2.1. Evaluation of the diffusio-osmotic slip

For an asymptotically thin interaction region ($\epsilon \ll 1$), we expand the field variables ($\boldsymbol {u},c, p, \psi$) as

(2.7)\begin{equation} f(x,y)=f^{(0)}(x,y)+\epsilon f^{(1)}(x,y)+\cdots. \end{equation}

Following Michelin & Lauga (Reference Michelin and Lauga2014), we divide the domain into an ‘inner’ ($0 \ll y \ll 1$) and ‘outer’ region ($y \gg 1$). We use MAE to replace the solution of the inner region with coarse-grained boundary conditions for the outer region.

2.1.1. Outer region

Neglecting the rapidly decaying interaction potential in the outer region, i.e. $\psi =0$ (Sharifi-Mood et al. Reference Sharifi-Mood, Koplik and Maldarelli2013; Michelin & Lauga Reference Michelin and Lauga2014), the leading-order equations read

(2.8a)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}^{(0)} = 0, \quad - \boldsymbol{\nabla} p^{(0)} + {\nabla}^{2} \boldsymbol{u}^{(0)} + De \, \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\mathsf{S}}^{(0)} = \boldsymbol{0}, \end{gather}

(2.8b)\begin{gather} Pe(\boldsymbol{u}^{(0)}\boldsymbol{\cdot} \boldsymbol{\nabla} c^{(0)})= {\nabla}^{2}c^{(0)}, \end{gather}

subject to the following boundary conditions:

(2.9a)\begin{gather} \boldsymbol{u}^{(0)} \rightarrow \boldsymbol{0} , \; p^{(0)} \rightarrow 0 \quad \text{and}\quad c^{(0)} \rightarrow 0 \quad \text{as } y \rightarrow \infty, \end{gather}

(2.9b)\begin{gather} \boldsymbol{n}\boldsymbol{\cdot} \boldsymbol{\nabla} c^{(0)} = \mathcal{K}(x) \quad \text{at } y \rightarrow 0. \end{gather}

The concentration boundary condition (2.9b) has been shown to be well-posed, provided $\epsilon Pe\ll 1$ (Michelin & Lauga Reference Michelin and Lauga2014, p. 580). To obtain the macroscale boundary condition for velocity at the surface (i.e. $y \rightarrow 0$), the outer solution must be matched with the inner solution.

2.1.2. Inner region

We first rescale the variables to derive the equations in the inner layer. The variables in the inner layer are defined as $\hat {f}$. In the thin interaction layer limit ($\epsilon \ll 1$), using boundary layer principles, we define the characteristic scales for $y^{*}$-direction, vertical velocity ($v^{*}$) and pressure ($p^{*}$) as $\lambda _{I}^{*}$, $\epsilon \, U_{ch}$ and $p_{ch} \, \epsilon ^{-2}$, respectively. Using (2.7) and (2.2), we obtain the leading-order equations in the inner region as

(2.10a)\begin{gather} \frac{\partial \hat{u}^{(0)}}{\partial \hat{x}} + \frac{\partial \hat{v}^{(0)}}{\partial \hat{y}} =0 , \end{gather}

(2.10b)\begin{gather} - \frac{\partial \hat{p}^{(0)}}{\partial \hat{x}} + \frac{\partial^2 \hat{u}^{(0)}}{\partial \hat{y}^2} +De \left( \frac{\partial \hat{S}_{xx}^{(0)}}{\partial \hat{x}} + \frac{\partial \hat{S}_{yx}^{(0)}}{\partial \hat{y}} \right) = (\hat{c}^{(0)} + C_{\infty})\frac{\partial \hat{\psi}^{(0)}}{\partial \hat{x}} , \end{gather}

(2.10c)\begin{gather} - \frac{\partial \hat{p}^{(0)}}{\partial \hat{y}} + De \left( \frac{\partial \hat{S}_{yy}^{(0)}}{\partial \hat{y}} \right) = (\hat{c}^{(0)}+C_{\infty}) \frac{\partial \hat{\psi}^{(0)}}{\partial \hat{y}} , \end{gather}

(2.10d)\begin{gather} \epsilon^{2}Pe \left( \hat{u}^{(0)} \frac{\partial \hat{c}^{(0)}}{\partial \hat{x}} + \hat{v}^{(0)} \frac{\partial \hat{c}^{(0)}}{\partial \hat{y}} \right) = \frac{\partial}{\partial \hat{y}} \left( \frac{\partial \hat{c}^{(0)}}{\partial \hat{y}} + (\hat{c}^{(0)}+C_{\infty}) \frac{\partial \hat{\psi}^{(0)}}{\partial \hat{y}} \right) , \end{gather}

subject to the following surface boundary conditions at the leading order:

(2.11)\begin{equation} \left. \boldsymbol{u}^{(0)} \right\vert_{\hat{y}=0} = \boldsymbol{0} \quad \text{and}\quad \left( \left.\frac{\partial \hat{c}^{(0)}}{\partial \hat{y}}\right\vert_{\hat{y}=0} +\left.(\hat{c}^{(0)}+C_{\infty}) \frac{\partial \psi}{\partial \hat{y}}\right\vert_{\hat{y}=0} \right) =0. \end{equation}

For low to moderate advective effects (i.e. $Pe \ll \epsilon ^{-2}$), we can neglect the left-hand side of (2.10d). This decouples the solute concentration field from the hydrodynamics. The pressure scaling in the inner region suggests a decay condition as $\hat {y} \rightarrow \infty$ (Michelin & Lauga Reference Michelin and Lauga2014, p. 579). The components of the polymeric stress tensor ($\boldsymbol{\mathsf{S}}$) are rescaled as

(2.12a)\begin{gather} {\mathsf S}_{xx} = \frac{\hat{{\mathsf S}}_{xx}}{\epsilon^{2}} =\frac{1}{ \epsilon^{2}}\left( \frac{\partial \hat{u}}{\partial \hat{y}} \right)^{2},\quad {\mathsf S}_{yy} =\frac{\hat{{\mathsf S}}_{yy}}{\epsilon^{2}} =\frac{1}{\epsilon^{2}} \left( 1 +2 \delta \right) \left( \frac{\partial \hat{u}}{\partial \hat{y}} \right)^{2}, \end{gather}

(2.12b)\begin{gather} {\mathsf S}_{xy}={\mathsf S}_{yx} = \dfrac{\hat{{\mathsf S}}_{xy}}{\epsilon} =\dfrac{\hat{{\mathsf S}}_{yx}}{\epsilon} =\dfrac{2 \delta}{ \epsilon}\left( \dfrac{\partial \hat{u}}{\partial \hat{y}} \dfrac{\partial \hat{u}}{\partial \hat{x}} + \dfrac{\hat{v} \dfrac{\partial^2 \hat{u}}{\partial \hat{y}^2} + \hat{u} \dfrac{\partial^2 \hat{u}}{\partial \hat{y} \partial \hat{x}} }{2} \right). \end{gather}

We now perform a perturbation expansion in Deborah number ($De$), i.e. accounting for weakly nonlinear viscoelastic effects such that $\epsilon \ll De \ll 1$. The field variables for each term in (2.7) can be further expanded as

(2.13)\begin{equation} f^{(i)}(x,y)=f^{(i)}_{0}(x,y)+De \, f^{(i)}_{1}(x,y)+\cdots. \end{equation}

Here, $f$ represents the velocity and pressure field. The concentration field is not expanded in $De$ as it is decoupled from velocity field in the inner region.

