2 Model development

In the present study, we focus on the 2-lobed branch of the deformation of levitated droplets in rotation, the branch on which angular velocity of rotation decreases as the deformation length increases. In this section, we develop a simple model describing the deforming behaviour of 2-lobed droplets in rotation and derive the governing equations of the progress of deformation, the stress-balance equation in the longitudinal direction of the 2-lobed droplet and the equation of angular momentum conservation.

We consider a levitated 2-lobed fluid drop rotating around a fixed axis that passes through its centre of gravity (figure 1 a). As the 2-lobed droplet has an axisymmetrical axis of the droplet shape that is perpendicular to the angular velocity vector, the droplet surface can be expressed as

(2.1) $$\begin{eqnarray}\boldsymbol{s}(x,\unicode[STIX]{x1D703})=(x,h(x)\cos \unicode[STIX]{x1D703},~h(x)\sin \unicode[STIX]{x1D703})^{\text{T}}(-R_{\mathit{Max}}\leqslant x\leqslant R_{\mathit{Max}},0\leqslant \unicode[STIX]{x1D703}\leqslant 2\unicode[STIX]{x03C0}),\end{eqnarray}$$

in Cartesian coordinates where the centre of gravity of the droplet is at the origin, the axisymmetrical axis of the droplet shape is along the $x$ -axis and the direction of the angular velocity vector is along the $z$ -axis. The $y$ -axis is set by following the right-hand rule. Here, $R_{Max}$ is the maximum deformation length of the droplet and $\unicode[STIX]{x1D703}$ is the auxiliary variable associated with the representation of droplet surface using a cylindrical system. Further, $h(x)$ is the radius of the cross-section at the position $x$ as shown in figure 1(a), which is a single-variable function of $x$ that satisfies $h^{\prime }(0)=0,~h(\pm R_{\mathit{Max}})=0,$ and $|h^{\prime }(\pm R_{\mathit{Max}})|=\infty$ . Hereafter, regarding the terms ‘axisymmetric’ or ‘axisymmetrical’ in this article, the axis of axial symmetry represents the longitudinal axis of the 2-lobed (‘dumbbell-like’ shaped) droplets, not the axis of rotational angular velocity $\unicode[STIX]{x1D74E}$ . In other words, the axisymmetrical axis is the $x$ -axis in figure 1(a), not the $z$ -axis.

In the rotational frame fixed to the rotating droplet, the Navier–Stokes (NS) equation can be written as

(2.2) $$\begin{eqnarray}{\displaystyle \frac{\text{D}\boldsymbol{u}}{\text{D}t}}={\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\unicode[STIX]{x1D748}}-\unicode[STIX]{x1D74E}\times (\unicode[STIX]{x1D74E}\times \boldsymbol{r})-2\unicode[STIX]{x1D74E}\times \boldsymbol{u}-\left({\displaystyle \frac{\text{D}\unicode[STIX]{x1D74E}}{\text{D}t}}\right)\times \boldsymbol{r},\end{eqnarray}$$

where $\text{D}/\text{D}t$ and $\tilde{\unicode[STIX]{x1D748}}$ denote the material derivative and stress tensor, respectively. In the NS equation above, the body forces except for the centrifugal force, Euler force and Coriolis force, such as gravity, are neglected. As the droplet consists of an incompressible fluid, the density and volume of the droplet are conserved during its deformation. In order to obtain the stress-balance equation in the direction of $\boldsymbol{e}_{x}$ , i.e. the longitudinal direction of the droplet, we calculate the volume integration of the NS equation over the right half of the droplet $(0\leqslant x\leqslant R_{\mathit{Max}},~0\leqslant r\leqslant h(x),~0\leqslant \unicode[STIX]{x1D703}\leqslant 2\unicode[STIX]{x03C0})$ , where $\unicode[STIX]{x1D74E}=(0,0,\unicode[STIX]{x1D714})^{\text{T}}$ and $\boldsymbol{r}=(x,r\cos \unicode[STIX]{x1D703},r\sin \unicode[STIX]{x1D703})^{\text{T}}$ . Owing to the axial symmetry of the shape with respect to the $x$ -axis and the assumption that the flow inside the droplet is axisymmetric, the integration of summation of the last two terms on the right-hand side of the NS equation within the cross-section of the droplet at the position $x$ , $y^{2}+z^{2}\leqslant h(x)^{2}$ , is along the $y$ direction. Accordingly, the stress-balance equation of the 2-lobed droplet in the direction of $\boldsymbol{e}_{x}$ is obtained by calculating the volume integration of the terms of inertial force, internal force and centrifugal force in the NS equation within the right half of the droplet $(0\leqslant x\leqslant R_{\mathit{Max}},~0\leqslant r\leqslant h(x),~0\leqslant \unicode[STIX]{x1D703}\leqslant 2\unicode[STIX]{x03C0})$ . Using the generalised Stokes’ theorem and the boundary condition, $\tilde{\unicode[STIX]{x1D70E}}\boldsymbol{\cdot }\boldsymbol{n}=-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\boldsymbol{n}$ on the surface, the stress-balance equation in the direction of $\boldsymbol{e}_{x}$ can be rewritten as

(2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\int _{V/2}{\displaystyle \frac{\text{D}^{2}}{\text{D}t^{2}}}\boldsymbol{r}~r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x & = & \displaystyle \int _{C_{0}}\tilde{\unicode[STIX]{x1D748}}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}S-\unicode[STIX]{x1D6FE}\int _{S_{f}}\boldsymbol{n}\unicode[STIX]{x1D705}\,\text{d}S+\unicode[STIX]{x1D70C}\int _{C_{0}}\boldsymbol{n}{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D714}^{2}(x^{2}+y^{2})\,\text{d}S\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D70C}\int _{S_{f}}\boldsymbol{n}{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D714}^{2}(x^{2}+y^{2})\,\text{d}S,\end{eqnarray}$$

where $C_{0}$ is the central cross-section, $S_{f}$ is the free surface of the right half of the droplet $(0\leqslant x\leqslant R_{\mathit{Max}},~0\leqslant \unicode[STIX]{x1D703}\leqslant 2\unicode[STIX]{x03C0})$ , $\boldsymbol{n}$ is the outward unit normal vector to each integral region of surface integration, $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D6FE}$ denote the density and coefficient of surface tension of the droplet, respectively, and $\unicode[STIX]{x1D705}$ is twice the mean curvature of the interface, which can be evaluated as $\unicode[STIX]{x1D705}=1/(h~(1+(\text{d}h/\text{d}x)^{2})^{1/2})-(\text{d}^{2}h/\text{d}x^{2})/(1+(\text{d}h/\text{d}x)^{2})^{3/2}$ . Supplemental explanations of the derivation of (2.3) are provided in appendix A. Although (2.3) is a vector equation, the $y$ and $z$ components of each integral over $S_{f}$ in (2.3) become 0 owing to the axial symmetry of the shape with respect to the $x$ -axis. The $y$ and $z$ components of each integral over $C_{0}$ in (2.3) are also 0 because $\boldsymbol{n}$ is equal to $-\boldsymbol{e}_{x}$ on $C_{0}$ . By substituting the stress tensor in the cylindrical coordinate (Landau & Lifshits Reference Landau and Lifshits1959) and following Eggers & Dupont (Reference Eggers and Dupont1994) and Eggers & Fontelos (Reference Eggers and Fontelos2005), the integral of $\tilde{\unicode[STIX]{x1D748}}\boldsymbol{\cdot }\boldsymbol{n}$ over $C_{0}$ can be evaluated by assuming that the internal flow is axisymmetric. The integrals over $S_{f}$ can be calculated using $\boldsymbol{n}=(-\text{d}h/\text{d}x,\cos \unicode[STIX]{x1D703},\sin \unicode[STIX]{x1D703})^{\text{T}}/(1+(\text{d}h/\text{d}x)^{2})^{1/2}$ and $\text{d}S=h\sqrt{1+(\text{d}h/\text{d}x)^{2}}\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x$ over the free surface. Using these, each term of (2.3) is explicitly calculated and the stress-balance equation in the direction of $\boldsymbol{e}_{x}$ , i.e. the longitudinal direction of the 2-lobed droplet in rotation, can be shown to be

(2.4) $$\begin{eqnarray}{\displaystyle \frac{\text{d}^{2}}{\text{d}t^{2}}}\left({\displaystyle \frac{1}{2}}mX_{G}\right)={\displaystyle \frac{1}{2}}mX_{G}\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x03C0}\left[\unicode[STIX]{x1D6FE}\left(H_{\mathit{mid}}^{2}\cdot \unicode[STIX]{x1D705}|_{H_{\mathit{mid}}}-2H_{\mathit{mid}}\right)+6\unicode[STIX]{x1D702}H_{\mathit{mid}}{\displaystyle \frac{\text{d}H_{\mathit{mid}}}{\text{d}t}}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is the coefficient of viscosity, $m$ is the total mass of the droplet, $H_{\mathit{mid}}$ is the radius of the central cross-section and $X_{G}$ denotes the distance between the central cross-section and the centre of gravity of each lobe, which can be evaluated as $mX_{G}/2=\int _{0}^{R_{\mathit{Max}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}xh(x)^{2}\,\text{d}x$ .

