3.1. Dynamics

The initial shape of the drop is assumed to be a torus with circular cross-section and major radius $R$ , as depicted in figure 1. Since for the chosen scaled variables, the volume of the torus equals $4{\rm\pi}/3$ , the initial minor radius of the drop is $a=\sqrt{2/(3{\rm\pi}R)}$ . Here, we can take advantage of the axial symmetry of the drop and introduce a cylindrical coordinate system $(r,{\it\varphi},z)$ with the basis $(\boldsymbol{e}_{r},\boldsymbol{e}_{{\it\varphi}},\boldsymbol{e}_{z})$ , where the $z$ -axis coincides with the $x_{3}$ -axis. Equation (2.1) takes the form $\boldsymbol{u}^{\infty }=r\,\boldsymbol{e}_{r}-2z\,\boldsymbol{e}_{z}$ , and the problem (2.2)–(2.4) can be considered to be axisymmetric, i.e. independent of the angular coordinate ${\it\varphi}$ , so that $\boldsymbol{u}^{\pm }$ and $p^{\pm }$ are represented by

In this case, the integral representation (2.6) can be integrated over the angular coordinate ${\it\varphi}$ and simplified to expressions containing curvilinear integrals over the cross-section of the drop interface $S$ . The expressions for the hydrodynamic variables can be found in, e.g., (3.3) or (3.5a) and (3.5b) in Zabarankin et al. (Reference Zabarankin, Smagin, Lavrenteva and Nir2013) for the velocity and (3.9) and (3.10) in Zabarankin & Nir (Reference Zabarankin and Nir2011) for the pressure $p^{\pm }$ and vorticity $u_{i,j}^{\pm }-u_{j,i}^{\pm }$ ; see also appendix A.

The drop dynamics is simulated for various values of $R$ and $Ca$ with $N=200$ and ${\rm\Delta}{\it\tau}=2\times 10^{-4}$ . The drop deformation is characterized by the major radius of the deformed torus, by the Taylor deformation factor of the torus and by a deformation factor of the torus cross-section,

respectively, where $R_{max}$ and $R_{min}$ denote maximum and minimum distances of the drop interface from the $z$ -axis and

It should be recalled that the Taylor deformation factor varies between 0 and 1. The limiting behaviour $D_{\mathit{torus}}\rightarrow 1$ corresponds to $R_{max}\gg z_{max}$ , i.e. to a highly expanded thin torus. The opposite limit $D_{\mathit{torus}}=0$ would mean that $R_{max}=z_{max}$ . The latter equality is impossible for circular tori. Thus, the initial value of this parameter in our computations is always positive,

Figure 2.

Time evolution of parameters of deforming initially circular tori. (a–c) Major radius of the toroid. (d–f) Taylor deformation factor of the torus. (g–i) Deformation factor of the torus cross-section. In (a,d,g), $Ca=0.18$ ; dashed curves, $R=0.82$ ; solid curves, $R=0.81$ ; dot-dashed curves, $R=0.8107$ . In (b,e,h), $Ca=0.1$ ; dashed curves, $R=1.2$ ; solid curves, $R=1$ ; dot-dashed curves, $R=1.09327$ . In (c,f,i), $Ca=0.04$ ; dashed curves, $R=1.9$ ; solid curves, $R=1.86$ ; dot-dashed curves, $R=1.86969$ .

Figure 3.

Dynamic deformation of the cross-section of an initially circular toroidal drop: $Ca=0.16$ ; (a) $R=0.86$ ; (b) $R=0.85236$ ; (c) $R=0.852$ .

Figure 2 shows the typical evolution of these parameters at relatively small values of $Ca$ . In all of these cases, $R({\it\tau})$ and $D_{\mathit{torus}}$ initially increase. If the initial major radius $R$ exceeds some critical value $R_{cr}$ , the torus expands indefinitely, $R({\it\tau})$ and $D_{\mathit{torus}}$ grow monotonically with time. In these cases, shown by the dashed curves, $D_{\mathit{torus}}\rightarrow 1$ and $R({\it\tau})$ exhibits exponential growth as ${\it\tau}\rightarrow \infty$ . If $R<R_{cr}$ , then after initial growth, $R({\it\tau})$ and $D_{\mathit{torus}}$ attain maximum values and then decrease (see the solid curves). Finally, for $R\sim R_{cr}$ , after initial increase, all of the parameters approach some ‘equilibrium’ value and remain visibly constant for a relatively long time (dot-dashed curves). The initial period of deformation grows rapidly with increase of $Ca$ . Remarkably, the dependence of the equilibrium value of $D_{\mathit{torus}}$ on $Ca$ is not monotonic: the ‘equilibrium’ shapes for both $Ca=0.04$ and $Ca=0.18$ are more deformed than that for $Ca=0.1$ ; see figure 2(d–f).

