2.1 Experimental set-up and materials

The experimental set-up shown in figure 1 consists of an acrylic glass inner and outer cuboidal vessels with the following dimensions: $(L\times W\times H)~7.3\times 7.3\times 9~\text{cm}^{3}$ and $8\times 8\times 8~\text{cm}^{3}$ , respectively (Volkova et al. Reference Volkova, Naletova and Turkov2013). The thickness of the glass is 2.5 mm. The inner vessel is inserted vertically into the other up to the required distance $d$ between their horizontal plates. The distance $d$ can be changed from 0 to 6 cm in steps of 1 mm. The cylindrical coordinates $(r,\unicode[STIX]{x1D719},z)$ are introduced with the origin in the middle of the upper plate. The $z$ axis coincides with the axis of symmetry of the coil, see figure 1.

The electromagnetic coil with the inserted ferrite core with soft magnetic properties of low coercivity is fixed in the inner vessel, so that the axis of symmetry of the coil is perpendicular to the upper horizontal plate through its centre. Parameters of the coil are: height 4.6 cm, inner and outer radii 0.4 cm and 1 cm, respectively. The winding of the coil is made of 270 turns of a copper wire with a diameter of 1.04 mm. The ferrite core has a radius of 0.4 cm and a height of 12.5 cm, the relative permeability of the material is 400. Its lower end extends out of the coil for 1 mm. The current source provides a range of output currents in the coil $i=0{-}2.6~\text{A}$ . The distribution of the magnetic field in the area between two horizontal plates is measured with an analogue Hall sensor in the absence of the MF. The maximum magnitude of the magnetic field strength $\boldsymbol{H}$ is up to 655 Oe for $i=2.6~\text{A}$ measured directly under the upper plate.

An MF volume $V_{0}$ is injected in the middle of the working area, which is the area between the horizontal plates $z=0$ and $z=-d$ of the experimental set-up. In order to reduce capillary and gravitational forces, the area was first filled with a non-magnetic liquid (NML) with a density smaller than the MF.

Experimental results were carried out using a water-based MF and polymethylsilicone liquid as an NML. This combination is chosen due to the fact that these fluids are immiscible, and the MF does not stain the walls of the vessel. The parameters of the MF are listed in table 1. This low concentration fluid has a small value of the saturation magnetisation and, therefore, the normal-field instability of the surface in applied magnetic fields was not observed. The polymethylsilicone liquid has the following parameters: the density is $\unicode[STIX]{x1D70C}_{2}=0.91~\text{g}~\text{cm}^{-3}$ , the kinematic viscosity 5.5 cSt. The surface tension coefficient on the interface between these fluids is measured by the pendant drop method: $\unicode[STIX]{x1D70E}=11.7~\text{g}~\text{s}^{-2}$ .

2.2 MF volume in a step-like increasing and decreasing magnetic field

The experimental results of the MF behaviour between the horizontal plates when the current in the coil increases and decreases quasi-statically are presented. In each experimental series, the distance between the plates is fixed to $d=1.2~\text{cm}$ . An MF volume $V_{0}$ is initially injected in the middle of the lower plate when the current in the coil is turned off. Then the current is increased in steps of 0.001–0.01 A to a maximal value of 2.6 A, and thereafter, it is decreased back to zero. Each current value is held for 10–20 s until the MF takes its equilibrium shape. Thus, the process of the MF transformation can be seen as quasi-static. The following values of $V_{0}$ are selected: $0.4~\text{cm}^{3}$ , $0.8~\text{cm}^{3}$ and $2~\text{cm}^{3}$ .

During experimental series with various MF volumes, it is shown that three different simply connected axisymmetric forms of the volume may exist, namely: a drop on the lower plate, a drop retained on the upper plate beneath the coil or an MFB between the plates. Their equilibrium surface shape may be described by a function $z=h(r)$ . A set of solutions split into two disconnected parts of drops on the lower and upper plates with the total MF volume is also possible.

As a characteristic parameter of the drops on the upper and lower plates, the coordinate $h_{0}=h(0)$ , $-d\leqslant h_{0}\leqslant 0$ , of the intersection point of the MF surface $z=h(r)$ with the $z$ axis is chosen, see figure 2(a,c). The MFB is characterised by the radius $r_{0}>0$ of the contact spot of the MF with the upper plate: $h(r_{0})=0$ , see figure 4(e). These coordinates are measured on the experimental photos and plotted against the current.

Figure 2. Equilibrium surface shapes of an MF volume $V_{0}=0.4~\text{cm}^{3}$ . Panels (a–i) correspond to the increasing current $i$ from 0 A to 2.6 A, whereas (i–l) conform to the decreasing current $i$ from 2.6 A to 0 A.

Figure 3. Dependence of $h_{0}$ on $i$ for $V_{0}=0.4~\text{cm}^{3}$ : ▫ – drop on the lower plate; ▵ – drop on the upper plate. Current values at which abrupt changes occur: $\uparrow i_{a}=0.84~\text{A}$ , $\uparrow i_{b}=1.2~\text{A}$ , $\uparrow i_{c}=1.64~\text{A}$ , $\uparrow i_{d}=2.49~\text{A}$ and $\downarrow i_{e}=0.28~\text{A}$ .

Figure 4. Equilibrium surface shapes of an MF volume $V_{0}=0.8~\text{cm}^{3}$ . Panels (a–h) correspond to the increasing current $i$ from 0 A to 2.6 A, whereas (h–l) conform to the decreasing current $i$ from 2.6 A to 0 A.