${O}$($De^{0}$) solution: Since our objective is to obtain a first-order correction to the diffusio-osmotic slip, for convenience, we temporarily drop the superscript $(0)$ from all the variables. The solution to leading-order (i.e. $O$($De^{0}$)) governing equations is obtained as

(2.14a)\begin{equation} \hat{c}(\hat{x},\hat{y})=\mathcal{I}(\hat{x}) {\rm e}^{-\hat{\psi}(\hat{x},\hat{y})} - C_{\infty} , \quad \hat{p}_{0}(\hat{x},\hat{y})=\mathcal{I}(\hat{x}) \left( {\rm e}^{-\hat{\psi}(\hat{x},\hat{y})} -1 \right), \end{equation}

and

(2.14b)\begin{equation} \hat{u}_{0}(\hat{x},\hat{y})=-\mathcal{I}'(\hat{x}) \int_{0}^{\hat{y}} \int_{t}^{\infty} \left( {\rm e}^{-\hat{\psi}(\hat{x},s)} -1 \right) {\textrm{d}}s \, {\textrm{d}}t + \mathcal{J}_{0}(\hat{x})\,\hat{y}. \end{equation}

Here, $\mathcal {I}(\hat {x})$ and $\mathcal {J}_{0}(\hat {x})$ are to be determined through matching, and $\mathcal {I}'(\hat {x})$ represents ${\textrm {d}} \mathcal {I}/{\textrm {d}} {\hat {x}}$. The leading-order velocity field ($\hat {u}_{0}$) is represented in a form different from that reported by Anderson et al. (Reference Anderson, Lowell and Prieve1982) and Michelin & Lauga (Reference Michelin and Lauga2014), as it helps in the evaluation of the higher-order velocity field. It can be seen in figure 2(a) that this expression is equivalent to that provided in the literature.

Figure 2.

(a) Velocity field of Newtonian fluid inside the interaction layer (for $\Phi _{0}=-1$). (b) Variation of the modification to the velocity field. The results are for an exponentially decaying solute-surface interaction ($\hat {\psi }(\xi )=\Phi _{0} {\rm e}^{-\xi }$), where a negative $\Phi _{0}$ represents an attractive interaction. Here, $c_{x}$ and $c_{xx}$ are the concentration gradients at the outer edge of the inner region i.e. $c_{x}\vert _{y=0}$ and $c_{xx}\vert _{y=0}$.

${O}$($De^{1}$) solution: For ease of calculation, we assume the interaction potential ($\hat {\psi }$) to be independent of the tangential direction ($x$). The flow field at ${O}(De)$ is governed by

(2.15a)\begin{gather} \frac{\partial \hat{u}_{1}}{\partial \hat{x}} + \frac{\partial \hat{v}_{1}}{\partial \hat{y}} =0 , \end{gather}

(2.15b)\begin{gather} - \frac{\partial \hat{p}_{1}}{\partial \hat{x}} + \frac{\partial^2 \hat{u}_{1}}{\partial \hat{y}^2} + \frac{\partial \hat{S}_{xx \, 0}}{\partial \hat{x}} + \frac{\partial \hat{S}_{xy \, 0}}{\partial \hat{y}} = 0 , \end{gather}

(2.15c)\begin{gather} - \frac{\partial \hat{p}_{1}}{\partial \hat{y}} + \frac{\partial \hat{S}_{yy \, 0}}{\partial \hat{y}} = 0. \end{gather}

Using the pressure decay condition ($\hat {p} \rightarrow 0$ as $\hat {y} \rightarrow \infty$), the solution to (2.15c) yields $\hat {p}_{1}=\hat {S}_{yy \, 0} - \mathcal {J}_{0}(\hat {x})^{2}(1+2\delta )$. Simplifying (2.15b), we obtain

(2.16)\begin{align} \frac{\partial^{2} \hat{u}_{1}}{\partial \hat{y}^{2}} = - \delta \left\{ -\frac{\partial \hat{u}_{0}}{\partial \hat{y}} \frac{\partial^{2} \hat{u}_{0}}{\partial \hat{y} \partial \hat{x}} + \frac{\partial \hat{u}_{0}}{\partial \hat{x}} \frac{\partial^{2} \hat{u}_{0}}{\partial \hat{y}^{2}} + \hat{v}_{0} \frac{\partial^{3} \hat{u}_{0}}{\partial \hat{y}^{3}} + \hat{u}_{0} \frac{\partial^{3} \hat{u}_{0}}{\partial \hat{x} \partial \hat{y}^{2}} \right\} - 2 \mathcal{J}_{0}(\hat{x}) \mathcal{J}_{0}'(\hat{x}) (1+2\delta) . \end{align}

Here, $\hat {v}_{0}$ is found by substituting (2.14b) in the continuity equation and integrating it over the $\hat {y}$ direction. Substituting (2.14b) and $\hat {v}_{0}$ into (2.16), integrating twice over the $\hat {y}$ direction and using the no-slip condition, we obtain

(2.17)\begin{align} \hat{u}_{1}&= \mathcal{J}_{1}(\hat{x}) \hat{y} - \delta \mathcal{I}'(\hat{x})\mathcal{I}''(\hat{x}) \int_{0}^{\hat{y}}\, \textrm{d}p \int_{p}^{\infty} \left\{ \left( \int_{r}^{\infty} \mathcal{F}(s) \, {\textrm{d}}s \right)^{2}\right.\nonumber\\ & \quad + \left. 2 \mathcal{F}(r) \left( \int_{0}^{r}s\mathcal{F}(s)\, {\textrm{d}}s + r \int_{r}^{\infty} \mathcal{F}(s)\, {\textrm{d}}s \right)\, {\textrm{d}}r \right. \nonumber\\ &\quad + \left. \hat{\psi}'(r) {\rm e}^{-\hat{\psi}(r)} \left( \int_{0}^{r} \left( r - \frac{s}{2} \right)s\mathcal{F}(s)\, {\textrm{d}}s \, + \frac{r^{2}}{2} \int_{r}^{\infty} \mathcal{F}(s)\, {\textrm{d}}s \right) \right. \nonumber\\ & \quad - \left. \left( \frac{\mathcal{J}_{0}(\hat{x})}{\mathcal{I}'(\hat{x})} + \frac{\mathcal{J}_{0}'(\hat{x})}{\mathcal{I}''(\hat{x})} \right) \left( r \mathcal{F}(r) + \int_{r}^{\infty} \mathcal{F}(s)\, \textrm{d}s \right) \right. \nonumber\\ &\quad + \left.\frac{\mathcal{J}_{0}(\hat{x})}{\mathcal{I}''(\hat{x})} \mathcal{F}'(r) \frac{r^{2}}{2} + \frac{\mathcal{J}_{0}(\hat{x}) \mathcal{J}_{(0)}'(\hat{x})}{\mathcal{I}'(\hat{x}) \mathcal{I}''(\hat{x})} - \frac{2\mathcal{J}_{0}(\hat{x}) \mathcal{J}_{0}'(\hat{x})(1+2\delta) }{\delta \mathcal{I}'(\hat{x}) \mathcal{I}''(\hat{x})} \right\} \,\textrm{d}r . \end{align}

Here, $\mathcal {F}(\xi )=-1+{\rm e}^{-\hat {\psi }(\xi )}$, $\mathcal {J}_{0},\, \mathcal {J}_{1}$ and $\mathcal {I}$ are to be determined through matching. Details of the derivation can be found in the supplementary material available at https://doi.org/10.1017/jfm.2020.428.

