Time dependences of the ferrofluid height and maximum reachable height of ferrofluid have been analyzed. The method of the study of ferrofluid capillary rise based on the use of magnetic measurements has been proposed. It has been demonstrated that porous material parameters can be extracted from the results of measurements of the inductances of the solenoid with porous medium inside and the small sensing coil within a single experiment.


INTRODUCTION
Capillary driven flow is an important field of research because it has many applications in science, industry and daily life [1][2][3]. Fluid flows in porous media are strongly influenced by capillarity effects. Examples where fluid flow in porous media plays an important role are drainage and irrigation of agricultural lands, groundwater movement, nuclear waste storage, oil and gas winning, drying of food materials, heat pipes, spacecraft propellant management devices, marker pens, etc. [4]. Investigations of a ferrofluid flow through the porous media have found some new features connected with the capability of influence of external magnetic fields [5]. A ferrofluid (or magnetic fluid) is a colloidal suspension of ultra-fine ferro-or ferrimagnetic nanoparticles suspended in a carrier fluid. Because of the small size of the ferromagnetic particles, ferrofluids can flow through porous media such as natural sediments or fractured rock due to gravitational, pressure gradient, capillary, and magnetic forces [6]. The ferrofluid flow can be manipulated through control of the external magnetic field and without any direct For the better understanding of the mechanisms of this flow control by the magnetic field, it is important to study ferrofluid behavior in a separate capillary tube without any porous material. Previously several aspects of ferrofluid motions in a separate capillary tube have been considered [7][8][9]. The stationary height (as a characteristic of capillary force) of a ferrofluid in a separate capillary under the action of uniform magnetic field has been analyzed in [7,8]. The effect of the uniform magnetic field on the dynamics of magnetic fluid ascension into a vertical cylindrical capillary has been analyzed in [9]. It was found that due to the fluid meniscus deformation, the surface pressure drop decreases in the magnetic field (directed transverse to the capillary) leading to the decrease in the height and the velocity of the capillary ascension. The horizontal and vertical displacements of some volume of ferrofluid in sandy porous media under the action of lateral magnetic force produced by a permanent magnet have been studied in [10,11]. It was shown that gravitational and magnetic forces result in a predictable configuration of the fluid in the porous medium. Different aspects of ferrofluid motions in porous media have been considered in theoretical studies [12][13][14][15][16][17][18][19]. The capillary ascension of ferrofluid in a porous medium has not been studied previously. The capillary ascension of ferrofluid affected by the non-uniform magnetic field producing the net force on the ferrofluid has also not been studied in the existing works. The work presented here extends the previous works by considering the dynamics of ferrofluid capillary rise in porous media under the action of non-uniform magnetic fields. The linear capillary transport and capillary rise experiments are commonly performed in cylindrical vertical columns filled with porous material, using visual methods to register the advance of the wetting front [20,21]. We will follow such methodology in our study.

MATERIALS AND METHODS
The experimental setup used in our experiments is shown in Fig. 1. It consists of a glass rectangular reservoir filled with ferrofluid. A stable ferrofluid level in the reservoir was kept by the overhead tank (not shown in Fig. 1). First we examined the ferrofluid rise in a single capillary. To do this, the cylindrical glass tube with the inner diameter of 200 μm was inserted into a reservoir with ferrofluid. The glass tube was supplied by OJSC "THERMOPRIBOR" (Russia). The tube was installed on a support mount (not shown in Fig. 1) and vertically oriented. The tube was also labeled with a millimeter scale. Once the bottom of the capillary tube contacted with ferrofluid surface, the capillary rise of ferrofluid was observed and recorded by a high-speed digital camera (Casio EXILIM EX-F1). An analogy exists between linear capillary transport in a cylindrical tube and linear capillary transport in a porous structure. The advantage of the capillary tube is that (in contrast to porous media) all of its corresponding pore structure parameters are known with the tube radius and the porosity equal to unity. Thus, as the physical processes occurring in both systems are the same, the capillary tubes can be considered as a good model medium for theoretical and experimental investigations of the flow in other porous structures. To study the ferrofluid rise in porous medium we used a cylindrical glass tube with the inner diameter of 5 mm filled with sand and inserted into a reservoir with ferrofluid. The length of the tube was larger than the maximum reachable fluid height.
