2.1. Swimming velocity

We wish to establish the formula for the swimming velocity U_i of the stick-slip squirmer in terms of a prescribed tangential squirming velocity $u_i^s$ on the squirmer's surface. Since U_i is determined by the driving squirming force $F_i^{squirm} ={-} {F_i}$ in balance with the drag force F_i on the squirmer, to determine $F_i^{squirm}$, we keep the squirmer fixed with the no-penetration condition and the slip boundary condition with a non-uniform slip length aλ(x) on its surface as

(2.2a)\begin{gather}{u^{\prime}_i}{n_i} = 0,\end{gather}

(2.2b)\begin{gather}{u^{\prime}_i} - u_i^s = \frac{{a\lambda (\boldsymbol{x})}}{\mu }{\sigma ^{\prime}_{jk}}{n_k}({\delta _{ij}} - {n_i}{n_j}).\end{gather}

In (2.2b), the effects of surface slip are assumed to be described by the Navier slip condition (Navier Reference Navier1823). This model states that the extent of fluid slippage can be reflected by the ratio of the slip velocity to the tangential stress. Since there is an additional tangential squirming here, the slip velocity ${u^{\prime}_i} - u_i^s$ has to be the fluid velocity relative to the prescribed squirming velocity $u_i^s$. Therefore, slip effects will lead ${u^{\prime}_i}$ to mismatch $u_i^s$, making the latter's momentum not fully transmitted into the fluid, in contrast to ${u^{\prime}_i} = u_i^s$ in the usual no-slip situation.

For the auxiliary problem, we take a uniform-slip sphere of identical size having the constant slip length a $\hat{\lambda }$ translating at speed ${\hat{U}_i}$ with the following boundary conditions on its surface:

(2.3a)\begin{gather}({\hat{u}^{\prime}_i} - {\hat{U}_i}){n_i} = 0,\end{gather}

(2.3b)\begin{gather}{\hat{u}^{\prime}_i} - {\hat{U}_i} = \frac{{a\hat{\lambda }}}{\mu }{\hat{\sigma ^{\prime}}_{jk}}{n_k}({\delta _{ij}} - {n_i}{n_j}).\end{gather}

For both problems, the velocity fields vanish as $\textrm{|}\boldsymbol{x}\textrm{|} \to \infty$ far away from the spheres. We first extract the driving squirming force $F_i^{squirm} = \int_{{S_p}} {{{\sigma ^{\prime}}_{ij}}{n_j}} \,\textrm{d}S$ by re-arranging the left-hand side of (2.1) to

(2.4)\begin{equation}\int_{{S_p}} {{{\hat{u}^{\prime}}_i}} {\sigma ^{\prime}_{ij}}{n_j}\,\textrm{d}S = {\hat{U}_i}\int_{{S_p}} {{{\sigma ^{\prime}}_{ij}}{n_j}} \,\textrm{d}S + \int_{{S_p}} {({{\hat{u}^{\prime}}_i}} - {\hat{U}_i}){\sigma ^{\prime}_{ij}}{n_j}\,\textrm{d}S.\end{equation}

Recognizing that the velocity jump $({\hat{u}^{\prime}_i} - {\hat{U}_i})$ on the right only acts tangentially along the sphere's surface in view of (2.3a), this jump can be replaced by the slip term in (2.3b), allowing us to recast the left-hand side of (2.1) to

(2.5)\begin{equation}\int_{{S_p}} {{{\hat{u}^{\prime}}_i}} {\sigma ^{\prime}_{ik}}{n_k}\,\textrm{d}S = {\hat{U}_i}\int_{{S_p}} {{{\sigma ^{\prime}}_{ik}}{n_k}} \,\textrm{d}S + (a/\mu )\int_{{S_p}} {\hat{\lambda }{{\hat{\sigma ^{\prime}}}_{jm}}{n_m}} ({\delta _{ij}} - {n_i}{n_j}){\sigma ^{\prime}_{ik}}{n_k}\,\textrm{d}S.\end{equation}

Similarly, for the right-hand side of (2.1), we extract the prescribed tangential squirming velocity $u_i^s$ and replace the velocity jump $({u^{\prime}_i} - u_i^s)$ with the slip term in (2.2b):

(2.6)\begin{align} & \int_{{S_p}} {{{u^{\prime}}_i}{{\hat{\sigma }^{\prime}}_{ij}}{n_j}\,\textrm{d}S} = \int_{{S_p}} {u_i^s{{\hat{\sigma }^{\prime}}_{ij}}{n_j}\,\textrm{d}S} + \int_{{S_p}} {({{u^{\prime}}_i} - u_i^s){{\hat{\sigma }^{\prime}}_{ik}}{n_k}\,\textrm{d}S} \nonumber\\ & \quad = \int_{{S_p}} {u_i^s{{\hat{\sigma }^{\prime}}_{ij}}{n_j}\,\textrm{d}S} + (a/\mu )\int_{{S_p}} {\lambda (\boldsymbol{x}){{\sigma ^{\prime}}_{jm}}{n_m}} ({\delta _{ij}} - {n_i}{n_j}){{\hat{\sigma }^{\prime}}_{ik}}{n_k}\,\textrm{d}S. \end{align}

Subtracting (2.5) from (2.6), we arrive at

(2.7)\begin{equation}{\hat{U}_i}F_i^{squirm} = \int_{{S_p}} {u_i^s{{\hat{\sigma }^{\prime}}_{ij}}{n_j}\,\textrm{d}S} + (a/\mu )\int_{{S_p}} {(\lambda (\boldsymbol{x}) - \hat{\lambda })} {\sigma ^{\prime}_{jm}}{n_m}({\delta _{ij}} - {n_i}{n_j}){\hat{\sigma }^{\prime}_{ik}}{n_k}\,\textrm{d}S.\end{equation}

In the above, the integral involving the slip variation $(\lambda (\boldsymbol{x}) - \hat{\lambda })$ still contains the unknown traction ${\sigma ^{\prime}_{jm}}{n_m}$ on the squirmer's surface. For this reason, we determine $F_i^{squirm}$ approximately by assuming that the magnitude of the slip variation is small compared with the average dimensionless slip length 〈λ(x)〉 by setting $\hat{\lambda } = \langle \lambda (\boldsymbol{x})\rangle $ in (2.7), i.e.

(2.8)\begin{equation}\varepsilon \equiv |\lambda (\boldsymbol{x}) - \langle \lambda (\boldsymbol{x})\rangle |\ll 1.\end{equation}

In other words, we assume a small slip anisotropy for the squirmer to allow us to determine $F_i^{squirm}$ accurate to O(ε) using (2.7) in which the slip length of the auxiliary uniform-slip sphere is taken to be the average slip length of the squirmer.

With (2.8), the unknown surface traction ${\sigma ^{\prime}_{jm}}{n_m}$ in the slip variation integral can be approximated as the surface traction on a uniform-slip squirmer, $\sigma ^{\prime (0)}_{jm}{n_m}$, plus an O(ε) correction as

(2.9)\begin{equation}{\sigma ^{\prime}_{jm}}{n_m} = \sigma ^{\prime (0)}_{jm}{n_m} + O(\varepsilon ).\end{equation}

Substituting (2.9) into (2.7) and further writing ${\hat{\sigma }^{\prime}_{ij}}{n_j}$ in terms of the translation resistance tensor $R_{ij}^T$ obtained previously for a uniform-slip sphere (Premlata & Wei Reference Premlata and Wei2020), we obtain

(2.10a)\begin{gather}{\hat{\sigma }^{\prime}_{ik}}{n_k} ={-} \mu R_{ij}^T{\hat{U}_j},\end{gather}

(2.10b)\begin{gather}R_{ij}^T = \frac{1}{{a(1 + 3\hat{\lambda })}}\left( {\frac{3}{2}{\delta_{ij}} + 9\hat{\lambda }{n_i}{n_j}} \right).\end{gather}

Equation (2.7) with $\hat{\lambda } = \langle \lambda (\boldsymbol{x})\rangle $ furnishes the squirming force for propelling a weakly stick-slip squirmer:

(2.11)\begin{equation}F_i^{squirm} ={-} \mu \int_{{S_p}} {u_k^sR_{ik}^T\,\textrm{d}S} - a\int_{{S_p}} {\mathrm{\Delta }\lambda (\boldsymbol{x})\textrm{ }} \sigma ^{\prime (0)}_{jm}{n_m}({\delta _{jk}} - {n_j}{n_k})R_{ik}^T\,\textrm{d}S,\end{equation}

where $\Delta \lambda (\boldsymbol{x}) = \lambda (\boldsymbol{x}) - \langle \lambda (\boldsymbol{x})\rangle$ and $\hat{\lambda }$ in $R_{ij}^T$ of (2.10) is taken to be the average dimensionless slip length 〈λ(x)〉 of the stick-slip squirmer. We restate that the driving squirming force described by (2.11) is accurate up to O(ε). It actually turns out that, by comparing the results of directly solving the flow field around a stick-slip squirmer (see Appendices A and B), (2.11) is only valid when 〈λ(x)〉 is small.