It is evident from figure 2(g–i) that, in all these cases, the cross-section initially elongates in the $r$ direction and $D_{\mathit{cross}\text{-}\mathit{section}}$ increases. Since initially the torus has a circular cross-section, $D_{\mathit{cross}\text{-}\mathit{section}}(0)=0$ . The initial period of elongation grows rapidly with increase of $Ca$ . Then, $D_{\mathit{cross}\text{-}\mathit{section}}$ either approaches some ‘equilibrium’ value, which remains visibly constant for a long time (dot-dashed curves), or attains a maximum and then decreases (solid curves). The ‘equilibrium’ deformation of the cross-section increases monotonically with $Ca$ . In the case of expansion (dashed curves), $D_{\mathit{cross}\text{-}\mathit{section}}\rightarrow 0$ as ${\it\tau}\rightarrow \infty$ , i.e. the shape of the drop becomes close to an expanding circular torus, and the maximum deformation is smaller than the ‘equilibrium’ one. In the case of contraction (solid curves), $D_{\mathit{cross}\text{-}\mathit{section}}$ attains a maximum, which is greater than the equilibrium value, and then decreases as the torus collapses towards the $z$ -axis (see figure 3 c).

Table 1.

Critical major radius, $R_{cr}$ , for various values of $Ca$ .

Figure 4.

Dynamic deformation of the cross-section of an initially circular toroidal drop: (a)  $Ca=0.04$ , $R=1.2$ ; (b) $Ca=0.1$ , $R=1$ ; (c) $Ca=0.15$ , $R=0.86$ .

Figure 5.

Deformation parameters of the torus in two cases: (a–c) $Ca=0.2$ and (d–f) $Ca=0.5$ . In (a–c), the dotted, dot-dashed, dashed and solid curves correspond to $R=1$ , 0.786, 0.780438260324 and 0.780438260323 respectively, whereas in (d–f), they correspond to $R=0.8$ , 0.7, 0.6846 and 0.6845 respectively.

While figure 2 summarizes three types of drop dynamic deformation behaviour observed for $Ca\lesssim 0.19$ , figure 3 exemplifies these types of behaviour and change of cross-section shapes for $Ca=0.16$ and for $R$ assuming supercritical, near-critical and subcritical values.

In summary, the circular cross-section of the torus initially elongates in the $r$ direction. Then, the torus either contracts to the $z$ -axis or expands indefinitely. For any $Ca$ below a critical capillary number $Ca_{cr}$ , there exists a critical major radius $R_{cr}$ of the initial circular torus, separating these two scenarios (see figure 3 a,c) and estimated from table 1 as a function of $Ca$ to be

with a mean absolute error of $8.4\times 10^{-4}$ . For $Ca<Ca_{cr}$ and $R\sim R_{cr}$ , after initial elongation, the drop maintains the same shape for a relatively long time (figure 3 b), which suggests the existence of unstable stationary shapes of viscous toroids – the subject of § 4. One can expect that this time can be made as long as desired with a very accurate choice of $R$ . However, in practical calculations, we are restricted by the computer accuracy (the number of digits). Here, we assume ‘relatively long time’ to be ${\it\tau}^{\ast }\sim 6$ . The values of $R_{cr}$ that are presented in table 1 satisfy the above definition for $Ca\leqslant 0.19$ . For $Ca=0.196$ , the maximal time of stay at ‘equilibrium’ achievable with 15 digits of $R$ equals 3. Table 1 shows that the needed accuracy of $R_{cr}$ increases with $Ca$ – the greater $Ca$ is, the more accurately $R_{cr}$ should be determined to sustain a visually stationary cross-section for a prolonged period of time. Moreover, the lower $Ca$ is, the longer value of ${\it\tau}^{\ast }$ is available. (The above interpolating function of $R_{cr}(Ca)$ provides only a rough (and quick) estimate of $R_{cr}$ .)