2.1.3. Matching

For any field variable $f$, the matching condition at ${O}$($\epsilon ^{0}$) is

(2.18)\begin{equation} \lim_{y \rightarrow 0} \left(f^{(0)}_{0}+De f^{(0)}_{1}+\cdots\right)=\lim_{\hat{y} \rightarrow \infty}\left(\hat{f}^{(0)}_{0}+De \hat{f}^{(0)}_{1}+\cdots\right). \end{equation}

The matching condition for the concentration field ($\hat {c}$) yields $\mathcal {I}(\hat {x}) = \lim _{y \rightarrow 0} c^{(0)}(x,y) + C_{\infty }$. We substitute (2.14b) and (2.17) in the above matching condition and obtain $\mathcal {J}_{0} = \mathcal {J}_{1}=0$ at the leading order. At the present order of approximation, the total slip velocity at the outer edge of the interaction layer is

(2.19)\begin{equation} \left. u^{(0)} \right\vert_{y=0} = \left. c^{(0)}_{x}\right\vert_{y=0} \left( M_{0} + De \, \delta \, M_{1} \left.c^{(0)}_{xx}\right\vert_{y=0} \right), \end{equation}

where $M_{0}$ and $M_{1}$ are the mobility coefficients representing the Newtonian and non-Newtonian contribution, respectively,

(2.20a,b)\begin{equation} M_{0} = - \int_{0}^{\infty} \int_{t}^{\infty} \mathcal{F}(s) \,\textrm{d}s \, \textrm{d}t \quad \text{and}\quad {M_{1}} = - {\displaystyle\int_{0}^{\infty} } \int_{p}^{\infty} \mathcal{G}(r)\, {\textrm{d}}r \, {\textrm{d}}p , \end{equation}

where

(2.21)\begin{align} \mathcal{G}(r)&= \left\{ \left( \int_{r}^{\infty} \mathcal{F}(s) \, {\textrm{d}}s \right)^{2} + 2\,\mathcal{F}(r) \left( \int_{0}^{r} s \mathcal{F}(s)\, {\textrm{d}} s + r \int_{r}^{\infty} \mathcal{F}(s)\, {\textrm{d}} s \right) \right. \nonumber\\ & \quad + \left. \hat{\psi}'(r) \, {\rm e}^{-\hat{\psi}(r)} \left( \int_{0}^{r} \left( r - \frac{s}{2}\right) s\, \mathcal{F}(s) \,{\textrm{d}}s + \frac{r^{2}}{2} \int_{r}^{\infty} \mathcal{F}(s) \,{\textrm{d}}s \right) \right\}. \end{align}

Modification to the diffusio-osmotic slip velocity is found to be proportional to the first and second tangential derivative of the bulk-scale concentration at the particle surface. The dependency of viscoelastic effects on the second derivative can be intuitively understood by realizing that the polymers stretch only when there exists a spatial variation (or gradient) in the flow. In the current physics, as the flow itself is generated by the chemical gradient, an appearance of a second derivative at the leading order is intuitive. The proportionality to $De \, \delta$ and the definitions described in (2.5a–d) suggests that the effect of viscoelasticity is solely due to first normal stress difference. Furthermore, the dimensional form of the slip velocity reveals that the first-order slip has a dependency on the characteristic length scale ($a^{*}$), whereas the Newtonian slip is independent of it.

2.1.4. Velocity profile in the inner region

Figure 2(a) shows the Newtonian component of the velocity field (2.14b) inside the thin interaction layer for an exponentially decaying solute-surface attraction, i.e. $\hat {\psi }(\xi )=\Phi _{0} {\rm e}^{-\xi }$ (Anderson et al. Reference Anderson, Lowell and Prieve1982; Sharifi-Mood et al. Reference Sharifi-Mood, Koplik and Maldarelli2013). The velocity monotonically grows away from the surface and approaches an asymptotic value, which upon multiplication with the tangential concentration gradient provides the Newtonian slip velocity. The viscoelastic component of the velocity field ($\hat {u}^{(0)}_{1}$), shown in figure 2(b), also grows monotonically away from the surface and attains an asymptotic value. The magnitude of this asymptotic value increases with the magnitude of interaction ($\Phi _{0}$).

The interactions between solute molecules and the particle surface can be either attractive or repulsive; if the solute is more attracted to the surface than the solvent, the interaction coefficient ($\Phi _{0}$) is negative and viceversa. To show that the current analysis is valid for other forms of potential, we evaluate the velocity field inside the interaction layer for (i) an exponentially repulsive and (ii) a long-range van der Waals type interaction (arising primarily from the dipolar forces ${\sim }1/r^{6}$). (i) For repulsive interactions, the thin ‘layer’ is characterized by an absence of solute concentration and the direction of motion (and slip) is opposite to that observed for an attractive interaction (Michelin & Lauga Reference Michelin and Lauga2014). Figure 3(a) shows that while the Newtonian slip is reversed (cf. figure 2a), the contribution from viscoelasticity is in the same direction as the case of attraction potential (cf. figure 2b). (ii) To incorporate long-range Lennard-Jones-type potential (attractive van der Waals interactions), we follow Anderson et al. (Reference Anderson, Lowell and Prieve1982, p. 112) and assume the dipolar interactions between the solute molecules to be pairwise additive. The potential is defined as

(2.22)\begin{equation} \hat{\psi}(\xi)=\Phi_{0} \left(- \frac{1}{(1+\xi)^{9}} + \frac{1}{(1+\xi)^{3}} \right). \end{equation}

The velocity fields (Newtonian and viscoelastic) remain qualitatively similar to that obtained from the attractive exponential interactions. As (2.22) decays slower than the exponential interaction, the asymptotic value (i.e. slip velocity) is obtained at larger distances from the surface.

Figure 3.

Velocity field inside the interaction layer for (a) repulsive exponential interactions ($\Phi _{0}=+1$) and (b) attractive long-range van der Waals interactions ($\Phi _{0}=-1$). Here, $c_{x}$ and $c_{xx}$ are the concentration gradients at the outer edge of the inner region i.e. $c_{x}\vert _{y=0}$ and $c_{xx}\vert _{y=0}$.

