The influence of a magnetic field on the mechanical behavior of a fluid interface

Abstract

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.

Appendix

In this appendix we present more details on the full expression of the asymptotic solution ${\mathcal {O}}(Bo_m)$.

$$\begin{aligned} Y_1(X)&= -\frac{\text{ csch }^4\left( \sqrt{Bo}\right) }{96 Bo^{5/2} D_0^3} \left\{ 12 \sqrt{Bo} D_0 \left[ 3 Bo^2 D_0^2 D_1+2 (\xi -\beta _0^{*}) \cot ^2(\alpha )\right] \right. \\&\quad - 24 \sqrt{Bo} D_0 \left[ 2 Bo^2 D_0^2 D_1+(\xi -\beta _0^{*}) \cot ^2(\alpha )\right] \cosh \left( 2 \sqrt{Bo}\right) \\&\quad +12 Bo^{5/2} D_0^3 D_1 \cosh \left( 4 \sqrt{Bo}\right) + \cot ^2(\alpha ) \left[ 8 (-\xi +\beta _0^{*}) \sqrt{Bo} D_0 \cosh \left( \alpha _4\right) \right. \\&\quad+ 4 \sqrt{Bo} \left\{ 2 \xi D_0-\beta _0^{*} \left[ 2 D_0+3 \cot (\alpha ) (1+X)\right] \right\} \cosh \left( \alpha _3\right) \\&\quad+4 \sqrt{Bo} D_0 \cosh \left( 2 \alpha _3\right) \left[ \xi - \beta _0^{*} \right] -8 \xi \sqrt{Bo} D_0 \cosh \left( 2 \sqrt{Bo} X\right) \\&\quad+8 \beta _0^{*} \sqrt{Bo} D_0 \cosh \left( 2 \sqrt{Bo} X\right) + 8 \sqrt{Bo} D_0 \cosh \left( \alpha _1\right) \left[ \xi -\beta _0^{*} \right] \\&\quad-12 \beta _0^{*} \sqrt{Bo} \cot (\alpha ) \cosh \left( \alpha _1\right) \left[ 1-X\right] +4 \xi \sqrt{Bo} D_0 \cosh \left( 2 \alpha _1\right) \\&\quad-4 \beta _0^{*} \sqrt{Bo} D_0 \cosh \left( 2 \alpha _1\right) -8 \sqrt{Bo} D_0 \cosh \left( \alpha _2\right) \left[ \xi - \beta _0^{*}\right] \\&\quad+\beta _0^{*} \cot (\alpha ) \sinh \left[ \sqrt{Bo} (1-3 X)\right] +3 \beta _0^{*} \cot (\alpha ) \left[ \sinh \left( \alpha _4\right) -\sinh \left( \alpha _2\right) \right] \\&\quad\left. \left. -12 \beta _0^{*} \cot (\alpha )\left[ \sinh \left( \alpha _1\right) - \sinh \left( \alpha _3\right) \right] +\beta _0^{*} \cot (\alpha )\sinh \left( \alpha _5\right) \right] \right\} \end{aligned}$$

with

$$\begin{aligned} \alpha _1&=\sqrt{Bo} (1+X) \quad \alpha _2= \sqrt{Bo} (3+X) \quad \alpha _3=\sqrt{Bo} (-1+X)\\ \alpha _4&=\sqrt{Bo} (-3+X) \quad \alpha _5=\sqrt{Bo} (1+3X) \quad \xi= \chi _0\left( 1+\chi _0\right) \end{aligned}$$

Fig. 1

A sketch of the problem used in the mathematical formulation

Fig. 2

A sketch of the free surface’s geometrical parameters

Fig. 3

Combination of $Bo_m$ and $\chi _0$ in which a magnetic fluid in the small gap between two parallel plates can rise against gravity for contact angles higher than $\pi /2$. This curve was plotted using the constant curvature theory, Eq. (39). The inserts in this figure show two possible configuration of the meniscus. In this case we consider $\beta = 1/10$ and $Bo = 1/10$

Fig. 4

Equilibrium height versus the magnetic Bond number. The black circles represent numerical results, the solid line denotes the exact solution for small values of the magnetic Bond number given by the ${\mathcal {O}}(Bo_m)$ asymptotic theory, while the dashed line denotes the ${\mathcal {O}}(Bo_m^2)$ solution. The insert in this figure shows the detail of the asymptotic for smaller values of the magnetic Bond number. For this plot: $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$ and $\alpha = \pi /2 -1/10$

Fig. 5

Equilibrium height as a function of the magnetic Bond number. The black circles represent numerical values, the solid line denotes the theoretical solution for the constant curvature condition. For this plot: $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$ and $\alpha = \pi /2 - 1/10$

