3.1 Spreading behaviour without a magnetic field

Without a magnetic field, as introduced in § 1, the interpolating scheme of (1.1) can be used to predict $\unicode[STIX]{x1D6FD}_{max}$ in the cross-over regime where viscous and capillary forces are both important. As a consequence, part of our numerical results, which are conducted with fixed $\unicode[STIX]{x1D703}=90^{\circ }$ and different liquid metals, are plotted in figure 3 to verify whether this interpolating scheme is applicable. In figure 3(a), all the data are observed to fall onto a single curve with the scaling law interpolated between $Re^{1/5}$ and $We^{1/2}$ , which is fitted accurately with (1.1). However, in figure 3(b), the interpolating law between $Re^{1/5}$ and $We^{1/4}$ does not collapse the data points onto a single curve – as was also found by Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010).

Figure 3.

The maximum spreading factor rescaled by different interpolating scheme when $\unicode[STIX]{x1D703}=90^{\circ }$ . (a) As a function of $We\,Re^{-2/5}$ . (b) As a function of $We\,Re^{-4/5}$ . The solid line is the Padé approximant function fitted with (1.1). It is observed that all data fall onto a single curve fitted with the scaling law of $\unicode[STIX]{x1D6FD}_{max}\propto Re^{1/5}f_{c}(We\,Re^{-2/5})$ , while contrary results are found in the scaling law of $\unicode[STIX]{x1D6FD}_{max}\propto Re^{1/5}f_{c}(We\,Re^{-4/5})$ .

However, equation (1.1) does not consider the influence of surface wettability. We have further investigated this by changing the contact angle to $\unicode[STIX]{x1D703}=60^{\circ }$ and $\unicode[STIX]{x1D703}=120^{\circ }$ , and the results are shown in figure 4(a,b). It is observed that all results still obey the scaling law of (1.1); however, the fitting constant $A$ is different, indicating that $A$ is determined by the surface wettability. To investigate this further, we simulate additional numerical cases with $\unicode[STIX]{x1D703}=75^{\circ }$ and $\unicode[STIX]{x1D703}=105^{\circ }$ to determine the correction between $A$ and $\unicode[STIX]{x1D703}$ ; the result is presented in figure 4(c), where a linear relation is observed between them.

Figure 4.

The maximum spreading factor rescaled with a function of parameter $We\,Re^{-2/5}$ for various contact angles: (a) $\unicode[STIX]{x1D703}=60^{\circ }$ , (b) $\unicode[STIX]{x1D703}=120^{\circ }$ . It is found that they conform to the same scaling law of (1.1), although the constant coefficient $A$ is different. (c) The variation trend of $A$ with different surface wettability, showing that $A$ increases almost linearly with $\unicode[STIX]{x1D703}$ .

3.2 Spreading behaviour under a vertical magnetic field

When a vertical magnetic field is applied, the induced Lorentz force, which acts as a resistance to the spreading, will decrease the maximum spreading radius. The evolution of the droplet shapes of liquid GaInSn during spreading is presented in figure 5, respectively without and with magnetic field, with other fixed parameters of $Re=13\,422$ , $We=96$ and $\unicode[STIX]{x1D703}=90^{\circ }$ . It is observed that, with $N=0$ , the spreading factor gradually increases to $\unicode[STIX]{x1D6FD}_{max}\approx 4$ while the height of the droplet keeps decreasing. In contrast, in the case of $N=5.660$ , the maximum spreading factor decreases to $\unicode[STIX]{x1D6FD}_{max}\approx 1.7$ ; thereafter, the interface of the droplet starts to fluctuate while the spreading radius remains almost unchanged.

Figure 5.

The evolution of the droplet shapes during the spreading process without and with magnetic field respectively, with other fixed parameters of $Re=13\,422$ , $We=96$ and $\unicode[STIX]{x1D703}=90^{\circ }$ . (a) $N=0$ , (b) $N=5.660$ . It is observed that $\unicode[STIX]{x1D6FD}_{max}$ decreases significantly under the influence of a magnetic field; moreover, the interface of the droplet fluctuates after attaining the maximum spreading radius.

