Ferrofluid Permeation into Three-Dimensional Random Porous Media: A Numerical Study Using the Lattice Boltzmann Method

Abstract

Colloidal suspensions containing magnetic nanoparticles placed in appropriate carrier liquids present strong magnetic dipoles. These suspensions, in general, exhibit normal liquid behaviour coupled with super paramagnetic properties. This leads to the possibility of remotely controlling the flow of such liquids with a moderate-strength external magnetic field. In this study, we numerically investigate the viability of controlling and steering a base-fluid with magnetic-sensitive nanoparticles into randomly structured fibrous porous media. Three dimensional flow simulations are performed using the lattice Boltzmann method. The simulation results for the flow front are presented, and the effect of the magnetic field strength on the rate of ferrofluid penetration is discussed. It is shown that the porosity of the porous medium and the size of the fibres have a strong effect on the ferrofluid penetration rate.

Appendix

The local coordinate system for a magnetic field produced by a permanent magnet is shown in Fig. 12. Note that in this figure the $z$ axis in local magnet coordinate system and that for the flow simulations are in different directions.

Fig. 12

Local coordinate system for the magnetic field produced by a permanent magnet (Oldenburg et al. 2000)

McCaig and Clegg (1987) presented the three-dimensional relations for the magnetic field strength inside a rectangular channel, as follows:

$$\begin{aligned} {\textit{Hx}}&= \frac{{B}_{{r}}}{4\pi \mu _0}\text{ ln }\left\{ {\frac{({y}+{b})+\left[ {\left( {{y}+{b}} \right) ^{2}+({x}-{a})^{2}+{z}^{2}} \right] ^{\frac{1}{2}}}{({y}-{b})+\left[ {\left( {{y}-{b}} \right) ^{2}+({x}-{a})^{2}+{z}^{2}} \right] ^{\frac{1}{2}}}\times \frac{({y}-{b})+\left[ {\left( {{y}-{b}} \right) ^{2}+({x}+{a})^{2}+{z}^{2}} \right] ^{\frac{1}{2}}}{({y}+{b})+\left[ {\left( {{y}+{b}} \right) ^{2}+({x}+{a})^{2}+{z}^{2}} \right] ^{\frac{1}{2}}}} \right\} \nonumber \\&\qquad \qquad \qquad -\frac{{B}_{{r}}}{4\pi \mu _0}\text{ ln }\left\{ \frac{({y}+{b})+\left[ {\left( {{y}+{b}} \right) ^{2}+\left( {{x}-{a}} \right) ^{2}+\left( {{z}+{L}} \right) ^{2}} \right] ^{\frac{1}{2}}}{\left( {{y}-{b}} \right) +\left[ {\left( {{y}-{b}} \right) ^{2}+\left( {{x}-{a}} \right) ^{2}+\left( {{z}+{L}} \right) ^{2}} \right] ^{\frac{1}{2}}}\right. \nonumber \\&\qquad \qquad \qquad \left. \times \frac{({y}-{b})+\left[ {\left( {{y}-{b}} \right) ^{2}+\left( {{x}+{a}} \right) ^{2}+({z}+{L})^{2}} \right] ^{1/2}}{\left( {{y}+{b}} \right) +\left[ {\left( {{y}+{b}} \right) ^{2}+\left( {{x}+{a}} \right) ^{2}+({z}+{L})^{2}} \right] ^{\frac{1}{2}}} \right\} \end{aligned}$$