Furthermore, by calculating the volume integration of the torque generated by the Coriolis and Euler forces of the NS equation around the $z$ axis, the conservation of angular momentum is expressed as

(2.5) $$\begin{eqnarray}I_{zz}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D714}}{\text{d}t}}+\unicode[STIX]{x1D714}{\displaystyle \frac{\text{d}I_{zz}}{\text{d}t}}=\unicode[STIX]{x0394}L(t)\simeq 0,\end{eqnarray}$$

where $I_{zz}$ is the moment of inertia of the right half of the droplet around the $z$ -axis, which is evaluated as $I_{zz}=\int _{V/2}\unicode[STIX]{x1D70C}(x^{2}+y^{2})\,\text{d}x\,\text{d}y\,\text{d}z=\int _{0}^{R_{\mathit{Max}}}\int _{0}^{2\unicode[STIX]{x03C0}}\int _{0}^{h(x)}\unicode[STIX]{x1D70C}(x^{2}+r^{2}\cos ^{2}\unicode[STIX]{x1D703})r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x$ . By calculating this integration with respect to $r$ and $\unicode[STIX]{x1D703}$ , $I_{zz}$ is written such that $I_{zz}=\int _{0}^{R_{\mathit{Max}}}\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}(x^{2}h(x)^{2}+h(x)^{4}/4)\,\text{d}x.$ $\unicode[STIX]{x0394}L(t)$ represents a slight increase in the angular momentum that corresponds to a small perturbation applied to the system, and $\unicode[STIX]{x0394}L(t)=0$ almost every time.

Following Brown & Scriven (Reference Brown and Scriven1980), we non-dimensionalise the angular velocity $\unicode[STIX]{x1D714}$ , time $t$ , the distance between the central cross-section and an arbitrary cross-section $x$ , the maximum deformation length $R_{\mathit{Max}}$ , and the radius of the central cross-section $H_{\mathit{mid}}$ using the reference constants $\unicode[STIX]{x1D714}_{0}$ and $R_{0}$ such that

(2.6a-e ) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\ast }\unicode[STIX]{x1D714}_{0},\quad t=t_{\ast }/\unicode[STIX]{x1D714}_{0},\quad x=R_{0}x_{\ast },\quad R_{\mathit{Max}}=R_{\mathit{Max}\ast }R_{0},\quad \text{and}\quad H_{\mathit{mid}}=H_{\mathit{mid}\ast }R_{0},\end{eqnarray}$$

where $R_{0}$ is the radius of a spherical drop of the same volume and $\unicode[STIX]{x1D714}_{0}$ is the smallest eigenfrequency of the spherical harmonic oscillation of the drop, which is defined as $\sqrt{8\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D70C}R_{0}^{3})}$ (Rayleigh Reference Rayleigh1879; Landau & Lifshits Reference Landau and Lifshits1959). Hereafter, the symbols with an asterisk $\ast$ denote the non-dimensionalised variables. As the differential operators $\text{d}/\text{d}t$ and $\text{d}/\text{d}x$ are rewritten as $\unicode[STIX]{x1D714}_{0}(\text{d}/\text{d}t_{\ast })$ and $(1/R_{0})(\text{d}/\text{d}x_{\ast })$ , respectively, the dimensionless forms of (2.4) and (2.5) can be obtained as follows:

(2.7) $$\begin{eqnarray}{\displaystyle \frac{\text{d}^{2}}{\text{d}t_{\ast }^{2}}}x_{g\ast }=x_{g\ast }\unicode[STIX]{x1D714}_{\ast }^{2}-H_{\mathit{mid}\ast }-H_{\mathit{mid}\ast }^{2}\left.{\displaystyle \frac{\text{d}^{2}h_{\ast }(x_{\ast })}{\text{d}x_{\ast }^{2}}}\right|_{x_{\ast }=0}+{\displaystyle \frac{6\unicode[STIX]{x1D702}R_{0}\unicode[STIX]{x1D714}_{0}}{\unicode[STIX]{x1D6FE}}}H_{\mathit{mid}\ast }{\displaystyle \frac{\text{d}H_{\mathit{mid}\ast }}{\text{d}t_{\ast }}}\end{eqnarray}$$

and

(2.8) $$\begin{eqnarray}I_{zz\ast }{\displaystyle \frac{\text{d}\unicode[STIX]{x1D714}_{\ast }}{\text{d}t_{\ast }}}+\unicode[STIX]{x1D714}_{\ast }{\displaystyle \frac{\text{d}I_{zz\ast }}{\text{d}t_{\ast }}}=0,\end{eqnarray}$$

where $x_{g\ast }=8\int _{0}^{R_{\mathit{Max}\ast }}x_{\ast }h_{\ast }(x_{\ast })^{2}\,\text{d}x_{\ast }$ .

As (2.7)–(2.8) include terms calculated using the entire shape of the droplet, the entire shape of the droplet should be explicitly determined throughout the entire process of deformation. In this paper, we determine the cross-sectional radius of the droplet $h_{\ast }(x_{\ast })$ as a mathematical function, i.e. the Cassinian oval curve, by focusing on the similarity of the shapes. A Cassinian oval is a quartic plane curve defined as

(2.9) $$\begin{eqnarray}(Ax_{\ast }^{2}+z_{\ast }^{2})^{2}-(Au_{\ast }^{2}-v_{\ast }^{2})(Ax_{\ast }^{2}-z_{\ast }^{2})-Au_{\ast }^{2}v_{\ast }^{2}=0~(Au_{\ast }>v_{\ast }>0).\end{eqnarray}$$

The shape of the curve, up to mathematical similarity, depends on two dimensionless parameters: $k=u_{\ast }/v_{\ast }$ and $A$ . When $k\geqslant 1$ , the curve is a single connected loop. It is convex for $1\leqslant k\leqslant \sqrt{3/A}$ and 2-lobed for $k>\sqrt{3/A}$ (Basset Reference Basset1901), as described in figure 1(b) (refer to appendix B for details). The limiting case of $k\rightarrow \infty$ of (2.9) is an $\infty$ -like shaped curve that has a singular point at the origin and the curve approaches $z_{\ast }=\pm \sqrt{A}x_{\ast }$ at the origin (the red curve in figure 1 b). In other words, when $k$ approaches $\infty$ and the neck region of the Cassinian droplet pinches off, the vicinity of the midpoint of the Cassinian droplet is approximated by cones with the apex angle $\unicode[STIX]{x1D719}=2\text{arctan}(\sqrt{A})$ , i.e. $A$ is a parameter expressing the degree of sharpness of the neck region at the moment the neck region of the Cassinian droplet pinches off. We infer that $A<1=(\text{arctan}45^{\circ })^{2}$ by observing the shape of actual 2-lobed droplets at the moment the droplet breaks up (figure 10h of Wang et al. (Reference Wang, Anilkumar, Lee and Lin1994) and figure 7G of Rhim et al. (Reference Rhim, Chung and Elleman1990)); however, we need not to limit the range of the value of $A$ here. The value of the parameter $A$ is fixed within the process of deformation and breakup of the Cassinian droplet, and the value is determined in § 3.