Figure 5. Dependencies of $h_{0}$ (left axis) and $r_{0}$ (right axis) on $i$ for $V_{0}=0.8~\text{cm}^{3}$ : ▫ – $h_{0}$ for the drop on the lower plate; ▵ – $h_{0}$ for the drop on the upper plate; ○ – $r_{0}$ for the MFB. Current values at which abrupt changes occur: $\uparrow i_{a}=0.74~\text{A}$ , $\uparrow i_{b}=0.85~\text{A}$ , $\uparrow i_{c}=1.16~\text{A}$ , $\downarrow i_{d}=0.72~\text{A}$ and $\downarrow i_{e}=0.38~\text{A}$ .

Figure 6. Equilibrium surface shapes of an MF volume $V_{0}=2~\text{cm}^{3}$ . Panels (a–e) correspond to the increasing current $i$ from 0 A to 2.6 A, whereas (e–j) conform to the decreasing current  $i$ from 2.6 A to 0 A.

Figure 7. Dependencies of $h_{0}$ (left axis) and $r_{0}$ (right axis) on $i$ for $V_{0}=2~\text{cm}^{3}$ . Measured data: ▫ – $h_{0}$ for the drop on the lower plate; ▵ – $h_{0}$ for the drop on the upper plate; ○ – $r_{0}$ for the MFB. Current values at which abrupt changes occur: $\uparrow i_{a}=0.69~\text{A}$ and $\downarrow i_{b}=0.37~\text{A}$ . Theoretical result is shown by the solid line.

For a MF volume of $V_{0}=0.4~\text{cm}^{3}$ , several stepwise changes in the MF surface shape take place when the current either increases or decreases quasi-statically from 0 A to 2.6 A, see figure 2. At four critical values of the increasing current $i$ , namely $i_{a}=0.84~\text{A}$ , $i_{b}=1.2~\text{A}$ , $i_{c}=1.64~\text{A}$ and $i_{d}=2.49~\text{A}$ , the drop of the MF on the lower plate becomes unstable, a part of it jumps suddenly up and initially forms, and subsequently merges with, a drop on the upper plate. The first such transition at $\uparrow i_{a}=0.84~\text{A}$ is shown in figure 2(b,c), the second one occurring at $\uparrow i_{b}=1.2~\text{A}$ in figure 2(d,e), etc. This is because, when the current reaches these critical values, the imbalance between gravitational, surface and magnetic energies occurs for the present volume of the drop on the lower plate. Note that, in general, the number of such transitions depends on the pattern of the current change and the dynamics of the process. However, in the described experiment, the current is changed in a quasi-static manner, and so these four transitions and their critical currents are most likely stable from the point of reproducibility. For each jump, the volume of the MF transferred from one plate to the other can be approximated by image analysis of experimental photos.

At $i_{max}=2.6~\text{A}$ almost the whole initial volume gathers into a drop on the upper plate, only a small droplet is left on the lower plate, see figure 2(i). When the current decreases, these two drops retain up to a value $\downarrow i_{e}=0.28~\text{A}$ , at which a part of the upper drop separates and falls abruptly onto the lower plate. At the end of the process, when $i=0~\text{A}$ , the initial simply connected MF volume consists of two separate drops, see figure 2(a,l). It should be noted that no MFB of $V_{0}=0.4~\text{cm}^{3}$ between the plates was observed for any values of the increasing and decreasing current. The dependence of $h_{0}$ on $i$ for $V_{0}=0.4~\text{cm}^{3}$ is shown in figure 3. Hereinafter, the white and grey markers correspond to the increasing and decreasing current, respectively. This graphic clearly shows that there is a hysteresis phenomenon of the MF surface shape.

Figure 4 shows equilibrium surface shapes of an MF volume $V_{0}=0.8~\text{cm}^{3}$ . In this case, the formation of the MFB occurs both for the increasing and decreasing current. As the current increases from $i=0~\text{A}$ , the initial volume on the lower plate rises and becomes unstable at $\uparrow i_{a}=0.74~\text{A}$ as it divides into two drops, see figure 4(b,c). At the next threshold value, $\uparrow i_{b}=0.85~\text{A}$ , the upper and lower drops gather into an MFB (figure 4 d,e), which at $\uparrow i_{c}=1.16~\text{A}$ breaks up into two drops (figure 4 f,g). These resulting drops remain with the increase in current to $i_{max}=2.6~\text{A}$ and its further decrease down to $\downarrow i_{d}=0.72~\text{A}$ . At this critical value, the MFB between the plates occurs again from the coalescence of the upper and lower drops, see figure 4(i,j). It exists for the current from $\downarrow i_{d}=0.72~\text{A}$ to $\downarrow i_{e}=0.38~\text{A}$ and then disintegrates into two drops again. So, the MFB may occur in completely different ranges of the increasing and decreasing current. The dependencies of the coordinates $h_{0}$ and $r_{0}$ of the observed surface shapes of $V_{0}=0.8~\text{cm}^{3}$ are plotted against the current $i$ in figure 5.

For the MF volume $V_{0}=2~\text{cm}^{3}$ equilibrium surface shapes are shown in figure 6. With the increase in current from $i=0~\text{A}$ , the initial drop on the lower plate becomes unstable at $\uparrow i_{a}=0.69~\text{A}$ and transforms abruptly to an MFB between the plates, see figure 6(b,c). The resulting bridge between the plates exists up till the value $i_{max}=2.6~\text{A}$ . Then, for the decreasing current the MFB remains up to $\downarrow i_{b}=0.37~\text{A}$ and disintegrates into two drops, as for all other described cases (figure 6 h,i). The dependencies of $h_{0}$ and $r_{0}$ on $i$ are presented in figure 7.