To facilitate the evaluation of the slip variation term in (2.11), we use the slip boundary condition (2.3b) for the leading-order uniform-slip problem to replace the tangential stress by the velocity jump on the squirmer's surface:

(2.12)\begin{equation}\sigma ^{\prime (0)}_{jm}n_{m}({\delta _{jk}} - {n_j}{n_k}) = \frac{\mu }{{a\langle \lambda \rangle }}(u^{\prime \textrm{(0)}}_k - u_k^s).\end{equation}

So (2.11) can be rewritten as

(2.13)\begin{equation}F_i^{squirm} ={-} \mu \int_{{S_p}} {u_k^sR_{ik}^T\,\textrm{d}S} - \mu \int_{{S_p}} {\frac{{\mathrm{\Delta }\lambda (\boldsymbol{x})}}{{\langle \lambda \rangle }}\textrm{ }} (u^{\prime (0)}_k - u_k^s)R_{ik}^T\,\textrm{d}S.\end{equation}

The velocity jump $(u^{\prime (0)}_k - u_k^s)$ can be obtained below by solving the leading-order flow field $u^{\prime (0)}_k$ in spherical polar coordinates (see Appendix A):

(2.14) \begin{equation}\frac{1}{{\langle \lambda \rangle }}(u^{\prime (0)}_k - u_k^s) ={-} \sin \theta \left[ {\frac{{3{B_1}}}{{1 + 3\langle \lambda \rangle }}P_1^{\prime}(\cos \theta ) + \frac{{(5/3){B_2}}}{{1 + 5\langle \lambda \rangle }}P_2^{\prime}(\cos \theta )} \right]{e_\theta }.\end{equation}

To determine the swimming velocity U_i for the squirmer, $F_i^{squirm}$ given by (2.13) has to be counterbalanced by the drag force F_i on the squirmer as it swims. For the latter, it can be obtained by either using the reciprocal theorem (Premlata & Wei Reference Premlata and Wei2021) or by solving the translation flow field around the squirmer (see Appendix B). This force also includes a correction due to the small slip anisotropy, taking the form (Premlata & Wei Reference Premlata and Wei2021)

(2.15)\begin{equation}{F_i} ={-} 6{\rm \pi}\mu a\textrm{ }\left( {\frac{{1 + 2\langle \lambda \rangle }}{{1 + 3\langle \lambda \rangle }}} \right)\left( {1 + \frac{{\mathcal Q}}{{(1 + 2\langle \lambda \rangle )(1 + 3\langle \lambda \rangle )}}} \right){U_i}.\end{equation}

Here, $\mathcal{Q}$ of O(ε) measures the strength of surface quadrupole according to the slip pattern as

(2.16)\begin{equation}{\mathrm{{\mathcal P}}_{2ij}} = \int_{{S_p}} {(\lambda (\boldsymbol{x}) - \langle \lambda (\boldsymbol{x})\rangle )(3{n_i}{n_j} - {\delta _{ij}})\,\textrm{d}S} /(4{\rm \pi}\mu {a^2}) = {\mathcal Q}(3{d_i}{d_j} - {\delta _{ij}}).\end{equation}

In (2.16), d_i is the stick-slip director pointing from the stick (or the less slippery) face to the slip (or the more slippery) face. This director can be either aligned to or opposite to the swimming direction e_i of the squirmer in the absence of slip anisotropy. Together with the force-free condition $F_i^{squirm} + {F_i} = 0$, we can obtain the formula for the swimming velocity of a weakly stick-slip squirmer:

(2.17)\begin{equation}{U_i} = \frac{1}{{6{\rm \pi}\mu a}}\textrm{ }\left( {\frac{{1 + 3\langle \lambda \rangle }}{{1 + 2\langle \lambda \rangle }}} \right){\left( {1 + \frac{{\mathcal Q}}{{(1 + 2\langle \lambda \rangle )(1 + 3\langle \lambda \rangle )}}} \right)^{ - 1}}F_i^{squirm},\end{equation}

where $F_i^{squirm}$ is provided by (2.13) in which $u_i^s$, $R_{ij}^T$ and $(u^{\prime \textrm{(0)}}_i - u_i^s)$ are given by (1.2), (2.10b) and (2.14), respectively. Note that $F_i^{squirm}$ also includes an O(ε) slip variation correction due to the second term in (2.13). Using (2.17), we are able to compute U_i for the squirmer with an arbitrary axisymmetric slip pattern. For simplicity, we consider a two-faced slip pattern in terms of a partition angle α, such as that sketched in figure 2(a), to specify

(2.18)\begin{equation}\lambda (\boldsymbol{x}) = \left\{\begin{array}{@{}ll} {{\lambda_R}}&{\textrm{0} \le \theta \le \alpha }\\ {{\lambda_L}}&{\textrm{otherwise}} \end{array} \right..\end{equation}

The slip length jumps from aλ_R on the right face to aλ_L on the left face so that the average dimensionless slip length and the strength of the surface quadrupole are

(2.19a)\begin{gather}\langle \lambda \rangle = (1/2)({\lambda _R} + {\lambda _L} - ({\lambda _R} - {\lambda _L})\cos \alpha ),\end{gather}

(2.19b)\begin{gather}{\mathcal Q} = (1/4)({\lambda _L}-{\lambda _R})({\cos ^3}\alpha - \cos \alpha ).\end{gather}

Carrying out the integrals in (2.13) (see Appendix C), we can use (2.18) to determine the swimming velocity below after taking a small ε expansion and keeping the terms to O(ε)

(2.20)\begin{equation}{U_i} = \frac{2}{3}\textrm{ }\frac{{{e_i}}}{{1 + 2\langle \lambda \rangle }}[{B_1} + ({\lambda _L} - {\lambda _R})(\,{f_1}(\alpha ){B_1} + {f_2}(\alpha ){B_2})].\end{equation}

Here, e_i is set to be the swimming direction (towards the right in figure 2a) of the squirmer without stick-slip disparity. The coefficient f ₁(α) is contributed from the surface quadrupole (2.19b), and f ₂(α) comes from a combination of the surface octupole and dipole (Premlata & Wei Reference Premlata and Wei2022). These coefficients vary with the stick-slip partition angle α and the average dimensionless slip length (2.19a) according to

(2.21a)\begin{gather}{f_1}(\alpha ) = \frac{{1/2}}{{1 + 2\langle \lambda \rangle }}({\cos ^3}\alpha - \cos \alpha ),\end{gather}

(2.21b)\begin{gather}{f_2}(\alpha ) = \frac{{1/2}}{{1 + 5\langle \lambda \rangle }}{\sin ^4}\alpha .\end{gather}

Note that α = 0 and α = π recover no-slip and uniform-slip squirmers, respectively. The swimming velocity given by (2.20) also agrees with that obtained by solving the squirming and the translation flow fields around the squirmer under the force-free condition (see (B9)–(B11) in Appendix B).

If the squirmer is uniform slip with (λ_L–λ_R) = 0, (2.20) is reduced to

(2.22)\begin{equation}{U_i} = \frac{2}{3}\textrm{ }\frac{{{B_1}}}{{1 + 2\langle \lambda \rangle }}{e_i}.\end{equation}

Here, $\langle \lambda \rangle = 0$ recovers the no-slip result U_i = (2/3)B ₁e_i, determined solely by the B ₁ mode (Lighthill Reference Lighthill1952; Blake Reference Blake1971). It is evident that surface slip tends to lower U_i because the less tangential stress can be transmitted from the squirming motion into the fluid. In the full slip limit with $\langle \lambda \rangle \to \infty$, there will be no swimming at all because the surface is stress-free, and no squirming force can be generated from such a surface.

If there exists a slip anisotropy on the squirmer's surface, however, (2.20) indicates that the swimming of the squirmer can be driven by an extensile/contractile squirming of the B ₂ mode without involving the B ₁ mode that usually provides the thrust for the swimming. Figure 3(a) illustrates physically why this can happen to a stick-slip squirmer. Even though the extensile/contractile squirming imposed by the B ₂ mode is symmetric, the generated squirming forces on the stick and the slip faces are asymmetric because of stick-slip disparity. This, in turn, generates a net squirming force to set the squirmer in motion.

Figure 3.

Schematic illustrations of how a stick-slip squirmer establishes its propulsion (blue arrow) when the squirming forces (pink arrows) by a single squirming mode are modified by a stick-slip disparity: (a) with only a symmetric extensile/contractile squirming by the B ₂ mode, an asymmetric squirming force can be developed on the stick and the slip (in shade) faces to give net propulsion; (b) for the propulsion by a unidirectional squirming through the B ₁ mode, the resulting asymmetric squirming force can be decomposed into a symmetric force dipole of strength (F_stick − F_slip) to form a stresslet plus an offset force. The direction of this disparity-induced stresslet under the B ₁ mode is controlled by the stronger squirming force on the stick side.

2.2. Stresslet

We now derive the stresslet formula for a stick-slip squirmer according to (Batchelor Reference Batchelor1970)

(2.23)\begin{equation}{S_{ij}} = \int_{{S_p}} {{x_j}} {\sigma ^{\prime}_{ik}}{n_k}\,\textrm{d}S - 2\mu \int_{{S_p}} {{{u^{\prime}}_i}} {n_j}\,\textrm{d}S.\end{equation}

For convenience, we consider the squirmer held fixed in a quiescent fluid with the following boundary conditions at its surface:

(2.24a)\begin{gather}{u^{\prime}_i}{n_i} = 0,\end{gather}

(2.24b)\begin{gather}{u^{\prime}_i} - u_i^s = \frac{{a\lambda (\boldsymbol{x})}}{\mu }{\sigma ^{\prime}_{jk}}{n_k}({\delta _{ij}} - {n_i}{n_j}).\end{gather}

For the auxiliary problem, we consider a uniform-slip sphere of identical size with its slip length taken to be the average slip length a〈λ〉 of the stick-slip squirmer and hold it fixed at the centre of a purely straining flow field $\hat{u}_i^B = \hat{E}_{ij}^B{x_j}$, with $\hat{E}_{ij}^B$ being the strain rate tensor. The boundary conditions at the sphere's surface are

(2.25a)\begin{gather}({\hat{u}^{\prime}_i} + \hat{u}_i^B){n_i} = 0,\end{gather}

(2.25b)\begin{gather}{\hat{u}^{\prime}_i} + \hat{u}_i^B = \frac{{a\langle \lambda \rangle }}{\mu }\textrm{(}{\hat{\sigma }^{\prime}_{jk}}{n_k} + \hat{\sigma }_{jk}^B{n_k}\textrm{)(}{\delta _{ij}} - {n_i}{n_j}\textrm{)},\end{gather}

where $\hat{\sigma }_{jk}^B{n_k} = 2\mu \hat{E}_{jk}^B{n_k}$ is the surface traction exerted by the auxiliary bulk straining field.