Figure 6.

Spreading behaviours under the influence of different vertical magnetic fields, the working medium is GaInSn and the fixed parameters are $Re=13\,422$ , $We=96$ and $\unicode[STIX]{x1D703}=90^{\circ }$ , while the magnetic intensity is varied from $N=0$ to $N=16$ for (a,b) or from $N=5.66$ to $N=40.251$ for (c). (a) The shape of the drop interface when the maximum spreading radius occurs. (b) The variation of the maximum spreading factor $\unicode[STIX]{x1D6FD}_{max}$ and the sum of the vorticities inside the drop at that moment. (c) The time histories of the spreading with a strong magnetic field, for which it is found that $t_{m}$ converges to $D_{0}/2V_{0}$ when $N$ is much larger.

The droplet shapes, at the moment when $\unicode[STIX]{x1D6FD}_{max}$ occurs, are plotted against $N$ in figure 6(a), as the magnetic intensity is varied from $N=0$ to $N=16$ . In the figure, the maximum spreading radius is restrained by stronger magnetic fields; the rim region, which bulges out of the spreading drop, is also narrowed. It is well known that there are recirculation eddies formed in the rim, even if the drop stops moving at maximum spreading radius. The maximum spreading radius, as well as the sum of the vorticities inside the drop at that moment, is shown in figure 6(b). It is observed that the spreading radius is dramatically reduced by the magnetic field within the range of smaller $N$ ; however, the reducing trend slows when $N$ becomes larger. That is because, under such a stronger magnetic field, the spreading velocity will be much slower, and the Lorentz force, whose value has a necessary limit based on the inertia, will also decrease to maintain this force balance. This variation trend is very similar to that found in the rising of bubbles (Zhang, Ni & Moreau Reference Zhang, Ni and Moreau2016). Meanwhile, the evolution of $\unicode[STIX]{x1D714}_{sum}$ at that moment obeys a similar law, indicating that vortices are greatly suppressed by the Lorentz force. In figure 6(c), we also plot the time histories of the spreading under stronger magnetic field ( $5.660<N<40.251$ ), in order to track the time when maximum spreading is reached, defined as $t_{m}$ . In the figure, it is observed that, with stronger magnetic field, $t_{m}$ gradually converges to $t_{m}=D_{0}/2V_{0}$ ; the detailed physics of this time scale will be presented in § 3.3.

Furthermore, to study in-depth the influence of a magnetic field on the spreading behaviour, the energy conversion process during spreading should be investigated. Under a strong magnetic field, as a result of the Joule dissipation, the initial kinetic energy will be dissipated in a very short time. Considering that the gravitational potential energy is negligible during the spreading, the energy conservation equation can be written as

(3.1) $$\begin{eqnarray}\displaystyle KE_{0}+SE_{0}=KE+VE+SE+JE, & & \displaystyle\end{eqnarray}$$

where $KE$ is the kinetic energy, $VE$ is the viscous dissipation energy and $SE$ is the surface energy – the expressions for which can be found in many papers (Bennett & Poulikakos Reference Bennett and Poulikakos1993; Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996). The subscript $0$ indicates the initial energy before impact. Under an external magnetic field, $JE$ is the Joule dissipation, taking the form

(3.2) $$\begin{eqnarray}\displaystyle JE=\int _{0}^{t}\int _{V}\boldsymbol{V}(r,z)\boldsymbol{\cdot }(\boldsymbol{j}(r,z)\times \boldsymbol{B})\,\text{d}V\,\text{d}t, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{j}$ is the induced electric current, given as $\boldsymbol{j}=\unicode[STIX]{x1D70E}_{e}\boldsymbol{V}\times \boldsymbol{B}$ .

Figure 7.

The influence of vertical magnetic field on different components of energy consumption; the fixed impact parameters are $Re=13\,422$ , $We=96$ and $\unicode[STIX]{x1D703}=90^{\circ }$ . (a) Evolutions of different energies with $N=10.063$ . (b) The final proportions of different energies with various values of $N$ .

