Ferrofluid droplets in planar extensional flows: Droplet shape and magnetization reveal novel rheological signatures of ferrofluid emulsions

We present a three-dimensional computational study of the impact of external magnetic fields on the dynamics of superparamagnetic ferrofluid droplets and rheology of dilute ferrofluid emulsions in planar extensional flows. Specifically, we show how the intensity and direction of uniform magnetic fields affect the planar extensional rheology of ferrofluid emulsions by changing the shape and magnetization of the constituent ferrofluid droplets in suspension. We find that the two traditional extensional viscosities associated with the normal stresses of the bulk emulsion in extension either remain constant or increase with the field intensity; the only exception occurs when the field direction is perpendicular to the extension plane, where increasing the field intensity keeps the planar extensional viscosity constant and modestly decreases the second extensional viscosity. We also find that the droplet tilts in the flow when the external field is not aligned with one of the flow main directions, which changes the recirculation pattern and flow topology inside the droplet. At the microscopic level, the droplet experiences a magnetic torque because of a small misalignment between its magnetization and the external field direction. At the macroscopic level, the bulk emulsion experiences a field-induced internal torque that leads to a nonsymmetric stress tensor with unexpected shear components in extension. To account for this unconventional stress-strain response, we introduce new extensional material functions such as shear and rotational viscosity coefficients that unveil novel rheological signatures of ferrofluid emulsions in planar extensional flows. This study offers new insights into applications based on the field-assisted manipulation of ferrofluid droplets and sheds light on the potential of ferrofluid emulsions as a model system for chiral fluids with internal rotational degrees of freedom that can be activated and controlled by coupling static magnetic fields with hydrodynamic flows.

Figure 1

Sketch of the problem (not to scale). A superparamagnetic ferrofluid droplet is suspended in a nonmagnetizable fluid. The two phases are incompressible Newtonian liquids of the same density and viscosity, and the dispersed-to-continuous phase magnetic permeability ratio is two. The droplet is initially spherical, has radius $a$, and the domain size (normalized by $a$) is 12.5, 10, and 7.5 in the $x, y$, and $z$ direction, respectively (the droplet volume fraction is $\beta \approx 0.45\%$). The system is subjected to a planar extensional flow in the $xy$ plane, and the far-field velocity ${\mathit{u}}_{\infty }$ of the continuous phase has a constant extension rate $\dot{\varepsilon }$ that defines the directions of flow extension ($x$ direction), flow compression ($y$ direction), and neutral ($z$ direction). The system is also subjected to a uniform magnetic field ${\mathit{H}}_{\mathit{0}}$ applied externally (the external field is in the $y$ direction in this sketch). The origin $\mathit{x}=\mathit{0}$ is fixed at the droplet center (the coordinate system at the bottom right is shown just for the sake of reference).

Figure 2

(a) Droplet distortion in the $xy$ plane ($D_{xy}$) as a function of Ca with no external magnetic field (${\mathrm{Ca}}_{\mathrm{mag}}=0$): present work (black circles), experimental results of Hsu and Leal [58] (red triangles), and simulation results of Hoang and Park [59] (blue squares). The insets show the droplet cross section in the $xy$ plane at $\mathrm{Ca}=0.02$ and $\mathrm{Ca}=0.115$. (b) Droplet distortion in the $xy$ plane ($D_{xy}$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $x$ direction with no external flow (${\mathit{u}}_{\infty }=\mathit{0}$): present work (black circles) and theoretical predictions of Afkhami et al. [21] (blue line). The insets show the droplet cross section in the $xy$ plane at ${\mathrm{Ca}}_{\mathrm{mag}}=2$ and ${\mathrm{Ca}}_{\mathrm{mag}}=20$.

Figure 3

Droplet distortion in the $xy$ plane ($D_{xy}$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the (a) $x$ direction and (b) $y$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds). The insets show the droplet cross section in the $xy$ plane at different conditions. The data set is not complete in (a) because the droplet does not achieve a steady shape when $\mathrm{Ca}=0.08$ and ${\mathrm{Ca}}_{\mathrm{mag}}\ge 6$ and when $\mathrm{Ca}=0.12$ and ${\mathrm{Ca}}_{\mathrm{mag}}\ge 0$ [see the discussion for Fig. 2].