To construct the stresslet from (2.1), we first extract the stress moment part $\int_{{S_p}} {{x_j}{{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} $ in (2.23) from the left-hand side of (2.1), as shown by (2.26) below. Specifically, we add and subtract the stress moment from a purely straining flow field $\hat{u}_i^B = \hat{E}_{ij}^B{x_j}$. The subtracted part gives $\int_{{S_p}} {{x_j}{{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} $. In the added part, $({\hat{u}^{\prime}_i} + \hat{u}_i^B)$ can be replaced by the slip term in (2.25b). The above manipulations lead to

(2.26)\begin{align} & \int_{{S_p}} {{{\hat{u}^{\prime}}_i}{{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} ={-} \int_{{S_p}} {\hat{u}_i^B{{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} + \int_{{S_p}} {({{\hat{u}^{\prime}}_i} + \hat{u}_i^B){{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} \nonumber\\ & \quad ={-} \hat{E}_{ij}^B\int_{{S_p}} {{x_j}{{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} + \dfrac{a}{\mu }\int_{{S_p}} {\langle \lambda \rangle ({{\hat{\sigma }^{\prime}}_{jm}}{n_m} + \hat{\sigma }_{jm}^B{n_m})({\delta _{ij}} - {n_i}{n_j}){{\sigma ^{\prime}}_{ik}}{n_k}\,\textrm{d}S} . \end{align}

Similarly, for the velocity moment part $- 2\mu \int_{{S_p}} {{{u^{\prime}}_i}{n_j}\,\textrm{d}S} $ of the stresslet, we can extract it from the subtracted term in the re-written form below of the right-hand side of (2.1) by working with the stress field $\hat{\sigma }_{ij}^B$ from $\hat{u}_i^B$ followed by replacing ${u^{\prime}_i}$ in the added term using (2.24b)

(2.27)\begin{align} \int_{{S_p}} {{{u^{\prime}}_i}{{\hat{\sigma }^{\prime}}_{ik}}{n_k}\,\textrm{d}S} &={-} \int_{{S_p}} {{{u^{\prime}}_i}\hat{\sigma }_{ij}^B{n_j}\,\textrm{d}S} + \int_{{S_p}} {{{u^{\prime}}_i}({{\hat{\sigma }^{\prime}}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k})\,\textrm{d}S} \nonumber\\ & \quad ={-} 2\mu \textrm{ }\hat{E}_{ij}^B\int_{{S_p}} {{{u^{\prime}}_i}{n_j}\,\textrm{d}S} + \int_{{S_p}} {u_i^s({{\hat{\sigma }^{\prime}}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k})\,\textrm{d}S} \nonumber\\ & \quad \quad + \dfrac{a}{\mu }\int_{{S_p}} {\lambda (\boldsymbol{x}){{\sigma ^{\prime}}_{jm}}{n_m}\textrm{(}{\delta _{ij}} - {n_i}{n_j}\textrm{)(}{{\hat{\sigma }^{\prime}}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k}\textrm{)}\,\textrm{d}S} . \end{align}

Combining (2.26) and (2.27) leads to

(2.28)\begin{align} \hat{E}_{ij}^B{S_{ij}} & ={-} \int_{{S_p}} {u_i^s({{\hat{\sigma }^{\prime}}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k})\,\textrm{d}S}\nonumber \\ & \quad - \dfrac{a}{\mu }\int_{{S_p}} {\textrm{(}\lambda (\boldsymbol{x}) - \langle \lambda \rangle \textrm{) }{{\sigma ^{\prime}}_{jm}}{n_m}\textrm{(}{\delta _{ij}} - {n_i}{n_j}\textrm{)(}{{\hat{\sigma }^{\prime}}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k}\textrm{)}\,\textrm{d}S} . \end{align}

Here, the total surface traction $({\hat{\sigma }^{\prime}_{ik}}{n_k} + \hat{\sigma }_{ik}^B{n_k})$ can be expressed in terms of the third-rank resistance tensor $\varSigma _{qij}$ in a purely straining field (Luo & Pozrikidis Reference Luo and Pozrikidis2008):

(2.29a)\begin{gather}{\hat{\sigma ^{\prime}}_{qk}}{n_k} + \hat{\sigma }_{qk}^B{n_k} = \mu {\varSigma _{qij}}\hat{E}_{ij}^B,\end{gather}

(2.29b)\begin{gather}{\varSigma _{qij}} = \frac{5}{{1 + 5\langle \lambda \rangle }}({\delta _{iq}}{n_j} + 8\langle \lambda \rangle {n_q}{n_i}{n_j}).\end{gather}

Eliminating the common $\hat{E}_{ij}^B$ in (2.28) renders the exact formula for the stresslet

(2.30)\begin{equation}{S_{ij}} ={-} \mu \int_{{S_p}} {u_q^s{\varSigma _{qij}}\,\textrm{d}S} - a\int_{{S_p}} {\textrm{(}\lambda (\boldsymbol{x}) - \langle \lambda \rangle \textrm{)}{{\sigma ^{\prime}}_{km}}{n_m}\textrm{(}{\delta _{kq}} - {n_k}{n_q}\textrm{)}{\varSigma _{qij}}\,\textrm{d}S} .\end{equation}

For the unknown traction $\sigma _{km}^{\prime}{n_m}$ on the squirmer, we follow the earlier treatment and limit an O(ε) slip variation under (2.8) to approximate this traction as the uniform-slip one (2.12). The stresslet on the squirmer can then be determined accurately to O(ε) as

(2.31)\begin{equation}{S_{ij}} ={-} \mu \int_{{S_p}} {u_q^s{\varSigma _{qij}}\,\textrm{d}S} - \mu \int_{{S_p}} {\textrm{ }\frac{{\mathrm{\Delta }\lambda }}{{\langle \lambda \rangle }}\textrm{(}u^{\prime \textrm{(0)}}_q - u_q^s\textrm{)}{\varSigma _{qij}}\,\textrm{d}S} .\end{equation}

Equation (2.31) without the slip variation term is identical to the stresslet formula derived by Lauga & Michelin (Reference Lauga and Michelin2016) for no-slip squirmers. In addition, (2.31) may be applicable to an arbitrary squirmer geometry when the length scale a is taken as the characteristic length scale for a given geometry.

For the stick-slip pattern (2.18) considered here, the stresslet calculated from (2.31) (with the details given in Appendix D) is found to be

(2.32)\begin{equation}{S_{ij}} = \textrm{ }\frac{{4{\rm \pi}\mu {a^2}}}{{1 + 5\langle \lambda \rangle }}\left( {{e_i}{e_j} - \frac{{{\delta_{ij}}}}{3}} \right)[{B_2} + ({\lambda _L} - {\lambda _R})({h_1}(\alpha ){B_1} + {h_2}(\alpha ){B_2})].\end{equation}

Here, the coefficient h ₁(α) comes from a combination of the surface octupole and dipole, and h ₂(α) is constituted by the surface hexadecapole and quadrupole (Premlata & Wei Reference Premlata and Wei2022). These coefficients vary with the stick-slip partition angle α and the average dimensionless slip length (2.19a) according to

(2.33a)\begin{gather}{h_1}(\alpha ) = \frac{{45/16}}{{1 + 3\langle \lambda \rangle }}{\sin ^4}\alpha ,\end{gather}

(2.33b)\begin{gather}{h_2}(\alpha ) = \frac{{5/4}}{{1 + 5\langle \lambda \rangle }}(3{\cos ^5}\alpha - 5{\cos ^3}\alpha + 2\cos \alpha ).\end{gather}

The limiting cases α = 0 and α = π in (2.33) recover the stresslets for no-slip and uniform-slip squirmers, respectively. In the absence of slip variations with (λ_L–λ_R) = 0, (2.32) is reduced to

(2.34)\begin{equation}{S_{ij}} = \frac{{4{\rm \pi}\mu {a^2}}}{{1 + 5\langle \lambda \rangle }}\left( {{e_i}{e_j} - \frac{{{\delta_{ij}}}}{3}} \right){B_2}.\end{equation}

Also, 〈λ〉 = 0 recovers the no-slip result driven by the B ₂ mode alone, as in (1.4) (Ishikawa & Pedley Reference Ishikawa and Pedley2007; Lauga & Michelin Reference Lauga and Michelin2016). Equation (2.34) reveals that the strength of the stresslet is diminished by slip effects. This is expected because less squirming work can be transmitted into the fluid when surface slip is present. In the full slip limit with $\langle \lambda \rangle \to \infty$, the stresslet will completely vanish because no tangential stress can be generated from the squirming motion.