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$$\begin{aligned} {\textit{Hy}}&= \frac{B_r}{4\pi \mu _0 }\text{ ln }\left\{ {\frac{(x+a)+\left[ {\left( {y-b} \right) ^{2}+(x+a)^{2}+z^{2}} \right] ^{\frac{1}{2}}}{(x-a)+\left[ {\left( {y-b} \right) ^{2}+(x-a)^{2}+z^{2}} \right] ^{\frac{1}{2}}}\times \frac{(x-a)+\left[ {\left( {y+b} \right) ^{2}+(x-a)^{2}+z^{2}} \right] ^{\frac{1}{2}}}{(x+a)+\left[ {\left( {y+b} \right) ^{2}+(x+a)^{2}+z^{2}} \right] ^{\frac{1}{2}}}} \right\} \nonumber \\&\qquad \qquad \qquad -\frac{B_r }{4\pi \mu _0} \text{ ln }\left\{ \frac{(x+a)+\left[ {\left( {y-b} \right) ^{2}+\left( {x+a} \right) ^{2}+\left( {z+L} \right) ^{2}} \right] ^{\frac{1}{2}}}{\left( {x-a} \right) +\left[ {\left( {y-b} \right) ^{2}+\left( {x-a} \right) ^{2}+\left( {z+L} \right) ^{2}} \right] ^{\frac{1}{2}}}\right. \nonumber \\&\qquad \qquad \qquad \left. \times \frac{(x-a)+\left[ {\left( {y+b} \right) ^{2}+\left( {x-a} \right) ^{2}+(z+L)^{2}} \right] ^{1/2}}{\left( {x+a} \right) +\left[ {\left( {y+b} \right) ^{2}+\left( {x+a} \right) ^{2}+(z+L)^{2}} \right] ^{\frac{1}{2}}} \right\} \end{aligned}$$

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$$\begin{aligned} {\textit{Hz}}&= \frac{B_r}{4\pi \mu _0}\left\{ \tan ^{-1}\left[ {\frac{\left( {x+a} \right) \left( {y+b} \right) }{z\left[ {\left( {y+b} \right) ^{2}+\left( {x+a} \right) ^{2}+z^{2}} \right] ^{\frac{1}{2}}}} \right] +\tan ^{-1}\left[ {\frac{\left( {x-a} \right) \left( {y-b} \right) }{z\left[ {\left( {y-b} \right) ^{2}+\left( {x-a} \right) ^{2}+z^{2}} \right] ^{\frac{1}{2}}}} \right] \right. \nonumber \\&\qquad \qquad \qquad -\left. \tan ^{-1}\left[ {\frac{\left( {x+a} \right) \left( {y-b} \right) }{z\left[ {\left( {y-b} \right) ^{2}+\left( {x+a} \right) ^{2}+z^{2}} \right] ^{\frac{1}{2}}}} \right] -\tan ^{-1}\left[ {\frac{\left( {x-a} \right) \left( {y+b} \right) }{z\left[ {\left( {y+b} \right) ^{2}+\left( {x-a} \right) ^{2}+z^{2}} \right] ^{\frac{1}{2}}}} \right] \right\} \nonumber \\&\qquad \qquad \qquad -\frac{B_r }{4\pi \mu _0}\left\{ \tan ^{-1}\left[ {\frac{(x+a)\left( {y\!+\!b} \right) }{\left( {z\!+\!L} \right) \left[ {\left( {x\!+\!a} \right) ^{2}\!+\!\left( {y\!+\!b} \right) ^{2}\!+\!\left( {z\!+\!L} \right) ^{2}} \right] ^{\frac{1}{2}}}} \right] \right. \nonumber \\&\qquad \qquad \qquad +\tan ^{-1}\left[ {\frac{(x-a)\left( {y-b} \right) }{\left( {z+L} \right) \left[ {\left( {x-a} \right) ^{2}+\left( {y-b} \right) ^{2}+\left( {z+L} \right) ^{2}} \right] ^{\frac{1}{2}}}} \right] \nonumber \\&\qquad \qquad \qquad -\tan ^{-1}\left[ {\frac{(x+a)(y-b)}{(z+L)[\left( {x+a} \right) ^{2}+\left( {y-b} \right) ^{2}+({z}+L)^{2}]^{1/2}}} \right] \nonumber \\&\qquad \qquad \qquad \left. -\tan ^{-1}\left[ {\frac{(x-a)(y+b)}{(z+L)[\left( {x-a} \right) ^{2}+\left( {y+b} \right) ^{2}+({z}+L)^{2}]^{1/2}}} \right] \right\} \end{aligned}$$

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