The above results demonstrate that the velocity fields inside the inner region approach an asymptotic value, the slip velocity. The Newtonian component of this slip is equal to the product of concentration gradient ($c_{x}\vert _{y=0}$) and mobility coefficient $M_{0}$. Similarly, the viscoelastic component of velocity field involves $M_{1}$ (see (2.19)). The magnitude of mobility coefficients depends on the nature of interaction (exponential or van der Waals) and magnitude of attraction or repulsion ($\Phi _{0}$). Figure 4 quantifies the effect of the type and magnitude of interaction on $M_{0}$ and $M_{1}$. Figure 4(a) shows that the sign of Newtonian mobility coefficient is opposite for repulsive interactions ($\Phi _{0} > 0$) as compared to attractive ones ($\Phi _{0} < 0$), for both the type of interactions, i.e. short-range exponential or long-range van der Waals solute-surface interaction. Figure 4(b) shows that the sign of $M_{1}$ is always negative for both types of interactions. Furthermore, for repulsive interactions, the magnitude of mobility coefficients ($M_{0}$ and $M_{1}$) is reduced because the adsorption coefficient (characterizing the amount of solute excess near surface) in such cases is generally lower than that of attractive interactions (Anderson Reference Anderson1989, p. 69). For similar reasons, the exponential potential has higher mobility coefficient than that arising from van der Waals type interactions (Anderson et al. Reference Anderson, Lowell and Prieve1982, p. 113).

Figure 4.

Variation of mobility coefficient (a) $M_{0}$ and (b) $M_{1}$ with respect to $\Phi _{0}$, for exponential and van der Waals (vdW) interactions.

The above results, valid for moderate advective effects ($\epsilon Pe \ll 1$), are also applicable to diffusio-osmotic flows of complex fluids in microchannels, arising from externally imposed concentration gradients or generated due to active ‘patches’ (Michelin et al. Reference Michelin, Montenegro-Johnson, De Canio, Lobato-Dauzier and Lauga2015; Michelin & Lauga Reference Michelin and Lauga2019).

Several experimental studies (Ke et al. Reference Ke, Ye, Carroll and Showalter2010; Ebbens & Howse Reference Ebbens and Howse2011; Baraban et al. Reference Baraban, Tasinkevych, Popescu, Sanchez, Dietrich and Schmidt2012) have shown that the particles are propelled with catalytic surface oriented at the rear-end, suggesting an attractive interaction between the solute molecules and active particle. Thus, we focus primarily on the exponentially attractive interaction because of its simplicity and ease of computation. We follow Sabass & Seifert (Reference Sabass and Seifert2012) and Sharifi-Mood et al. (Reference Sharifi-Mood, Koplik and Maldarelli2013) and assume $| \Phi _{0} |=1$ in the results which follow.

2.2. Diffusio-osmotic slip on an active particle

We now extend the results of the previous section (2.19) to an axisymmetric Janus particle. Here, the direction normal to the surface is $r$ and the tangential direction is polar angle $\theta$, varying from $0$ to ${\rm \pi}$. The surface activity $\mathcal {K}$ follows a step function (Golestanian et al. Reference Golestanian, Liverpool and Ajdari2007; Michelin & Lauga Reference Michelin and Lauga2014; Natale et al. Reference Natale, Datt, Hatzikiriakos and Elfring2017)

(2.23)\begin{equation} \mathcal{K} (\theta)= \begin{cases} 1, & \theta < \theta_{c},\\ 0, & \theta > \theta_{c}, \end{cases} \end{equation}

where $\theta _{c}$ is the angle at which the activity undergoes a step change; it represents the surface coverage of activity. To evaluate the slip velocity, we require the solution to the concentration field ($c$) in the outer region. In the absence of advection ($Pe\rightarrow 0$), the solution to (2.8b) is sought in terms of an expansion in Legendre polynomials. Following Golestanian et al. (Reference Golestanian, Liverpool and Ajdari2007), we obtain the solution to bulk-scale concentration field as

(2.24)\begin{equation} c(r,\theta)=\displaystyle\sum_{n=0}^{\infty} {\frac{-\mathcal{K}_{n}}{(n+1)}}\, \frac{P_{n}(\cos \theta)}{r^{n+1}}, \end{equation}

where $P_{n}$ is the $n$th-order Legendre polynomial. Here, $\mathcal {K}_{n}$ are the spectral coefficients of the activity distribution

(2.25)\begin{equation} \mathcal{K}(\theta)=\sum_{n=0}^{\infty} \mathcal{K}_{n} P_{n}(\cos\theta) . \end{equation}

These coefficients are found by taking an inner product of (2.25) with the Legendre polynomials (Michelin & Lauga Reference Michelin and Lauga2014), and are obtained as

(2.26a,b)\begin{equation}\mathcal{K}_{0}=\frac{(1-\cos \theta_{c})}{2} \quad \text{and}\quad \mathcal{K}_{n}=\frac{-1}{2} (P_{n+1}(\cos \theta_{c}) - P_{n-1}(\cos \theta_{c})) \quad \text{for } n \geq 1. \end{equation}

Using the expression (2.24) for disturbance concentration, we obtain the total tangential slip velocity ($M_{0} c_{\theta }(1,\theta ) + De \delta M_{1} c_{\theta }(1,\theta ) c_{\theta \theta }(1,\theta )$)

(2.27)\begin{align} \left.\boldsymbol{u}\right\vert_{r=1} = M_{0} \sum_{n=1}^{\infty} \frac{-\mathcal{K}_{n}}{n+1} \frac{\partial P_{n}}{\partial \theta} \boldsymbol{e}_{\theta} + De \, \delta M_{1} \left(\sum_{n=1}^{\infty} \frac{-\mathcal{K}_{n}}{(n+1)} \frac{\partial P_{n}}{\partial \theta} \right) \left(\sum_{n=1}^{\infty} \frac{-\mathcal{K}_{n}}{(n+1)} \frac{\partial^{2} P_{n}}{\partial \theta^{2}}\right) \boldsymbol{e}_{\theta}. \end{align}

Figure 5(a) shows the Newtonian slip velocity varying across the polar angle for three different surface coverages ($\theta _{c}={\rm \pi} /4, \, {\rm \pi} /2, \, 3{\rm \pi} /4$). The negative value shows that the diffusio-osmotic flow is towards the active region, and is highest in magnitude at the point of discontinuity of activity ($\theta =\theta _{c}$). (These results are validated with those reported by Michelin & Lauga (Reference Michelin and Lauga2014) and are provided in the supplementary material.) For a second-order fluid, figure 5(b) shows a localized reversal of slip velocity across $\theta _{c}$ for all three surface coverages, i.e. the slip velocity exhibits sharp nonlinear gradients across the point of discontinuity of activity. Natale et al. (Reference Natale, Datt, Hatzikiriakos and Elfring2017) also reported such behaviour in viscoelastic stresses at $\theta _{c}$ in their FEM simulations. They reported a viscoelasticity triggered extensional flow, across the point of discontinuity in surface activity.