We used fine deposited river sand with a low content of organic matter provided by local supplier. Sand grain diameters were ranged from 0.03 to 0.3 mm. The microscopic image of the grains of the sand sample used in the experiments is shown in Fig. 2. Sand samples were cleaned using the distilled water, dried in an oven at 105 °C and then cooled at room temperature. A moving interface formed between the wet and non-wet regions of the sandy porous medium was observed and recorded by a digital camera.
An electromagnet driven by a dc current source has provided a non-uniform magnetic field. To produce the non-uniform magnetic field with downward directed gradient the electromagnet poles were placed at the bottom of the glass tube ( Fig. 1). To produce the upward directed magnetic field gradient the electromagnet poles were placed at the top of the glass tube. The tube has been mounted along the centerline between the electromagnet poles. The external magnetic field strength H produced by the electromagnet has been measured along the glass tube. It has been revealed that the magnetic field variation with height should only be considered. For the tube centrally located between the two poles, the magnetic field change within the horizontal cross section of a glass tube can be neglected. The measurements have shown that the magnetic field changes roughly linearly with height. For the simplicity of calculations and experimental results analysis we will assume that the magnetic field gradient in vertical direction is constant and its absolute value is equal to 10 6 A/m 2 . The corresponding results of measurements of the magnetic field strength as a function of the distance from the electromagnet poles are shown in Fig. 3.
The fact that a ferrofluid has magnetic properties makes it possible to magnetically sense flow in porous media. To this end we used a solenoid with the sand inside instead of just a glass tube. The solenoid inner diameter was 5 mm and its length was 6.5 cm. The tube used in this experiment is like that used in the experiment on the visual observation of ferrofluid motion in sand.
All the described measurements of the wetting front location have been repeated three times and the mean value was plotted on the graphs. The upper and lower bounds are shown as error bars.
In our experiments we used a kerosene-based ferrofluid with dispersed magnetite nanoparticles of about 10 nm diameter stabilized with oleic acid. The properties of the ferrofluid are: density is ρ = 1000 kg·m −3 , dynamic viscosity is η = 5 mPa·s, initial magnetic permeability is μ = 1.85, interfacial tension at the ferrofluid-air interface is σ = 0.027 N/m. The ferrofluid used was made by OJSC "NIPIgaspererabotka" (Russia). The magnetization curve of ferrofluid is presented in Fig. 4. The magnetization curve has been obtained by the use of a vibrating sample magnetometer (Lake Shore 7410-S). Here the magnetization is plotted as a function of internal magnetic field which has been calculated in a standard way from the applied magnetic field using the demagnetizing factor of the ferrofluid sample shape. Magnetic force acting on the ferrofluid depends not only on the magnetic field gradient but also on the magnetic field strength as the magnetization of ferrofluid grows with the magnetic field increasing. Figure 6 shows the experimental dependences of the ferrofluid maximum (static) height h max in a cylindrical capillary on the magnetic field strength at the bottom of capillary H 0 obtained by visual observations. As it can be seen, the maximum height decreases with the magnetic field increasing when the magnetic field gradient is down directed and increases when the magnetic field gradient is upward directed. In these experiments the magnetic field gradient has changed only slightly during the magnetic field strength H 0 changing as it can be seen from Fig. 3.

ANALYSIS OF THE RESULTS
The momentum balance of a ferrofluid inside a single capillary tube in the presence of a non-uniform magnetic field gives Here R is the inner tube radius, θ is the contact angle formed between solid and liquid, where N is the demagnetizing factor, which was set equal to 0.5 for the cylindrical ferrofluid column under study [24]. Here the external magnetic field can be considered to be perpendicular to the capillary. The results of numerical solution of Eqs. (1) and (2) are presented in Fig. 5. As is seen from Fig. 5, the experimental and theoretical results are in a good agreement.
The static balance clearly yields the maximum rise height, i.e. the extended Jurin's law [25]. One can obtain Equation (3) numerical solution is presented in Fig. 6. A qualitative agreement between Eq. (3) and the experimental data can be seen from Fig. 6.
