4.1 $\unicode[STIX]{x1D6F4}\ll 1$

A regular perturbation expansion in the parameter $\unicode[STIX]{x1D6F4}$ can be used in (2.5)–(2.7) to obtain,

(4.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{\ast } & = & \displaystyle 1-\unicode[STIX]{x1D6F4}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})M_{I}(\unicode[STIX]{x1D6FD})+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6F4}^{2}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})\nonumber\\ \displaystyle & & \displaystyle \times \,[2M_{I}(\unicode[STIX]{x1D6FD})M_{I}^{\prime }(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})+(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}(3(M_{I}(\unicode[STIX]{x1D6FD}))^{2}-(M_{R}(\unicode[STIX]{x1D6FD}))^{2})]\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D74E}}=1-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6F4}^{2}(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})(M_{R}(\unicode[STIX]{x1D6FD})^{2}+M_{I}(\unicode[STIX]{x1D6FD})^{2})\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6FA}}_{\Vert } & = & \displaystyle \hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}\left\{\unicode[STIX]{x1D6F4}M_{I}(\unicode[STIX]{x1D6FD})\right.+\unicode[STIX]{x1D6F4}^{2}\left[M_{I}\right.(\unicode[STIX]{x1D6FD})^{2}(1-2(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})\nonumber\\ \displaystyle & & \displaystyle -\,M_{I}(\unicode[STIX]{x1D6FD})M_{I}^{\prime }(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})+(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}M_{R}(\unicode[STIX]{x1D6FD})^{2}\left.\left.\!\!\right]\right\}\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6FA}}_{\bot } & = & \displaystyle (\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}\left\{-\unicode[STIX]{x1D6F4}M_{R}(\unicode[STIX]{x1D6FD})\right.+\unicode[STIX]{x1D6F4}^{2}M_{I}(\unicode[STIX]{x1D6FD})\nonumber\\ \displaystyle & & \displaystyle \times [(3(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}-1)M_{R}(\unicode[STIX]{x1D6FD})+(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})M_{R}^{\prime }]\left.\!\!\right\},\end{eqnarray}$$

where

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle M_{R}^{\prime }=\left.\frac{\text{d}M_{R}(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FA}^{\ast })}{\text{d}\unicode[STIX]{x1D6FA}^{\ast }}\right|_{\unicode[STIX]{x1D6FA}^{\ast }=1}, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle M_{I}^{\prime }=\left.\frac{\text{d}M_{I}(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FA}^{\ast })}{\text{d}\unicode[STIX]{x1D6FA}^{\ast }}\right|_{\unicode[STIX]{x1D6FA}^{\ast }=1}. & \displaystyle\end{eqnarray}$$

The couple stress coefficients are obtained by substituting the solutions (4.1)–(4.4), into (3.5)–(3.7),

(4.7) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{c}^{(1)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)} & = & \displaystyle -\unicode[STIX]{x1D6F4}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})M_{I}(\unicode[STIX]{x1D6FD})+\unicode[STIX]{x1D6F4}^{2}\left[\right.\!(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})(M_{I}(\unicode[STIX]{x1D6FD})^{2}-M_{R}(\unicode[STIX]{x1D6FD})^{2})\nonumber\\ \displaystyle & & \displaystyle +\,(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})^{2}M_{I}(\unicode[STIX]{x1D6FD})M_{I}^{\prime }\left.\!\right],\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{c}^{(3)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)} & = & \displaystyle -\unicode[STIX]{x1D6F4}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}M_{R}(\unicode[STIX]{x1D6FD})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6F4}^{2}\left[2\right.\!(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{3}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}M_{R}(\unicode[STIX]{x1D6FD})M_{I}(\unicode[STIX]{x1D6FD})\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})^{3/2}M_{I}(\unicode[STIX]{x1D6FD})M_{R}^{\prime }\left.\!\right].\end{eqnarray}$$

The couple stress coefficients (4.7)–(4.8) can be simplified in the limits $\unicode[STIX]{x1D6FD}\ll 1$ and $\unicode[STIX]{x1D6FD}\gg 1$ using (2.10)–(2.13) for a uniform particle and (2.14) and (2.15) for a thin shell. The $O(\unicode[STIX]{x1D6F4})$ contributions to the couple stress coefficients for $\unicode[STIX]{x1D6F4}\ll 1$ are summarised in table 1. These are proportional to the asymptotic expressions for $M_{R}(\unicode[STIX]{x1D6FD}_{p})$ and $M_{I}(\unicode[STIX]{x1D6FD}_{p})$ in (2.10)–(2.13) for the regime $\unicode[STIX]{x1D6FD}_{p}\ll 1$ and $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6FD}_{p}\gg 1$ and $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ for uniform particles, where the largest contribution to the couple stress coefficients is due to the $O(\unicode[STIX]{x1D6F4})$ term in (4.7)–(4.8). Similarly, for thin shells, the couple stress coefficients are proportional to the asymptotic expressions for (2.14)–(2.15) for $\unicode[STIX]{x1D6FD}_{s}\ll 1$ and $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6FD}_{s}\gg 1$ and $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{s}^{-1}$ . In both the aforementioned regimes, the first couple stress coefficient is proportional to $(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})$ , and the third couple stress coefficient is proportional to $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}$ .

Table 1.

The $\unicode[STIX]{x1D6F4}\ll 1$ limiting values of the couple stress coefficients, equations (4.7)–(4.8), for a uniform particle and a thin shell.