In contrast, for a stick-slip squirmer, its stresslet depends not only on the B ₂ mode but also on the B ₁ mode, as indicated by (2.32). This means that the stresslet in this case can be sustained purely by a unidirectional tangential squirming through the B ₁ mode without requiring the B ₂ mode, as in the no-slip case. Recall in figure 1(a) for a no-slip squirmer that no force dipole can form with the B ₁ mode alone since the squirming force here is unidirectional and symmetric about the squirmer's equator at which the maximum squirming force occurs. On the contrary, for a stick-slip squirmer illustrated in figure 3(b), when it is subjected to unidirectional squirming on its surface through the B ₁ mode, the tangential squirming force distribution will become skewed with a stronger (weaker) squirming force on the stick (slip) side because of stick-slip disparity. This will not only produce a net squirming force to propel the squirmer but also form a stresslet, as the asymmetric squirming force distribution can be decomposed into a symmetric force dipole plus an offset unidirectional force distribution like the B ₁ mode. Note that the strength of this slip asymmetry induced stresslet is determined by the O(ε) squirming force difference between the stick and the slip faces. Also, the direction of such a stresslet is controlled by the stronger squirming force (which is on the stick side in this case). In the present case with B ₁ > 0, this induced stresslet is of contractile type (with respect to the equator of the squirmer). But the overall stresslet (2.32), by combining the stresslet generated by the B ₂ mode, can be of either extensile type or contractile type. This is determined by the sign of (2.32) through the parenthesis or through the stresslet coefficient $\mathbb{S}$ (defined in (4.1b)) to indicate the direction of the stresslet. Whether a squirmer is a puller or pusher can also be classified according to the sign of $\mathbb{S}$: $\mathbb{S}$ > 0 is a puller and $\mathbb{S}$ < 0 is a pusher, true for both homogeneous and stick-slip squirmers.

4.1. Positive stick-slip polarity (λ_R−λ_L) > 0 case

4.1.1. The case β = B ₂/B ₁ > 0

Starting with an originally no-slip puller squirmer with β = B ₂/B ₁ > 0, we look at how $\mathbb{V}$ and $\mathbb{S}$ are modified by adding a slip face on its front (right) with (λ_R−λ_L) > 0. Figure 8(a) plots how $\mathbb{V}$ varies with the partition α/π of the slip face for various values of β and the corresponding plot for $\mathbb{S}$ is shown in figure 8(b).

Figure 8.

Plots of how the swimming coefficient $\mathbb{V}$ in (4.1a) and the stresslet coefficient $\mathbb{S}$ in (4.1b) vary with the partition α/π of the slip face and β = B ₂/B ₁ for a right-slip squirmer with (λ_L, λ_R) = (0, 0.2). Panels(a) and (b) show the results with β > 0. For a small β = 0.1, $\mathbb{V}$ decreases monotonically from the no-slip value (α/π = 0) to the uniform-slip value (α/π = 1) in (a) but the squirmer can change to a pusher with $\mathbb{S}$ < 0 due to stresslet inversion by the much stronger B ₁ mode (like in figure 4a) in (b). Increasing β not only lowers $\mathbb{V}$ but also makes the squirmer return to the degraded puller state due to the backward puller action by the B ₂ mode (as in figure 5a). For a large β = 10, the backward pulling can dominate to reverse its swimming direction with $\mathbb{V}$ < 0 in (a) like a backward puller with $\mathbb{S}$ > 0 in (b). Panels (c) and (d) plot $\mathbb{V}$ and $\mathbb{S}$ with β < 0 under which the squirmer always acts as a pusher. Apparent swimming enhancement from $\mathbb{V}$ < 1 to $\mathbb{V}$ > 1 by increasing |−β| is observed in (c) while $\mathbb{V}$ > 1 comes with a diminishment of $|\mathbb{S}|\ (\mathbb{S}$ < 0) in (d). The swimming states are marked according to figure 7(a).

As revealed in figure 8(a), for a small value of β such as β = 0.1 under which the B ₁ mode is more dominating than the B ₂ mode, $\mathbb{V}$ remains positive and decreases monotonically with α from the no-slip value with α = 0 to the uniform-slip value with α = π. Hence, the stick-slip squirmer does not change the swimming direction. This squirmer swims slightly more slowly than the no-slip squirmer due to the swimming diminishment by the dominating B ₁ mode through the f ₁ term and by the drag reduction factor ${(1 + 2\langle \lambda \rangle )^{ - 1}}$ in (4.1a) due to an increase of 〈λ〉 with α according to (2.19a).

However, it is also this dominating B ₁ mode that enhances the O(ε) pusher-like stresslet (of $\mathbb{S}$ < 0) (see figure 4a) to compete with the co-existing puller stresslet (of $\mathbb{S}$ > 0) sustained by the B ₂(>0) mode. When β is small, the former can even outweigh the latter into a total stresslet inversion since the slip disparity effect on $\mathbb{S}$ through (λ_L−λ_R) < 0 in (4.1b) is greatly amplified by the ${\beta ^{ - 1}}{h_1} > 0$ term from the B ₁ mode. This explains why the $\mathbb{S}$-α profile at β = 0.1 resembles the profile of $({\lambda _L}\unicode{x2013}{\lambda _R}){\beta ^{ - 1}}{h_1}\sim{-} {\beta ^{ - 1}}{\sin ^4}\alpha$ to display a negative minimum at around α/π = 0.5 in figure 8(b). Increasing β decreases the magnitude of this stresslet minimum until $\mathbb{S}$ changes its sign to indicate a stresslet/swimmer-type inversion from a pusher of $\mathbb{S}$ < 0 to a puller of $\mathbb{S}$ > 0, corresponding to the type change when entering the β > ε/2 regime on the phase diagram in figure 7(a). While such a stresslet/swimmer type inversion effect due to the B ₁ mode becomes sluggish by raising β for diminishing the ${\beta ^{ - 1}}{h_1}$ term in (4.1b), $\mathbb{V}$ is further slowed down in figure 8(a). This velocity diminishment is due to the amplified (λ_L−λ_R)f ₂β < 0 term in (4.1a) through the O(ε) backward puller propulsion by the B ₂ mode (see figure 5a). When reaching β ∼1 where the B ₂ mode becomes comparable to the B ₁ mode, $\mathbb{V}$ starts to display a minimum at around α/π = 0.5 due to $({\lambda _L}\unicode{x2013}{\lambda _R}){f_2}\beta \sim{-} \beta {\sin ^4}\alpha$ in (4.1a). The corresponding $\mathbb{S}$ no longer changes its sign and remains positive because the co-existing B ₂-driven puller stresslet of $\mathbb{S}$ > 0 outshines the O(ε) B ₁-induced pusher stresslet of $\mathbb{S}$ < 0.

For a large enough β > 1, the f ₂β term from the B ₂ mode dominates and amplifies the backward puller propulsion effect on the swimming velocity (4.1a) to even change its sign to reverse the swimming direction, as shown in figure 8(a). Meanwhile, the diminishment of $\mathbb{S}$ by the B ₁-induced pusher effect is reduced by increasing β so that $\mathbb{S}$ remains positive to preserve the swimmer type, as shown in figure 8(b). The dominant B ₂ mode effect due to large β also explains why the $\mathbb{V}$–α profile at β = 10 in figure 8(a) resembles the profile of $({\lambda _L}\unicode{x2013}{\lambda _R}){f_2}\beta \sim{-} \beta {\sin ^4}\alpha$ to display a negative minimum at around α/π = 0.5. This corresponds to the swimming state with β > 2/ε in figure 6(a) due to counteracting propulsion, and is classified as a backward puller on the phase diagram in figure 7(a). In the regime between small and large values of β, such as 1 < β < 8, the squirmer simply swims slower than both the no-slip and uniform-slip squirmers without changing its swimming direction and swimmer type, as shown in figures 8(a) and 8(b) for $\mathbb{V}$ and $\mathbb{S}$. The squirmer in this intermediate regime behaves as a degraded puller with ε/2 < β < 2/ε on the phase diagram in figure 7(a).

As such, for a squirmer having a comparable stick-slip partition to α = π/2, we observe that $\mathbb{V}$ changes from positive to negative when β is increased while the corresponding $\mathbb{S}$ goes from negative to positive. The opposite trends of $\mathbb{V}$ and $\mathbb{S}$ with β are the reflections of the different propulsion scenarios illustrated in figure 6(a) or on the β > 0 plane of the phase diagram in figure 7(a), which are summarized below:

1. (i) small 0 < β < ε/2 gives a forward pusher with $\mathbb{V}$ > 0 and $\mathbb{S}$ < 0, where the stresslet inversion results from the dominating B ₁-induced stresslet;

2. (ii) intermediate ε/2 < β < 2/ε gives a degraded forward puller with $\mathbb{V}$ > 0 and $\mathbb{S}$ > 0, which is just a slower squirmer of the same type as the no-slip case;

3. (iii) large β > 2/ε gives a backward puller with $\mathbb{V}$ < 0 and $\mathbb{S}$ > 0, where the reverse motion is a result of the winning B ₂-induced counteracting propulsion.

4.2. Negative stick-slip polarity (λ_R−λ_L) < 0 case

We now move to the negative stick-slip polarity d_k = −e_k case when a slip face is placed on the rear (left) of an originally no-slip squirmer with (λ_R−λ_L) < 0. In contrast to the d_k = e_k case in § 4.1, the squirming forces of the B ₁ mode and the B ₂ mode on the front-stick face are stronger than those on the rear-slip face. But they may reinforce or counteract each other according to the sign of β as illustrated in figures 6(c) and 6(d) to give rich variations in the swimmer velocity and type as summarized on the phase diagram in figure 7(b). Quantitative features of $\mathbb{V}$(α′/π) and $\mathbb{S}$(α′/π) are shown in figure 9 in terms of the flipped partition angle α′ = (π − α) measured from the pole of the slip side.

Figure 9.