As the Cassinian oval curve expressed by (2.9) intersects the $x_{\ast }$ -axis and $z_{\ast }$ -axis at $(\pm u_{\ast },0)$ and $(0,\pm v_{\ast })$ , respectively, the parameters $u_{\ast }$ and $v_{\ast }$ correspond to $R_{\mathit{Max}\ast }$ and $H_{\mathit{mid}\ast }$ , respectively. In the present study, the shape of the interface of a 2-lobed droplet is determined as a solid of revolution generated by a Cassinian oval curve in order to have axial symmetry with respect to the $x_{\ast }$ -axis such that $(Ax_{\ast }^{2}+y_{\ast }^{2}+z_{\ast }^{2})^{2}-(Au_{\ast }^{2}-v_{\ast }^{2})(Ax_{\ast }^{2}-y_{\ast }^{2}-z_{\ast }^{2})-Au_{\ast }^{2}v_{\ast }^{2}=0.$ By substituting $\sqrt{y_{\ast }^{2}+z_{\ast }^{2}}=h_{\ast }(x_{\ast })$ into this equation and solving for $h_{\ast }(x_{\ast })$ , we obtained the dimensionless radius profile of the Cassinian droplet such that

(2.10) $$\begin{eqnarray}h_{\ast }(x_{\ast })=\sqrt{{\textstyle \frac{1}{2}}(-Au_{\ast }^{2}+v_{\ast }^{2}-2Ax_{\ast }^{2}+\sqrt{(Au_{\ast }^{2}+v_{\ast }^{2})^{2}+8A(Au_{\ast }^{2}-v_{\ast }^{2})x_{\ast }^{2}})}.\end{eqnarray}$$

Subsequently, $x_{g\ast }$ is calculated as

(2.11) $$\begin{eqnarray}x_{g\ast }=8\int _{0}^{u_{\ast }}x_{\ast }\cdot h_{\ast }(x_{\ast })^{2}\,\text{d}x_{\ast }={\displaystyle \frac{1}{3A}}(A^{2}u_{\ast }^{4}+4Au_{\ast }^{2}v_{\ast }^{2}+v_{\ast }^{4}),\end{eqnarray}$$

and the dimensionless moment of inertia $I_{zz\ast }$ can be expressed as a function of $u_{\ast }$ and $v_{\ast }$ (refer to appendix C). Substituting the formulas (2.10) and (2.11) into (2.7), the dimensionless form of the stress-balance equation of a Cassinian drop is derived as

(2.12) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}^{2}}{\text{d}t_{\ast }^{2}}}\left({\displaystyle \frac{1}{3A}}(A^{2}u_{\ast }^{4}+4Au_{\ast }^{2}v_{\ast }^{2}+v_{\ast }^{4})\right) & = & \displaystyle \unicode[STIX]{x1D714}_{\ast }^{2}~{\displaystyle \frac{1}{3A}}(A^{2}u_{\ast }^{4}+4Au_{\ast }^{2}v_{\ast }^{2}+v_{\ast }^{4})\nonumber\\ \displaystyle & & \displaystyle -\,{\displaystyle \frac{v_{\ast }(A(1+A)u_{\ast }^{2}-(3A-1)v_{\ast }^{2})}{Au_{\ast }^{2}+v_{\ast }^{2}}}\nonumber\\ \displaystyle & & \displaystyle +\,{\displaystyle \frac{6\unicode[STIX]{x1D702}R_{0}\unicode[STIX]{x1D714}_{0}}{\unicode[STIX]{x1D6FE}}}\cdot v_{\ast }{\displaystyle \frac{\text{d}v_{\ast }}{\text{d}t_{\ast }}}.\end{eqnarray}$$

As there are three dependent variables, $u_{\ast },~v_{\ast },~\text{and}~\unicode[STIX]{x1D714}_{\ast }$ , in the system, the deformation of a 2-lobed droplet can be quantitatively evaluated using (2.12), the conservation of angular momentum, and the volume conservation equation.

As expressed in (2.1), the droplet is determined as a solid of revolution generated by a Cassinian oval curve, and the shape of an arbitrary cross-section of the droplet is a circle. Therefore, the 2-lobed shape of the elliptical cross-section observed immediately after the transition point from the equilibrium shape that is axisymmetric with respect to the axis of rotation (e.g. figure 10 in Wang et al. (Reference Wang, Anilkumar, Lee and Lin1994) and figure 7 in Rhim et al. (Reference Rhim, Chung and Elleman1990)) is not expressed by the proposed model.

3 The stationary solution and its linear stability

In this section, we demonstrate the existence of stationary solutions of (2.8) (2.12) and perform a linear stability analysis of the solutions. Henceforth, $k=u_{\ast }/v_{\ast }$ , which was introduced in the previous section and represents the ratio of the maximum deformation length to the radius of the central cross-section of a Cassinian droplet, is treated as one of the dependent variables of the governing equations. Substituting $u_{\ast }/k$ for $v_{\ast }$ into the integral $\int _{0}^{u_{\ast }}h_{\ast }(x_{\ast })^{2}\,\text{d}x_{\ast }$ , we obtain the equation of dimensionless volume conservation:

(3.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{2}{3}} & = & \displaystyle \int _{0}^{u_{\ast }}h_{\ast }(x_{\ast })^{2}\,\text{d}x_{\ast }={\displaystyle \frac{1}{12}}u_{\ast }(-Au_{\ast }^{2}+3v_{\ast }^{2})\nonumber\\ \displaystyle & & \displaystyle +\,{\displaystyle \frac{(Au_{\ast }^{2}+v_{\ast }^{2})^{2}}{8\sqrt{2A(Au_{\ast }^{2}-v_{\ast }^{2})}}}\text{log}_{e}\left({\displaystyle \frac{3Au_{\ast }^{2}-v_{\ast }^{2}+2u_{\ast }\sqrt{2A(Au_{\ast }^{2}-v_{\ast }^{2})}}{Au_{\ast }^{2}+v_{\ast }^{2}}}\right)\nonumber\\ \displaystyle & = & \displaystyle u_{\ast }^{3}\left[{\displaystyle \frac{1}{12}}\left(-A+{\displaystyle \frac{3}{k^{2}}}\right)+{\displaystyle \frac{1}{8\sqrt{2}}}{\displaystyle \frac{(Ak^{2}+1)^{2}}{k^{3}\sqrt{A(Ak^{2}-1)}}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\times \,\log _{\mathit{e}}\left({\displaystyle \frac{3Ak^{2}-1+2\sqrt{2}k\sqrt{A(Ak^{2}-1)}}{Ak^{2}+1}}\right)\right].\end{eqnarray}$$

Solving (3.1) for $u_{\ast }$ , each variable, $u_{\ast }=R_{\mathit{Max}}/R_{0}$ and $v_{\ast }=H_{\mathit{mid}}/R_{0}=u_{\ast }/k$ , can be expressed as a single-variable function of $k$ with the parameter $A$ .