Figure 4

Droplet distortion in the (a) $xy$ plane ($D_{xy}$) and (b) $xz$ plane ($D_{xz}$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $z$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds). The insets show the droplet cross section in the (a) $xy$ plane and (b) $xz$ plane at different conditions.

Figure 5

Three-dimensional view of oblate droplets at $\mathrm{Ca}=0.12$ when the external magnetic field is applied in the (a) $y$ direction (at ${\mathrm{Ca}}_{\mathrm{mag}}={\mathrm{Ca}}_{\mathrm{mag}}^{*}\approx 10.03$) and (b) $z$ direction (at ${\mathrm{Ca}}_{\mathrm{mag}}={\mathrm{Ca}}_{\mathrm{mag}}^{*}\approx 5.74$). In (a) $L_x=L_y$ is the equatorial radius in the $xy$ plane and $p_r$ is the polar radius in the $z$ axis; in (b) $L_x=L_z$ is the equatorial radius in the $xz$ plane and $p_r$ is the polar radius in the $y$ axis. For the sake of visualization, the droplet shape is projected on each plane (black), flow streamlines outside the droplet are projected on the $xy$ plane (red), and magnetic field lines outside the droplet are projected on the $yz$ and $xz$ plane (blue).

Figure 6

Magnetic capillary number ${\mathrm{Ca}}_{\mathrm{mag}}^{*}$ at which the droplet becomes an oblate ellipsoid (circles, left axis) and the corresponding droplet polar radius $p_r$ (squares, right axis) as a function of $\mathrm{Ca}$. The results are for external magnetic fields applied in the $y$ direction (red symbols) and $z$ direction (blue symbols). The solid lines are linear fits with fixed intercepts (${\mathrm{Ca}}_{\mathrm{mag}}^{*}=0$ and $p_r=1$ at $\mathrm{Ca}=0$): ${\mathrm{Ca}}_{\mathrm{mag}}^{*}\approx 80\mathrm{Ca}$ and $p_r\approx 1-1.7\mathrm{Ca}$ when the external field is in the $y$ direction (red); ${\mathrm{Ca}}_{\mathrm{mag}}^{*}\approx 44\mathrm{Ca}$ and $p_r\approx 1-2.6\mathrm{Ca}$ when the external field is in the $z$ direction (blue). The coefficient of determination of all adjustments is $R^2>0.99$.

Figure 7

$z$ component of the flow vorticity ($\mathit{\xi }=\mathbf{\nabla }\times \mathit{u}$) and flow streamlines in the $xy$ plane. The results are for $\text{Ca}=0.04$ when (a) there is no external magnetic field (${\mathrm{Ca}}_{\mathrm{mag}}=0$) and when the external magnetic field is applied in the (b) $x$ direction, (c) $y$ direction, and (d) $z$ direction at ${\mathrm{Ca}}_{\mathrm{mag}}=16$.

Figure 8

Magnitude of the magnetic field and magnetic field lines in the $xy$ plane. The results are for ${\mathrm{Ca}}_{\mathrm{mag}}=12$ when (a) the external magnetic field is applied in the $x$ direction with no external flow (${\mathit{u}}_{\infty }=\mathit{0}$) and when the external magnetic field is applied in the (b) $x$ direction, (c) $y$ direction, and (d) $z$ direction with the external flow at $\mathrm{Ca}=0.04$.

Figure 9

Magnitude of the droplet magnetization ($M$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds).

Figure 10

Droplet contribution to the planar extensional viscosity (${\eta }_p$) of the ferrofluid emulsion as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds). Note that ${\eta }_p$ is normalized by the droplet volume fraction ($\beta$).

Figure 11

Droplet contribution to the second extensional viscosity (${\eta }_2$) of the ferrofluid emulsion as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds). Note that ${\eta }_2$ is normalized by the droplet volume fraction ($\beta$).

Figure 12

Droplet distortion in the $yz$ plane ($D_{zy}$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.08$ (red triangles), and $\mathrm{Ca}=0.12$ (green diamonds). The insets show the droplet cross section in the $yz$ plane at different conditions.

Figure 13

Three-dimensional view of the prolatelike droplet at $\mathrm{Ca}=0.02$ when the external magnetic field is applied in the $x=y$ direction at ${\mathrm{Ca}}_{\mathrm{mag}}=20$. For the sake of visualization, the droplet shape is projected on each plane (black), flow streamlines outside the droplet are projected on the $xy$ plane (blue), and magnetic field lines outside the droplet are projected on the $xz$ and $yz$ plane (red). The angle $\theta$ is determined by the droplet major axis and the $x$ axis.