Figure 3(c) shows that for $R<R_{cr}$ , the torus cross-section is deformed in the vicinity of the $z$ -axis when the interface approaches it. The inner part of the drop surface forms a thin almost circular cylinder centred at the $z$ -axis. It is anticipated that it will collapse, the drop will regain a simply connected shape and, with the passage of time, it will settle on the stationary shape studied in Zabarankin et al. (Reference Zabarankin, Smagin, Lavrenteva and Nir2013). However, the code stops when any point of the torus comes closer than $10^{-4}$ to the $z$ -axis (this is reflected by the fact that the solid curves in figure 2 do not span the full ${\it\tau}$ range), so that detailed investigation of the drop coalescence calls for a separate work. Figure 4(a–c) shows sequential stages of torus contraction for three pairs: $Ca=0.04$ , $R=1.2$ ; $Ca=0.1$ , $R=1$ ; $Ca=0.15$ , $R=0.86$ respectively. In all these cases, when the torus contracts to the $z$ -axis, the inner part of its cross-section straightens first and then develops a region of converted curvature at the centre. Thus, the closest approach of the interface to the $z$ -axis is not at the $z=0$ plane but rather at some distance from it. It should be noted that the concave and convex portions of the cross-section perimeter, depicted in figure 4, are connected by narrow regions of very high local curvature, although the interface is regular and continuous. With increase of $Ca$ , the torus cross-section becomes more elongated in the $r$ direction and the size of the straightened and dimpled region decreases. It is observed numerically that if $R<R_{cr}$ and $R\not \approx R_{0}=(3{\rm\pi}/2)^{-1/3}$ , where $R_{0}$ corresponds to a closed circular torus, the developed stage of contraction does not depend on $R$ .

Figure 6.

Dynamic deformation of the initially circular cross-section of a toroidal drop for $Ca=0.2$ in three cases: (a) $R=0.780438260323$ , (b) $R=0.780438260324$ , (c) $R=0.8$ .

For $Ca>Ca_{cr}$ , no ‘equilibrium’ shapes that persist for a relatively long time were found, and the drop dynamics simulations suggest that $Ca_{cr}\in (0.19,0.2)$ . Figure 5 shows typical examples of the time evolution of $R({\it\tau})$ , $D_{\mathit{torus}}$ and $D_{\mathit{cross}\text{-}\mathit{section}}$ for $Ca>Ca_{cr}$ . In figures 5(a–c), $Ca=0.2$ and the dotted, dot-dashed, dashed and solid curves correspond to $R=1$ , $0.7806$ , 0.780438260324 and 0.780438260323 respectively. One can observe that, during the initial period, the cross-section elongates rapidly in the $r$ direction, the major radius and both deformation factors grow; then, $D_{\mathit{cross}\text{-}\mathit{section}}$ slows down and, if the initial $R>R_{cr}$ , it eventually begins to decrease. Moreover, after the initial rapid increase, $D_{\mathit{torus}}$ slows down but continues to grow, and then, after the cross-section begins to contract in the $r$ direction, it accelerates again. As time increases, the torus expands indefinitely, with the major radius increasing exponentially with time and with the cross-section approaching a circular shape. When $R<R_{cr}$ , the interface approaches the proximity of the $z$ -axis and the code stops. These two types of behaviour are illustrated in figure 6(a,c), where cross-section shapes corresponding to figure 5(a–c) are shown at several time moments. In figure 6(a), where the process starts at $R=0.780438260323$ (solid curves in figure 5 a,b), the drop elongates in the $r$ direction, approaches the $z$ -axis, becomes thinner in the centre region and collapses. In figure 6(b), where $R=0.780438260324$ (dashed curves in figure 5 a,b) is greater than $R$ in figure 6(a) by just $10^{-12}$ , the drop behaves initially as in figure 6(a): it also approaches the $z$ -axis near the $z=0$ plane. However, its interface does not collapse, and the initial approach to the $z$ -axis is followed by contraction of the cross-section and expansion of the torus. In figure 6(c), where $R=0.8$ , no close approach to the $z$ -axis occurs, and the dynamic behaviour is similar to that described in the supercritical case for $Ca<0.19$ (see, e.g., figure 3 a). With the passage of time, the cross-section of the drop assumes an almost circular shape, and the torus expands indefinitely.