When $u_{\ast }(k;A)$ and $v_{\ast }(k;A)$ are substituted into (2.8) and (2.12), each component is also expressed as a single-variable function of $k$ . Hereafter, the symbols with $(k;A)$ denote variables in which $u_{\ast }(k;A)$ and $v_{\ast }(k;A)$ have been substituted. Subsequently, the first-order coupled differential equations of $k$ , $\unicode[STIX]{x1D714}_{\ast }$ , and $\ell ~(=\text{d}k/\text{d}t_{\ast })$ can be obtained as follows:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}k}{\text{d}t_{\ast }}}=\ell , & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D714}_{\ast }}{\text{d}t_{\ast }}}=-\unicode[STIX]{x1D714}_{\ast }{\displaystyle \frac{I_{zz\ast }^{\prime }(k;A)}{I_{zz\ast }(k;A)}}\ell , & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\ell }{\text{d}t_{\ast }}}=-{\displaystyle \frac{x_{g\ast }^{\prime \prime }(k;A)}{x_{g\ast }^{\prime }(k;A)}}\ell ^{2}+{\displaystyle \frac{x_{g\ast }(k;A)}{x_{g\ast }^{\prime }(k;A)}}\unicode[STIX]{x1D714}_{\ast }^{2}-{\displaystyle \frac{K(k;A)}{x_{g\ast }^{\prime }(k;A)}}+N\cdot {\displaystyle \frac{v_{\ast }(k;A)v_{\ast }^{\prime }(k;A)}{x_{g\ast }^{\prime }(k;A)}}\ell , & \displaystyle\end{eqnarray}$$

where $K(k;A)$ denotes the second term of the right-hand side of (2.12) and the prime symbols $^{\prime }$ and $^{\prime \prime }$ are the first and second derivatives with respect to $k$ , respectively. $N$ is a dimensionless constant expressed as

(3.5) $$\begin{eqnarray}N={\displaystyle \frac{6\unicode[STIX]{x1D702}R_{0}\unicode[STIX]{x1D714}_{0}}{\unicode[STIX]{x1D6FE}}}=12\sqrt{2}{\displaystyle \frac{\unicode[STIX]{x1D702}}{\sqrt{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}R_{0}}}}=12\sqrt{2}Oh,\end{eqnarray}$$

where $Oh$ is the Ohnesorge number calculated using $R_{0}$ , which is the radius of a spherical drop of the same volume.

Solving (3.2)–(3.4) with the left-hand side of each equation set to 0, the set of stationary solutions can be obtained as

(3.6a-c ) $$\begin{eqnarray}\forall k\in [\sqrt{3/A},\infty ),\quad \unicode[STIX]{x1D714}_{\ast }=\unicode[STIX]{x1D714}_{\ast ,s}(k;A),\quad \ell =0,\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\ast ,s}(k;A)$ is the positive solution of $x_{g\ast }(k;A)\cdot \unicode[STIX]{x1D714}_{\ast }^{2}-K(k;A)=0$ , which is expressed as

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\ast ,s}(k;A)=\left.\sqrt{{\displaystyle \frac{3Av_{\ast }(A(1+A)u_{\ast }^{2}-(3A-1)v_{\ast }^{2})}{(Au_{\ast }^{2}+v_{\ast }^{2})(A^{2}u_{\ast }^{4}+4Au_{\ast }^{2}v_{\ast }^{2}+v_{\ast }^{4})}}}~\right|_{\substack{ u_{\ast }=u_{\ast }(k;A) \\ v_{\ast }=v_{\ast }(k;A)}}.\end{eqnarray}$$

$\unicode[STIX]{x1D714}_{\ast ,s}(k;A)$ and $u_{\ast }(k;A)$ are functions of $k$ with the parameter $A$ , and full forms are expressed in appendix C. When the value of parameter $A$ is determined, we can calculate one corresponding value of $k$ for a given value of $u_{\ast }(k;A)$ , and corresponding value of $\unicode[STIX]{x1D714}_{\ast ,s}(k;A)$ is calculated. In other words, using (3.7) we can draw a plot of parametric equations $(\unicode[STIX]{x1D714}_{\ast ,s}(k;A),u_{\ast }(k;A))$ by changing the values of $k$ within the range of $k>\sqrt{3/A}$ , the range of $k$ within which a Cassinian droplet has a 2-lobed shape. The graphs are described in figure 2.

In order to investigate the stability of the stationary solution against a small perturbation, we now perform a linear stability analysis. An example of a perturbation is a slight increase in the angular momentum, i.e. $\unicode[STIX]{x0394}L(t)$ in (2.5). Linearising (3.2)–(3.4) at $(k,\unicode[STIX]{x1D714}_{\ast },\ell )=(k,\unicode[STIX]{x1D714}_{\ast ,s}(k;A),0)$ , we obtain the Jacobian matrix $\unicode[STIX]{x1D645}$ such that

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D645} & = & \displaystyle \left.\left(\begin{array}{@{}ccc@{}}0 & 0 & 1\\ -\unicode[STIX]{x1D714}_{\ast }\left({\displaystyle \frac{I_{zz\ast }^{\prime }}{I_{zz\ast }}}\right)^{\prime }\ell & -\left({\displaystyle \frac{I_{zz\ast }^{\prime }}{I_{zz\ast }}}\right)\ell & -\left({\displaystyle \frac{I_{zz\ast }^{\prime }}{I_{zz\ast }}}\right)\unicode[STIX]{x1D714}_{\ast }\\ {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k}}(\text{RHS of (3.4)}) & 2\unicode[STIX]{x1D714}_{\ast }\left({\displaystyle \frac{x_{g\ast }}{x_{g\ast }^{\prime }}}\right) & -2\ell \left({\displaystyle \frac{x_{g\ast }^{\prime \prime }}{x_{g\ast }^{\prime }}}\right)+N{\displaystyle \frac{v_{\ast }v_{\ast }^{\prime }}{x_{g\ast }^{\prime }}}\end{array}\right)\right|_{\substack{ \unicode[STIX]{x1D714}_{\ast }=\unicode[STIX]{x1D714}_{\ast ,s}(k;A), \\ \ell =0}}\nonumber\\ \displaystyle & = & \displaystyle \left(\begin{array}{@{}ccc@{}}0 & 0 & 1\\ 0 & 0 & -\left({\displaystyle \frac{I_{zz\ast }^{\prime }}{I_{zz\ast }}}\right)\unicode[STIX]{x1D714}_{\ast ,s}\\ \unicode[STIX]{x1D714}_{\ast ,s}^{2}\left({\displaystyle \frac{x_{g\ast }}{x_{g\ast }^{\prime }}}\right)^{\prime }-\left({\displaystyle \frac{K}{x_{g\ast }^{\prime }}}\right)^{\prime } & 2\unicode[STIX]{x1D714}_{\ast ,s}\left({\displaystyle \frac{x_{g\ast }}{x_{g\ast }^{\prime }}}\right) & N{\displaystyle \frac{v_{\ast }v_{\ast }^{\prime }}{x_{g\ast }^{\prime }}}\end{array}\right).\end{eqnarray}$$

The eigenvalues of $\unicode[STIX]{x1D645}$ are expressed as

(3.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{0}=0,\quad \unicode[STIX]{x1D706}_{\pm }={\displaystyle \frac{\unicode[STIX]{x1D611}_{33}\pm \sqrt{\unicode[STIX]{x1D611}_{33}^{2}+4(\unicode[STIX]{x1D611}_{31}+\unicode[STIX]{x1D611}_{23}\unicode[STIX]{x1D611}_{32})}}{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D611}_{ij}$ is the $(i,j)$ component of the matrix $\unicode[STIX]{x1D645}$ . If both $\text{Re}[\unicode[STIX]{x1D706}_{+}]$ and $\text{Re}[\unicode[STIX]{x1D706}_{-}]$ are negative, the stationary solution is stable. As $x_{g\ast }(k;A)$ increases and $v_{\ast }(k;A)$ decreases with an increase in $k$ , $\unicode[STIX]{x1D611}_{33}<0$ for an arbitrary value of $k$ . Therefore, the condition of linear stability is equivalent to