Plots of how the swimming coefficient $\mathbb{V}$ in (4.1a) and the stresslet coefficient $\mathbb{S}$ in (4.1b) vary with the partition (1−α/π) of the slip face and β = B ₂/B ₁ for a left-slip squirmer with (λ_L,λ_R) = (0.2,0). Panels (a) and (b) plot the results with β > 0. Panel (a) shows an apparent swimming enhancement by increasing β due to the puller action by the B ₂ mode (like in figure 5c). By increasing β, the squirmer acts as a puller when $\mathbb{V}$ < 1 or an enhanced puller when $\mathbb{V}$ > 1 in (a), along with a diminishment of $\mathbb{S}$ > 0 in (b). Panels (c) and (d) plot the results with β < 0. When|−β| is small, $\mathbb{V}$ decreases monotonically with (1−α/π) when the squirmer switches from a no-slip pusher to a puller with $\mathbb{S}$ > 0 in (d) due to stresslet inversion by the much stronger B ₁ mode (like in figure 4b). Increasing |−β| lowers $\mathbb{V}$ due to the backward pusher action by the B ₂ mode (like in figure 5d). Raising |−β| also diminishes the puller stresslet induced by the B ₁ mode so that the squirmer can return to the pusher state with $\mathbb{S}$ < 0 in (d) when |−β| is increased to a certain value. Further lowering $\mathbb{V}$ at large|−β| turns the squirmer into a backward pusher with $\mathbb{V}$ < 0 as displayed in (c). The swimming states are marked according to figure 7(b).

4.3. Symmetry relationships in $\mathbb{V}$ and $\mathbb{S}$

With the foregoing discussions on how the two modes work together under different stick-slip polarities and partitions, we detect similarity in reinforcing or counteracting propulsion. The profiles of $\mathbb{V}$ in figures 8(a) and 8(c) for d_k = e_k having the slip face on the front are identical to those displayed in figures 9(c) and 9(a) for d_k = −e_k having the slip face on the rear. The behaviour of $\mathbb{S}$ seen in figure 9(b) is an inversion of that displayed in figure 8(d) with sign change. Flipping $\mathbb{S}$ in figure 8(b) with respect to $\mathbb{S}$ = −1 leads to the profile of $\mathbb{S}$ seen in figure 9(d) with respect to $\mathbb{S}$ = 1. If all the particular features – sign change and enhancement of $\mathbb{V}$ and $\mathbb{S}$ – are marked accordingly on the β–ε phase plane in figure 7, symmetry with respect to the signs of β and the stick-slip polarities is observed. In fact, such symmetry on the β–ε phase plane can be directly derived from (4.1a) and (4.1b) in terms of the stick-slip polarity factor $\varLambda$ = |λ_R–λ_L|d_ke_k, the partition angle α, and β according to

(4.2a)\begin{gather}\mathbb{V}(\varLambda ,\alpha ,\beta ) = \mathbb{V}( - \varLambda ,{\rm \pi} - \alpha , - \beta ),\end{gather}

(4.2b)\begin{gather}\mathbb{S}(\varLambda ,\alpha ,\beta ) = -\mathbb{S}( - \varLambda ,{\rm \pi} - \alpha , - \beta ),\end{gather}

(4.2c)\begin{gather}[\mathbb{S}(1 + 5\langle \lambda \rangle ) - \textrm{sign}(\beta )](\varLambda ,\alpha ,\beta ) ={-} [\mathbb{S}(1 + 5\langle \lambda \rangle ) - \textrm{sign}( - \beta )]( - \varLambda ,{\rm \pi} - \alpha , - \beta ).\end{gather}

Equation (4.2b) can also be derived from (4.2c). These symmetry relationships hold when the stick-slip polarity and the imposed squirming actions are reversed. They are consequences of the reversibility of Stokes flow.

5.1. Squirming flow structures

In the preceding sections, we demonstrate how the combined impacts of the B ₁ mode and the B ₂ mode give rise to new features for a squirmer due to stick-slip disparity. To gain more insights into these features from a flow structure point of view, we visualize how these squirming modes individually generate and jointly modify the corresponding flow structures. To do so, we derive the squirming flow field to the O(ε) accuracy in Appendix A. The flow solution, $\boldsymbol{u} = {\boldsymbol{u}^{(0)\,}} + {\boldsymbol{u}^{(1)}}$, is derived as a leading-order flow field u⁽⁰⁾ generated by a uniform-slip squirmer with the average slip length a〈λ〉 plus an O(ε) correction flow u⁽¹⁾ arising from stick-slip disparity. Here, we consider a squirmer with an equal stick-slip partition (α = π/2) and a fixed stick-slip strength (λ_L, λ_R) = (0, 0.2) so that 〈λ〉 = 0.1. The flow field is plotted with respect to the squirmer's centre by keeping the squirmer fixed, and the front (right) always refers to as the swimming direction e_i of a no-slip squirmer with B ₁ > 0.

We first look at the flow structures driven solely by the B ₁ mode with (B ₁, B ₂) = (1, 0). Figure 10(a) plots the leading-order flow field around the uniform-slip squirmer, showing a typical point-force flow field ${\boldsymbol{u}^{(0)}}\sim \varepsilon {B_1}(a/r)$ from the right to the left. The O(ε) correction flow field caused by stick-slip disparity shown in figure 10(b) exhibits a stresslet-like flow ${\boldsymbol{u}^{(1)}}\sim{-} \varepsilon {B_1}{(a/r)^2}$, drawing the fluid towards the equator and ejecting it away from the front and rear poles. The corresponding stresslet is thus of contractile type with $\mathbb{S}$ < 0, whose sign is in accordance with (2.32) when B ₁ > 0 and B ₂ = 0, and that read in figure 8(b) with β = 0.1. Figure 10(c) is the total flow field $\boldsymbol{u} = {\boldsymbol{u}^{(0)\,}} + {\boldsymbol{u}^{(1)}}$, resembling the leading-order flow in figure 10(a). This is because this O(ε) correction stresslet field is not only weak but also decays like 1/r ², giving way to the dominating 1/r point-force flow field imposed by the B ₁ mode.

Figure 10.

Squirming flow structures around a half-faced stick-slip squirmer when subjected to unidirectional tangential squirming with (B ₁, B ₂) = (1,0). The squirmer has a slip face on the right with (λ_L, λ_R) = (0,0.2) and 〈λ〉 = 0.1. Panel (a) shows the leading-order flow with the uniform slip length a〈λ〉, displaying a typical point-force field towards the left generated by the B ₁ mode. Panel (b) displays the O(ε) correction flow field due to a stick-slip disparity truncated at the 100th spherical harmonic in (A5), displaying a symmetric stresslet flow field induced by the B ₁ mode. Panel (c) is the overall flow field by combining (a) and (b), revealing that the flow is still dominated by the point-force field in (a) because the induced stresslet field in (b) is much weaker and decays much faster. The ‘+’ and ‘−’ signs labelling the slip and the stick faces in (b) indicate the corresponding positive and negative slip variations with respect to the average value 〈λ〉.

If the flow field is generated purely by an extensile squirming due to the B ₂ mode with (B ₁, B ₂) = (0, 5), the leading-order uniform-slip result displays a typical symmetric straining flow structure in figure 11(a) due to the imposed stresslet, drawing the fluid from the poles and ejecting it at the squirmer's equator. When there is a stick-slip disparity with a slip face on the right, the asymmetric squirming force distribution along the squirmer generates a net propulsion force towards the left on the squirmer to make it a backward puller, as illustrated in figure 5(a). Hence, an O(ε) point-force correction flow is generated towards the right in figure 11(b). After superposition in figure 11(c), the total flow field reveals a clockwise swirl on the front-slip side (right), as marked by the left box in figure 11(c). A closer look is provided in figure 11(d), revealing that such a swirl is a result of the competition between the locally counterflowing stresslet field towards the left and the point-force field towards the right. While the leading-order stresslet field ${\boldsymbol{u}^{(0)}}\sim {B_2}{(a/r)^2}$ is strong near the squirmer, it decays rapidly as (a/r)² away from the squirmer. On the other hand, the correction point-force field ${\boldsymbol{u}^{(1)}}\sim \varepsilon {B_2}(a/r)$ is weak at small r but decays more slowly as a/r so that it dominates in the far field. It follows, therefore, that u⁽⁰⁾ will begin to be suppressed by u⁽¹⁾ at around r/a ∼1/ε beyond which the latter ultimately governs the far-field region, as shown in figure 11(e).

Figure 11.

Squirming flow structures around a half-faced stick-slip squirmer when subjected to a purely extensile/contractile squirming with (B ₁, B ₂) = (0,5). The squirmer has a slip face on the right with (λ_L, λ_R) = (0, 0.2) and 〈λ〉 = 0.1. Panel (a) shows the leading-order flow with the uniform slip length a〈λ〉, displaying a typical symmetric stresslet flow structure. Panel (b) shows the O(ε) correction flow field arising from a stick-slip disparity truncated at the 100th spherical harmonic in (A5), showing a point-force flow field towards the right due to the B ₂ mode. Panel (c) is the overall flow field combining (a) and (b), revealing a swirl on the right-slip face when the stresslet field is counteracted by the induced point-force field in (b), as closely examined in panel (d). The strong stresslet field decays rapidly as (a/r)² and cannot compete with the slowly decaying (a/r) point-force correction field at around r /a∼1/ε and beyond, as shown in panel (e). Here, r is the distance to the centre of the squirmer and the velocity magnitude has been enlarged by 5 times for visualization. The ‘+’ and ‘−’ signs labelling the slip and the stick faces in (b) indicate the corresponding positive and negative slip variations with respect to the average value 〈λ〉.