For the first couple stress coefficient, there is an intermediate regime, $\unicode[STIX]{x1D6FD}_{p}\gg 1$ and $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ for uniform particles and $\unicode[STIX]{x1D6FD}_{s}\gg 1$ and $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{s}^{-1}$ for thin shells, where the $O(\unicode[STIX]{x1D6F4}^{2})$ term in (4.7) is larger than the $O(\unicode[STIX]{x1D6F4})$ term. For a uniform particle, $M_{I}(\unicode[STIX]{x1D6FD}_{p})\propto \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ (2.13), and $M_{R}(\unicode[STIX]{x1D6FD}_{p})$ tends to $-(1/2)$ (2.11). The $O(\unicode[STIX]{x1D6F4})$ term on the right in the series in (4.7) is smaller than the $O(\unicode[STIX]{x1D6F4}^{2})$ term for $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})\ll \unicode[STIX]{x1D6F4}^{2}$ , or $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ . For a thin shell, $M_{I}(\unicode[STIX]{x1D6FD}_{s})\propto \unicode[STIX]{x1D6FD}_{s}^{-1}$ and $M_{R}(\unicode[STIX]{x1D6FD}_{s})=-(1/2)$ for $\unicode[STIX]{x1D6FD}_{s}\gg 1$ (equations (2.14)–(2.15)). The $O(\unicode[STIX]{x1D6F4})$ term on the right in (4.7) is smaller than the $O(\unicode[STIX]{x1D6F4}^{2})$ term for $(\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D6FD}_{s})\ll \unicode[STIX]{x1D6F4}^{2}$ , or $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{s}^{-1}$ . In the two aforementioned regimes, it can be shown that the $O(\unicode[STIX]{x1D6F4}^{2})$ term is the largest contribution to the first couple stress coefficient, and this contribution is proportional to $(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})$ . The expressions for the first couple stress coefficient in this regime are provided in table 1.

In contrast to the first couple stress coefficient, in (4.8) for the third couple stress coefficient, the $O(\unicode[STIX]{x1D6F4})$ term is dominant even for $\unicode[STIX]{x1D6FD}_{p}\gg 1$ and $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{s}\gg 1$ and $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{s}^{-1}$ for a thin shells. Therefore there is one common expression for the third couple stress coefficient for $\unicode[STIX]{x1D6FD}\gg 1$ in table 1.

In table 1, the couple stress coefficients are explicitly expressed in terms of dimensional variables, in order to highlight their vorticity dependence. For $\unicode[STIX]{x1D6FD}\ll 1$ , the first and second couple stress coefficients are independent of the magnitude of the vorticity, and the third couple stress coefficient is proportional to the vorticity magnitude $|\unicode[STIX]{x1D74E}|$ . Interestingly, for $\unicode[STIX]{x1D6FD}\gg 1$ , the couple stress coefficients are proportional to a negative power of the vorticity. The vorticity exponent for the first and third couple stress coefficients is different for uniform particles and thin shells. This is because the high $\unicode[STIX]{x1D6FD}$ limit of $M_{I}$ for a uniform particle in (2.13) is different from that for a thin shell in (2.15).

4.2 $\unicode[STIX]{x1D6F4}\gg 1$

An examination of (2.5)–(2.7) indicates that $(1-\hat{\unicode[STIX]{x1D6FA}}_{\boldsymbol{H}}^{2})$ is necessarily small for $\unicode[STIX]{x1D6F4}\gg 1$ , in order to obtain a balance between the terms on the left and right. This implies that the angular velocity is nearly aligned with the magnetic field, and the angle between the two is proportional to $\unicode[STIX]{x1D6F4}^{-1}$ for $\unicode[STIX]{x1D6F4}\gg 1$ . Expansions for $\unicode[STIX]{x1D6FA}^{\ast }$ , $\hat{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D74E}}$ , $\hat{\unicode[STIX]{x1D6FA}}_{\Vert }$ and $\hat{\unicode[STIX]{x1D6FA}}_{\bot }$ of the form,

(4.9) $$\begin{eqnarray}\displaystyle \star =\star ^{(0)}+\unicode[STIX]{x1D6F4}^{-1}\star ^{(1)}+\,\unicode[STIX]{x1D6F4}^{-2}\star ^{(2)}, & & \displaystyle\end{eqnarray}$$

are inserted into the conservation (2.5)–(2.7) and the normalisation condition (2.19). The results, correct to $O(\unicode[STIX]{x1D6F4}^{-2})$ , are,

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}^{\ast }=\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}+\frac{\unicode[STIX]{x1D6F4}^{-2}(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}{2(M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})},\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D74E}} & = & \displaystyle \hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}+\frac{\unicode[STIX]{x1D6F4}^{-1}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})}{M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6F4}^{-2}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}{2(M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})},\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6FA}}_{\Vert } & = & \displaystyle \sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}-\frac{\unicode[STIX]{x1D6F4}^{-1}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})}{M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x1D6F4}^{-2}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}(1+(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}{2(M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})},\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FA}}_{\bot }=-\frac{\unicode[STIX]{x1D6F4}^{-1}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})}{M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}.\end{eqnarray}$$