To summarise, in the experimental series with various MF volumes and a fixed distance between the plates, the abrupt behaviour and hysteresis of the surface shape are studied as the current changes quasi-statically. The current values at which these abrupt transformations take place are determined. Moreover, the wetting hysteresis phenomenon is observed, that is the difference between the contact angle values for the increasing and decreasing current. To take this effect into account in further numerical calculations of an MF equilibrium shape, the contact angles $\unicode[STIX]{x1D703}_{0}$ on the upper and lower plates are measured through image analysis of experimental photos. In the case of $V_{0}=2~\text{cm}^{3}$ , these dependencies $\unicode[STIX]{x1D703}_{0}(i)$ will be given in § 4.3. As a result, the MF equilibrium shape exhibits two types of hysteresis when the current changes cyclically. First, there is a bistability of simply connected shapes in certain ranges of currents (e.g. figure 7 at $i=0.5~\text{A}$ ), and it depends on the initial conditions and pattern of the current change as to which one is observed. Second, due to the contact angle hysteresis, the MF volume can have different surface forms, see figure 6(d,f) at $i=1~\text{A}$ .

2.3 Disintegration of the MFB between the plates

The realisation of an MFB depends mainly on the current and the relation of the MF volume to the distance between the plates. To determine ranges of parameters for which the MFB exists, the following experiment has been carried out. At a certain value of the current $i>0~\text{A}$ , an MF volume is injected so that it takes a form of the axisymmetric MFB. The distance $d$ between the horizontal plates is fixed. In the first attempt, the current in the coil increases quasi-statically and decreases in the second one. In both cases, critical current values $i_{cr}=i_{cr}(d)$ at which the MFB breaks abruptly into two drops are measured.

Figure 8. Stability curves $i_{cr}=i_{cr}(d)$ of the MFB of $V_{0}=0.6~\text{cm}^{3}$ , $0.75~\text{cm}^{3}$ and $1~\text{cm}^{3}$ . Surface shapes of the MFB for $V_{0}=0.75~\text{cm}^{3}$ and $d=1~\text{cm}$ : (a) $i=0.5~\text{A}$ ; (b) $\uparrow i_{cr}=0.98~\text{A}$ ; (c) $\downarrow i_{cr}=0.25~\text{A}$ .

The experiments were carried out for three different MF volumes $V_{0}$ : $0.6~\text{cm}^{3}$ , $0.75~\text{cm}^{3}$ and $1~\text{cm}^{3}$ . The distance between the plates was changed in the range from 0.5 cm to 1.6 cm with a step size 1 mm. The dependencies of the critical current $i_{cr}=i_{cr}(d)$ are presented in figure 8. One can see that these lines have the same qualitative character. For each value $V_{0}$ , the dependence $i_{cr}=i_{cr}(d)$ is the so-called stability curve of the MFB between the plates. Since the curve divides the plane of parameters $i$ and $d$ into two parts, the region to the left of it corresponds to the values, for which a stable MFB can exist, see figure 8(a). On the contrary, for the values of $i$ and $d$ from the right part of the plane the MFB cannot be created and does not exist at all.

3.1 Formulation of the problem

Consider a heavy incompressible MF and a surrounding NML lying between horizontal plates, which are fixed at a distance  $d$ from each other, in the presence of a non-uniform magnetic field. In the axisymmetric case, all parameters are assumed to depend on the cylindrical coordinates $r$ and $z$ , see figure 1. As shown in § 2.2, a given MF volume $V_{0}$ may take three different simply connected axisymmetric forms, namely: a drop on the lower plate, a drop on the upper plate or an MFB. Consider that the curve $z=h(r)$ determines the equilibrium surface shape of the interface between the MF drop on the lower (or upper) plate and the NML. For the MFB between the plates, it is convenient to describe its axisymmetric surface shape with a function $r=\unicode[STIX]{x1D701}(z)$ .

For the theoretical considerations, the non-inductive approximation is assumed, that is, the MF is weakly magnetisable so that it does not distort the strength $\boldsymbol{H}$ of an applied magnetic field. The subscripts 1 and 2 denote the parameters of the MF and the NML, respectively.

The energy of a finite MF volume surrounded by an NML consists of the surface energy, the energy in the gravity field and the magnetostatic energy (Landau & Lifshitz Reference Landau and Lifshitz1980):

Here, $\unicode[STIX]{x1D70E}=\text{const.}$ is the surface tension coefficient, $\text{d}s$ is the element of the interface $S$ , $\unicode[STIX]{x1D70C}_{1}$ and $\unicode[STIX]{x1D70C}_{2}$ are the densities of the MF and NML, wherein $\unicode[STIX]{x1D70C}_{1}>\unicode[STIX]{x1D70C}_{2}$ , $g$ is the gravity acceleration, $\text{d}v$ is the element of the MF volume $V$ and $H(r,z)$ is the strength of the axisymmetric magnetic field.

The magnetisation of the MF $M_{1}(H)$ with a small concentration of ferromagnetic particles can be described by the Langevin formula

Here, $T=\text{const.}$ is the temperature of the fluids, $k$ is the Boltzmann constant and $M_{1S}$ is the MF saturation magnetisation, which is linked with the magnetic moment $m$ of a single ferromagnetic particle and the number of ferromagnetic particles $n$ .

The MF energy $F$ described by formula (3.1) is a nonlinear functional defined on the class of continuously differentiable functions $z=h(r)$ (or $r=\unicode[STIX]{x1D701}(z)$ for the MFB) with variable end points. In the static equilibrium state, the energy $F$ must have minimum for all surface shape variations satisfying the condition of the constant MF volume:

3.2 Variation of the energy functional

In what follows, without loss of generality, we will only consider the variation of the functional $F$ for the simply connected MF drop on the lower plate. For the drop on the upper plate and the MFB, the analysis procedure is the same, with the differences given in appendices A and B, respectively.