When both modes are present with (B ₁, B ₂) = (1, 5), the leading-order flow with uniform slip displays a skewed straining flow with a counterclockwise swirl on the rear (left) face in figure 12(a) due to a similar competition between the mode-driven flows. Here, the right-flowing skewed stresslet field from the prevailing B ₂ mode (as in figure 11a) dominates near the squirmer but decays with r, and eventually losses to the left-flowing point-force field from the B ₁ mode (as in figure 10a) to generate the swirling motion. As for the O(ε) correction field shown in figure 12(b), the dominating B ₂ mode induces a point-force field towards the right (like in figure 11b), whereas the stresslet field induced by the much weaker B ₁ mode (like in figure 10b) is almost undetectable. Such a point-force-like correction flow from the B ₂ mode interacts with the leading-order flow, assisting the lower part of the swirl on the rear (left) of the squirmer but opposing the upper part of the swirl on the front (right). The assisting action thus strengthens the swirl and it is advected further towards the equator of the squirmer in figure 12(c). This implies a weaker point force to drive the squirmer due to a stronger backward pulling action on the rear against the leading-order forward pulling on the front. This, in turn, results in a slower swimming, in accordance with the swimming diminishment with $\mathbb{V}$ < 1 at β = 5 shown in figure 8(a). Such slower swimming resulting from the backward pulling action on the rear of the squirmer is also consistent with counteracting propulsion under 1 < β < 2/ε in figure 6(a), making the squirmer swim like a degraded puller as classified in figure 7(a). In fact, the shift of the rear swirl towards the equator of the squirmer due to the weakening of the skewed straining field of u⁽⁰⁾ by the opposite forcing from u⁽¹⁾ is characteristic of a degraded puller.

Figure 12.

Squirming flow structures around a half-faced stick-slip puller squirmer with (B ₁, B ₂) = (1,5). The squirmer has a slip face on the right with (λ_L, λ_R) = (0,0.2) and 〈λ〉 = 0.1. Panel (a) shows the leading-order flow with the uniform slip length a〈λ〉, exhibiting a swirl on the left of the squirmer. Panel (b) is the O(ε) correction flow field truncated at the 100th spherical harmonic in (A5), showing a prevailing point-force field towards the right due to the stick-slip disparity through the stronger B ₂ mode. The two counter-flowing fields add up to advect the swirl towards the equator in (c) for the overall flow field. The ‘+’ and ‘−’ signs labelling the slip and the stick faces in (b) indicate the corresponding positive and negative slip variations with respect to the average value 〈λ〉.

However, if the squirmer changes to possess a contractile stresslet with (B ₁, B ₂) = (1, −5), the overall flow structure shown in figure 13(c) is intrinsically different from figure 12(c) with (B ₁, B ₂) = (1, 5). In this case, u⁽⁰⁾ with uniform slip still displays a counterclockwise swirl in figure 13(a) but on the front (right) side. The corresponding O(ε) correction field u⁽¹⁾ shown in figure 13(b) also exhibits a prevailing point-force-like flow towards the left due to the stronger negative B ₂ mode. Compared with the flow in figure 12, this unidirectional u⁽¹⁾ now interacts with the swirl of u⁽⁰⁾ in an opposite manner – opposing the lower part of the swirl on the front but assisting the upper part of the swirl and the rest – so that the swirl becomes suppressed. Such swirl suppression by a reinforced point-force field towards the left implies a stronger point force to propel the squirmer from behind, thereby leading to a swimming enhancement with $\mathbb{V}$ > 1 at β = −5 in figure 8(c) as a result of reinforcing propulsion under β < −1 in figure 6(b). In contrast to figure 12(c) for a degraded puller showing a shift of the rear swirl towards the equator, the suppression of the front swirl in u⁽⁰⁾ due to the flow promotion by u⁽¹⁾ is a trademark of an enhanced pusher as classified in figure 7(a).

Figure 13.

Squirming flow structures around a half-faced stick-slip pusher squirmer with (B ₁, B ₂) = (1, −5). The squirmer has a slip face on the right with (λ_L, λ_R) = (0,0.2) and 〈λ〉 = 0.1. Panel (a) shows the leading-order flow with the uniform slip length a〈λ〉, displaying a swirl on the right of the squirmer. Panel (b) displays the O(ε) correction flow field truncated at the 100th spherical harmonic in (A5), revealing a strong point-force field towards the left due to the stick-slip disparity through the dominating B ₂ mode. Panel (c) is the overall flow field, revealing a suppression of the swirl by the induced point-force field in (b) when the two modes reinforce each other on the left-stick face. The ‘+’ and ‘−’ signs labelling the slip and the stick faces in (b) indicate the corresponding positive and negative slip variations with respect to the average value 〈λ〉.

5.2. Swimming power and swimming efficiency

As a stick-slip squirmer can swim faster or slower than the no-slip one, we study whether the squirmer spends more or less power in its swimming under a specific swimming scenario. This requires the knowledge of the overall flow field when the squirmer is in motion at U_i. In addition to the squirming flow field ${u^{\prime}_i}$ derived in Appendix A for a squirmer held fixed, we need to further include the flow field due to a constant squirmer motion. This translation-induced flow, $u_i^T$, is solved in Appendix B as if there were a uniform flow passing a fixed squirmer at −U_i. The two are superposed to give the desired total flow ${v_i} = {u^{\prime}_i} + u_i^T$ around the squirmer, where $u_i^T$ is obtained by selecting U_i to eliminate the 1/r point force Stokeslet contribution to ensure force free on the squirmer.

The swimming power is defined as the rate of viscous work dissipated by the squirmer. For a purely tangential squirming motion where the radial velocity $v_r=0$ on the squirmer's surface at r = a, the swimming power can be calculated according to (Blake Reference Blake1971; Pak & Lauga Reference Pak and Lauga2014)

(5.1)\begin{equation}\mathrm{{\mathcal P}} ={-} \int_{{S_p}} {{n_i}{\sigma _{ij}}{v_j}\,\textrm{d}S} = -2{\rm \pi}{a^2}\int_0^{\rm \pi} {{{({\sigma _{r\theta }}{v_\theta })}_{r = a}}\sin \theta \,\textrm{d}\theta } .\end{equation}

In the above, only the tangential stress $\sigma_{r\theta}=\mu r \partial_r(v_\theta/r)$ contributes and not the normal stress because there is no radial fluid motion on the squirmer's surface. Using Lamb's general solution for the overall field $v_i$ with $v_r\ (r=a)=0$, we show in Appendix E that (5.1) takes the following form:

(5.2) \begin{equation}\mathrm{{\mathcal P}} = {\rm \pi} a\mu \Bigg[ {48\frac{{\hat{B}_{01}^2}}{{{a^6}}} + 24\frac{{\hat{B}_{02}^2}}{{{a^8}}} + 16\sum\limits_{n = 3}^\infty {\left( {\frac{{n + 1}}{n}} \right)\frac{{\hat{B}_{0n}^2}}{{{a^{2(n + 2)}}}}} } \Bigg].\end{equation}

These coefficients ${\hat{B}_{0n}} = \hat{B}_{0n}^{(0)} + \hat{B}_{0n}^{(1)} + O({\varepsilon ^2})$ are the contributions from the uniform-slip part $\hat{B}_{0n}^{(0)}$ plus those from the O(ε) slip variation part $\hat{B}_{0n}^{(1)}$. Noticing that $\hat{B}_{0n}^{(0)}(n \ge 3) = O(\varepsilon )$ contribute to O(ε ²) in (5.2), we may truncate (5.2) to n = 2 to approximate $\mathcal{P}$ as the uniform-slip contribution, $\mathcal{P}$⁽⁰⁾, plus the O(ε) correction from slip disparity, $\mathcal{P}$⁽¹⁾, as

(5.3a)\begin{gather}{\mathcal P} = {{\mathcal P}^{(0)}} + {{\mathcal P}^{(1)}} + O({\varepsilon ^2}),\end{gather}

(5.3b)\begin{gather}{{\mathcal P}^{(0)}} = 24{\rm \pi}\mu a\Bigg[ {2{{\left( {\frac{{\hat{B}_{01}^{(0)}}}{{{a^3}}}} \right)}^2} + {{\left( {\frac{{\hat{B}_{02}^{(0)}}}{{{a^4}}}} \right)}^2}} \Bigg],\end{gather}

(5.3c)\begin{gather}{{\mathcal P}^{(1)}} = 48{\rm \pi}\mu a\left[ {2\left( {\frac{{\hat{B}_{01}^{(0)}}}{{{a^3}}}} \right)\left( {\frac{{\hat{B}_{01}^{(1)}}}{{{a^3}}}} \right) + \left( {\frac{{\hat{B}_{02}^{(0)}}}{{{a^4}}}} \right)\left( {\frac{{\hat{B}_{02}^{(1)}}}{{{a^4}}}} \right)} \right],\end{gather}

where

(5.4a)\begin{gather}\frac{{\hat{B}_{01}^{(0)}}}{{{a^3}}} ={-} \frac{{{B_1}}}{{3(1 + 2\langle \lambda \rangle )}},\end{gather}

(5.4b)\begin{gather}\frac{{\hat{B}_{02}^{(0)}}}{{{a^4}}} ={-} \frac{{{B_2}}}{{3(1 + 5\langle \lambda \rangle )}},\end{gather}

(5.4c)\begin{gather}\frac{{\hat{B}_{01}^{(1)}}}{{{a^3}}} ={-} \frac{{{B_1}}}{{5(1 + 2\langle \lambda \rangle )}}{{\mathsf{g}}_2} + \frac{{{B_2}}}{{2(1 + 5\langle \lambda \rangle )}}\left[ {{{\mathsf{g}}_1} - \frac{3}{7}{{\mathsf{g}}_3}} \right],\end{gather}

(5.4d)\begin{gather}\frac{{\hat{B}_{02}^{(1)}}}{{{a^4}}} ={-} \frac{{(2/3){B_1}}}{{(1 + 2\langle \lambda \rangle )}}\left[ {{{\mathsf{g}}_1} - \frac{3}{7}{{\mathsf{g}}_3}} \right] + \frac{{(5/21){B_2}}}{{(1 + 5\langle \lambda \rangle )}}\left[ {{{\mathsf{g}}_2} - \frac{4}{3}{{\mathsf{g}}_4}} \right],\end{gather}

(5.4e)\begin{gather}{{\mathsf{g}}_n}(n \ge 1) = {\textstyle{1 \over 2}}({\lambda _L} - {\lambda _R})[{P_{n + 1}}(\cos \alpha ) - {P_{n - 1}}(\cos \alpha )].\end{gather}

The ${{\mathsf{g}}_n}$ values are the coefficients given by (A7b) in the Legendre expansion of the slip variation (A7a) and hence only appear in $\hat{B}_{0n}^{(1)}$.