In particular, considering a test case of $N=10.063$ , the time histories of different components of the energy are presented in figure 7(a). As shown in the figure, the kinetic energy is dissipated rapidly once the impact starts. During the spreading process, almost all the initial kinetic energy is transferred into Joule dissipation, which ultimately takes 87 % of the total energy. In contrast, the viscous dissipation effect and capillary effect are negligible throughout the spreading. To study the influence of the magnetic strength on energy conversion, the final proportions of different energy components versus $N$ have been plotted at the time when the maximum spreading radius is reached, as shown in figure 7(b). It is observed that, with stronger magnetic field, the proportion of Joule dissipation increases correspondingly, while the other energies decrease monotonously. Therefore, besides the viscous regime and capillary regime in which the viscous force and capillary force dominate respectively, we have another ‘Joule regime’, in which the Joule dissipation is dominant over the other two effects, such as for $N>2.515$ in figure 7(b).

Consequently, we are able to interpret why $\unicode[STIX]{x1D6FD}_{max}$ becomes smaller with larger $N$ – because more kinetic energy is dissipated by Joule dissipation, hardly any residual energy is left for the drop to spread on the surface.

3.3 Scaling law for predicting the maximum spreading factor

To derive the maximal spreading radius under the combination of the viscous, capillary and Joule effects, according to our routine outline described in § 1, we will first consider the extreme Joule regime in which the kinetic energy is completely dissipated by Joule dissipation. Under such circumstances, the initial kinetic energy is estimated as:

(3.3) $$\begin{eqnarray}\displaystyle KE_{0}\sim \unicode[STIX]{x1D70C}V_{0}^{2}D_{0}^{3}. & & \displaystyle\end{eqnarray}$$

According to (3.2), when the drop is at its maximum extension, Joule dissipation is calculated as

(3.4) $$\begin{eqnarray}\displaystyle JE=\int _{0}^{t}\int _{V}\boldsymbol{V}(r,z)\boldsymbol{\cdot }(\boldsymbol{j}(r,z)\times \boldsymbol{B})\,\text{d}V\,\text{d}t=\unicode[STIX]{x1D70E}_{e}\overline{V_{r}}^{2}B_{z}^{2}\unicode[STIX]{x1D6FA}t_{m}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}$ is the drop volume, scaling as $\unicode[STIX]{x1D6FA}\sim D_{0}^{3}$ , leaving the scale of $t_{m}$ to be considered further. For hydrodynamic spreading of the droplet, Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) and Josserand & Zaleski (Reference Josserand and Zaleski2003) have proved theoretically that in the very first stage of the impact, the pressure at the centre of the impact is very high, and this strong pressure gradient drives the liquid to spread. In this pressure-driven stage, the time scale is $t\sim D_{0}/2V_{0}$ . After that, the pressure decreases rapidly and the fluid flow becomes similar to a time-dependent hyperbolic flow, in which the viscous effect and capillary effect gradually increase until maximum spreading is reached.

When imposing the magnetic field, the first pressure-driven stage should still keep a time scale of $t\sim D_{0}/2V_{0}$ because the pressure gradient is very strong and the Lorentz force, which is dependent on the spreading velocity, is still increasing. However, in the second stage, with a rapidly decaying pressure, the inertial force of the liquid is almost balanced by the Lorentz force, and the spreading decelerates according to

(3.5) $$\begin{eqnarray}\displaystyle \frac{\text{D}V_{r}}{\text{D}t}=-\frac{1}{\unicode[STIX]{x1D70C}}(\boldsymbol{j}\times \boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{e}_{r}=-(\unicode[STIX]{x1D70E}_{e}B_{z}^{2}/\unicode[STIX]{x1D70C})V_{r}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{r}$ is the radial direction vector. Clearly, the Lorentz force acts on a time scale of

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70C}/(\unicode[STIX]{x1D70E}_{e}B_{z}^{2}).\end{eqnarray}$$