Since the angular velocity is nearly aligned with the magnetic field for $\unicode[STIX]{x1D6F4}\gg 1$ , the components $\hat{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D74E}}$ and $\hat{\unicode[STIX]{x1D6FA}}_{\Vert }$ differ from $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ and $\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}$ respectively by $O(\unicode[STIX]{x1D6F4}^{-1})$ . The magnitude of the angular velocity tends to $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ in the limit $\unicode[STIX]{x1D6F4}\gg 1$ ; the difference $(\unicode[STIX]{x1D6FA}^{\ast }-\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})$ is proportional to $\unicode[STIX]{x1D6F4}^{-2}$ . The component $\hat{\unicode[STIX]{x1D6FA}}_{\bot }$ of the angular velocity unit vector perpendicular to the $\hat{\unicode[STIX]{x1D74E}}{-}\hat{\boldsymbol{H}}$ plane is $O(\unicode[STIX]{x1D6F4}^{-1})$ . The first and third couple stress coefficients, obtained by substituting the expressions (4.10)–(4.13) into (3.5)–(3.7), are,

(4.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{c}^{(1)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)}=-(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})+\frac{\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})}{\unicode[STIX]{x1D6F4}(M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}, & & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{c}^{(3)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)}=-\frac{\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})}{\unicode[STIX]{x1D6F4}(M_{R}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}+M_{I}(\unicode[STIX]{x1D6FD}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}. & & \displaystyle\end{eqnarray}$$

Equation (4.14) indicates that the scaled first couple stress coefficient, $(\unicode[STIX]{x1D702}_{c}^{(1)}/(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2))$ , attains a constant value $-(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})$ for $\unicode[STIX]{x1D6F4}\gg 1$ ; from (3.10), the scaled couple stress coefficient $(\unicode[STIX]{x1D702}_{c}^{(2)}/(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2))$ approaches $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}$ in this limit. The third couple stress coefficient decreases proportional to $\unicode[STIX]{x1D6F4}^{-1}$ for $\unicode[STIX]{x1D6F4}\gg 1$ .

The asymptotic limits for the couple stress coefficients for $\unicode[STIX]{x1D6FD}\ll 1$ and $\unicode[STIX]{x1D6FD}\gg 1$ are listed in table 2. For a thin shell, by coincidence, the asymptotic expressions in the limits $\unicode[STIX]{x1D6FD}_{s}\ll 1$ and $\unicode[STIX]{x1D6FD}_{s}\gg 1$ are identical, even though the functions $M_{R}(\unicode[STIX]{x1D6FD}_{s}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})$ and $M_{I}(\unicode[STIX]{x1D6FD}_{s}\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})$ in (4.14)–(4.15) are different. For a uniform particle, the third couple stress coefficient is proportional to $\unicode[STIX]{x1D6F4}^{-1}$ and independent of $\unicode[STIX]{x1D6FD}_{p}$ for both small and large $\unicode[STIX]{x1D6FD}_{p}$ , although there is a small difference in the numerical prefactor. The first couple stress coefficient is independent of $|\unicode[STIX]{x1D74E}|$ and the third couple stress coefficient is proportional to $|\unicode[STIX]{x1D74E}|$ for a thin shell for small and large $\unicode[STIX]{x1D6FD}_{s}$ . For a uniform particle, the third couple stress coefficient is proportional to $|\unicode[STIX]{x1D74E}|$ for small and large $\unicode[STIX]{x1D6FD}_{p}$ .

Table 2.

The $\unicode[STIX]{x1D6F4}\gg 1$ limiting values of the couple stress coefficients, equations (4.14)–(4.15), for a uniform particle and a thin shell.

Figure 3.

The scaled angular velocity $\unicode[STIX]{x1D6FA}^{\ast \ast }$ (a), negative of the scaled first couple stress coefficient $-\unicode[STIX]{x1D702}_{1}^{\ast }$ (b) and third couple stress coefficient $\unicode[STIX]{x1D702}_{3}^{\ast }$ (c), defined in (4.16)–(4.18), as a function of $\unicode[STIX]{x1D6F4}$ for a uniform particle with $\unicode[STIX]{x1D6FD}_{p}=1$ (dashed lines), $\unicode[STIX]{x1D6FD}_{p}=10$ (solid lines) and $\unicode[STIX]{x1D6FD}_{p}=1000$ (dotted lines) for ○  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/3)$ , ▵  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/2)$ , ▿  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/\sqrt{2})$ and ♢  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=0.99$ .

4.3 Numerical results

The variation of the scaled particle angular velocity and couple stress coefficients with $\unicode[STIX]{x1D6F4}$ for different values of $\unicode[STIX]{x1D6FD}$ and $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ are shown in figure 3 for a uniform particle and figure 4 for a thin shell. In these figures, the asymptotic analysis of §§ 4.1 and 4.2 has been used to rescale the coordinates. The magnitude of the angular velocity $\unicode[STIX]{x1D6FA}^{\ast }$ approaches $1$ for $\unicode[STIX]{x1D6F4}\ll 1$ (4.1), and tends towards $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ for $\unicode[STIX]{x1D6F4}\gg 1$ (4.10). A natural definition of the rescaled angular velocity is,

(4.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{\ast \ast }=\frac{\unicode[STIX]{x1D6FA}^{\ast }-\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}}{1-\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}}, & & \displaystyle\end{eqnarray}$$

which varies from $1$ for $\unicode[STIX]{x1D6F4}\ll 1$ to $0$ for $\unicode[STIX]{x1D6F4}\gg 1$ . The scaled first couple stress coefficient $(\unicode[STIX]{x1D702}_{c}^{(1)}/(3\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}/2))$ is $0$ for $\unicode[STIX]{x1D6F4}=0$ (4.7), and it increases to a maximum value of $(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})$ for $\unicode[STIX]{x1D6F4}\gg 1$ (4.14). An appropriate definition of the scaled first couple stress coefficient is,

(4.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{1}^{\ast }=\frac{\unicode[STIX]{x1D702}_{c}^{(1)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})}, & & \displaystyle\end{eqnarray}$$

which varies in the range 0–1. The scaled third couple stress coefficient is proportional to $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}$ for $\unicode[STIX]{x1D6F4}\ll 1$ (4.8) and for $\unicode[STIX]{x1D6F4}\gg 1$ (equation 4.15); a suitable scaled third couple stress coefficient is,

(4.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{3}^{\ast }=\frac{\unicode[STIX]{x1D702}_{c}^{(3)}}{(3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})}. & & \displaystyle\end{eqnarray}$$

Figure 4.