For the uniform-slip part at leading order, the swimming power (5.3b) is

(5.5)\begin{equation}{{\mathcal P}^{(0)}} = \frac{{16{\rm \pi}}}{3}\frac{{\mu aB_1^2}}{{{{(1 + 2\langle \lambda \rangle )}^2}}}\Bigg[ {1 + \frac{{{\beta^2}}}{2}{{\left( {\frac{{1 + 2\langle \lambda \rangle }}{{1 + 5\langle \lambda \rangle }}} \right)}^2}} \Bigg].\end{equation}

When 〈λ〉 = 0, (5.5) is reduced to the no-slip result obtained by Blake (Reference Blake1971) but a non-zero 〈λ〉 always decreases $\mathcal{P}$⁽⁰⁾ from the no-slip value. Since 〈λ〉 increases with the portion of the slip face according to (2.19a), $\mathcal{P}$⁽⁰⁾ decreases with the slip partition α/π as shown in figure 14 for a right-slip squirmer. Other than the slip portion effect, figure 14 also depicts that $\mathcal{P}$⁽⁰⁾ increases with β ² due to the B ₂ mode in addition to the swimming power $(16{\rm \pi}/3){(1 + 2\langle \lambda \rangle )^{ - 2}}\mu aB_1^2$ for a neutral uniform-slip squirmer without the B ₂ mode. This might explain why a squirmer-like microorganism such as Volvox carteri typically swims with β < 1 (Lauga Reference Lauga2020) to reduce the swimming power. Also, because of this β ² dependence, $\mathcal{P}$⁽⁰⁾ is invariant when β changes sign so that both puller and pusher squirmers having the same uniform slip length consume an identical amount of swimming power under the same value of |β|.

Figure 14.

Plot of the leading-order swimming power $\mathcal{P}$⁽⁰⁾ described by (5.5) against the partition α/π of the slip face for a stick-slip squirmer with (λ_L, λ_R) = (0,0.2) and the average value 〈λ〉 varying with α according to (2.19a). For a given value of β = B ₂/B ₁, $\mathcal{P}$⁽⁰⁾ typically decreases monotonically with α from the no-slip value at α = 0 to the uniform-slip value at α = π. Increasing β increases the power.

Differences in the swimming powers between puller (β > 0) and pusher (β < 0) squirmers are reflected by the swimming power correction $\mathcal{P}$⁽¹⁾ arising from stick-slip disparity. Judging from (5.3c), $\mathcal{P}$⁽¹⁾ < 0 is possible due to the counteracting B ₁ and B ₂ modes in (5.4c) and (5.4d). For this reason, a squirmer may exploit a stick-slip disparity to save energy with a properly selected slip partition in some ranges of α bounded by particular values of α determined by $\mathcal{P}$⁽¹⁾(α) = 0.

Figure 15 plots how the overall swimming power $\mathcal{P}$ described by (5.3a) varies with the partition α/π of the slip face for both puller and pusher squirmers when including the contribution (5.3c) from $\mathcal{P}$⁽¹⁾. We first consider a squirmer whose slip face is on the right. In comparison with an originally no-slip puller with α = 0 and β > 0 in figure 15(a), adding a slip face on the front can raise the power consumption with a small α to a local maximum at α/π ≈ 0.2, after which a further increase of α can reduce $\mathcal{P}$ to a local minimum at α/π ≈ 0.5. Increasing β raises the power consumption as the squirmer changes from a pusher for a small β to a degraded puller with a moderate value of β, and eventually becomes a backward puller when β is large. These results imply that adding a front-slip face of α/π ≈ 0.2 to 0.5 to a no-slip puller may save the power most to boost its swimming efficiency, as will also be shown later. On the other hand, if one wishes to keep the swimming power as low as possible, β will need to be taken as small as possible. The smallness of β limits the power undulation with α and does not show much difference from the no-slip case.

Figure 15.

Dependence of the total swimming power $\mathcal{P}$ described by (5.3) on the partition of the slip face for a right-slip squirmer with (λ_L, λ_R) = (0,0.2) in (a) and (b), and for a left-slip squirmer with (λ_L, λ_R) = (0.2,0) in (c) and (d). For the original puller with β = B ₂/B ₁ > 0 shown in (a), $\mathcal{P}$ can exhibit a minimum at around α/π = 0.5, lower than the no-slip and uniform-slip values. This minimum power increases with rising β as the squirmer changes from a pusher to a degraded puller, and then converts to a backward puller. For the original pusher case with β < 0 in (b), while increasing |−β| increases $\mathcal{P}$ by changing the squirmer from a pusher to an enhanced pusher, the squirmer can spend less power than the no-slip and uniform-slip squirmers to exhibit a minimum swimming power at around α/π = 0.8. Panels (c) and (d) plot the left-slip case, displaying the features identical to those in (b) and (a), respectively. In (c) for β > 0, $\mathcal{P}$ of a puller or an enhanced puller is typically the lowest at around (1−α/π) = 0.8. In (d) for β < 0 under which the squirmer can be a puller, a degraded puller, or a backward pusher, $\mathcal{P}$ can be the lowest at around (1−α/π) = 0.5.

Figure 15(b) shows the swimming power curves for an originally no-slip pusher with β < 0 after adding a slip face on the right. We also observe an undulation of $\mathcal{P}$(α/π) but in a somewhat opposite increase/decrease trend to the β > 0 case seen in figure 15(a); $\mathcal{P}$ still increases with |−β| as the squirmer gradually changes from a regular pusher to an enhanced pusher. Similar to the result with β > 0, the undulation of $\mathcal{P}$ with α is minute when the squirmer swims like a pusher when |−β| is small. Increasing |−β| to a moderate value such as β = −1, $\mathcal{P}$ is found to decrease with α from the no-slip value to a minimum at around α/π = 0.25 and peaks at around α/π = 0.5 to a value slightly below the no-slip value. Further increasing |−β| to a value such as β = −3 or beyond changes the stick-slip squirmer to an enhanced pusher whose swimming power is always lower than the maximum power expended by the no-slip pusher with α = 0. However, the partition that creates the minimum power is around α/π = 0.75 for small |−β| and slightly above for large |−β|.

Figures 15(c) and 15(d) plot the case when the stick-slip director is flipped with the slip face on the rear (left) of a squirmer. Here, we plot the swimming power against the partition of the slip face, (1 − α/π). The curves in figure 15(c) for β > 0 match those in figure 15(b). The differences in the swimming power between the stick-slip squirmer and the no-slip squirmer start to become apparent at β = 1 or larger when the stick-slip squirmer acts like a regular puller or towards an enhanced puller. In this case, the swimming power is always lower than the maximum swimming power occurring to the no-slip puller. Similarly, the partition for the lowest swimming power is again found to be around (1 − α/π) = 0.75 and slightly above, as shown in figure 15(c).

Figure 15(d) displays the power consumptions for the originally no-slip pusher (β < 0) case, showing that the maximum and minimum swimming powers come about at around (1 − α/π) = 0.2 and (1 − α/π) = 0.5, respectively, identical to the trends in figure 15(a). When the squirmer acts as a degraded pusher or a pusher when |−β| is greater than a certain value like β = −1, it can save the power most with the slip partition of around (1−α/π) = 0.5. However, if |−β| is below that value, the squirmer will become a puller due to stresslet inversion (see figure 9d) and its swimming power will not have much difference than that of the no-slip pusher.

Having obtained the swimming power and understood its behaviours above, we put forth to derive the swimming efficiency and analyse how it behaves. This efficiency is defined as the ratio of the rate of work required to drag the squirmer to the swimming power that produces the same swimming velocity (Lighthill Reference Lighthill1952):

(5.6)\begin{equation}\zeta ={-} {F_i}{U_i}/{\mathcal P}.\end{equation}

Here, the drag force F_i and the swimming velocity U_i are given by (2.15) and (2.20), respectively, and both contain O(ε) slip variation corrections. Using (2.15) together with (2.20) and writing ${U_i} = U_i^{(0)} + U_i^{(1)} + O({\varepsilon ^2})$, the rate of work in (5.6) is

(5.7)\begin{equation}- {F_i}{U_i} = 6{\rm \pi}\mu a|U_i^{(0)}{|^{\textrm{ }2}}\left( {\frac{{1 + 2\langle \lambda \rangle }}{{1 + 3\langle \lambda \rangle }}} \right)\textrm{ }\Bigg[ {1 + \frac{{\mathcal Q}}{{\textrm{(}1 + 2\langle \lambda \rangle \textrm{)(}1 + 3\langle \lambda \rangle \textrm{)}}} + 2\frac{{U_i^{(1)}}}{{U_i^{(0)}}}} \Bigg] + O({\varepsilon ^2}),\end{equation}

where $U_i^{(0)}$ is given by (2.22),$U_i^{(1)}$ is the O(ε) slip variation correction in (2.20) and $\mathcal{Q}$ is the O(ε) slip correction to the drag force due to the surface quadrupole given by (2.19b). Combining (5.3) and collecting the terms to O(ε), (5.6) is reduced to