In figure 5(d–f), where $Ca=0.5$ , during the initial period the cross-section elongates rapidly in the $r$ direction and then slows down. If the initial $R\lesssim 0.6845$ , the torus collapses. For higher initial $R$ , $D_{\mathit{torus}}$ continues to grow rapidly, which contrasts the two stages evident in the case of $Ca=0.2$ . Figure 7 shows an evolving cross sections shape at this relatively high $Ca=0.5$ . In figure 7(a), where $R=0.8$ , considerable initial elongation occurs simultaneously with torus expansion. The cross-section first assumes an elliptic shape, then flattens, and forms a dimple. With the passage of time and as the torus expands further, the cross-section contracts and assumes an elliptic shape that becomes more and more circular. In figure 7(b), where $R=0.7$ , the deformation pattern is qualitatively similar to that observed for higher values of $R$ . However, in this case, the cross-section becomes more elongated and the dimple is more developed, so that the cross-section becomes very thin in its centre region and is even more susceptible to a potential breakup. Moreover, thin highly expanded tori may become unstable to disturbances, involving the azimuthal angle ${\it\varphi}$ , and break up in a non-axisymmetric manner. Nevertheless, if breaking does not occur, with further expansion of the torus, its cross-section contracts again and assumes an almost circular shape. Table 2 summarizes outcomes of deformation of circular tori with initial radius  $R$ .

Figure 7.

Dynamic deformation of the initially circular cross-section of a toroidal drop for $Ca=0.5$ in two cases: (a) $R=0.8$ and (b) $R=0.7$ .

3.2. Dynamics in the absence of external flow

The case of the absence of external flow corresponds to $Ca=0$ . The representation (2.6) is not directly applicable in this case. However, multiplying (2.6) by $Ca$ , rescaling the velocity and taking $Ca=0$ results in

The rescaling implies that the velocity and time are scaled by $U^{\ast }={\it\gamma}/{\it\mu}^{-}$ and $t^{\ast }=l{\it\mu}^{-}/{\it\gamma}$ respectively. It should be noted that (3.6) contains no parameters.

Table 2.

Outcomes of deformation of circular tori with initial radius $R$ .

We employed (3.6) to simulate the dynamics of a torus in a quiescent fluid for initial conditions corresponding to those in Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009) and Yao & Bowick (Reference Yao and Bowick2011). Figure 8(a) demonstrates typical dynamic deformations of initially circular fat tori with various initial aspect ratios, $R/a$ . Here, we chose the aspect ratio as a characteristic of the initial state (instead of $R$ which is used in the rest of the paper) in order to facilitate comparison with the available experimental data. This dynamics is qualitatively similar to that for a small capillary (see figure 4 a): the droplet first shrinks keeping an almost circular cross-section, then, as its inner part contracts to the $z$ -axis, considerable deformation takes place. The inner part of the drop surface forms a thin almost circular cylinder centred at the $z$ -axis and then develops a region of converted curvature in the centre. In contrast to the case of deformation in the presence of external compressional flow, no initial elongation in the $r$ direction occurs. It should be noted also that the developed stages of deformation are almost independent of the initial aspect ratio, unless the latter is very close to one (the case of a closed torus). In this case, no region of converted curvature is formed and the drop assumes an almost spherical shape with a thin cylindrical hole around the $z$ -axis.

Figure 8.

(a) Dynamic deformation of the initially circular cross-section of a toroidal drop for $Ca=0$ in four cases: $R/a=1.009$ , $1.4$ , $1.9$ and $2.1$ . (b) The inner radius $R_{in}=\min _{\boldsymbol{x}\in S}r$ (dotted and solid curves) and $R_{in,\,z=0}=\min _{\boldsymbol{x}\in S,\,z=0}r$ (see inset) of the droplet normalized by its initial value for initial aspect ratios of 1.4 (dotted and dashed curves) and 1.9 (solid and dot-dashed curves). The physical parameters are taken from Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009):  ${\it\mu}=30\,000~\text{cp}$ , ${\it\sigma}=4~\text{mN}~\text{m}^{-1}$ .