We now compare the time scale with the characteristic time of $T_{0}=D_{0}/V_{0}$ , that is

(3.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70F}}{T_{0}}=\frac{\unicode[STIX]{x1D70C}V_{0}}{\unicode[STIX]{x1D70E}_{e}B_{z}^{2}D_{0}}=\frac{1}{N}.\end{eqnarray}$$

In the Joule regime, where $N$ is much larger, it is easy to deduce $\unicode[STIX]{x1D70F}/T_{0}\ll 1$ . It means the kinetic energy will be dissipated by the Joule effect within a very short time after the pressure drops. Under such circumstances, the time is ignored in this stage. Therefore, $t_{m}$ will converge to the first stage, given as $t_{m}\sim D_{0}/2V_{0}$ . This is also validated in our numerical simulations, as described previously in figure 6(c).

Therefore, the average radial velocity scales as $\overline{V_{r}}\sim D_{max}V_{0}/D_{0}$ . Taking all of them into (3.4), and supposing $KE_{0}=JE$ , by which the kinetic energy is balanced by the Joule dissipation, we finally obtain

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}V_{0}^{2}D_{0}^{3}\sim \unicode[STIX]{x1D70E}_{e}D_{max}^{2}D_{0}^{2}B_{z}^{2}V_{0}\Rightarrow \unicode[STIX]{x1D6FD}_{max}=\frac{D_{max}}{D_{0}}\sim N^{-1/2}. & & \displaystyle\end{eqnarray}$$

To validate the correctness of this scaling law in the Joule regime, we present the evolution of $\unicode[STIX]{x1D6FD}_{max}$ from our numerical results with $Re=13\,422$ , $We=96$ , $\unicode[STIX]{x1D703}=90^{\circ }$ and $N>2.515$ , and plot the comparative results in figure 8. We observe that the numerical results are predicted accurately by the scaling law (3.8), which shows a positive correction between $\unicode[STIX]{x1D6FD}_{max}$ and $N^{-1/2}$ in the Joule regime.

Figure 8.

A test of the scaling law of (3.8) in the Joule regime, using $\unicode[STIX]{x1D6FD}_{max}$ calculated from our numerical simulations within fixed $Re=13\,422$ , $We=96$ , $\unicode[STIX]{x1D703}=90^{\circ }$ and varied $N$ from $2.515{-}15.723$ . It is observed that all results collapse onto a single curve fitting with $\unicode[STIX]{x1D6FD}_{max}\sim N^{-1/2}$ , indicating that (3.8) is totally applicable in the Joule regime.

After obtaining $\unicode[STIX]{x1D6FD}_{max}\propto N^{-1/2}$ in the Joule regime, we should consider a more general situation in which all the viscous, capillary and Joule effects can be condensed into one scaling relation depending on the group of variables $\{Re^{1/5},We^{1/2},N^{-1/2},\unicode[STIX]{x1D703}\}$ . Although (1.1) is proved to be effective by constructing an interpolating scheme in the cross-over regime between the viscous and capillary regimes, it will be much more difficult to put $N^{-1/2}$ in, because there will be three variables contained in the interpolation function. Moreover, if the contact angle is also taken into consideration for a more universal scaling law, things will become even more complex.

Alternatively, we can define a new variable, $\unicode[STIX]{x1D6FD}_{0}$ , which represents the maximal spreading factor without magnetic field, namely under $N=0$ . In this way, the viscosity effect, the capillary effect and the influence of wettability are all integrated into $\unicode[STIX]{x1D6FD}_{0}$ , which can be calculated by (1.1). Consequently, the problem of constructing an interpolation scheme is simplified into interpolating between $\unicode[STIX]{x1D6FD}_{0}$ and $N^{-1/2}$ scaling. To test this possibility, we adopt an approach similar to Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) and Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) by constructing a function

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{max}\propto N^{-1/2}f_{c}(\unicode[STIX]{x1D6FD}_{0}^{2}N), & & \displaystyle\end{eqnarray}$$

where $f_{c}$ is a function of the parameter $\unicode[STIX]{x1D6FD}_{0}^{2}N$ ; we also introduce the impact parameter $L=\unicode[STIX]{x1D6FD}_{0}^{2}N$ to identify the relative influence between the three effects. According to this function, we plot all numerical results obtained with $\unicode[STIX]{x1D703}=90^{\circ }$ , as shown in figure 9(a). From the figure, we observe this approach indeed succeeds in collapsing all data points for different liquid metals and impact velocities onto a single curve. It indicates that (3.9) is applicable in describing the maximum spreading radius in the cross-over regime between the viscous, capillary and Joule regimes.