The scaled angular velocity $\unicode[STIX]{x1D6FA}^{\ast \ast }$ (a), the scaled first couple stress coefficient $\unicode[STIX]{x1D702}_{1}^{\ast }$ (b) and third couple stress coefficient $\unicode[STIX]{x1D702}_{3}^{\ast }$ (c), defined in (4.16)–(4.18), as a function of $\unicode[STIX]{x1D6F4}$ for a thin shell with $\unicode[STIX]{x1D6FD}_{s}=0.3$ (dashed lines), $\unicode[STIX]{x1D6FD}_{s}=3$ (solid lines) and $\unicode[STIX]{x1D6FD}_{s}=100$ (dotted lines) for ○  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/3)$ , ▵  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/2)$ , ▿  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/\sqrt{2})$ and ♢  $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=0.99$ .

The magnitude of the angular velocity $\unicode[STIX]{x1D6FA}^{\ast \ast }$ (figures 3 a and 4 a) decreases from $1$ to $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ over a range of approximately two decades in $\unicode[STIX]{x1D6F4}$ . The functional form of $\unicode[STIX]{x1D6FA}^{\ast \ast }$ shows a relatively small dependence on $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ for small $\unicode[STIX]{x1D6FD}$ , but there is a significant dependence on $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ for $\unicode[STIX]{x1D6FD}_{p}=300$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{p}=100$ for a thin shell. It is observed that the magnitude of the slope of the $\unicode[STIX]{x1D6FA}^{\ast \ast }-\unicode[STIX]{x1D6F4}$ curve increases and appears to diverge at $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/3)$ for $\unicode[STIX]{x1D6FD}_{p}=300$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{s}=100$ for a thin shell. This aspect is analysed in detail in § 6.

In contrast to the angular velocity, $\unicode[STIX]{x1D702}_{1}^{\ast }$ and $\unicode[STIX]{x1D702}_{3}^{\ast }$ exhibit a remarkable lack of dependence on $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ when $\unicode[STIX]{x1D6FD}$ is small; all the results for the first and third couple stress coefficients collapse on to $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ -independent curves over the entire range of $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{p}\leqslant 10$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{s}\leqslant 3$ for a thin shell. The scaled first couple stress coefficient $\unicode[STIX]{x1D702}_{1}^{\ast }$ increases proportional to $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6F4}\ll 1$ , and tends to a limiting value of $1$ for $\unicode[STIX]{x1D6F4}\gg 1$ . For low $\unicode[STIX]{x1D6F4}$ and high $\unicode[STIX]{x1D6FD}$ , two distinct regions are visible in figures 3(b) and 4(b) for the first couple stress coefficient. These correspond to the two distinct regimes identified in table 1. For a uniform particle, $\unicode[STIX]{x1D702}_{1}^{\ast }$ is found to be independent of $\unicode[STIX]{x1D6FD}_{p}$ for $\unicode[STIX]{x1D6F4}\lesssim 3\times 10^{-2}$ in figure 3(b); this is in agreement with the prediction in the parameter regime $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ in table 1. For a thin shell, $\unicode[STIX]{x1D702}_{1}^{\ast }$ appears to become independent of $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ only for $\unicode[STIX]{x1D6F4}\lesssim 10^{-2}$ ; this is due to the more stringent requirement that $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{s}^{-1}$ for a thin shell in table 1. There is an intermediate region, $\unicode[STIX]{x1D6F4}\gtrsim 3\times 10^{-2}$ for a uniform particle and $\unicode[STIX]{x1D6F4}\gtrsim 10^{-2}$ for a thin shell, where $\unicode[STIX]{x1D702}_{1}^{\ast }$ does depend on $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ . In this regime, the variation of $\unicode[STIX]{x1D702}_{1}^{\ast }$ with $\unicode[STIX]{x1D6F4}$ in agreement with the asymptotic result $\unicode[STIX]{x1D702}_{1}^{\ast }\propto \unicode[STIX]{x1D6F4}^{2}$ in table 1. Table 1 also shows that the first couple stress coefficient is proportional to $(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})$ in this regime, and therefore the scaled first couple stress coefficient (4.17) does depend on $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}$ in figures 3(b) and 4(b).

The scaled third couple stress coefficient (figures 3 c and 4 c) increases proportional to $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6F4}\ll 1$ , and decreases proportional to $\unicode[STIX]{x1D6F4}^{-1}$ for $\unicode[STIX]{x1D6F4}\gg 1$ . Consistent with the results in table 1, there is only one regime for $\unicode[STIX]{x1D6F4}\ll 1$ for the third couple stress coefficient, and $\unicode[STIX]{x1D702}_{3}^{\ast }\propto \unicode[STIX]{x1D6F4}$ in this regime. Due to this, there is a remarkable data collapse in the limits $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$ for both low and high $\unicode[STIX]{x1D6FD}$ . The appearance of an inflection point close to the maximum in the $\unicode[STIX]{x1D702}_{3}^{\ast }$ graph for $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=1/3$ is noteworthy; the reason for this is discussed in § 6.