(3.10) $$\begin{eqnarray}D(k;A):=\unicode[STIX]{x1D611}_{31}+\unicode[STIX]{x1D611}_{23}\unicode[STIX]{x1D611}_{32}<0,\end{eqnarray}$$

regardless of the value of $N$ . As $D(k;A)$ is a monotonically increasing function of $k$ (refer to the latter part of appendix C), there exists a critical value $k_{\mathit{crit},A}$ that satisfies $D(k_{\mathit{crit},A};A)=0$ for each value of the parameter $A$ . Subsequently, according to the proposed model, if and only if $k>k_{\mathit{crit},A}$ , the stationary solution is unstable for the corresponding $A$ . In other words, the condition of linear stability only depends on the ratio of the maximum deformation length to the radius of the central cross-section of the droplet, regardless of its physical properties, i.e. $\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x1D6FE}$ , $\unicode[STIX]{x1D70C}$ and $R_{0}$ . By numerically solving the inequality (3.10), $k_{\mathit{crit},A}$ is calculated for each value of $A$ . We discuss the rate of growing of a minute shift of $(\unicode[STIX]{x1D714}_{\ast },u_{\ast })$ from an equilibrium point under the condition that the stationary solution (3.6) is unstable. The Taylor expansion of $\unicode[STIX]{x1D706}_{+}=(\unicode[STIX]{x1D611}_{33}+\sqrt{\unicode[STIX]{x1D611}_{33}^{2}+4D(k;A)})/2$ at $k=k_{\mathit{crit},A}$ that satisfies $D(k_{\mathit{crit},A};A)=0$ is calculated as $\unicode[STIX]{x1D706}_{+}=-D/\unicode[STIX]{x1D611}_{33}+(1/\unicode[STIX]{x1D611}_{33})(D/\unicode[STIX]{x1D611}_{33})^{2}-(2/\unicode[STIX]{x1D611}_{33}^{2})(D/\unicode[STIX]{x1D611}_{33})^{3}+O((D/\unicode[STIX]{x1D611}_{33})^{4})$ . As $\unicode[STIX]{x1D611}_{33}$ is negative and proportional to $N$ , $\text{Re}[\unicode[STIX]{x1D706}_{+}]$ becomes larger as $N=12\sqrt{2}\unicode[STIX]{x1D702}/\sqrt{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}R_{0}}=12\sqrt{2}Oh$ becomes smaller under the condition that the stationary solution (3.6) is unstable. Therefore, a minute shift of $(\unicode[STIX]{x1D714}_{\ast },u_{\ast })$ begins to grow at a rate inversely proportional to the Ohnesorge number.

Figure 2.

Results of least squares fittings using the ascending part of the 2-lobed branch of Brown and Scriven’s numerical calculation (figure 6 of Brown & Scriven (Reference Brown and Scriven1980), red lines shown in panels (b) and (d)) and the experimentally obtained scatter plot (figure 4 of Tanaka et al. Reference Tanaka, Matsumoto, Kaneko and Abe2013). (a,b) Least squares regression between (3.11) and Brown and Scriven’s numerical solution. The plot of Brown and Scriven’s solution is fitted best when $A=0.821$ . The coefficient of determination, $R^{2}$ , is approximately 0.984. (c,d) Result of the least squares fitting using the experimentally obtained scatter plot as the dataset. The experimental result is very well approximated by (3.11) when $A=0.8185$ . The coefficient of determination is approximately 0.928. Panel (d) is created by using the data provided by authors of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013). The entire data are shown in figure 4 of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013).

In order to determine the optimal value of the parameter $A$ , we perform a least square regression with two preceding studies: Brown and Scriven’s numerical calculation and an experimental study (Tanaka et al. Reference Tanaka, Matsumoto, Kaneko and Abe2013). The dataset used for regression with (Brown & Scriven Reference Brown and Scriven1980) is a part of Brown and Scriven’s solution of 2-lobed equilibrium shapes satisfying $0.305\leqslant \unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}<0.506$ , which belongs to the region where the angular momentum does not decrease as the angular velocity of rotation decreases (the graph displayed in figure 6 of Brown & Scriven (Reference Brown and Scriven1980)). The dataset used for the regression with the experimental study is a part of the experimental data presented in figure 4 of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013), an experimental study on measuring viscosity by using the breakup behaviour of a levitated droplet, and is provided by the authors of the paper. The dataset used for the regression with the experimental study consists of points on the 2-lobed branch of experimentally obtained scatter plot of $R_{\mathit{Max}}/R_{0}$ versus $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}$ of $\unicode[STIX]{x1D702}=0.0026~\text{Pa}~\text{s}$ satisfying $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}<0.525$ and these of $R_{\mathit{Max}}/R_{0}$ versus $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}$ of $\unicode[STIX]{x1D702}=9.3~\text{Pa}~\text{s}$ satisfying $0.35<\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}<0.525$ that belongs to the stable region in the proposed model for all the values of $A$ tested. Here, in this study, the 2-lobed branch of the experimentally obtained scatter plot is a set of points that are located close to the ascending part of the 2-lobed brunch of Brown and Scriven’s solution and that are not close to the transition point to the 2-lobed shape out of the points on experimentally obtained scatter plot (shown in figure 4 of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013)). According to the above definition, the dataset used for the regression with the experimental study is obtained by removing the four circles that are located in the bottom right and the triangles that are located in the range of $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}<0.35$ from the scatter plot shown in 2(d). The regression function is defined as

(3.11) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}_{\ast }=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D714}_{\ast ,s}(k;A)\quad & (k\leqslant k_{\mathit{crit},A}),\\ \unicode[STIX]{x1D714}_{\ast ,s}(k_{\mathit{crit},A};A)I_{zz\ast }(k_{\mathit{crit},A};A)/I_{zz\ast }(k;A)\quad & (k>k_{\mathit{crit},A}),\end{array}\right.\\ R_{\mathit{Max}\ast }=u_{\ast }(k;A).\end{array}\right\}\end{eqnarray}$$

Here, $\unicode[STIX]{x1D714}_{\ast ,s}(k;A)$ is the stationary solution of the rotation rate that is expressed in (3.7). This regression function corresponds to the situation where the maximum deformation length of the droplet increases while following the locus of a linearly stable stationary solution by quasi-statically adding the angular momentum, and the angular momentum is conserved when $R_{\mathit{Max}\ast }=u_{\ast }>u_{\ast \mathit{crit},A}$ .

We show the results of the least squares regression in figure 2. As shown in figure 2(a,b), Brown and Scriven’s solution of 2-lobed equilibrium shapes satisfying $0.305\leqslant \unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}<0.506$ is fitted best by (3.11) when $A=0.8210$ . On the other hand, the experimentally obtained scatter plot (figure 4 of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013)) is fitted best by (3.11) when $A=0.8185$ (figure 2 c,d). For each least squares regression, the coefficient of determination, which is defined as $R^{2}=1-\sum _{\mathit{data}}(\unicode[STIX]{x1D714}_{\ast }(k;A)-\unicode[STIX]{x1D714}_{\mathit{data}})^{2}/\sum _{\mathit{data}}(\overline{\unicode[STIX]{x1D714}}_{\mathit{data}}-\unicode[STIX]{x1D714}_{\mathit{data}})^{2}$ , is maximised at the value of $A$ shown above. Here, $\unicode[STIX]{x1D714}_{\ast }(k;A)$ is the angular velocity expressed in (3.11). $\unicode[STIX]{x1D714}_{\mathit{data}}$ is the $\unicode[STIX]{x1D714}$ -component of each element of the dataset for regression and $\overline{\unicode[STIX]{x1D714}}_{\mathit{data}}$ is the mean of $\unicode[STIX]{x1D714}_{\mathit{data}}$ . The set of $\unicode[STIX]{x1D714}_{\mathit{data}}$ is defined for each regression dataset. The ascending region of the 2-lobed brunch of Brown and Scriven’s solution is fitted best by (3.11) and takes a maximum value of 0.984 when $A=0.8210$ , and experimental data used for regression provided by Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013) are fitted best by (3.11) with a maximum value of 0.928 when $A=0.8185$ . As the coefficient of determination defined in the present study becomes larger, a better fit between the data and the result obtained from the Cassinian droplet model is provided. As mentioned above, the maximum value of $R^{2}$ obtained from the regression with Brown and Scriven’s solution ( $R^{2}\simeq$ 0.984) is larger than the maximum value of $R^{2}$ obtained from the regression with experiment (Tanaka et al. Reference Tanaka, Matsumoto, Kaneko and Abe2013) ( $R^{2}\simeq$ 0.928). Besides, as shown in figure 2(d) and mentioned in the original paper (Tanaka et al. Reference Tanaka, Matsumoto, Kaneko and Abe2013), the experimentally obtained dataset used for the regression is very consistent with Brown and Scriven’s solution. In other words, most of points of the experimental dataset lie very close to the ascending part of the 2-lobed brunch of Brown and Scriven’s solution (figure 2 d). From above two reasons, we adopt the value of $A$ that is determined form Brown and Scriven’s solution as the optimal value of $A$ , $A_{\mathit{opt}}$ . Therefore, in this study, the optimal value of the parameter $A$ is determined as