The energy functional $F[h]$ of the drop on the lower plate with the surface $z=h(r)$ can be calculated by formula (3.1) accounting for (3.2)

Here, $(r_{1},-d)$ , $r_{1}>0$ , is the point of intersection of the surface $z=h(r)$ with the lower plate, a prime denotes the derivative with respect to $r$ . The surface tension forces $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{12}$ , $\unicode[STIX]{x1D70E}_{s1}$ and $\unicode[STIX]{x1D70E}_{s2}$ act between the phases, where the subscripts 1, 2 and $s$ correspond to the MF, NML and the solid plate, respectively. In accordance with the Young equation, they are linked by the relation $\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D70E}_{s2}-\unicode[STIX]{x1D70E}_{s1}$ , where $\unicode[STIX]{x1D703}_{0}$ is the contact angle, $0\leqslant \unicode[STIX]{x1D703}_{0}\leqslant \unicode[STIX]{x03C0}$ .

According to the Euler theorem in the calculus of variations, to find the minimum of the energy functional (3.4) the following auxiliary functional $\unicode[STIX]{x1D6F9}[h]$ is considered:

where $p_{0}$ is a constant to be determined. The forms of $\unicode[STIX]{x1D6F9}$ in the case of the drop on the upper plate and the MFB are given in appendices A and B, respectively.

The functional $\unicode[STIX]{x1D6F9}[h]$ is defined by (3.5) in the class of continuously differentiable functions $z=h(r)$ with variable end points $(0,h_{0})$ and $(r_{1},-d)$ . Here, $h_{0}$ , $-d<h_{0}<0$ , and $r_{1}>0$ are unknown coordinates to be determined.

Let $u(r)$ be a certain variation of the drop surface shape $z=h(r)$ in the plane $\unicode[STIX]{x1D719}=\text{const}$ . End points of the variable curve $h(r)+u(r)$ are denoted by $(0,h_{0}+\unicode[STIX]{x1D6FF}h_{0})$ and ( $r_{1}+\unicode[STIX]{x1D6FF}r_{1},d$ ). The variation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}[u]$ of the functional $\unicode[STIX]{x1D6F9}[h]$ is defined as an expression, which is linear in increments $u(r)$ , $u^{\prime }(r)$ , $\unicode[STIX]{x1D6FF}h_{0}$ , $\unicode[STIX]{x1D6FF}r_{1}$ and which differs from the total increment $\unicode[STIX]{x1D6F9}[h+u]-\unicode[STIX]{x1D6F9}[h]$ by a quantity higher by more than an order of magnitude as compared to the distance between the functions $h(r)$ and $h(r)+u(r)$ (Gelfand & Fomin Reference Gelfand and Fomin2000):

where the prime and the subscripts $h$ and $h^{\prime }$ denote partial derivatives with respect to the corresponding arguments.

In order for the energy functional $F[h]$ extremum to be reached on the curve $h(r)$ , it is necessary that the variation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}[u]$ vanishes for $z=h(r)$ and all admissible $u(r)$ (Gelfand & Fomin Reference Gelfand and Fomin2000). From the requirement that the integrand in (3.6) be equal to zero, in view of (3.4) and (3.5), we have the following Euler–Lagrange equation

Since $u(r)$ is an arbitrary function, the terms in front of the independent increments $\unicode[STIX]{x1D6FF}h_{0}$ and $\unicode[STIX]{x1D6FF}r_{1}$ in (3.6) should be zero. Therefore, the following boundary conditions at the variable end points of the admissible curve $z=h(r)$ must be satisfied:

Thus, to find the minimum of the functional $F[h]$ on the curve $z=h(r)$ with variable end points, it is required first to solve the Euler–Lagrange equation (3.7). It is a nonlinear second-order differential equation for the curve $h(r)$ , which determines the axisymmetric surface shape of the MF drop on the lower plane. Then, the values of two arbitrary constants appearing in the general solution of (3.7), the constant $p_{0}$ , and the coordinates $h_{0}$ and $r_{1}$ of the variable end points, are determined from the boundary conditions (3.8) and the condition $V_{0}=\text{const}$ .

It can be shown that (3.7) also determines the surface shape of the MF drop on the upper plate with one correction, with the requirement of replacing the minus by a plus on the right-hand side of the equation. Then, the boundary conditions at the variable end points are: $h(0)=h_{0}$ , $h^{\prime }(0)=0$ , and $h(r_{1})=0$ , $h^{\prime }(r_{1})=\tan \unicode[STIX]{x1D703}_{0}$ with $-d<h_{0}<0$ and $r_{1}>0$ .

In order to determine the shape $r=\unicode[STIX]{x1D701}(z)$ of the MFB between the plates, the Euler–Lagrange equation $f_{\unicode[STIX]{x1D701}}^{\prime }-(\text{d}/\text{d}z)f_{\unicode[STIX]{x1D701}^{\prime }}^{\prime }=0$ should be solved with the corresponding form of the function $f(z,\unicode[STIX]{x1D701},\unicode[STIX]{x1D701}^{\prime })$ given in appendix B. In this case, the boundary conditions are $\unicode[STIX]{x1D701}(0)=r_{0}$ , $\unicode[STIX]{x1D701}^{\prime }(0)=\cot \unicode[STIX]{x1D703}_{0}$ , and $\unicode[STIX]{x1D701}(-d)=r_{1}$ , $\unicode[STIX]{x1D701}^{\prime }(-d)=-\cot \unicode[STIX]{x1D703}_{0}$ , where $r_{0},r_{1}>0$ are variable coordinates to be determined.

It should be noted that the solutions of the Euler–Lagrange equation do not always realise a local minimum or maximum of the MF energy since this equation gives only a necessary condition for the functional to have an extremum.

3.3 Sufficient conditions of the minimum energy

To find sufficient conditions for a functional minimum, it is necessary to introduce a new concept, namely, the second variation of a functional. Denote by $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ the terms of the second order relative to the end variations $\unicode[STIX]{x1D6FF}h_{0}$ and $\unicode[STIX]{x1D6FF}r_{1}$ , the increment of $u(r)$ and its derivative, which are determined by the expansion of $\unicode[STIX]{x1D6F9}[h+u]-\unicode[STIX]{x1D6F9}[h]$ in a Taylor series. The expression $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ is a quadratic functional and is called the second variation of the functional $\unicode[STIX]{x1D6F9}[h]$ .