Figure 5.

(a) The dashed line represents the Newtonian slip velocity along the polar angle for three different surface coverages ($\theta _{c}={\rm \pi} /4, {\rm \pi} /2, 3{\rm \pi} /4$). The empty circular markers represent the maximum magnitude of the slip velocity for a range of surface coverage: $\theta _{c} \in (0,{\rm \pi} )$. (b) The solid line represents the total slip velocity (for three different surface coverages) along the polar angle for $\hat {\psi }(\xi )=- {\rm e}^{-\xi }$, $De=0.01$ and $\delta =-0.5$ (corresponds to $\Psi _{2}^{*}=0$). The empty square markers represent the maximum magnitude of the slip velocity for a range of surface coverage: $\theta _{c} \in (0,{\rm \pi} )$. (c) Concentration profile and its gradient along the polar angle (on particle surface) for $\theta _{c}={\rm \pi} /2$. Panel (d) shows the profile for $c_{\theta \theta }$. First 700 modes were used to describe the concentration field and its gradients.

To analyse this slip reversal or the extensional flow across the point of transition in activity, we examine the behaviour of the concentration field around the particle, as the slip (2.19) depends on the first and second polar gradient of the concentration field. In figure 5(c,d), $c$, $c_{\theta }$ and $c_{\theta \theta }$ profiles are shown for $\theta _{c}={\rm \pi} /2$. A step change in surface activity causes the second tangential gradient to undergo a sharp reversal at $\theta =\theta _{c}$, which is also reflected in the slip velocity (in figure 5b). The profile of $c_{\theta \theta }$, despite accounting for first 700 modes, exhibits oscillations near the point of discontinuity of surface activity. This seemingly divergent behaviour is because of the step function representation of surface activity (2.23), which is widely employed in the literature (Golestanian et al. Reference Golestanian, Liverpool and Ajdari2007; Michelin & Lauga Reference Michelin and Lauga2014; Natale et al. Reference Natale, Datt, Hatzikiriakos and Elfring2017). This oscillatory behaviour indicates a violation of scaling in the inner region: in § 2.1.2 the length scale in the tangential direction was assumed to be $a^{*}$, which is not the true characteristic of the system as variations in the tangential direction are rapid for a step discontinuity in surface activity. Since such step discontinuities are unlikely to be realized in experiments, we now consider a smooth variation in the activity and demonstrate that the oscillatory behaviour of $c_{\theta \theta }$ for step activity is a mathematical artefact.

2.3. Diffusio-osmotic slip on an active particle: sigmoidal function approximation

The step change in surface activity is approximated as a logistic function

(2.28)\begin{equation} \mathcal{K}(\theta)={1}/\!\left( {1+ \exp\left[{-\zeta \left(\theta_{c}- \theta \right)}\right]}\right). \end{equation}

Here, $\zeta$ (transition parameter) determines the sharpness of the transition ofnon-dimensional activity from 1 to 0. We have chosen $\zeta =16$ such that the transition length is large enough to maintain the consistency of the analysis with respect to the scaling used in § 2.1.2 (i.e. $l_{transition}^{*} \sim a^{*}$). (For a particle of $a^{*}=5\ {\rm \mu} \textrm {m}$, $\zeta =16$ corresponds to a circumferential length of ${\sim } 2 \ {\rm \mu} \textrm {m}$, which is much larger than the typical interaction layer thickness.) The expressions for concentration and slip velocity are identical to (2.24) and (2.27), respectively. The spectral modes can be found by substituting (2.28) into (2.25) and taking the inner product with respect to $P_{n}$ on both sides

(2.29)\begin{equation} \mathcal{K}_{n}=\frac{2n+1}{2} \int_{0}^{{\rm \pi}} \mathcal{K}(\theta) P_{n}(\cos \theta)\, {\textrm{d}}\theta. \end{equation}

For activity represented by (2.28), the concentration field and its tangential gradients (at the surface) are depicted in figure 6(a). The variation of concentration field and its gradients along $\theta$ is qualitatively similar to that reported for the step change; the magnitudes, however, are significantly reduced. Figures 6(b) and 6(c) show the variation of activity and its faster convergence in comparison to the step function. Figure 6(d) compares the slip velocity for a second-order fluid with that of a Newtonian fluid.

Figure 6.

(a) Profile of concentration and its first and second gradient along the polar angle on particle surface for $\theta _{c}={\rm \pi} /2$. (b) Variation of surface activity as a step (dotted line) and smooth function (solid line). (c) Convergence of spectral modes: solid (blue) line shows fast convergence for the sigmoidal activity; and dotted (black) line depicts slow convergence for step activity. (d) Different curves represent the total slip velocity along the polar angle for $\hat {\psi }(\xi )=-1 {\rm e}^{-\xi }$, $\zeta =16$, $\delta =-0.5$ and different Deborah numbers. The inset shows the slip velocities for Newtonian and second-order fluid for the case of repulsive interactions: $\Phi _{0}=+1$, $De = 0$ (, black) and $De = 0.2$ (, blue).

We observe a reduction and an increase in slip velocity (relative to the Newtonian slip) across $\theta _{c}$. The magnitude of modification in the slip is significantly reduced (as compared with that of step change in activity) which is a consequence of smooth transition of activity. The above observations are for attractive interactions between solute and particle surface; for repulsive interactions (as shown in the inset of figure 6d), the effect of viscoelasticity is qualitatively reversed.

Figure 7 demonstrates the effect of the transition parameter ($\zeta$) on the convergence of spectral coefficients, concentration gradients and slip velocity. Figure 7(a) shows that as $\zeta$ decreases, the transition from active to passive surface becomes smoother. Figure 7(b) shows that the convergence is faster for lower $\zeta$. As the transition becomes smoother the magnitude of tangential gradients of concentration field reduces (figure 7c,d). As a result, the slip velocity for Newtonian and second-order fluid reduces (shown in figure 7(e–f), respectively).

Figure 7.

Analysis of activity, spectral modes, concentration gradients and slip velocity for three different transition parameters: $\zeta =4$ (, pink), $\zeta =8$ (, black), and $\zeta =16$ (, blue). (a) Variation of activity with polar angle $\theta$. (b) Convergence of spectral modes. (c) Tangential concentration gradient. (d) Second tangential derivative of concentration field. (e) Slip velocity of a Newtonian fluid. (f) Slip velocity of the second-order fluid. Parameters: $De=0.2$, $\delta =-0.5$, $\Phi _{0}=-1$.