Figure 9.

The rescaled maximum spreading ratio as a function of $\unicode[STIX]{x1D6FD}_{0}^{2}N$ , with the solid line showing the interpolating function (3.10) fitted to the numerical results. (a) The numerical results obtained within fixed $\unicode[STIX]{x1D703}$ of $90^{\circ }$ when using different liquid metals as working medium. (b) The numerical results obtained within different $\unicode[STIX]{x1D703}$ of $60^{\circ }$ , $90^{\circ }$ and $120^{\circ }$ . It is observed that no matter what liquid metal we use, and no matter what contact angle we set, all results collapse onto the single curve given by (3.10).

To make (3.9) more quantitative, we also adopt the Padé approximant by constructing the formulation of $f_{c}(L)$ , equation (3.9) is transformed to

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{max}N^{1/2}=L^{1/2}/(1+BL^{1/2}),\quad L=\unicode[STIX]{x1D6FD}_{0}^{2}N, & & \displaystyle\end{eqnarray}$$

where $B$ is the fitting constant, and $f_{c}(L)$ satisfies $\lim _{L\rightarrow 0}\unicode[STIX]{x1D6FD}_{max}=\unicode[STIX]{x1D6FD}_{0}$ , $\lim _{L\rightarrow \infty }\unicode[STIX]{x1D6FD}_{max}\propto N^{-1/2}$ .

The fit of (3.10) agrees excellently with the numerical results, as shown in figure 9(a), where the determination coefficient $R^{2}$ is 0.988, indicating that (3.10) is very effective in predicting the maximum spreading radius in the cross-over regime.

Moreover, as we stated, the influence of $\unicode[STIX]{x1D703}$ is included in $\unicode[STIX]{x1D6FD}_{0}$ , so (3.10) should be applicable for different contact angles. To validate this, all numerical results, calculated with $\unicode[STIX]{x1D703}=60^{\circ }$ , $\unicode[STIX]{x1D703}=90^{\circ }$ and $\unicode[STIX]{x1D703}=120^{\circ }$ respectively, are plotted in figure 9(b) with the formulation of (3.9). From the figure, it is observed that all the data are collapsed onto a single curve, which can be fitted well by the scaling law of (3.10). Therefore, the analytical solution, which is based on the energy conservation principle, describes the spreading behaviour very well with various liquids, impact velocities, external magnetic fields and contact angles.

Regarding the influence of a dynamic contact angle $\unicode[STIX]{x1D703}_{d}$ in practical droplet spreading problems, we think it can be divided into two aspects: (i) The influence of $\unicode[STIX]{x1D703}_{d}$ on $\unicode[STIX]{x1D6FD}_{0}$ . According to Pasandideh-Fard et al. (Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996), it is less accurate to predict $\unicode[STIX]{x1D6FD}_{max}$ by using a constant equilibrium contact angle instead of using $\unicode[STIX]{x1D703}_{d}$ . However, this discrepancy between the two results will be very limited when the two contact angles are close. Also, Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) have proved that (1.1) fits very well with the experimental results, where the contact angle is definitely dynamic. (ii) The influence of $\unicode[STIX]{x1D703}_{d}$ on $N^{-1/2}$ . As presented in the derivation of (3.8), $t_{m}$ is almost dependent on the first pressure-driven stage, in which the influence of $\unicode[STIX]{x1D703}_{d}$ is also limited. Therefore, our model should be applicable in practical situations even if the contact angle is dynamic.