In order that the minimum of the energy functional $F[h]$ together with the subsidiary condition of the constancy of the volume $V_{0}[h]$ be reached on the curve $z=h(r)$ , it is necessary and sufficient that on the extremal $z=h(r)$ the differential $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}[u]$ be equal to zero and the second variation $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ be positive definite for all admissible surface perturbations $u(r)$ :

Accounting for formula (3.4) and the boundary conditions (3.8) and $f_{hh^{\prime }}^{\prime \prime }=0$ , the expression for the second variation of the drop on the lower plate takes the following form:

Notice that $f_{h^{\prime }h^{\prime }}^{\prime \prime }=\unicode[STIX]{x1D70E}r(1+h^{\prime 2})^{-3/2}>0$ for $r>0$ .

Let us introduce two continuously differentiable functions $\unicode[STIX]{x1D714}_{1}(r)$ and $\unicode[STIX]{x1D714}_{2}(r)$ , which are the solutions of the following equation

and satisfy the respective boundary conditions

The sufficient condition for the second variation $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ to be positive definite for all admissible variations $u(r)$ is that the functions $\unicode[STIX]{x1D714}_{1}(r)$ and $\unicode[STIX]{x1D714}_{2}(r)$ fulfil the condition $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}>0$ for any point of the segment $[0,r_{1}]$ .

This is due to the fact that, in this case, the following transformation of $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ can be made. Divide the integral in (3.10) into two parts: from $0$ to $r_{\ast }$ and from $r_{\ast }$ to $r_{1}$ , where $0<r_{\ast }<r_{1}$ . Then, we add to and subtract from each integrand a term of the form $\text{d}(\unicode[STIX]{x1D714}_{i}u^{2})$ , $i=1,2$ . According to (3.11), the reformed integrands are both perfect squares. In view of $u(0)=\unicode[STIX]{x1D6FF}h_{0}$ and $u(r_{1})=-h^{\prime }(r_{1})\unicode[STIX]{x1D6FF}r_{1}$ , the expression (3.10) for $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ is simplified to

Since the first two terms are zero according to (3.12), it is obvious that if $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}>0$ on $[0,r_{1}]$ , the second variation $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6F9}[u]$ will be positive definite for all admissible $u(r)$ . This is the sufficient condition for the minimum of the energy functional defined on the class of axisymmetric continuously differentiable functions $z=h(r)$ describing the MF equilibrium surface shape.

For the MF drop on the lower plate (3.11) and conditions (3.12) have the following forms:

Equation (3.14) can be solved numerically taking into account that the shape of the MF surface $z=h(r)$ is determined by (3.7).

The analysis of the minimum energy problem for the MF drop on the upper plate is similar to the case above. The characteristic $\unicode[STIX]{x1D714}$ functions are defined by the same equation (3.11) with the boundary conditions (3.12) taking into account the corresponding form of the function $f(r,h,h^{\prime })$ given in appendix A. The sufficient condition of the minimum energy is identical: $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}>0$ on $[0,r_{1}]$ , where $r_{1}$ is determined by $h(r_{1})=0$ .

For the MFB between the plates with the axisymmetric surface shape $r=\unicode[STIX]{x1D701}(z)$ , the sufficient condition of the minimum energy is $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}>0$ on $[-d,0]$ . The equation for the determination of the functions $\unicode[STIX]{x1D714}_{i}(z)$ , $i=1,2$ and their boundary conditions are given in appendix B.

3.4 Necessity and sufficiency of the minimum conditions

Consider the case where the sufficient conditions of the minimum energy are not satisfied for an extremal solution $z=h(r)$ of the MF surface shape. Then, it may be shown that there exists another continuously differentiable function in the weak neighbourhood of $h(r)$ , which provides a smaller value for the energy functional than the extremal $h(r)$ . This will reveal that the curve $z=h(r)$ cannot realise a local minimum for the energy functional.

Let us suppose that the characteristic $\unicode[STIX]{x1D714}$ functions are defined on the whole segment $[0,r_{1}]$ , but $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}<0$ . Using the substitution $\unicode[STIX]{x1D714}=-f_{h^{\prime }h^{\prime }}^{\prime \prime }u^{\prime }/u$ , we may introduce two perturbation functions $u_{1}(r)$ and $u_{2}(r)$ on $[0,r_{1}]$ :

Note that these functions are the solutions of (3.11) written in the form: $(f_{h^{\prime }h^{\prime }}^{\prime \prime }u^{\prime }(r))^{\prime }=f_{hh}^{\prime \prime }u(r)$ . They satisfy the following boundary conditions: $u_{1}(0)=\unicode[STIX]{x1D6FF}h_{0}$ , $u_{1}^{\prime }(0)=0$ and $u_{2}(r_{1})=-h^{\prime }(r_{1})\unicode[STIX]{x1D6FF}r_{1}$ , $u_{2}^{\prime }(r_{1})=(\unicode[STIX]{x1D714}_{2}h^{\prime }/f_{h^{\prime }h^{\prime }}^{\prime \prime })|_{r=r_{1}}\unicode[STIX]{x1D6FF}r_{1}$ . So, it is possible to define them on the whole segment $[0,r_{1}]$ , even when the functions $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ do not exist everywhere on it. Moreover, it can be shown that the expression $f_{h^{\prime }h^{\prime }}^{\prime \prime }(u_{1}^{\prime }u_{2}-u_{1}u_{2}^{\prime })=C$ is constant for $0<r\leqslant r_{1}$ . This means that the difference $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}$ does not change sign within the common domain of these functions.