The $\unicode[STIX]{x1D6FD}$ dependence of the couple stress coefficients is examined in further detail for $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/\sqrt{2})$ in figures 5 and 6. For a uniform particle, the asymptotic expression for $\unicode[STIX]{x1D702}_{1}^{\ast }$ in table 1 is proportional to $\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{p}\ll 1$ , and proportional to $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})$ for $\unicode[STIX]{x1D6FD}_{p}\gg 1$ and $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ . In figure 5, the dashed lines show the first couple stress plotted against $(\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6F4})$ , and the dotted lines show the asymptotic expression for $\unicode[STIX]{x1D6FD}_{p}\ll 1$ . It is observed that the asymptotic expression provides an excellent fit for the $(\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6F4})\lesssim 1$ and $\unicode[STIX]{x1D6FD}_{p}\leqslant 3$ . There is some deviation from the asymptotic expression for $\unicode[STIX]{x1D6FD}_{p}\geqslant 10$ . The solid lines in figure 5 are graphs of $\unicode[STIX]{x1D702}_{1}^{\ast }$ versus $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})$ . These provide good fits for the first couple stress coefficient for $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})\leqslant 10^{-3}$ and $\unicode[STIX]{x1D6FD}_{p}\geqslant 30$ . There is an intermediate range of $3<\unicode[STIX]{x1D6FD}_{p}<30$ where the asymptotic expressions are not in quantitative agreement with the numerical results. In addition, there is a range of $10^{-3}\leqslant \unicode[STIX]{x1D6F4}\leqslant 10^{-1}$ , corresponding to the limit $\unicode[STIX]{x1D6FD}_{p}\gg 1$ and $1\gg \unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6FD}_{p}^{-1/2}$ in table 1, where the there is a dependence on $\unicode[STIX]{x1D6FD}_{p}$ . In this range, we find that a graph of $\unicode[STIX]{x1D702}_{1}^{\ast }$ versus $\unicode[STIX]{x1D6F4}$ results in data collapse for $\unicode[STIX]{x1D6FD}_{p}\geqslant 30$ ; this result is not shown here for conciseness. For a thin shell, the qualitative behaviour of $\unicode[STIX]{x1D702}_{1}^{\ast }$ is similar for a uniform particle. The asymptotic expression for $\unicode[STIX]{x1D702}_{1}^{\ast }$ in table 1 is proportional to $\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{s}\ll 1$ . This is in agreement with the numerical result in figure 5(b) for $\unicode[STIX]{x1D6FD}_{s}\leqslant 1$ and $\unicode[STIX]{x1D6F4}\leqslant 1$ . For $\unicode[STIX]{x1D6FD}_{s}\gg 1$ and $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{s}^{-1}$ , $\unicode[STIX]{x1D702}_{1}^{\ast }$ is proportional to $(\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D6FD}_{s})$ in table 1. The numerical result in figure 5 is in agreement with the asymptotic expression for $\unicode[STIX]{x1D6FD}_{s}\geqslant 10$ in the limit $\unicode[STIX]{x1D6F4}\ll 1$ . However, the asymptotic expression is applicable for a smaller range in $\unicode[STIX]{x1D6F4}$ in comparison to that for a uniform particle, due to the stronger condition $\unicode[STIX]{x1D6F4}\ll \unicode[STIX]{x1D6FD}_{s}^{-1}$ . In the intermediate regime $\unicode[STIX]{x1D6FD}_{s}^{-1}\ll \unicode[STIX]{x1D6F4}\ll 1$ , there is collapse of the data when $\unicode[STIX]{x1D702}_{1}^{\ast }$ is plotted as a function of $\unicode[STIX]{x1D6F4}$ ; this regime is not shown here for conciseness.

Figure 5.

The negative of the scaled first couple stress coefficient $-\unicode[STIX]{x1D702}_{1}^{\ast }$ for $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/\sqrt{2})$ for a uniform particle (a) as a function of $\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6F4}$ (dashed lines) and $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})$ (solid lines) for $\unicode[STIX]{x1D6FD}_{p}=1$ (○), $\unicode[STIX]{x1D6FD}_{p}=3$ (▵), $\unicode[STIX]{x1D6FD}_{p}=10$ (▿), $\unicode[STIX]{x1D6FD}_{p}=30$ (◃), $\unicode[STIX]{x1D6FD}_{p}=100$ (▹) and $\unicode[STIX]{x1D6FD}_{p}=300$ (♢); and a thin shell (b) as a function of $(\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x1D6F4})$ (dashed lines) and $(\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D6FD}_{s})$ (solid lines) for $\unicode[STIX]{x1D6FD}_{s}=0.3$ (○), $\unicode[STIX]{x1D6FD}_{s}=1$ (▵), $\unicode[STIX]{x1D6FD}_{s}=3$ (▿), $\unicode[STIX]{x1D6FD}_{s}=10$ (◃), $\unicode[STIX]{x1D6FD}_{s}=30$ (▹) and $\unicode[STIX]{x1D6FD}_{s}=100$ (♢).