(3.12) $$\begin{eqnarray}A_{\mathit{opt}}=0.8210.\end{eqnarray}$$

In terms of $A_{\mathit{opt}}$ , the boundary of the linear stability of the stationary solutions is derived as

(3.13) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}k_{\mathit{crit},A_{\mathit{opt}}}\simeq 5.379,~\unicode[STIX]{x1D714}_{\ast ,\mathit{crit},A_{\mathit{opt}}}\simeq 0.337,~u_{\ast ,\mathit{crit},A_{\mathit{opt}}}\simeq 2.050,~v_{\ast ,\mathit{crit},A_{\mathit{opt}}}\simeq 0.381,\\ 2\unicode[STIX]{x03C0}I_{zz\ast }(k_{\mathit{crit},A_{\mathit{opt}}};A_{\mathit{opt}})\cdot \unicode[STIX]{x1D714}_{\ast ,\mathit{crit},A_{\mathit{opt}}}\simeq 2.046,\end{array}\right\}\end{eqnarray}$$

by solving the inequality (3.10).

The horizontal broken line drawn in figure 2(d) is the boundary of the linear stability of the stationary solution obtained in the present study, and the angular momentum of the Cassinian droplet rotating at the rate corresponding to the stationary solution becomes maximum at $k_{\mathit{crit},A_{\mathit{opt}}}$ (refer to appendix C), and the dimensionless critical values of the rotation rate and deformation length are displayed above. In the graph obtained in Brown & Scriven (Reference Brown and Scriven1980), the dimensionless angular momentum is maximised at $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })\approx (0.31,2.10)$ . The maximum value is approximately equal to $2.02$ (Brown & Scriven Reference Brown and Scriven1980), which is close to the value shown in (3.13). In actual situations, as shown in figure 2(d), the scatter plots of $R_{\mathit{Max}}/R_{0}$ versus $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}$ appear to protrude slightly from Brown and Scriven’s solution around the point $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })=(0.332,2.091)$ . A similar deforming behaviour is shown more clearly in figure 7 of Wang et al. (Reference Wang, Anilkumar, Lee and Lin1994), which describes the deformation and breakup of a droplet consisting of low-viscosity silicone oil owing to rotation, and the critical point is $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })\approx (0.33,2.07).$ Moreover, as mentioned in Baldwin et al. (Reference Baldwin, Butler and Hill2015) and Butler et al. (Reference Butler, Stauffer, Sinha, Lilly and Spiteri2011), artificial tektites are obtained numerically or experimentally by solidifying steadily rotating deformed drops, and if an artificial tektite belongs to the 2-lobed family, the value of $H_{\mathit{mid}}/R_{\mathit{Max}}$ of each tektite, which is the same as $1/k$ , is in the range of 0.2 or more. In the present study, the stability limit of $H_{\mathit{mid}}/R_{\mathit{Max}}$ is $1/k_{\mathit{crit},A_{\mathit{opt}}}=1/5.379\simeq 0.186$ . Hence, the result of stability analysis of the present study is consistent with those of the aforementioned theoretical and experimental studies. Thus, it can be concluded that, even when the deformation length increases quasi-statically via the addition of angular momentum to the droplet, if it becomes slightly larger than $R_{0}\cdot u_{\ast ,\mathit{crit},A_{\mathit{opt}}}$ , the deformation length of the Cassinian droplet increases unsteadily and breakup occurs. By utilising the Cassinian droplet model, if the angular momentum $I_{zz\ast }\cdot \unicode[STIX]{x1D714}_{\ast }$ is conserved when the dimensionless deformation length $u_{\ast }=R_{\mathit{Max}\ast }>2.05$ , the breakup limit of $R_{\mathit{Max}\ast }=u_{\ast }$ and $\unicode[STIX]{x1D714}_{\ast }$ are calculated. When $k$ approaches infinity under the above assumption, $\unicode[STIX]{x1D714}_{\ast }(k;A_{\mathit{opt}})$ , $u_{\ast }(k;A_{\mathit{opt}})$ , and $v_{\ast }(k;A_{\mathit{opt}})$ approach the breakup limit:

(3.14a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\ast ,\mathit{break},A_{\mathit{opt}}}\simeq 0.249,\quad u_{\ast ,\mathit{break},A_{\mathit{opt}}}\simeq 2.24,\quad v_{\ast ,\mathit{break},A_{\mathit{opt}}}=0.\end{eqnarray}$$

We compared the Cassinian oval curve of $A_{\mathit{opt}}$ and the outline shape of an experimentally stretched low-viscosity droplet ( $\unicode[STIX]{x1D702}=0.0026~\text{Pa}~\text{s}$ , which is provided by the authors of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013)). As shown in figure 3(a), the Cassinian oval curve of $A_{\mathit{opt}}$ fits the outline shape of the actual low-viscosity droplet very well. At the moment the snapshot (a) is captured, the values of the dimensionless rotation rate and dimensionless deformation length of the Cassinian droplet are $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })=(0.331,2.06)$ , and the closest point in the experimental scatter plot of $\unicode[STIX]{x1D702}=0.0026~\text{Pa}~\text{s}$ is $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })=(0.332,2.091)$ . Slightly before the low-viscosity droplet breaks up (figure 3 b), the Cassinian oval curve and the outline shape of the actual droplet also overlap well, although there is a slight misalignment at the vicinity of the central cross-section. For snapshot (b), $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })$ of the corresponding Cassinian droplet of $A_{\mathit{opt}}$ is $(0.275,2.177)$ , and the closest point in the experimental scatter plot of $\unicode[STIX]{x1D702}=0.0026~\text{Pa}~\text{s}$ (displayed in figure 2 d) is $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })=(0.292,2.153)$ and the closest point in the plot of Brown and Scriven’s solution is $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })\approx (0.28,2.15)$ , which is located close to the summit of Brown and Scriven’s solution. Thus, we have demonstrated that the process of deformation of the Cassinian droplet is consistent with both the experimental observations and Brown and Scriven’s numerical calculation.

Figure 3.

The outline shape of an experimentally stretched low-viscosity 2-lobed droplet ( $\unicode[STIX]{x1D702}=0.0026~\text{Pa}~\text{s}$ ) is compared with a corresponding Cassinian oval curve $z_{\ast }=h_{\ast }(x_{\ast };k,A_{\mathit{opt}})$ . (a) The Cassinian oval curve of $A_{\mathit{opt}}=0.821$ (solid lines) fits the outline shape of the experimentally stretched 2-lobed droplet very well. The rotation rate and deformation length of the Cassinian droplet are calculated as $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })\simeq (0.331,2.06)$ using (3.11), and this calculated result closely approximates the deformation state $(\unicode[STIX]{x1D714}_{\ast },R_{\mathit{Max}\ast })=(0.332,2.091)$ in the experiment. (b)–(d) Three snapshots capturing the low-viscosity droplet immediately before its neck region pinches off. Although the outline shape of the low-viscosity droplet is not fitted well by the Cassinian oval curve itself for snapshots (c,d), the outline shape of the droplet is expressed by the combination of the breakup limit of the Cassinian oval curve and the interpolating hyperbolic cosine function (the broken line) smoothly connected with the Cassinian oval. Each snapshot is provided by the authors of Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013), and each snapshot is binarised by authors of this paper in order to clarify the outline shape of the droplet.

Figure 4.