Let $r=r_{\ast }$ be a point of the segment $[0,r_{1}]$ . Assuming the end point variations $\unicode[STIX]{x1D6FF}h_{0}$ and $\unicode[STIX]{x1D6FF}r_{1}$ are small and linked so that $u_{1}(r_{\ast })=u_{2}(r_{\ast })$ , the piecewise differentiable perturbation $\hat{u} (r)$ is defined as $\hat{u} (r)=u_{1}(r)$ for $0\leqslant r\leqslant r_{\ast }$ and $\hat{u} (r)=u_{2}(r)$ for $r_{\ast }\leqslant r\leqslant r_{1}$ . Using (3.12), (3.13) and (3.16), it is possible to show that in the weak neighbourhood of $h(r)$ the functional value for the curve $h+\hat{u}$ is smaller than $\unicode[STIX]{x1D6F9}[h]$ :

We can now construct a continuously differentiable perturbation $u(r)$ on $[0,r_{1}]$ so that the difference $\unicode[STIX]{x1D6F9}[h+u]-\unicode[STIX]{x1D6F9}[h+\hat{u} ]$ is infinitesimally small. For any $\unicode[STIX]{x1D700}>0$ , consider an $\unicode[STIX]{x1D700}$ neighbourhood of the point $r=r_{\ast }$ , where the derivative $\hat{u} ^{\prime }$ has a jump discontinuity, since $u_{1}^{\prime }(r_{\ast })\neq u_{2}^{\prime }(r_{\ast })$ . The smooth curve $u(r)$ is derived from $\hat{u} (r)$ by replacing the part in the interval $[r_{\ast }-\unicode[STIX]{x1D700},r_{\ast }+\unicode[STIX]{x1D700}]$ . For this, let us join the points $\hat{u} ^{\prime }(r_{\ast }-\unicode[STIX]{x1D700})$ and $\hat{u} ^{\prime }(r_{\ast }+\unicode[STIX]{x1D700})$ by a continuous curve $\unicode[STIX]{x1D710}(r)$ in such a way that

Therefore, we may define $u(r)$ as follows:

It is easy to verify that the function $u(r)$ is continuously differentiable on $[0,r_{1}]$ and satisfies the boundary conditions $u(0)=\unicode[STIX]{x1D6FF}h_{0}$ and $u(r_{1})=-h^{\prime }(r_{1})\unicode[STIX]{x1D6FF}r_{1}$ . At the points $r=r_{\ast }\pm \unicode[STIX]{x1D700}$ , we have $u=\hat{u}$ and $u^{\prime }=\hat{u} ^{\prime }$ .

Considering (3.5), the difference between the functional values on the curves $h+u$ and $h+\hat{u}$ may be simplified as:

If $\unicode[STIX]{x1D700}$ is sufficiently small, the curve $h+u$ lies in the weak neighbourhood of $h(r)$ . So, for an appropriate choice of $\unicode[STIX]{x1D700}$ and according to (3.17), the difference $\unicode[STIX]{x1D6F9}[h+u]-\unicode[STIX]{x1D6F9}[h]$ can be made negative:

Hence, if the conditions of the minimum energy are not satisfied, the admissible surface perturbation $u(r)$ defined by (3.19) is found such that the functional value $\unicode[STIX]{x1D6F9}[h+u]$ is smaller than $\unicode[STIX]{x1D6F9}[h]$ . This means that on the extremal solution $z=h(r)$ , the local minimum of the energy functional cannot be reached. In other words, the surface shape $z=h(r)$ is unstable.

Without going into the details, it should be noted that the reasoning will be the same also in the case where the characteristic functions $\unicode[STIX]{x1D714}_{1}(r)$ and $\unicode[STIX]{x1D714}_{2}(r)$ are not defined on the whole segment $[0,r_{1}]$ , and even where they do not have a common domain. Anyway, it is possible to choose the point $r=r_{\ast }$ so that the expression (3.17) is negative.

To summarise, the condition $\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{1}>0$ on the whole segment $[0,r_{1}]$ (or $[-d,0]$ ) is necessary and sufficient for a local minimum of the energy functional to be reached on the curve $z=h(r)$ (or $r=\unicode[STIX]{x1D701}(z)$ ). The proposed theoretical method makes is possible for the given parameters of the problem to calculate an MF equilibrium surface shape and examine its stability relative to small axisymmetric surface perturbations.

4.1 Modelling of the stationary magnetic field of the electromagnetic coil

When solving numerically, e.g. (3.7) and (3.14) for the drop on the lower plate, the problem of magnetic field distribution $H(r,z)$ in the area between the horizontal plates of the set-up occurs. The modelling and the calculation of the stationary magnetic field $H(r,z)$ of the axisymmetric electromagnetic coil with the inserted ferrite core were done numerically using a finite element software package (ANSYS^®). The geometry and material properties of the model were defined according to the parameters of the coil and the core used in the experiment, see § 2.1. For numerical computation, the PLANE53 element was chosen, which allows two-dimensional modelling of the magnetic field in axisymmetric problems. Far-field elements are defined by INFIN110 element type. The domain is meshed with elements of 0.05 cm edge length. Figure 9(a) shows the distribution of the magnetic field strength $H(r,z)$ in the vicinity of the lower end of the core obtained numerically. The results are compared with the data measured for the experimental coil with an analogue Hall sensor, see figure 9(b). The discrepancy in the results may be caused by the singularity of the magnetic field at the edge of the core and the real geometrical dimensions of the Hall sensor surface.

Figure 9. (a) Magnetic field distribution $H(r,z)$ (in Oe) of the electromagnetic coil with the ferrite core used in the experiment for $i=0.5~\text{A}$ ; (b) comparison between numerical and measured values of $H(r,z)$ for $i=0.5~\text{A}$ .