To analyze the ferrofluid rise in a sandy porous medium we will follow the works [21,23,26]. The momentum balance can also be given for the capillary rise of ferrofluid in a porous medium; here the viscous term is replaced by the Darcy law where φ is the porosity of the sand, and K its permeability, R is the pore mean radius.
There are different approaches to determine these parameters. Here we will use the experimental results described above to extract the values of the pore structure parameters. The pore radius can be calculated from the maximum obtainable (static) height h max . The corresponding calculations show that R = 55 μm. The porosity can be calculated from the maximum relative inductance of the solenoid with sand inside using Bruggeman's formula for the magnetic permeability of disperse system [27]: where μ ef is the effective magnetic permeability of medium, μ i is the magnetic permeability of internal phase, μ e is the magnetic permeability of external phase.
Bruggeman's formula is valid in a wide range of concentrations of disperse system (up to 100 vol. %). In our case the internal phase is presented by sand μ i = 1, the external phase is presented by ferrofluid e 1.85     , the effective magnetic permeability is equal to the maximum relative inductance of solenoid μ ef = L max /L 0 . Here we assume that the porosity is equal to the saturation, which is only the case if the pore space is completely filled with ferrofluid. The porosity is calculated to be φ = 0.38 in our case. The permeability K can be obtained by fitting the calculated according to Eq. (4) ferrofluid rise curve to experimental values (Fig. 7). Numerical solution of Eq. (4) at the absence of external magnetic field and comparison with the corresponding experimental values give K = 1.5·10 -11 m 2 . Using the obtained values of the pore structure parameters we can also calculate the ferrofluid rise curve in the presence of external non-uniform magnetic field (Fig. 7). As is seen from Fig. 7, in the case of downward directed magnetic field gradient the difference between the experimental and theoretical results is notable, on the other hand in the case of upward directed magnetic field gradient the experimental and theoretical results are in a good agreement (the difference is no more than 10%). When the ferrofluid penetrates into the porous medium the strong magnetic field can influence the medium microstructure. It is known that nonmagnetic particles immersed in the ferrofluid (sand grains in our case) can be arranged along the magnetic field lines. The force of interaction between non-magnetic particles in ferrofluid leading to the structure formation is proportional to the magnetic field strength, not to the magnetic field gradient. When the magnetic field gradient is downward directed, the region of penetration of ferrofluid into the porous medium is located in a strong magnetic field. So the structure formation will be more intensive in this case than in the case of upward directed gradient, although the absolute values of field gradient are the same in the both cases. This can lead to the resulting difference of experimental and theoretical results when the magnetic field gradient is down-directed.
In the above calculations it has been assumed that the magnetic field does not change the shape of a ferrofluid meniscus. Let us estimate the validity of this assumption. the magnetic pressure makes less than 0.1 of the capillary pressure for the porous medium of sand, and less than 0.2 for the single capillary tube studied. It can be concluded that the change of the meniscus shape caused by the magnetic field should take place but it will be not so notable. The presented estimation demonstrates the possible value of error contributed by the neglect of the magnetic field effect on the ferrofluid meniscus shape.
Consider the change of the solenoid inductance during the ferrofluid capillary rise.
The inductance can be presented as a sum of inductances of wetted and dry regions: L = L 1 + L 2 . We have: L 1 = μ 0 μ ef n 2 hS, L 2 = μ 0 n 2 lS, where μ 0 is the magnetic constant μ 0 = 4π·10 -7 H/m, n is the number of turns per unit length, S is the solenoid cross-section area, l is the length of dry region. Taking into account that: l 0 = h + l, L 0 = μ 0 n 2 l 0 S, L max = μ 0 μ ef n 2 l 0 S, it is easy to obtain where l 0 is the solenoid total length. Figure 10 shows the ferrofluid rise curve calculated according to Eq. (6) from the data presented in Fig. 8 in comparison with the corresponding results obtained by visual observations in zero field. As is seen, the results obtained by means of magnetic measurements are in a good agreement with the results of visual observations (the difference is no more than 10%).

DISCUSSION AND CONCLUSIONS
The study of liquid capillary rise in a cylindrical column of porous material is used to extract the pore structure parameters [20,21,23,26].