Fig. 6

Meniscus shape for different values of n. a Stands for $n=1$, b for $n=2$, c for $n=3$ and d for $n=4$. For this plot: $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $\alpha = \pi /2 - 1/10$, $\varepsilon =1$ and magnetic Bond numbers varying from 0 to 1.5

Fig. 7

Meniscus shape for $n = 100$. The solid line represents $Bo_m = 0$ while the dashed line considers $Bo_m = 3/2$. For this plot: $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $\alpha = \pi /2 - \beta_{0}^{*}$, $\varepsilon =1$

Fig. 8

Meniscus shape for three different cases. The inserts in this figure show the Fourier transform of the meniscus shape represented by the amplitude as a function of the wavenumber. The solid line represents $Bo_m = 1/10$, the dashed one features $Bo_m = 1/4$ and the dotted one represents $Bo_m = 1/2$. a The FFT for $Bo_m = 1/10$, b $Bo_m = 1/4$ and c $Bo_m = 1/2$. For this plot: $\varepsilon =1/2$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 + \frac{1}{8}$, $k/\pi = 2$

Fig. 9

Meniscus shape for a non magnetic case $Bo_m =0$ (a) and $Bo_m = 1/5$ (b). The inserts in (a) and (b) giving the amplitute as a function of the wavenumber represent the Fourier transform of the meniscus shape for both cases studied. For this plot: $\varepsilon =3$, $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $\alpha = \pi /2 - \frac{1}{10}$, $k/\pi = 2$

Fig. 10

Meniscus Haar wavelet transform. a $k/\pi =2$, $Bo_M=7/5$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\varepsilon = 1.0$ and $\alpha = \frac{\pi }{2}-\frac{1}{10}$, b $k/\pi = 4$, $Bo_M=1$, $\chi _0=1/10$, $\beta _0^{*}=1/5$, $\varepsilon = 7/10$ and $\alpha = \frac{\pi }{2}-\frac{1}{10}$

Fig. 11

Source of meniscus shape used to compute the wavelet transform. a Meniscus shape for $k/\pi =2$, $Bo_M=7/5$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\varepsilon = 1.0$ and $\alpha = \frac{\pi }{2}-\frac{1}{10}$. b Correspondent FFT. c $k/\pi = 4$, $Bo_M=1$, $\chi _0=1/10$, $\beta _0^{*}=1/5$, $\varepsilon = 7/10$ and $\alpha = \frac{\pi }{2}-\frac{1}{10}$. d Correspondent FFT

Fig. 12

Comparison between the amplitude of the first harmonic in the wavenumber spectrum and the Magnetic Bond number. The dotted curve can be approximated as second order polynomial given by $A = 408.38 + 1288.59Bo_m + 199.65Bo_m^2$. For this plot: $\varepsilon =1$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 - \frac{1}{8}$, $k/\pi = 1$

Fig. 13

Comparison between the meniscus patterns formed by different controlling parameters. a $\varepsilon =1$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 - \frac{1}{8}$, $k/\pi = 2$, $Bo_m = 3/2$. b $\varepsilon =1$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 - \frac{1}{8}$, $k/\pi = 1$, $Bo_m = 3/2$. c $\varepsilon =3$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 - \frac{1}{8}$, $k/\pi = 1$, $Bo_m = 13/10$. d $\varepsilon =3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $\alpha = \pi /2 - \frac{1}{5}$, $k/\pi = 1$, $Bo_m = 1/10$. e $\varepsilon =1$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 + \frac{1}{4}$, $k/\pi = 1$, $Bo_m = 1/2$. f $\varepsilon =3$, $\chi _0=1/10$, $\beta _0^{*}=0$, $\alpha = \pi /2 - \frac{1}{6}$, $k/\pi = 6$, $Bo_m = 2$

Fig. 14

Equilibrium rise D as a function of the contact angle $\alpha$. The solid line represents the theory with constant curvature and magnetic effects, Eq. (39), while the dashed line is the numerical solution. For this plot: $\varepsilon =0$, $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $Bo_m = 1/10$

Fig. 15

Equilibrium rise D as a function of $\varepsilon$. The insert in this figure shows two different meniscus shapes. In the insert the solid line represents $\varepsilon = 0$ while the dashed line denotes $\varepsilon = 5/4$. For this plot: $n =1$, $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $Bo_m = 1.0$. The solid line is well fitted by the relation: $D = c_0 + c_1 \varepsilon + c_2 \varepsilon ^2 + c_3 \varepsilon ^3$, where $c_0= 593/1000, c_1 = 3/1000 , c_2 = 199/500$ and $c_3=-3/1000$

Fig. 16

Meniscus shape considering symmetrical boundary conditions (a) and non-symmetrical boundary conditions (b). For this plot: $n =1$, $Bo=3/10$, $\chi _0=1/10$, $\beta _0^{*}=1/10$, $\varepsilon = 1.0$ and $\alpha = \frac{\pi }{2}-1/10$