The third couple stress coefficient is shown as a function of $\unicode[STIX]{x1D6F4}$ in figure 6. In table 1, the third couple stress coefficient is proportional to $\unicode[STIX]{x1D6FD}^{2}\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6FD}\ll 1$ . Figure 6 shows that this relationship is valid for a uniform particle for $\unicode[STIX]{x1D6FD}_{p}\leqslant 3$ and $\unicode[STIX]{x1D6F4}\lesssim 1$ , and for a thin shell for $\unicode[STIX]{x1D6FD}_{s}\leqslant 1$ and $\unicode[STIX]{x1D6F4}\lesssim 1$ . In table 1, the third couple stress coefficient proportional to $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6FD}\gg 1$ . Figure 6 indicates that this relationship is valid for $\unicode[STIX]{x1D6FD}_{p}\geqslant 30$ and $\unicode[STIX]{x1D6F4}\lesssim 1$ for a uniform particle, and for $\unicode[STIX]{x1D6FD}_{s}\geqslant 10$ and $\unicode[STIX]{x1D6F4}\lesssim 1$ for a thin shell. There is an intermediate interval of a decade, $3\lesssim \unicode[STIX]{x1D6FD}_{p}\lesssim 30$ for a uniform particle and $1\lesssim \unicode[STIX]{x1D6FD}_{s}\lesssim 10$ for a thin shell, where the asymptotic expressions are not quantitatively accurate. In the limit $\unicode[STIX]{x1D6F4}\gg 1$ , the asymptotic expressions for $\unicode[STIX]{x1D6FD}\ll 1$ and $\unicode[STIX]{x1D6FD}\gg 1$ in table 2 numerically close for a uniform particle. Figure 6(a) shows the numerical results are in agreement with the predicted $\unicode[STIX]{x1D6F4}^{-1}$ behaviour, and it is even possible to discern the small numerical difference in the coefficients for the limits $\unicode[STIX]{x1D6FD}_{p}\ll 1$ and $\unicode[STIX]{x1D6FD}_{p}\gg 1$ for a uniform particle. For a thin shell, the asymptotic expressions for $\unicode[STIX]{x1D6F4}\gg 1$ are identical in the limits $\unicode[STIX]{x1D6FD}_{s}\ll 1$ and $\unicode[STIX]{x1D6FD}_{s}\gg 1$ , and these are in quantitative agreement with the numerical results. For high $\unicode[STIX]{x1D6FD}$ , there is a transition between the low and high $\unicode[STIX]{x1D6F4}$ regimes at $\unicode[STIX]{x1D6F4}\approx 3$ for a uniform particle and $\unicode[STIX]{x1D6F4}\approx 2$ for a thin shell. For low $\unicode[STIX]{x1D6FD}$ , there is a transition from low to high $\unicode[STIX]{x1D6F4}$ regime at $\unicode[STIX]{x1D6F4}\approx 15\unicode[STIX]{x1D6FD}_{p}^{-1}$ for a uniform particle, and $\unicode[STIX]{x1D6F4}\approx 15\unicode[STIX]{x1D6FD}_{s}^{-1}$ for a thin shell.

Figure 6.

The scaled third couple stress coefficient $\unicode[STIX]{x1D702}_{3}^{\ast }$ for $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=(1/\sqrt{2})$ as a function of $\unicode[STIX]{x1D6F4}$ for a uniform particle (a) for $\unicode[STIX]{x1D6FD}_{p}=1$ (○), $\unicode[STIX]{x1D6FD}_{p}=3$ (▵), $\unicode[STIX]{x1D6FD}_{p}=10$ (▿), $\unicode[STIX]{x1D6FD}_{p}=30$ (◃), $\unicode[STIX]{x1D6FD}_{p}=100$ (▹) and $\unicode[STIX]{x1D6FD}_{p}=300$ (♢), and a thin shell (b) for $\unicode[STIX]{x1D6FD}_{s}=0.3$ (○), $\unicode[STIX]{x1D6FD}_{s}=1$ (▵), $\unicode[STIX]{x1D6FD}_{s}=3$ (▿), $\unicode[STIX]{x1D6FD}_{s}=10$ (◃), $\unicode[STIX]{x1D6FD}_{s}=30$ (▹) and $\unicode[STIX]{x1D6FD}_{s}=100$ (♢). In (a) the dashed lines, from right to left, are the relations $\unicode[STIX]{x1D702}_{3}^{\ast }=(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p}^{2}/315)$ (see table 1) for $\unicode[STIX]{x1D6FD}_{p}=1$ , $3$ and $10$ respectively. In (b) the dashed lines, from right to left, are the relations $\unicode[STIX]{x1D702}_{3}^{\ast }=(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{s}^{2}/18)$ (see table 1) for $\unicode[STIX]{x1D6FD}_{s}=0.3$ , $1$ and $3$ respectively.

5.1 Uniform particle

The left side of (5.1) is shown as a function of $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{p}=200$ in figure 7. When $\unicode[STIX]{x1D6F4}$ is increased from 5 to 6, there is first a transition from one solution to three solutions. As $\unicode[STIX]{x1D6F4}$ is further increased from 6 to 7, there is a second transition from three real solutions to one real solutions. Thus, there is an finite range of $\unicode[STIX]{x1D6F4}$ for which there are three solutions for a fixed value of $\unicode[STIX]{x1D6FD}_{p}$ ; there is only one solution outside this range.

Figure 7.

The left side of (5.1) shown as a function of $\unicode[STIX]{x1D6FA}^{\ast }$ for $\unicode[STIX]{x1D6FD}_{p}=200$ and $\unicode[STIX]{x1D6F4}=5$ (○), $6$ (▵) and $7$ (▿).

Figure 8.