The effect of viscoelasticity on the tangential velocity field inside the interaction layer is shown in figure 8 (for $\theta _{c}={\rm \pi} /2$). Figure 8(a) depicts the symmetry of velocity field across $\theta _{c}$ for Newtonian fluid. Addition of viscoelasticity breaks this symmetry: the contours of the SOF in figure 8(b) depict that the magnitude of velocity field is reduced before the transition of activity ($\theta < \theta _{c}$) and increased after it ($\theta > \theta _{c}$). These results suggest that the elastic effects (characterized by $De$) tend to locally reverse the slip velocity. At higher Deborah numbers, a reversal in slip velocity can generate a local extensional flow near the transition of activity (as shown by Natale et al. (Reference Natale, Datt, Hatzikiriakos and Elfring2017)), resembling a puller or pusher-type flow field. Interestingly, there have been similar observations in the context of electrophoresis. Recently, Li & Koch (Reference Li and Koch2020) reported that, in weakly viscoelastic fluids, polymer elasticity in the inner region changes the electrophoretic particle into a puller-type squirmer: the strong shear flow in the double layer causes the polymers to be stretched tangentially, resulting in a pulling flow at the front and rear end, and a pushing flow from the sides. Since electrophoretic and diffusiophoretic mechanisms share qualitative similarities (such as strong shear inside the thin layer), it is plausible that the effects of elasticity, observed in electrophoresis, emerge in diffusiophoresis as well.

Figure 8.

(a) Newtonian tangential velocity contours inside the interaction layer for $\theta _{c}={\rm \pi} /2$. (b) Tangential velocity contours corresponding to the second-order fluid for $\hat {\psi }(\xi )=- {\rm e}^{-\xi }$, $\zeta =16$, $\delta =-0.5$ and $De=0.2$. Arrows depict the velocity vectors ($\hat {\boldsymbol {u}}_{1}^{(0)}$).

2.4. Swimming velocity

The motion of the Janus sphere is found by using the reciprocal theorem. Following Stone & Samuel (Reference Stone and Samuel1996) and Ho & Leal (Reference Ho and Leal1976), we obtain the swimming velocity as

(2.30)\begin{equation} U=\frac{-1}{4{\rm \pi}} \int_{S_{p}} \left.\left( \boldsymbol{u}_{0} + De \,\boldsymbol{u}_{1} \right)\right\vert_{r=1} \boldsymbol{\cdot} \boldsymbol{e}_{z} \, \textrm{d}S - \frac{1}{6{\rm \pi}} De \int_{V_{f}} \boldsymbol{\mathsf{S}}_{\,0}\boldsymbol{:} \boldsymbol{\nabla}{\boldsymbol{u}^{t}}\, \textrm{d}V. \end{equation}

Here, $\boldsymbol {u}_{0}$ is the Newtonian slip velocity ($M_{0} c_{\theta }(1,\theta ) \, \boldsymbol {e}_{\theta }$), $De \, \boldsymbol {u}_{1}$ is the modification to slip due to viscoelasticity ($De \, M_{1} \delta \, c_{\theta }(1,\theta ) \, c_{\theta \theta }(1,\theta ) \, \boldsymbol {e}_{\theta }$), $\boldsymbol {u}^{t}$ is the test flow field which governs the motion of a rigid sphere in $z$-direction with unit velocity in a quiescent Newtonian medium (Michelin & Lauga Reference Michelin and Lauga2014; Datt et al. Reference Datt, Zhu, Elfring and Pak2015) and $\boldsymbol{\mathsf{S}}_{\,0}$ is the polymeric stress. The first integral ($U_{\lambda }$) denotes the contribution to swimming velocity arising from the slip, which contains a Newtonian ($U_{\lambda _0}$) and a non-Newtonian ($U_{\lambda _1}$) component

(2.31a,b)\begin{align} U_{\lambda_{0}} = \frac{M_{0}}{2} \int_{0}^{{\rm \pi}} c_{\theta}(1,\theta) \sin^{2}{\theta} \, {\textrm{d}}\theta; \quad U_{\lambda_{1}} = \frac{De\, \delta M_{1}}{2} \int_{0}^{{\rm \pi}} c_{\theta}(1,\theta) c_{\theta \theta}(1,\theta) \sin^{2}{\theta} \, {\textrm{d}}\theta. \end{align}

We evaluate the above components and compare them in figure 9 for different surface coverages. For attractive interactions between the solute molecules and particle, the Newtonian swimming velocity is in the negative $z$-direction (see figure 9a). Figures9(b) and 9(c) show that the modification to the swimming velocity ($U_{\lambda_{1}}$ contribution) is relatively small and changes sign with the surface coverage: the swimming velocity increases for $\theta _{c}<{\rm \pi} /2$ and reduces for $\theta _{c}>{\rm \pi} /2$.

Figure 9.

(a) Variation of swimming velocity due to Newtonian surface slip ($U_{\lambda _{0}}$) for various surface coverages. (b) Perturbation to swimming velocity due to modification in the slip ($U_{\lambda _{1}}$). (c) Ratio of the swimming velocity components, i.e. (b) and (a). (d) Variation of modification to the swimming velocity due to bulk polymeric stresses ($U_{B}$). (e) Ratio of (d) and (c): comparison of two components arising due to viscoelasticity. Parameters: $De=0.1$, $\delta =-0.5$, $\zeta =16$, $\Phi _{0}=-1$ (attractive; , blue) and $\Phi _{0}=+1$ (repulsive; , black).

For a repulsive interaction, $U_{\lambda _{0}}$ changes sign (depicting swimming in the opposite direction) as also reported by earlier studies (Sharifi-Mood et al. Reference Sharifi-Mood, Koplik and Maldarelli2013; Michelin &Lauga Reference Michelin and Lauga2014; Natale et al. Reference Natale, Datt, Hatzikiriakos and Elfring2017). The contribution of $U_{\lambda _{1}}$ is such that the swimming velocity is reduced for $\theta _{c} < {\rm \pi} /2$ and enhanced for $\theta _{c} > {\rm \pi} /2$. Here, $U_{\lambda _{1}}$ is in the same direction as that for attractive interaction. This outcome can be understood by comparing the flow field inside the interaction layer for attractive and repulsive cases, i.e. figures 2 and 3(a), respectively. In comparison to the attractive interaction, Newtonian velocity field was reversed for repulsive interactions, whereas the modification (arising from viscoelasticity) was found to be in the same direction. It is also interesting to note that, contrary to $U_{\lambda _{0}}$, $U_{\lambda _{1}}$ is an odd function (antisymmetric about ${\rm \pi} /2$) which is due to the proportionality to double tangential derivative of surface concentration, whereas the former is solely proportional to the single tangential derivative.