4.2 Stability of the equilibrium surface shape of MF drops and MFB for a constant contact angle

In this part, the numerical calculations are carried out assuming a constant contact angle of $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ at the MF–NML–solid plate interface. Therefore, the obtained results can be compared only qualitatively with experimental results of § 2.

Solution branches. Here, we consider only simply connected equilibrium shapes of an MF drop on the lower plate, an MF drop on the upper plate or an MFB. For these three shape types, the dependencies of the characteristic coordinates $h_{0}$ and $r_{0}$ on $i$ are obtained numerically for an MF volume $V_{0}=2~\text{cm}^{3}$ and a distance between the plates $d=1.2~\text{cm}$ , see figure 10. It can be seen that for a certain range of currents, the surface shape of the MF cannot be determined unambiguously.

Figure 10. Dependencies of $h_{0}$ (left axis) and $r_{0}$ (right axis) on $i$ obtained numerically in the case of a constant contact angle $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ for $V_{0}=2~\text{cm}^{3}$ and $d=1.2~\text{cm}$ : ▫ – $h_{0}$ for the drop on the lower plate; ▵ – $h_{0}$ for the drop on the upper plate; ○ – $r_{0}$ for the MFB. Stable solutions, on which the local minimum of the energy functional is reached are shown by black symbols.

Figure 11. (a) Unstable ( $h_{0}=-0.26~\text{cm}$ , $r_{1}=1.24~\text{cm}$ ) and stable ( $h_{0}=-0.58~\text{cm}$ , $r_{1}=1.31~\text{cm}$ ) solutions for the drops on the lower plate at $i=1~\text{A}$ for $V_{0}=2~\text{cm}^{3}$ , $d=1.2~\text{cm}$ and $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ ; (b) characteristic $\unicode[STIX]{x1D714}$ functions corresponding to these drops.

For example, there are two different branches of solutions for the drop on the lower plate: more lifted and flattened, see figure 11(a) at $i=1~\text{A}$ . For the obtained solutions, equation (3.14) with the boundary conditions (3.15) is solved numerically. It shows that for all drops, which correspond to the branch of the more lifted solutions, the necessary and sufficient conditions of the minimum energy are not satisfied, since $\unicode[STIX]{x1D714}_{2}(r)-\unicode[STIX]{x1D714}_{1}(r)<0$ on the intersection of the domains of these functions. An example is shown in figure 11(b) for $h_{0}=-0.26~\text{cm}$ at $i=1~\text{A}$ . According to § 3.4, it can be concluded that the local minimum of the energy functional for the obtained solutions is not reached and they are unstable. For more flattened solutions of the drop on the lower plate, the local energy minimum is reached for currents from $i=0~\text{A}$ to $i=1.02~\text{A}$ , as $\unicode[STIX]{x1D714}_{2}(r)-\unicode[STIX]{x1D714}_{1}(r)>0$ on the whole segment $[0,r_{1}]$ , see figure 11(b) for $h_{0}=-0.58~\text{cm}$ at $i=1~\text{A}$ . Hence, they are surface shapes stable relative to small axisymmetric surface perturbations. At currents from $i=1.02~\text{A}$ to the value $i=1.07~\text{A}$ , at which two solution branches of the drop on the lower plate intersect, the necessary and sufficient conditions of the minimum energy for the flattened drops are not satisfied, i.e. these surface shapes are unstable.

Figure 12. (a) Unstable ( $r_{0}=0.55~\text{cm}$ , $r_{1}=1.27~\text{cm}$ ) and stable ( $r_{0}=0.59~\text{cm}$ , $r_{1}=1.08~\text{cm}$ ) solutions for the MFB at $i=0.8~\text{A}$ for $V_{0}=2~\text{cm}^{3}$ , $d=1.2~\text{cm}$ and $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ ; (b) characteristic $\unicode[STIX]{x1D714}$ functions corresponding to these shapes.

The numerical calculation of the surface shape of the MF drop retained on the upper plate beneath the coil shows that one solution branch with $V_{0}=2~\text{cm}^{3}$ exists on the range of currents $i\geqslant 1.8~\text{A}$ , see figure 10. By checking the necessary and sufficient conditions of the energy minimum, it can be shown that from $i=1.8~\text{A}$ to $i=1.87~\text{A}$ they are not satisfied on the shapes of the upper drop. The solutions obtained for $i\geqslant 1.87~\text{A}$ meet all necessary and sufficient minimum conditions, so they are stable relative to small axisymmetric surface perturbations.

For the MFB of $V_{0}=2~\text{cm}^{3}$ , three solution branches exist, two of which do not satisfy the necessary and sufficient minimum conditions, see figure 10. The main branch is obtained for currents from $i=0.64~\text{A}$ to $i=2.55~\text{A}$ . The calculations show that MFB equilibrium surface shapes corresponding to the main branch realise the local energy minimum within the range from $i_{cr}=0.72~\text{A}$ to $i_{cr}=2.3~\text{A}$ . These marginal values are the critical currents of the MFB for $d=1.2~\text{cm}$ , at which it breaks up. Moreover, the volumes of the resulting drops on the lower and upper plates can be estimated approximately. Figure 12(a) illustrates two solutions of the MFB existing at $i=0.8~\text{A}$ . The one assigned with points $r_{0}=0.59~\text{cm}$ and $r_{1}=1.08~\text{cm}$ is stable relative to small axisymmetric surface perturbations since on it the necessary and sufficient conditions of the minimum energy are satisfied: $\unicode[STIX]{x1D714}_{2}(z)-\unicode[STIX]{x1D714}_{1}(z)>0$ on the whole segment $[-d,0]$ , see figure 12(b). Recall that these characteristic functions of the MFB are determined by equations given in appendix B. On the contrary, for the other solution with $r_{0}=0.55~\text{cm}$ and $r_{1}=1.27~\text{cm}$ , the local energy minimum is not reached because no common domain of $\unicode[STIX]{x1D714}_{1}(z)$ and $\unicode[STIX]{x1D714}_{2}(z)$ exists on $[-d,0]$ , see figure 12(b). So this MFB having a narrow neck in the middle is unstable and it would break up into two drops where the volume of each drop can be estimated from the height of the neck.