In the experiments of Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009), the drop to ambient fluid viscosity ratio was very small, ${\it\lambda}=1/30\,000$ , while in the theory developed in this paper ${\it\lambda}=1$ . A direct quantitative comparison between theory and experiments is, thus, not possible. However, assuming that the relevant time scale for drop deformation in viscous fluid is $l({\it\mu}^{+}+{\it\mu}^{-})/{\it\gamma}=(1+{\it\lambda})t^{\ast }$ , we concluded that the time scale for the experiment of Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009) is approximately half of what it would be for ${\it\lambda}=1$ . In figure 8(b), the inner radius of the droplet normalized by its initial value is plotted versus $0.5t^{\ast }{\it\tau}$ for initial aspect ratios of 1.4 (dotted and dashed curves) and 1.9 (solid and dot-dashed curves). The physical parameters taken from Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009) are ${\it\mu}=30\,000~\text{cp}$ , ${\it\sigma}=4~\text{mN}~\text{m}^{-1}$ and a volume of the drop of $272~\text{mm}^{3}$ . Two definitions of the inner radius are used: $R_{in}=\min _{\boldsymbol{x}\in S}r$ (dotted and solid curves) and $R_{in,\,z=0}=\min _{\boldsymbol{x}\in S,\,z=0}r$ (solid and dot-dashed curves), see the inset in figure 8(b). These two definitions coincide for circular tori, and, as it is evident in the figure, also during the initial period of torus deformation before the region of converted curvature is formed. Experimental data points taken from Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009) are presented by triangle and diamond symbols. It is evident that, during the initial period, the curves in figure 8(b) are almost linear, and the agreement of the experimental data and computations is excellent. However, as the torus contracts, considerable deformations result in slowing down of the inner radius change, and the linearity of the curves is lost. At this stage, the computed results deviate from the experimental data. The predicted final contraction time is approximately 1.5 times as much as that observed experimentally. This difference can be attributed to the axial asymmetry of both the flow and the drop shape in the experiments reported by Pairam & Fernández-Nieves (Reference Pairam and Fernández-Nieves2009), which, evidently, enhances the self-coalescence of the torus.

Table 3.

Parameters in (3.7) approximating the cross-sections of the pointwise stationary-state shapes with $K=1$ (fitted ellipses), the corresponding fit error (3.9) and $\Vert u_{n}\Vert _{S}$ defined by (3.8).

Table 4.

Parameters in (3.7) approximating the cross-sections of the pointwise stationary-state shapes with $K=5$ , the corresponding fit error (3.9) and $\Vert u_{n}\Vert _{S}$ defined by (3.8).

3.3. Stationary shapes

In this subsection, the cross-sections of the slowly changing deformed shapes, resulting from dynamic simulations via (2.6) of initially circular tori with $R\simeq R_{cr}$ (see, e.g., figure 3 b) and referred to further as pointwise stationary shapes, are fitted by smooth curves parametrized by

where $R$ , $a$ , $b_{1},\ldots ,b_{k}$ are parameters, and $T_{k}(t)$ is a $k\text{th}$ -order Chebyshev polynomial of the first kind. For $K=1$ , (3.7) describes an ellipse. Tables 3 and 4 present parameters for (3.7) approximating the cross-sections of pointwise stationary-state shapes with $K=1$ and $K=5$ respectively. The tables also present the mean absolute value $\Vert u_{n}\Vert _{S}$ defined by

where $u_{n}^{+}$ is determined by (2.10) and $\text{d}s=\sqrt{(r^{\prime }(t))^{2}+(z^{\prime }(t))^{2}}\,\text{d}t$ is the curve length element, and the fit error introduced by

where $N=200$ is the number of points in the cross-section of a pointwise stationary-state shape.

Figure 9.

Cross-sections of the pointwise stationary-state shapes (solid curves) and their approximations by (3.7) with $K=1$ (fitted ellipses, dashed curves) and with $K=5$ (circles) in two cases: (a) $Ca=0.16$ and (b) $Ca=0.19$ . Triangles mark the cross-sections of corresponding ‘stationary’ elliptic tori found from the inverse problem (4.18) with $w=r^{-1}$ .