(a) Relationship between angular momentum and rotation rate of the droplet that has the equilibrium shape belonging to the 2-lobed branch computed in Heine (Reference Heine2006) and that of the proposed ‘Cassinian droplet’ of $A_{\mathit{opt}}=0.821$ are compared. The scatter plot shown in panel (a) is created by reading off values from the 2-lobed branch of the computational result that is presented in figure 12 of Heine (Reference Heine2006). (b) The deformation of the Cassinian droplet is compared with the deformation of the actual droplet in rotation presented in Baldwin et al. (Reference Baldwin, Butler and Hill2015). The solid line shows $v_{\ast }^{3}/(4\unicode[STIX]{x03C0}/3)$ of the Cassinian droplet of $A_{\mathit{opt}}=0.821$ , which models 2-lobed droplets of circular cross-section, and is consistent with the deforming behaviour of the 2-lobed droplet obtained from Baldwin et al. (Reference Baldwin, Butler and Hill2015). The scatter plot shown in panel (b) is created by reading off values from figure 4 of Baldwin et al. (Reference Baldwin, Butler and Hill2015).

Furthermore, the relationship between angular momentum and rotation rate of rotating Cassinian droplet, i.e. the parametric plot of $2\unicode[STIX]{x03C0}I_{zz\ast }(k;A_{\mathit{opt}})\cdot \unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ versus $\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ , is compared with the numerical calculation presented in Heine (Reference Heine2006), which reproduced the relationship between the angular momentum and rotation rate of the droplet that has the equilibrium shape presented in figure 4 of Brown & Scriven (Reference Brown and Scriven1980). As shown in figure 4(a), the parametric plot of $(\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}}),2\unicode[STIX]{x03C0}I_{zz\ast }(k;A_{\mathit{opt}})\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}}))$ obtained by the Cassinian droplet model well reproduces the qualitative tendency of the diagram: $2\unicode[STIX]{x03C0}I_{zz\ast }(k;A_{\mathit{opt}})\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ also change increase to decrease as $\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ decreases, and the maximum value of $2\unicode[STIX]{x03C0}I_{zz\ast }(k;A_{\mathit{opt}})\cdot \unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ is almost equal to the maximum value of the angular momentum calculated in Heine (Reference Heine2006). Although the plot of $2\unicode[STIX]{x03C0}I_{zz\ast }(k;A_{\mathit{opt}})\cdot \unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ as a function of $\unicode[STIX]{x1D714}_{\ast ,s}(k;A_{\mathit{opt}})$ does not overlap quantitatively and precisely with the corresponding results presented in the previous studies, the relationship between the characteristic quantities of 2-lobed ‘dumbbell-like’ droplets calculated in this study are consistent with the result of the experimental study (Baldwin et al. Reference Baldwin, Butler and Hill2015) (figure 4 b). According to these observations and the really good approximate consistency observed between the result of the present study and results of the previous studies as expressed in figure 2 and figure 3, we conclude that the Cassinian oval curve is an appropriate approximate solution of the entire shape of the interface of rotating 2-lobed droplets.

Particularly in the case of low-viscosity droplets, the entire process of deformation of 2-lobed droplets including immediately before breakup is very well reproduced by the proposed model. If $N=(6\unicode[STIX]{x1D702}R_{0}\unicode[STIX]{x1D714}_{0})/\unicode[STIX]{x1D6FE}\rightarrow 0$ , and if the angular momentum $L_{\ast }=I_{zz\ast }\unicode[STIX]{x1D714}_{\ast }$ is conserved under the condition $u_{\ast }>u_{\mathit{crit}\ast }$ , the exact solutions of (3.2)–(3.4) can be derived. As the ratio $k~(=R_{\mathit{Max}}/H_{\mathit{mid}})$ is a monotonically increasing function of $t_{\ast }$ , the inverse function $t_{\ast }(k)$ exists and (3.2)–(3.4) can be transformed into

(3.15) $$\begin{eqnarray}-{\displaystyle \frac{\text{d}^{2}t_{\ast }}{\text{d}k^{2}}}=-{\displaystyle \frac{x_{g\ast }^{\prime \prime }}{x_{g\ast }^{\prime }}}\left({\displaystyle \frac{\text{d}t_{\ast }}{\text{d}k}}\right)+\left({\displaystyle \frac{x_{g\ast }}{x_{g\ast }^{\prime }}}\left({\displaystyle \frac{L_{\ast \mathit{crit}}}{I_{zz\ast }}}\right)^{2}-{\displaystyle \frac{K}{x_{g\ast }^{\prime }}}\right)\left({\displaystyle \frac{\text{d}t_{\ast }}{\text{d}k}}\right)^{3},\end{eqnarray}$$

where $L_{\ast \mathit{crit}}:=I_{zz\ast }(k_{\mathit{crit},A};A)\unicode[STIX]{x1D714}_{\ast ,s}(k_{\mathit{crit},A};A)$ is the angular momentum of a stably rotating Cassinian drop whose dimensionless deformation length is $u_{\ast }(k_{\mathit{crit},A};A)$ . The above equation is the Bernoulli differential equation, and therefore, its exact solution can be derived as

(3.16) $$\begin{eqnarray}t_{\ast }(k)=\int _{k_{0}}^{k}{\displaystyle \frac{x_{g\ast }^{\prime }(\tilde{k};A)}{\sqrt{\displaystyle \int _{k_{0}}^{\tilde{k}}\left[2x_{g\ast }^{\prime }(\hat{k};A)\left(x_{g\ast }(\hat{k};A)(L_{\ast \mathit{crit},A}^{2}/I_{zz\ast }^{2}(\hat{k};A))-K(\hat{k};A)\right)\right]\,\text{d}\hat{k}+C}}}\,\text{d}\tilde{k}.\end{eqnarray}$$

Figure 5.

The solution of (3.16) is plotted for different values of the integral constant $C$ in case of $A_{\mathit{opt}}=0.821$ . For each value of $C$ , the limit $\lim _{k\rightarrow \infty }t_{\ast }(k)$ , the time required for the extremely low-viscosity Cassinian droplet to reach breakup, is a finite value.

From the above equation, the time required to achieve a certain value of $k~(=R_{\mathit{Max}}/H_{\mathit{mid}})$ from the stability limit of the stationary solution can be evaluated. In figure 5, $t_{\ast }(k)$ is plotted for different values of $C$ . $C$ is a parameter related to the initial velocity of the centre of gravity of each lobe, and assumes a very small value in order to make $\text{d}k/\text{d}t_{\ast }$ close to 0 at $t_{\ast }=0$ . By using (3.16), it is possible to estimate the time required for the extremely low-viscosity droplet to reach breakup from the stability limit of the deformation length by evaluating $t_{\ast ,\mathit{break}}=\lim _{k\rightarrow \infty }t_{\ast }(k)$ . As an example, $t_{\ast ,\mathit{break}}\approx 35.5$ in the case of $C=1.0\times 10^{-5}$ . The reference constant of non-dimensionalisation is $\unicode[STIX]{x1D714}_{0}\approx 533~\text{s}^{-1}$ if the physical properties of a low-viscosity droplet are $R_{0}=1.0~\text{mm}$ , $\unicode[STIX]{x1D70C}=1210~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D6FE}=43~\text{mN}~\text{m}^{-1}$ , the time required for breakup is approximately equal to $0.066~\text{s}$ . It is demonstrated that the time during which the 2-lobed droplet of $\unicode[STIX]{x1D702}=1.0~\text{Pa}~\text{s}$ deforms unstably until breakup occurs is less than $0.1~\text{s}$ in Tanaka et al. (Reference Tanaka, Matsumoto, Kaneko and Abe2013). This fact is consistent with the calculation result presented here.

This breakup behaviour closely resembles the ‘finite time singularity’ in the jet breakup theory (e.g. Papageorgiou Reference Papageorgiou1995); however, it appears that the breakup of rotating extremely low-viscosity droplets in finite time occurs via a different mechanism. Referring to the stress-balance equation ((2.4) or (2.7)), under the situation where extremely low-viscosity droplets in rotation deform unstably and significantly and break up, among the terms of inertia, centrifugal force and curvature of the droplet interface, the term of centrifugal force is relatively large, and the stress-balance equation is modified as $(\text{A}~\text{function of }k\;\text{that takes a finite value})=(\unicode[STIX]{x1D702}/\sqrt{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}R_{0}})H_{\mathit{mid}\ast }(\text{d}H_{\mathit{mid}\ast }/\text{d}t_{\ast })$ . Therefore, if $N=12\sqrt{2}\unicode[STIX]{x1D702}/\sqrt{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}R_{0}}\rightarrow 0$ , $|\text{d}H_{\mathit{mid}\ast }/\text{d}t_{\ast }|\rightarrow \infty$ must occur. This mechanism, i.e. the balance of the centrifugal force and the term related to time derivative of the radius of the central cross-section, causes a sharp decrease in the central cross-section in the process by which an extremely low-viscosity droplet breaks up owing to rotation.