The solution $\unicode[STIX]{x1D6FA}^{\ast }$ of (5.1) for a uniform particle (a) and the negative of the scaled first couple stress coefficient $-\unicode[STIX]{x1D702}_{1}^{\ast }$ (b) as a function of $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{p}=50$ (○), 100 (▵), 200 (▿), 400 (♢) and 800 (▫). The dotted lines at the right in (a) show the limiting value $30(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p})^{-1}$ for $\unicode[STIX]{x1D6F4}\gg 1$ (5.3), and the dotted lines on the left in (b) show the limiting value $\unicode[STIX]{x1D6F4}M_{I}(\unicode[STIX]{x1D6FD}_{p})$ .

The solutions $\unicode[STIX]{x1D6FA}^{\ast }$ for (5.1) are shown as a function of $\unicode[STIX]{x1D6F4}$ for different values of $\unicode[STIX]{x1D6FD}_{p}$ in figure 8(a), and the corresponding scaled first couple stress coefficient (equation (4.17)) is shown in figure 8(b). It is evident that (5.1) has a unique solution,

(5.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6FA}^{\ast }=1-\unicode[STIX]{x1D6F4}M_{I}(\unicode[STIX]{x1D6FD}_{p}),\\ \displaystyle \unicode[STIX]{x1D702}_{1}^{\ast }=-\unicode[STIX]{x1D6F4}M_{I}(\unicode[STIX]{x1D6FD}_{p}),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

for $\unicode[STIX]{x1D6F4}\ll 1$ . The asymptotic results for $\unicode[STIX]{x1D702}_{1}^{\ast }$ for $\unicode[STIX]{x1D6F4}\ll 1$ , shown by the dotted lines on the left in figure 8(b), are in good agreement with the numerical results. For $\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p}\gg 1$ , the solution $\unicode[STIX]{x1D6FA}^{\ast }$ decreases proportional to $(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p})^{-1}$ , as shown by the dotted lines on the right in figure 8(a). This solution is inferred from (5.1), if we take the limit $\unicode[STIX]{x1D6FA}^{\ast }\ll 1$ , $M_{I}(\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6FA}^{\ast })=(\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6FA}^{\ast }/30)$ for $\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6FA}^{\ast }\ll 1$ (2.12), and $\unicode[STIX]{x1D6FA}^{\ast }\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p}\sim 1$ ,

(5.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{\ast }=\frac{30}{30+\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p}}. & & \displaystyle\end{eqnarray}$$

For $\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p}\gg 30$ , the solution (5.3) is approximately $30(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{p})^{-1}$ , shown by the dashed lines on the right in figure 8(a). Thus, asymptotic analysis can be used to determine the values of $\unicode[STIX]{x1D6FA}^{\ast }$ and $\unicode[STIX]{x1D702}_{1}^{\ast }$ for $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$ .

For $\unicode[STIX]{x1D6FD}_{p}=50$ , figure 8(a) shows that the solution for $\unicode[STIX]{x1D6FA}^{\ast }$ monotonically decreases as $\unicode[STIX]{x1D6F4}$ increases. When $\unicode[STIX]{x1D6FD}_{p}$ is greater than $100$ , there is a range of values of $\unicode[STIX]{x1D6F4}$ for which there are three solutions for $\unicode[STIX]{x1D6FA}^{\ast }$ . There are two turning points of infinite slope where the solution turns back on itself. There is a stable solution for $\unicode[STIX]{x1D6FA}^{\ast }$ which is a continuation of the solution for $\unicode[STIX]{x1D6F4}\ll 1$ (5.2), and another stable solution which is a continuation of the solution for $\unicode[STIX]{x1D6F4}\gg 1$ , both shown by the solid lines. In between, there is a solution shown by the dashed line, which is an unstable solution.

A salient feature of the turning points in figure 8(a) are the values of $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6FA}^{\ast }$ at right and left turning points. The value of $\unicode[STIX]{x1D6F4}$ at the right turning point shows a significant increase as $\unicode[STIX]{x1D6FD}_{p}$ is increased, whereas the value of $\unicode[STIX]{x1D6FA}^{\ast }$ is approximately the same. At the left turning point, the value of $\unicode[STIX]{x1D6F4}$ is approximately independent of $\unicode[STIX]{x1D6FD}_{p}$ , while the value of $\unicode[STIX]{x1D6FA}^{\ast }$ decreases significantly as $\unicode[STIX]{x1D6FD}_{p}$ is increased. The value of $\unicode[STIX]{x1D6F4}$ at the right and left turning points are shown in figure 9(a), while the values of $\unicode[STIX]{x1D6FA}^{\ast }$ are shown in figure 9(b). For $\unicode[STIX]{x1D6FD}_{p}<93.02$ , there is only one solution for $\unicode[STIX]{x1D6FA}^{\ast }$ for all values of $\unicode[STIX]{x1D6F4}$ . For $\unicode[STIX]{x1D6FD}_{p}>93.02$ , there are three real solutions when $\unicode[STIX]{x1D6F4}$ has a value in between a minimum, which tends to a constant value of 5.63 for $\unicode[STIX]{x1D6FD}_{p}\gg 1$ , and a maximum which tends to $(4\sqrt{2\unicode[STIX]{x1D6FD}_{p}}/9\sqrt{3})$ for $\unicode[STIX]{x1D6FD}_{p}\gg 1$ .

Figure 9.