Figure 9(a,b) shows the cross-sections of the pointwise stationary-state shapes (solid curves) and their approximations by (3.7) with $K=1$ (fitted ellipses, dashed curves) and with $K=5$ (circles) in two cases: (a) $Ca=0.16$ and (b) $Ca=0.19$ . They also depict the cross-sections of the corresponding ‘stationary’ elliptic tori (triangles) found from the inverse problem (4.18) in § 4. In figure 9(a), all of the shapes are visually indistinguishable – elliptic tori are a remarkably good approximation for stationary-state shapes up to $Ca=0.16$ . In figure 9(b), the cross-section of the pointwise stationary-state shape is no longer elliptic – the deviation from the fitted ellipse is quite visible. In this case, the cross-section of the of ‘stationary’ elliptic torus differs considerably not only from that of the pointwise stationary shape, but also from the fitted ellipse, which yields a much smaller mean absolute value $\Vert u_{n}\Vert _{S}$ . (The reason why $\Vert u_{n}\Vert _{S}$ is smaller for the fitted elliptic torus is because the ‘stationary’ elliptic torus is found from the inverse problem rather than from direct minimization of $\Vert u_{n}\Vert _{S}$ ; see § 4.1.) However, (3.7) with $K=5$ still provides an excellent approximation – the red circles firmly trace the solid curve.

Figure 10.

Streamlines about the stationary-state toroidal shape for $Ca=0.05$ .

Figure 11.

Streamlines about the stationary-state toroidal shape for $Ca=0.15$ .

Figure 12.

Streamlines about the stationary-state toroidal shape for $Ca=0.19$ .

Figures 10–12 show the streamline patterns inside and outside the cross-sections of the stationary-state drops in the $rz$ plane for $Ca=0.05$ , 0.15 and 0.19 respectively – the interface in all of these cases is a streamline, confirming that the drops are indeed at steady state. However, continuation of the dynamic computation for an extended time (this time grows with decrease of $Ca$ and with the accuracy of the initial radius $R$ ) always results in destruction of these shapes, reflecting their possible instability. Thus, the search for stationary tori by dynamic computations is hindered by the need for extremely accurate initial conditions. In the next section we describe an alternative much more efficient method of finding these shapes.

4.1. Inverse problem for finding approximate stationary shapes

Suppose that a family of shapes, $\mathscr{F}$ , is determined by a single free parameter, e.g. the major radius $R$ in circular tori of volume $4{\rm\pi}/3$ , and suppose that for a given $Ca$ , a ‘stationary’ shape minimizes $\mathscr{E}_{2}^{2}(u_{n}^{+},w;S)$ subject to $S\in \mathscr{F}$ . In this case, numerical optimization finds optimal $R$ iteratively, e.g. by the golden section method. (The complexity of the golden section method is the same as for the ordinary bisection method for finding a zero of a monotone function.) Then, in order to plot $R=R(Ca)$ , numerical optimization should be repeated for an array of values of $Ca$ , which is a computationally expensive procedure.

To find the curve $(Ca,R(Ca))$ approximately but with considerably less computational effort, Zabarankin (Reference Zabarankin2013) proposed to solve the inverse problem that interchanges the roles of $Ca$ and $R$ : for a given $R$ , minimize an error functional, e.g. $\mathscr{E}_{1}$ or $\mathscr{E}_{2}$ , with respect to $Ca$ . In this case, $\mathscr{E}_{2}^{2}(u_{n}^{+},w;S)$ is a quadratic function of $1/Ca$ ,

where

so that an optimal $Ca$ can be determined analytically,

which yields $Ca=Ca(R)$ with no iteration procedure. In general, the curves $(Ca,R(Ca))$ and $(Ca(R),R)$ do not coincide. However, they do coincide when $\mathscr{E}_{2}(u_{n}^{+},w;S)=0$ and $R(Ca)$ and $Ca(R)$ are uniquely determined. When the corresponding optimal values of $\mathscr{E}_{2}(u_{n}^{+},w;S)$ are small, these curves are expected to be close.

For a family of shapes having two free parameters, one of the parameters can be assumed to be fixed, $Ca$ determined by (4.5) and the second parameter found by minimizing $\mathscr{E}_{2}^{2}(u_{n}^{+},w;S)$ with (4.5) by the golden section method. This approach is demonstrated in § 4.4.