(5.8)\begin{equation}\zeta = {\zeta _0}\Bigg\{ {1 + \Bigg[ {\frac{{\mathcal Q}}{{\textrm{(}1 + 2\langle \lambda \rangle \textrm{)(}1 + 3\langle \lambda \rangle \textrm{)}}} + 2\frac{{U_i^{(1)}}}{{U_i^{(0)}}} - \frac{{{{\mathcal P}^{\textrm{(1)}}}}}{{{{\mathcal P}^{\textrm{(0)}}}}}} \Bigg]} \Bigg\} + O({\varepsilon ^2}),\end{equation}

where ζ₀ is the swimming efficiency for the uniform-slip case and given by

(5.9)\begin{equation}{\zeta _0} = \frac{1}{2}\left( {\frac{{1 + 2\langle \lambda \rangle }}{{1 + 3\langle \lambda \rangle }}} \right){\Bigg[ {1 + \frac{1}{2}{\beta^2}{{\left( {\frac{{1 + 2\langle \lambda \rangle }}{{1 + 5\langle \lambda \rangle }}} \right)}^2}} \Bigg]^{ - 1}}.\end{equation}

Not surprisingly, the presence of slip can cause non-trivial impacts on the swimming efficiency without and with stick-slip disparity. We first plot ζ ₀ against 〈λ〉 for different values of β in figure 16(a), showing that ζ ₀ at a given 〈λ〉 in general declines with increasing β as prescribed by (5.9). However, how ζ ₀ varies with the average dimensionless slip length 〈λ〉 is not monotonic. This is because the rate of the dissipation work −F_iU_i∝(1 + 2〈λ〉)/(1 + 3〈λ〉) decreases with 〈λ〉 at a rate different than that of the swimming power ${\mathcal P} \propto 1 + ({\beta ^2}/2){[(1 + 2\langle \lambda \rangle )/(1 + 5\langle \lambda \rangle )]^2}$. This makes ζ ₀ increase or decrease with 〈λ〉, depending on the range of 〈λ〉 and the value of β.

Figure 16.

Plot of the leading-order swimming efficiency ζ ₀ against the average dimensionless slip length 〈λ〉 according to (5.9). ζ ₀ can decrease with 〈λ〉 for β < 0.7, display a maximum in the range of 〈λ〉 for β = 0.7–1.2 shown in (b), or increase with 〈λ〉 for β > 1.2. For each of these cases, ζ ₀ approaches a constant as 〈λ〉→∞.

With a closer look at figure 16(b), ζ ₀ decreases with 〈λ〉 and saturates to a constant plateau whose value decreases with rising β when β < 0.7. For β = 0.7−1.2, however, ζ ₀ can exhibit a maximum at a specific 〈λ〉 that shifts to a larger value of 〈λ〉 (≤0.25) as β is increased, as shown in figure 16(b). For β > 1.5, ζ ₀ grows monotonically with 〈λ〉 for small 〈λ〉 and levels off when 〈λ〉 is large, as displayed in both panels in figure 16.

Unlike a uniform-slip squirmer, how the swimming efficiency ζ described by (5.8) behaves for a stick-slip squirmer further depends on the type of squirmer, the stick-slip polarity and the slip partition, as shown in figure 17. Figure 17(a) plots how ζ varies with the partition α/π of the slip face on the right of a squirmer with β > 0. When β is small, such as β = 0.1, the originally no-slip puller can change to a right-slip pusher due to stresslet inversion (figure 8b) and its swimming efficiency decreases monotonically with α/π. Increasing β to a moderate value like β = 0.5–2 at the degraded puller state, ζ first decreases with α/π and then is boosted to a local maximum at some value of α/π between 0.25 and 0.5. Such local efficiency maximum seems to be reminiscent of the minimum swimming power observed in figure 15(a). A closer examination of $\mathcal{P}$ and $\mathcal{P}$⁽⁰⁾ with β = 1 around α/π = 0.5 in figures 15(a) and 14 reveals that this minimum total $\mathcal{P}$ is smaller than the leading-order $\mathcal{P}$⁽⁰⁾ and hence we must encounter an energy-saving $\mathcal{P}$⁽¹⁾ < 0 by the stick-slip disparity. This in turn boots the swimming efficiency to give a greater power-saving ratio −$\mathcal{P}$⁽¹⁾/$\mathcal{P}$⁽⁰⁾(>0) in (5.8). However, such an energy-saving swimming strategy quickly loses its advantage when β is further increased to make ζ even lower. At a sufficiently large β, such as β = 4, the swimming efficiency can become the lowest at around α/π = 0.5. This is likely due to the much stronger efficiency depression by the swimming diminishment $U_i^{(1)}/U_i^{(0)} < 0$ in (5.8) in view of figure 8(a).

Figure 17.

Plot of the swimming efficiency ζ described by (5.8) against the partition of the slip face for a right-slip squirmer with (λ_L, λ_R) = (0,0.2) in (a) and (b), and for a left-slip squirmer with (λ_L, λ_R) = (0.2, 0) in (c) and (d). For the original puller with β = B ₂/B ₁ > 0 in (a), adding a right-slip face makes ζ lower than the no-slip value. While ζ can display a local maximum in a range of α/π for β = 0.5–2, a sufficiently large β such as β = 4 can reduce ζ to the lowest at around α/π = 0.5. For the original puller case with β < 0 in (b), the maximum efficiency occurs at around α/π = 0.25 for a puller with |−β| < 1. Interestingly, near α/π = 0.25, the swimming efficiency at β  = −0.5 is higher than that at β = −0.1. Increasing |−β| towards a large value at the enhanced pusher state shifts the maximum efficiency towards α/π = 0.5. Panels (c) and (d) present the left-slip case, displaying the features the same as those of (b) and (a), respectively. In (c) for β > 0, a stick-slip puller generally swims most efficiently with the corresponding slip partition (1−α/π) shifting from 0.25 to 0.5 as β is increased towards a large value at the enhanced puller state. In (d) for β < 0, while ζ with |−β| = 0.5–2 can display a local maximum in a range of (1−α/π), it becomes the lowest at around (1−α/π) = 0.5 when |−β| becomes sufficiently large like β = −4.

Figure 17(b) shows the calculated swimming efficiency behaviour for the right-slip squirmer with β < 0. A maximum efficiency is found to occur at around α/π = 0.25 for a pusher with |−β| < 1 (according to figure 8d) except for β = −0.1 for which ζ decreases monotonically with α/π. Interestingly, near α/π = 0.25, the swimming efficiency at β = −0.5 is higher than that at β = −0.1, against the general trend that ζ is typically lowered by increasing |β| due to an increase of the swimming power $\mathcal{P}$. Increasing |−β| towards a large value at the enhanced pusher state makes the occurrence of the maximum efficiency shift towards α/π = 0.5. It is interesting to note that an enhanced pusher with a large |−β| like β = −1 displays a local power maximum at around α/π = 0.5, as shown in figure 15(b). Judging from figures 14 and 15(b), the fact that $\mathcal{P}$ is much larger than $\mathcal{P}$⁽⁰⁾ should suggest a large power wasting ratio −$\mathcal{P}$⁽¹⁾/$\mathcal{P}$⁽⁰⁾ < 0 in (5.8) to depreciate the efficiency. The reason why the corresponding swimming efficiency is the highest in the range of α/π probably results from the much stronger efficiency elevation by the swimming enhancement $U_i^{(1)}/U_i^{(0)} > 0$ in (5.8) in view of figure 8(c).

Figures 17(c) and 17(d) are plotted for the negative stick-slip polarity case with the slip face on the left of a squirmer for both β > 0 and β < 0. Here we use (1−α/π) to present the effects of the slip partition on the swimming efficiency. Figure 17(c) plots the β > 0 case for the squirmer being either a regular puller or an enhanced puller (according to figure 9a). The results look exactly identical to those plotted in figure 17(b) for the right-slip pusher squirmer with β < 0. That is, the most efficient swimming occurs at around (1−α/π) = 0.25 for a puller with β < 2 (except for a very small β such as β = 0.1), and the maximum efficiency at β = 0.5 can be made higher than that at β = 0.1. For an enhanced puller when β is large, such as β = 10, it swims most efficiently with the slip partition around (1−α/π) = 0.5. This maximum swimming efficiency likely results from the strong swimming enhancement $U_i^{(1)}/U_i^{(0)} > 0$ in (5.8), as suggested by figure 9(a).

As for the β < 0 case shown in figure 17(d), we find that a left-slip pusher squirmer is generally less efficient than a no-slip pusher whose swimming is the most efficient. While the swimming efficiency can exhibit a local maximum in a range of (1−α/π) for |−β| = 0.5–2, it can drop to the lowest at around (1−α/π) = 0.5 when driven at a sufficiently large |−β|, such as β = −4. These features are the reflections of the competition between the efficiency elevation by the power saving −$\mathcal{P}$⁽¹⁾/$\mathcal{P}$⁽⁰⁾ > 0 (from figure 15d) and the efficiency depression by the swimming diminishment $U_i^{(1)}/U_i^{(0)} < 0$ (from figure 9c) in (5.8), similar to the mechanisms discussed before figure 17(a) for the right-slip squirmer with β > 0.

Once again, it is not coincident to observe identical swimming efficiency trends found between figures 17(a) and 17(d) as well as those between figures 17(b) and 17(c) when reversing the stick-slip polarity and the sign of β. Such similar pairing can also be found in the corresponding swimming power variations between figures 16(a) and 16(d) and between figure 16(b) and 16(c). These similarities in swimming performance are closely connected to the swimmer type and the stick-slip polarity: a right-slip puller and a left-slip pusher swim in a similar power spending fashion, so do a right-slip pusher and a left-slip puller. In fact, such similarities can be tracked back to the resemblances in the associated swimming velocity and stresslet behaviours in figures 8 and 9, owing to the symmetries between the swimming states in figure 7(a) and those in figure 7(b) by conforming to the kinematic reversibility of Stokes flow.