4 Summary and discussion

In this study, we proposed a simple model of rotating 2-lobed droplets by representing their outline shapes using the mathematical plane curves of Cassinian ovals (the Cassinian drop model). By continuously changing the values of their parameters, these curves can form various shapes from a peanut-like shape to an $\infty$ -like shape, which are observed in the process of deformation of a 2-lobed droplet stretched in laboratories. We obtained the governing equation of deformation of a Cassinian drop and demonstrated that a stationary solution of the angular velocity exists for an arbitrary maximum deformation length. As a result of the linear stability analysis, it was demonstrated that the condition of linear stability depends only on the ratio of the maximum deformation length to the central cross-section of the Cassinian drop, regardless of the physical properties of the droplet, i.e. volume, density, coefficient of surface tension and viscosity. Furthermore, via comparisons with an experimental observation (Tanaka et al. Reference Tanaka, Matsumoto, Kaneko and Abe2013), it was observed that the set of linearly stable stationary solutions of $A_{\mathit{opt}}$ was consistent with the behaviour of experimentally stretched 2-lobed droplets for a wide range of physical properties.

Further, we discuss the applicability of the Cassinian drop model to the breakup behaviour of 2-lobed droplets. In general, when a high-viscosity droplet breaks up, a liquid filament is formed around its midpoint as its deformation length rapidly increases. In case of low-viscosity droplets, a liquid filament is not formed, and the radius of the central cross-section decreases sharply. Conversely, in the Cassinian drop model, when the maximum deformation length approaches the breakup limit, the radius of the central cross-section rapidly decreases and the contour shape approaches a shape with a sharp singular point around its midpoint. Therefore, it is considered that the deformation behaviour of the proposed Cassinian droplet is very close to the deformation behaviour of a low-viscosity droplet. As mentioned in the latter part of § 3, the contour shape of an experimentally stretched low-viscosity 2-lobed droplet is very accurately approximated by the Cassinian oval curve of the optimum parameter $A_{\mathit{opt}}$ , except at the moment of breakup. The inconsistency between the breakup behaviour of the Cassinian droplet and that of an actual low-viscosity droplet occurs because the surfaces of actual droplets are physically required to be smooth. Although the Cassinian function alone does not express the shape of a low-viscosity droplet at the moment of breakup, by interpolating the vicinity of the central cross-section with a hyperbolic function, it can be represented by a combination of the narrow interpolation part and the breakup limit of the Cassinian oval curve (figure 3 c,d). As a low-viscosity 2-lobed droplet has a very weak restoring force owing to its viscosity, the radius of the interpolated hyperbolic region at the vicinity of its midpoint will decrease rapidly and will be broken owing to the centrifugal stress and fluid instabilities. Therefore, the maximum deformation length of low-viscosity droplets at the moment of breakup is slightly smaller than that of the Cassinian droplet at the breakup limit; $u_{\ast ,\mathit{break}}$ .

So far, we have demonstrated that the Cassinian droplet model is a suitable model for low-viscosity 2-lobed droplets. In the case of a high-viscosity droplet, even though a liquid filament can form immediately before the droplet breaks up, the liquid filament can be approximated by the hyperbolic cosine interpolation function. Therefore, it is considered that the distribution of the sectional radius of the high-/moderate-viscosity droplet immediately before the breakup is represented by the combination of the elongated hyperbolic interpolation function and the Cassinian oval curve smoothly connected to the interpolation function. As mentioned in the model development section, as the stress-balance equation can be applied to a 2-lobed droplet with an arbitrary distribution of the sectional radius (equations (2.7) and  (2.8)), it is expected that the breakup behaviour of high-viscosity droplets can be reproduced well by introducing an appropriate interpolation function.

Furthermore, in situations where a droplet separates from the main flow (for instance, a pinch-off process of a pendant drop consisting of low-viscosity fluid (e.g. figure 3 of Tirtaatmadja, McKinley & Cooper-White (Reference Tirtaatmadja, McKinley and Cooper-White2006), figure 4 of Clasen et al. (Reference Clasen, Phillips, Palangetic and Vermant2012)) or a process of releasing bubbles from a nozzle in the water (Czerski & Deane Reference Czerski and Deane2010)), there is a possibility that the radius profile of the interface of axial symmetrical flow can be closely approximated by the Cassinian oval curve. From the above examples, it is considered that the outline shape of the droplet can be well approximated by the Cassinian oval curve when the axisymmetrical axis of the shape of the droplet and the direction of the driving force of the deformation are parallel. Therefore, the proposed ‘Cassinian droplet model’ can be applied to large deformation of droplets in such a system. In actual situations where a drop separates from high-/moderate-/low-viscosity fluid, the fluid interface has an asymmetrical complicated shape at the neck region at the moment the liquid filament breaks up (e.g. Eggers Reference Eggers1993; Day, Hinch & Lister Reference Day, Hinch and Lister1998; Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), whereas there are situations where a smooth liquid filament is observed until the moment that the drop separates (e.g. the figures in Papageorgiou Reference Papageorgiou1995). In the former situations, however, the shape of the interface maintains axial symmetry and appears as a smooth liquid filament around the neck region, except for the moment when the neck collapses. It is highly possible that the radius profile of the interface of such liquid filaments can be approximated well by hyperbolic cosine interpolation function, by appropriately setting parameters of the function. Ultimately, whether the breakup of rotating moderate-/low-viscosity droplets belongs to the former or latter situations, in the process of extension and collapse of the liquid filament formed between the two lobes immediately before the droplet breaks up, the interface of the liquid filament is axially symmetrical and smooth and modelled by using the hyperbolic cosine interpolating function. Subsequently, as mentioned in this section, by modelling the shape of the interface of the stretched moderate-/low-viscosity droplet by the combination of the breakup limit of the Cassinian oval curve and an elongated hyperbolic cosine interpolation function smoothly connected to the Cassinian oval, the dimensionless deformation length is expected to be obtained as a function of the dimensionless angular velocity of rotation. Moreover, in Bhat et al. (Reference Bhat, Appathurai, Harris, Pasquali, McKinley and Basaran2010) authors have shown beads on a string for visco-elastic fluid. Since the bead is formed only at the moment that a droplet breaks up, it seems that the liquid filament formed in the process by which a drop separates from visco-elastic fluid can be modelled in the same way. For the moment of breakup, we should also consider responses to various perturbations, as represented by Rayleigh Plateau instability. The model of rotating droplets incorporating this effect is currently under consideration.

Although there is a very slight difference between the Cassinian oval curve and the outline shape of 2-lobed droplets in rotation, the characteristic quantities obtained as functions of rotating angular velocity by using the Cassinian droplet model approximate the deforming behaviour of actual low-viscosity droplets very well, whereas the Cassinian oval curve is a simple and easy-to-handle mathematical function with only three parameters: $u_{\ast },v_{\ast }$ and $A$ . Therefore, we conclude that the Cassinian oval curve is an appropriate approximate solution of the entire shape of the interface of 2-lobed droplets and the proposed Cassinian droplet model is suitable to model the deforming behaviour of rotating 2-lobed droplets. Namely, we could have successfully proposed a quite simple but physically reliable mathematical model by setting the outline shape of the droplet to the Cassinian oval curve. As the series of Cassinian oval curves matches the contour shape of rotating 2-lobed droplets very well, the Cassinian oval function appears to have a fundamental relationship with the deformation mechanism of rotating 2-lobed droplets. By applying various extensions to the Cassinian droplet model, we can handle more complicated or unstable deformation behaviour; for example, the relaxation process in the coalescence of binary droplet collision or the formation of viscous droplets from a jet ejected from a nozzle.