4.2. ‘Stationary’ circular tori

In the $rz$ half-plane of the cylindrical coordinates $(r,{\it\varphi},z)$ , the cross-section of a circular torus having the volume of a unit sphere, i.e. $4{\rm\pi}/3$ , can be parametrized by

where $R$ and $a$ are the major and minor torus radii respectively. In this case, $R\geqslant R_{0}=(3{\rm\pi}/2)^{-1/3}$ ${\approx}0.596$ , where $R_{0}$ corresponds to a closed torus (with no opening), and

Section 3.1 shows that circular tori are never stationary: for $R<R_{cr}$ , a circular torus contracts to the $z$ -axis and regains a simply connected shape, whereas for $R>R_{cr}$ , it will expand indefinitely, where the critical radius $R_{cr}$ is found by the trial-and-error method (see table 1). Nonetheless, when $R=R_{cr}(Ca)$ and $Ca\rightarrow 0$ , the stationary toroid approaches a circular torus, and the circular torus analysis can provide accurate estimates of $R_{cr}$ . (A streamfunction satisfying the boundary conditions (2.3) and (2.4) for circular tori can be constructed in the toroidal coordinates $({\it\xi},{\it\eta})$ as in Zabarankin & Krokhmal (Reference Zabarankin and Krokhmal2007), where $({\it\xi},{\it\eta})$ are related to the cylindrical coordinates $(r,z)$ as $r=c\sinh {\it\xi}/(\cosh {\it\xi}-\cos {\it\eta})$ and $z=c\sin {\it\eta}/(\cosh {\it\xi}-\cos {\it\eta})$ , with ${\it\xi}\in [0,\infty )$ , ${\it\eta}\in [-{\rm\pi},{\rm\pi}]$ and $c$ being a positive metric parameter, see also Pell & Payne (Reference Pell and Payne1960) and Khuri & Wazwaz (Reference Khuri and Wazwaz1997).)

Figure 13.

Circular tori: $(Ca,R(Ca))$ with $R(Ca)$ minimizing $\mathscr{E}_{1}(u_{n}^{+},r^{-1};S)$ (solid curve), $(Ca(R),R)$ with $Ca(R)$ determined by (4.9) (dot-dashed curve) and $(Ca,R_{cr}(Ca))$ with the critical radius $R_{cr}(Ca)$ tabulated in table 1 (dashed curve). The dashed and dot-dashed curves are visually indistinguishable.

Given that circular tori do not satisfy (2.10) identically, one of the approaches to find a ‘stationary’ torus is to minimize a kinematic condition error, e.g. $\mathscr{E}_{1}\left(u_{n}^{+},w;S\right)$ or $\mathscr{E}_{2}\left(u_{n}^{+},w;S\right)$ defined in (4.1), with respect to $R$ . Remarkably, the curve $(Ca,R(Ca))$ with

is quite close to $(Ca(R),R)$ , with

and to the curve $(Ca,R_{cr}(Ca))$ , where $R_{cr}(Ca)$ is tabulated in table 1, see figure 13. In fact, $(Ca(R),R)$ with $Ca(R)$ determined by (4.9) is visually indistinguishable from $(Ca,R_{cr}(Ca))$ , and the inverse of $Ca(R)$ is estimated to be

with a mean absolute error of $10^{-4}$ based on 26 tabulations.

In figure 14(b), triangles mark the curve $(Ca(R),a(R))$ , with $a(R)=\sqrt{2/(3{\rm\pi}R)}$ and $Ca(R)$ determined by (4.9).

4.3. Relationship between $Ca$ and $R$ for ‘stationary’ circular tori as $Ca\rightarrow 0$

Section 4.2 suggests that $R(Ca)\rightarrow \infty$ as $Ca\rightarrow 0$ . In this case, the asymptotic behaviour of $Ca(R)$ as $R\rightarrow \infty$ can be obtained as follows. With (2.6) and (4.7), $u_{i}^{+}(\boldsymbol{x})$ takes the form

Figure 14.