The second integral ($U_{B}$) accounts for the contribution from polymeric stresses in the bulk. In their evaluation of swimming velocity, Datt et al. (Reference Datt, Natale, Hatzikiriakos and Elfring2017) accounted for the bulk viscoelastic effects ($U_{B}$), and provided the following analytical expression (re-expressed here using the scales in (2.5a–d)):

(2.32)\begin{equation} \frac{U_{B}}{U_{\lambda_{0}}} = - De (1+\delta) \sum_{n=1}^{\infty} \frac{6n}{(n+1)^{2} (n+2)} \frac{\alpha_{n} \alpha_{n+1}}{\alpha_{1}^{2}}, \end{equation}

where $\alpha _{n} = {n \mathcal {K}_{n}}/{(2n+1)}$ and $\mathcal {K}_{n}$ is determined using (2.28) and (2.29). The modification due to the bulk stresses ($U_{B}/U_{\lambda _{0}}$) is plotted in figure 9(d) for smooth activity. The figure shows that $U_{B}$ can significantly enhance (or impede) the active swimming, depending upon the surface coverage ($\theta _{c}$) being more (or less) than ${\rm \pi} /2$. This was also reported by Datt et al. (Reference Datt, Natale, Hatzikiriakos and Elfring2017) for step activity. A comparison of the two effects of viscoelasticity, $U_{B}$ and $U_{\lambda _{1}}$ (shown in figure 9e), reveals that the contribution to the swimming velocity from the modification in slip ($U_{\lambda _{1}}$) is comparable to that from $U_{B}$ for $5{\rm \pi} /6 < \theta _{c}<{\rm \pi} /6$.

Figure 10 shows the effect of transition parameter ($\zeta$) on the components of swimming velocity for various surface coverages. As $\zeta$ decreases the Newtonian swimming velocity reduces (figure 10a) because the slip velocity decreases (shown in figure 7). However, for $\theta _{c}<{\rm \pi} /4 \,\hbox{and}\, \theta _{c}>3{\rm \pi} /4$, $U_{\lambda _{0}}$ is greatest for $\zeta =4$. This is because, for such coverages, the total area of catalytic activity is more for $\zeta =4$ than the steeper activity transitions ($\zeta =8,16$). Figure 10(b) shows the area under the $\mathcal {K}-\theta$ curve for $\zeta =4,8,16$, representing the total area of activity for $\theta _{c}={\rm \pi} /6$. For ${\rm \pi} /4 < \theta _{c} < 3{\rm \pi} /4$, the area of catalytic activity is independent of $\zeta$ and thus a direct comparison can be made. Figures 10(c) and 10(d) show that the contribution from $U_{\lambda _{1}}$ reduces as $\zeta$ decreases, which is due to significant reduction in slip velocity. Similarly, the contribution from bulk polymeric stresses reduces as $\zeta$ decreases, as shown in figure 10(e,f).

Figure 10.

Analysis of different components of swimming velocity for three different transition parameters: $\zeta =4$ (, pink), $\zeta =8$ (, black), and $\zeta =16$ (, blue). (a) Variation of $U_{\lambda _{0}}$ with different surface coverages. (b) Variation of activity $\mathcal {K}$ with polar angle $\theta$. (c) Variation of $U_{\lambda _{1}}$ for different surface coverages. (d) Contribution from the slip modification to the swimming velocity for different surface coverages. (e) Bulk stress contribution to the swimming velocity for different surface coverages. (f) Comparison of (e) with (d) (i.e. $U_{B}$ with $U_{\lambda _{1}}\!$) for different surface coverages. Parameters: $De=0.1$, $\delta =-0.5$, $\Phi _{0}=-1$.

3.1. Evaluation of the diffusio-osmotic slip

Similar to § 2.1, we use matched asymptotic expansion (assuming $Pe \ll \epsilon ^{-2}$ and $\epsilon \ll 1$) and perturbation expansion (in $\epsilon$) to obtain the leading-order slip for a generalized weakly shear-thinning fluid.

3.1.1. Outer region

At the leading order, the equations in the outer region are

(3.2)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}^{(0)} = 0, \quad - \boldsymbol{\nabla} p^{(0)} + {\mu(\gamma_{0})} {\nabla}^{2} \boldsymbol{u}^{(0)} + \boldsymbol{\nabla} {\mu(\gamma_{0})} \boldsymbol{\cdot} \boldsymbol{\mathsf{A}}^{(0)} = \boldsymbol{0}, \end{equation}

subject to the following boundary conditions:

(3.3)\begin{equation} \boldsymbol{u}^{(0)} \rightarrow \boldsymbol{0} , \; p^{(0)} \rightarrow 0 \quad \text{as } y \rightarrow \infty. \end{equation}

Here, $\mu (\gamma _{0}) = \mu ^{*} / \mu ^{*}_{0} = 1+\chi \mu _{1}(\gamma _{0})$.

3.1.2. Inner region

We rescale the variables (similar to § 2.1.2) and obtain the governing equations in the inner region (at the leading order) as

(3.4a)\begin{gather} \frac{\partial \hat{u}^{(0)}}{\partial \hat{x}} + \frac{\partial \hat{v}^{(0)}}{\partial \hat{y}} =0 , \end{gather}

(3.4b)\begin{gather} - \frac{\partial \hat{p}^{(0)}}{\partial \hat{x}} + {\mu(\gamma_{0})} \frac{\partial^2 \hat{u}^{(0)}}{\partial \hat{y}^2} + \frac{\partial \mu(\gamma_{0})}{\partial \hat{y}} \frac{\partial \hat{u}^{(0)}}{\partial \hat{y}} = (\hat{c}^{(0)} + C_{\infty})\frac{\partial \hat{\psi}^{(0)}}{\partial \hat{x}} , \end{gather}

(3.4c)\begin{gather} - \frac{\partial \hat{p}^{(0)}}{\partial \hat{y}} = (\hat{c}^{(0)}+C_{\infty}) \frac{\partial \hat{\psi}^{(0)}}{\partial \hat{y}} , \end{gather}

subject to the pressure decay and no-slip boundary condition

(3.5)\begin{equation} \hat{p} \rightarrow 0 \ \textrm{{as}} \ \hat{y} \rightarrow \infty; \quad \left. \hat{u}^{(0)} \right\vert_{\hat{y}=0} = 0. \end{equation}

Assuming that the difference between zero shear and infinite shear viscosity is small (i.e. $\chi \ll 1$), we perform a regular perturbation expansion in $\chi$ and represent the velocity and pressure fields as

(3.6)\begin{equation} f^{(i)}(x,y)=f^{(i)}_{0}(x,y)+\chi \, f^{(i)}_{1}(x,y)+\cdots . \end{equation}

Substituting the perturbed field variables in (3.4)–(3.5) gives the equations governing the system at $O(1)$ and $O(\chi )$.