Figure 13. Numerically obtained stability curves $i_{cr}=i_{cr}(d)$ of the MFB of $V_{0}=0.5~\text{cm}^{3}$ , $1~\text{cm}^{3}$ , $1.5~\text{cm}^{3}$ and $2~\text{cm}^{3}$ for the case of a constant contact angle $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ .

Stability curves. The theoretical method of finding stable/unstable MF surface shapes can be applied to obtain stability curves of the MFB numerically. This is accomplished as follows: for the fixed parameters $d$ and $V_{0}$ , MFB equilibrium shapes are calculated depending on the current $i$ , that is, similar dependencies $r_{0}(i)$ as shown in figure 10 are computed. By checking the necessary and sufficient conditions of the energy minimum, the surface shapes stable relative to small axisymmetric surface perturbations are marked out by black on the main solution branch. Taking the marginal stable points, the critical current values $i_{cr}(d)$ of the MFB with volume $V_{0}$ are determined. The resulting stability curves $i_{cr}=i_{cr}(d)$ are presented in figure 13 for four different MF volumes. It can be seen that, although these lines are carried out for a constant contact angle of $\unicode[STIX]{x1D703}_{0}=80^{\circ }$ , they are qualitatively similar to the experimental dependencies in figure 8. This can be seen as a good proof of the model validity.

Thus, based on the numerical calculation of MF equilibrium shapes for $\unicode[STIX]{x1D703}_{0}=\text{const.}$ , it is possible to determine stability regions of the simply connected solutions and to predict qualitatively how the transformation of a given MF volume can proceed. As seen in figure 10, for a large volume of $V_{0}=2~\text{cm}^{3}$ , the MF drop initially placed on the lower plate will become unstable at increasing $\uparrow i=1.02~\text{A}$ and most likely will abruptly form an MFB since the stability regions of these solutions have a wide intersection there. The same transformation process was observed in the experiment. However, it occurred at the other critical value $\uparrow i=0.69~\text{A}$ due to the presence of the wetting hysteresis phenomenon, see figure 6. For a lower value of $V_{0}$ , if no MFB exists (figure 13), the initial drop on the lower plate will divide into two parts for increasing $i$ . The obtained results reveal that the bistability of simply connected solutions in certain current ranges leads to the hysteresis of the MF surface.

4.3 Calculation of MF equilibrium surface shapes under consideration of the wetting hysteresis phenomenon

In order to compare stable equilibrium surface shapes of the MF obtained experimentally and theoretically, the wetting hysteresis phenomenon is taken into account. This means that the experimental dependencies of the contact angle $\unicode[STIX]{x1D703}_{0}(i)$ on current are used as constraints in the simulations.

Figure 14. Contact angles $\unicode[STIX]{x1D703}_{0}(i)$ on the lower (a) and upper (b) plates for the MF equilibrium surface shapes of $V_{0}=2~\text{cm}^{3}$ shown in figure 6. Measured data: ▫ – drop on the lower plate; ▵ – drop on the upper plate; ○ – MFB. Current values at which abrupt changes occur: $\uparrow i_{a}=0.69~\text{A}$ and $\downarrow i_{b}=0.37~\text{A}$ . The solid line shows the corresponding values $\unicode[STIX]{x1D703}_{0}(i)$ used in numerical calculations.

Since, in terms of technical applications the stable MFB between the plates existing in wide current ranges is of particular interest, the comparison of the experimental and numerical results is made for an MF volume of $V_{0}=2~\text{cm}^{3}$ . As was mentioned, the MF volume can have different surface forms due to the contact angle hysteresis. For example, figure 6(d,f) shows two different MFBs for the increasing and decreasing $i=1~\text{A}$ . Obviously, this is caused by the varying wetting/non-wetting contact angles, where an MF–NML interface meets the solid plate. These angles are measured by image analysis of experimental photos and plotted depending on the current in figure 14. When calculating an MF surface shape, these dependencies are applied as follows. For each current value $i$ , the form of numerical solution to be sought is defined according to the experiment, see figure 7. Starting with the simply connected drop on the lower plate for the increasing $i$ and then the MFB, the corresponding contact angles used in calculations are assigned values equal (or close) to the experimentally measured values $\unicode[STIX]{x1D703}_{0}(i)$ , so that the obtained solution has a volume of $V_{0}=2~\text{cm}^{3}$ . At the decreasing current $\downarrow i_{b}=0.37~\text{A}$ , when the MFB disintegrates, the volumes of the resulting upper and lower drops are estimated on the basis of the photo to be $0.13~\text{cm}^{3}$ and $1.87~\text{cm}^{3}$ , respectively. Then, the form of these drops is calculated separately from each other taking experimental values $\unicode[STIX]{x1D703}_{0}(i)$ .

Thus, figure 15 shows equilibrium surface shapes of this MF volume calculated considering the contact angle hysteresis for the increasing and decreasing current cases. It may be seen that the obtained surface shapes coincide with the experimental forms presented in figure 6. The theoretical dependencies of the characteristic coordinates $h_{0}$ and $r_{0}$ of the drops on the upper or lower plates and of the MFB are plotted against the current $i$ in figure 7 (bold line). It has been shown that, according to the theoretical analysis, the MF surface shapes observed in the experiment realise a local minimum of the MF energy for practically all values of the current.

Figure 15. Equilibrium surface shapes of an MF volume $V_{0}=2~\text{cm}^{3}$ obtained numerically by taking into account the wetting hysteresis phenomenon for the increasing (a–e) and decreasing (e–j) current $i$ .