The curves $(Ca(R),R)$ and $(Ca(R),a(R))$ for ‘stationary’ circular tori determined by (4.9) (triangles), for the ‘stationary’ elliptic tori found from the inverse problem (4.18) with $w=r^{-1}$ (dashed curves), and for (3.7) approximating the cross-sections of the pointwise stationary-state shapes with $K=1$ (fitted ellipses, red circles) and with $K=5$ (solid curves).

For arbitrary shape $S$ (not just a circular torus), the following identity holds:

see, e.g., Pozrikidis (Reference Pozrikidis1992), with which $u_{i}^{+}(\boldsymbol{x})$ simplifies to

Now let $\boldsymbol{x}=r\,\boldsymbol{e}_{1}+z\,\boldsymbol{e}_{3}$ and $\boldsymbol{y}=r_{1}\,\boldsymbol{e}_{1}\cos {\it\varphi}+r_{1}\,\boldsymbol{e}_{2}\sin {\it\varphi}+z_{1}\,\boldsymbol{e}_{3}$ , then $u_{i}^{+}(\boldsymbol{x})$ can be rewritten in the form (A 1); see appendix A. Further, with $(r,z)=(R+a\cos s,a\sin s)$ , $s\in [-{\rm\pi},{\rm\pi}]$ , $(r_{1},z_{1})=(R+a\cos t,a\sin t)$ , $t\in [-{\rm\pi},{\rm\pi}]$ and $a=\sqrt{2/(3{\rm\pi}R)}$ , the asymptotic behaviour of (A 1) as $R\rightarrow \infty$ can be obtained by Mathematica 10:

which implies that for ‘stationary’ circular tori, $Ca$ and $R$ are related by

4.4. ‘Stationary’ elliptic tori

Figure 3(b) shows that, for $Ca\leqslant 0.16$ , the cross-section of an initially circular torus with $R=R_{cr}(Ca)$ approaches an approximately elliptic shape. This section finds ‘stationary’ elliptic tori through the inverse problem outlined in § 4.1.

In the $rz$ half-plane in the cylindrical coordinates, the cross-section of an elliptic torus having the volume of a unit sphere can be parametrized by

where $R$ and $a$ can still be referred to as major and minor torus radii respectively, and $b=2/(3{\rm\pi}Ra)$ follows from the volume constraint. In this case,

The inverse problem in § 4.1 implies that for each $R$ and $a$ , $Ca$ is expressed by (4.5), and $\mathscr{E}_{2}^{2}(u_{n}^{+},w;S)$ with (4.5) is minimized with respect to $a$ over the interval $[\sqrt{2/(3{\rm\pi}R)},R]$ ,

which is found by the golden section method. (The golden section method requires identification of an interval on which a minimized function is convex. In this case, an appropriate subinterval of $[\sqrt{2/(3{\rm\pi}R)},R]$ can be identified visually by plotting the minimized function.) Optimal $Ca$ , denoted as $Ca(R)$ , is then determined by (4.5) with given $R$ and $a(R)$ found as in (4.18).

Table 5 presents $Ca(R)$ , $a(R)$ and $b(R)=2/(3{\rm\pi}\,R\,a(R))$ for ‘stationary’ elliptic tori found from the inverse problem (4.18) for $R\in [0.65,5]$ with $w=r^{-1}$ and also provides the mean absolute value $\Vert u_{n}\Vert _{S}$ defined by (3.8). The increase of $\Vert u_{n}\Vert _{S}$ for $Ca>0.16$ is quite noticeable. Figure 14 compares the parameters $R$ and $a$ as functions of $Ca$ for ‘stationary’ elliptic tori found from the inverse problem (4.18) with $w=r^{-1}$ (dashed curves) and for (3.7) which fits the cross-sections of the stationary toroids with $K=1$ (fitted ellipses, red circles) and with $K=5$ (solid curves). For $Ca\leqslant 0.16$ , all three curves are visually indistinguishable. The triangles correspond to $(Ca(R),R)$ with $Ca(R)$ determined by (4.9) for ‘stationary’ circular tori, which approximate the stationary toroids quite well for $Ca\leqslant 0.05$ .

Table 5.

Parameters of the ‘stationary’ elliptic torus (4.16) found from the inverse problem (4.18) with $w=r^{-1}$ and the corresponding mean absolute value $\Vert u_{n}\Vert _{S}$ defined by (3.8).