For a uniform particle, as a function of $\unicode[STIX]{x1D6FD}_{p}$ , the lower limit (○) and upper limit (▵) of the range of $\unicode[STIX]{x1D6F4}$ for the existence of multiple steady states (a), and the angular velocity $\unicode[STIX]{x1D6FA}^{\ast }$ (b) on the upper and lower branches for the values of $\unicode[STIX]{x1D6F4}$ shown in (a). The dashed line in (a) is $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})=(4\sqrt{2}/9\sqrt{3})$ . The value $\unicode[STIX]{x1D6FA}^{\ast }=(1/3)$ at the boundary $(\unicode[STIX]{x1D6F4}/\sqrt{\unicode[STIX]{x1D6FD}_{p}})=(4\sqrt{2}/9\sqrt{3})$ is shown by the dashed line in (b). The dotted line in (a) shows $\unicode[STIX]{x1D6F4}=5.63$ , and in (b) shows the asymptote $\unicode[STIX]{x1D6FA}^{\ast }=(11.61/\unicode[STIX]{x1D6FD}_{p})$ .

In appendix B, the $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6FD}$ dependence of the right and left turning points in figure 9(a), and the reason for the multistability, are examined using asymptotic analysis in the limit $\unicode[STIX]{x1D6FD}_{p}\unicode[STIX]{x1D6FA}^{\ast }\gg 1$ .

5.2 Thin shell

The results for the scaled angular velocity of a thin shell are briefly presented, since these are very similar to those for a uniform particle. The torque balance equation at steady state is,

(5.4) $$\begin{eqnarray}\displaystyle 1-\unicode[STIX]{x1D6FA}^{\ast }-\frac{3\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x1D6FA}^{\ast }}{2(9+\unicode[STIX]{x1D6FD}_{s}^{2}\unicode[STIX]{x1D6FA}^{\ast ^{2}})}=0. & & \displaystyle\end{eqnarray}$$

The solutions for the angular velocity are shown as a function of $\unicode[STIX]{x1D6F4}$ for different values of $\unicode[STIX]{x1D6FD}_{s}$ in figure 10(a), and the scaled first couple stress coefficient in (b). The transition between a unique solution and multiple solutions takes place at a value of $\unicode[STIX]{x1D6FD}_{s}$ that is about 15.7, which is approximately 6 times smaller than the value of $\unicode[STIX]{x1D6FD}_{p}$ at which the transition takes place for a uniform particle. For $(\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x1D6FA}^{\ast })\ll 1$ and $\unicode[STIX]{x1D6FA}^{\ast }\sim 1$ , there is a unique solution for (5.4),

(5.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{\ast }=\frac{1}{1+(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{s}/6)}. & & \displaystyle\end{eqnarray}$$

This solution predicts the decrease in $\unicode[STIX]{x1D6FA}^{\ast }$ from $1$ for small $\unicode[STIX]{x1D6F4}$ . In the limit $(\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x1D6FA}^{\ast })\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$ , the unique solution for $\unicode[STIX]{x1D6FA}^{\ast }$ is similar to the solution (5.3) for a uniform particle,

(5.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}^{\ast }=6(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{s})^{-1}. & & \displaystyle\end{eqnarray}$$

This solution is shown by the dotted lines on the right side of figure 10(a).

Figure 10.

The solution $\unicode[STIX]{x1D6FA}^{\ast }$ of (5.4) for a thin shell (a) and the negative of the scaled first couple stress coefficient $-\unicode[STIX]{x1D702}_{1}^{\ast }$ (b) as a function of $\unicode[STIX]{x1D6F4}$ for $\unicode[STIX]{x1D6FD}_{s}=10$ (○), 20 (▵), 40 (▿), 80 (♢) and 160 (▫). The dotted lines on the right in (a) show the limiting value $6(\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FD}_{s})^{-1}$ (5.6). The dotted lines on the left in (b) show the limiting value $\unicode[STIX]{x1D6F4}M_{I}(\unicode[STIX]{x1D6FD}_{s})$ .

The values of $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6FA}^{\ast }$ at the left and right turning points in figure 10(a) are shown in figure 11(a,b). At the right turning point, $\unicode[STIX]{x1D6F4}$ tends to $(\unicode[STIX]{x1D6FD}_{s}/6)$ and $\unicode[STIX]{x1D6FA}^{\ast }$ tends to $0.5$ for $\unicode[STIX]{x1D6FD}_{s}\gg 1$ . These are examined using asymptotic analysis in appendix B.

Figure 11.

For a thin shell, as a function of $\unicode[STIX]{x1D6FD}_{s}$ , the lower limit (○) and upper limit (▵) of the range of $\unicode[STIX]{x1D6F4}$ for which multiple steady states exists for the particle angular velocity (a), and the angular velocity $\unicode[STIX]{x1D6FA}^{\ast }$ (b) on the upper and lower branches for the $\unicode[STIX]{x1D6F4}$ shown in (a). The asymptotic expression $(\unicode[STIX]{x1D6F4}=\unicode[STIX]{x1D6FD}_{s}/6)$ (B 3) is the region below the dashed line in (a). The value $\unicode[STIX]{x1D6FA}^{\ast }=(1/2)$ for $\unicode[STIX]{x1D6FD}_{s}\gg 1$ and $\unicode[STIX]{x1D6F4}=(\unicode[STIX]{x1D6FD}_{s}/6)$ is shown by the dashed line in (b). The dotted line in (a) shows $\unicode[STIX]{x1D6F4}=4$ , and in (b) shows $\unicode[STIX]{x1D6FA}^{\ast }=(3/\unicode[STIX]{x1D6FD}_{s})$ .