Figure 14

(a) Droplet distortion ($D$) and (b) orientation ($\theta$, measured with respect to the $x$ axis and normalized by $\pi /4$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $x=y$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.06$ (red triangles), $\mathrm{Ca}=0.08$ (green diamonds), and $\mathrm{Ca}=0.1$ (magenta stars). The insets show the droplet cross section in the $xy$ plane at different conditions.

Figure 15

Magnitude of the magnetic field and magnetic field lines in the $xy$ plane. The result is for $\mathrm{Ca}=0.1$ when the external magnetic field is applied in the $x=y$ direction at ${\mathrm{Ca}}_{\mathrm{mag}}=4$. The arrows (not to scale) indicate the direction of the external field ${\mathit{H}}_{\mathit{0}}$ and the direction of the system magnetization $\langle \mathit{M}\rangle =\beta \mathit{M}$; ${\theta }_{\mathrm{mag}}$ is the angle between them.

Figure 16

$z$ component of the flow vorticity ($\mathit{\xi }=\mathbf{\nabla }\times \mathit{u}$) and flow streamlines in the $xy$ plane. The result is for $\mathrm{Ca}=0.04$ when the external magnetic field is applied in the $x=y$ direction at ${\mathrm{Ca}}_{\mathrm{mag}}=16$.

Figure 17

(a) Magnitude of the droplet magnetization ($M$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.06$ (red triangles), $\mathrm{Ca}=0.08$ (green diamonds), and $\mathrm{Ca}=0.1$ (magenta stars). (b) Angle between the ferrofluid emulsion bulk magnetization and the external field direction (${\theta }_{\mathrm{mag}}$, in degrees) as a function of $\mathrm{Ca}$. The results are for ${\mathrm{Ca}}_{\mathrm{mag}}=2$ (black circles), ${\mathrm{Ca}}_{\mathrm{mag}}=4$ (blue squares), ${\mathrm{Ca}}_{\mathrm{mag}}=8$ (red triangles), ${\mathrm{Ca}}_{\mathrm{mag}}=12$ (green diamonds), ${\mathrm{Ca}}_{\mathrm{mag}}=16$ (magenta stars), and ${\mathrm{Ca}}_{\mathrm{mag}}=20$ (yellow pentagons). In both cases, the external magnetic field is applied in the $x=y$ direction.

Figure 18

Magnitude of the field-induced internal torque in the ferrofluid emulsion (${\tau }_{\mathrm{mag}}$) as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $x=y$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.06$ (red triangles), $\mathrm{Ca}=0.08$ (green diamonds), and $\mathrm{Ca}=0.1$ (magenta stars). Note that ${\tau }_{\mathrm{mag}}$ is normalized by the droplet volume fraction ($\beta$).

Figure 19

Droplet contribution to the (a) planar extensional viscosity (${\eta }_p$) and (b) second extensional viscosity (${\eta }_2$) of the ferrofluid emulsion as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $x=y$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.06$ (red triangles), $\mathrm{Ca}=0.08$ (green diamonds), and $\mathrm{Ca}=0.1$ (magenta stars). Note that both ${\eta }_p$ and ${\eta }_2$ are normalized by the droplet volume fraction ($\beta$).

Figure 20

Droplet contribution to the (a) shear viscosity (${\eta }_s$) and (b) rotational viscosity (${\eta }_r$) of the ferrofluid emulsion as a function of ${\mathrm{Ca}}_{\mathrm{mag}}$ when the external magnetic field is applied in the $x=y$ direction. The results are for $\mathrm{Ca}=0.02$ (black circles), $\mathrm{Ca}=0.04$ (blue squares), $\mathrm{Ca}=0.06$ (red triangles), $\mathrm{Ca}=0.08$ (green diamonds), and $\mathrm{Ca}=0.1$ (magenta stars). Note that both ${\eta }_s$ and ${\eta }_r$ are normalized by the droplet volume fraction ($\beta$).

Figure 21

Force balance in a control volume of width $2e$ around a portion $\mathrm{\Gamma }$ of the droplet surface. The superscripts $i$ and $o$ distinguish the inner and the outer phases, respectively.