At $O(1)$, the equations and their solutions are identical to (2.14). The hydrodynamics at $O(\chi )$ is governed by

(3.7a)\begin{gather} \frac{\partial \hat{u}_{1}^{(0)}}{\partial \hat{x}} + \frac{\partial \hat{v}_{1}^{(0)}}{\partial \hat{y}} =0 , \end{gather}

(3.7b)\begin{gather} - \frac{\partial \hat{p}_{1}^{(0)}}{\partial \hat{x}} + \frac{\partial^2 \hat{u}_{1}^{(0)}}{\partial \hat{y}^2} + \mu_{1}(\gamma_{0}) \frac{\partial^2 \hat{u}_{0}^{(0)}}{\partial \hat{y}^2} + \frac{\partial \mu_{1}(\gamma_{0})}{\partial \hat{y}} \frac{\partial \hat{u}_{0}^{(0)}}{\partial \hat{y}} = 0 , \end{gather}

(3.7c)\begin{gather} - \frac{\partial \hat{p}_{1}^{(0)}}{\partial \hat{y}} = 0. \end{gather}

Integrating (3.7c) and using the pressure decay condition, we get $\hat {p}_{1}^{(0)} = 0$. (3.7b) thus yields

(3.8)\begin{equation} \frac{\partial^{2} \hat{u}_{1}^{(0)}}{\partial \hat{y}^{2}} = - \frac{\partial}{\partial \hat{y}} \left( \mu_{1}(\gamma_{0}) \frac{\partial \hat{u}_{0}^{(0)}}{\partial \hat{y}} \right). \end{equation}

Integrating the above equation twice gives

(3.9)\begin{align} \hat{u}_{1}^{(0)} = \mathcal{I}'(\hat{x}) \int_{0}^{\hat{y}} \mu_{1}(\gamma_{0}) \int_{t}^{\infty} \mathcal{F}(s)\,\textrm{d}s \, \textrm{d}t - \mathcal{J}_{0}(\hat{x}) \int_{0}^{\hat{y}} \mu_{1} (\gamma_{0})\,\textrm{d}s \; + \left(\mathcal{J}_{1}(\hat{x}) - \mathcal{J}_{0} (\hat{x}) \mu_{1}\right) \hat{y}, \end{align}

where $\mathcal {F}(s) = -1 + {\rm e}^{-\hat {\psi }(s)}$.

3.1.3. Matching

As in § 2.1.3, upon matching the inner and outer solutions, we find that $\mathcal {J}_{0}=\mathcal {J}_{1}=0$ and obtain

(3.10)\begin{equation} \left. u^{(0)} \right\vert_{y=0} = \left. c^{(0)}_{x}\right\vert_{y=0} \left( M_{0} + \chi \, M_{1} \right), \end{equation}

where $M_{0}$ is the mobility coefficient representing the Newtonian contribution and is expressed in (2.20a,b). The expression for $M_{1}$ is

(3.11)\begin{equation} M_{1} = {\displaystyle\int_{0}^{\infty} } \mu_{1}\left(\gamma_{0}^{(0)}\right) \int_{t}^{\infty} \mathcal{F}(s)\,\textrm{d}s \, \textrm{d}t . \end{equation}

Since the power-law model diverges for asymptotically small shear-rates (Bird et al. Reference Bird, Armstrong and Hassager1987), we choose the Carreau model to capture viscosity variation: $\mu _{1}=( 1+ \tau ^{*\,2} |\gamma _{0}^{*}|^{2} )^{({n-1})/{2}} - 1$ (Bird et al. Reference Bird, Armstrong and Hassager1987). Here, $\tau ^{*}$ is the fluid relaxation time scale and $n$ characterizes the degree of shear-thinning ($n<1$). For Carreau fluid, the non-Newtonian mobility coefficient ($M_{1}$)is

(3.12)\begin{equation} M_{1} = {\displaystyle\int_{0}^{\infty} } \left\{ \left[ 1 + \left( C\!u_{\lambda} \, \left. c_{x}^{(0)}\right\vert_{y=0} \int_{t}^{\infty} \mathcal{F}(s)\, {\textrm{d}}s \right)^{2} \right]^{({n-1})/{2}} -1 \right\} \int_{t}^{\infty} \mathcal{F}(s) \,\textrm{d}s \, \textrm{d}t , \end{equation}

where $C\!u_{\lambda }$ (based on shear in the interaction layer) is the ratio of time scales associated with relaxation ($\tau ^{*}$) to that of shear in the flow ($\lambda _{I}^{*}/U_{ch}$), $C\!u_{\lambda } = \tau ^{*}/(\lambda _{I}^{*}/U_{ch})$. This is different than the Carreau number corresponding to shear in the bulk fluid (i.e. outer region): $C\!u_{B} = \tau ^{*}/(a^{*}/U_{ch})$. The ratio of the two Carreau numbers ($C\!u_{\lambda }/C\!u_{B} = a^{*}/\lambda _{I}^{*}$) provides the comparison of shear rate inside to that outside the interaction layer, i.e. $O(\epsilon ^{-1})$.

3.2. Diffusio-osmotic slip on an active particle

As in § 2.2, we now apply the above results to an axisymmetric Janus sphere. Since the shear in the thin interaction layer is generally high (i.e. $U_{ch}/\lambda _{I}^{*} \sim 10^{2} \ \textrm {s}^{-1}$) and relaxation time scales of the biological fluids are ${\sim }1\,{\rm s}$ (Zare, Park & Rhee Reference Zare, Park and Rhee2019), large Carreau numbers are possible ($C\!u_{\lambda } \sim 10^{2}$). Contrary to the second-order fluid, $M_{1}$ for Carreau fluid depends both on the magnitude of interaction ($\Phi _{0}$) as well as the concentration gradient (determined by the activity transition parameter $\zeta$). Since the concentration field varies in the tangential direction, $M_{1}$ varies tangentially. Thus, we analyse the effect of $\Phi _{0}$ and $\zeta$ on $M_{1}$ for $\theta =\theta _{c}={\rm \pi} /2$ in figure 11. For both short-range exponential and long-range van der Waals interactions, $M_{1}$ is a weak function of $\zeta$. Here, $M_{1}$ for Carreau fluid is of a different sign for attractive ($\Phi _{0}<0$) and repulsive interactions ($\Phi _{0}>0$), as opposed to the case of second-order fluid (figure 4b).

Figure 11.

Non-Newtonian mobility ($M_{1}$) at $\theta ={\rm \pi} /2$ for two interaction potentials: (a) exponential type and (b) van der Waals type. Other parameters: $n=0.25$, $C\!u_{\lambda }=10^{2}$, $\theta _{c}={\rm \pi} /2$.

Using (2.24), (2.28), (3.10) and (3.12), we numerically evaluate the slip for shear-thinning fluid ($n=0.25$) and depict its variation along the surface of Janus particle in figure 12(a). For large Carreau numbers ($C\!u_{\lambda } \sim 10^{2}$) and attractive interactions, we find a marginal increase in the total slip velocity. For repulsive interactions, the enhancement in slip velocity is qualitatively similar (see the Appendix). The modification in slip (in both attractive and repulsive cases) is maximum at the transition of activity, depicting a local symmetry about $\theta _{c}$.

Figure 12.

(a) The dashed black line represents the Newtonian slip velocity. The solid green line represents the variation of total slip velocity for a shear thinning fluid along the polar angle for $\hat {\psi }(\xi )=- {\rm e}^{-\xi }$ and $C\!u_{\lambda }=100$. (b) Variation of non-Newtonian contribution to the swimming velocity (arising from the surface slip) with $C\!u_{\lambda }$, for three different surface coverage ($\theta _{c}$) and $\hat {\psi }(\xi ) = -{\rm e}^{-\xi }$. Solid (, black) and dashed lines (}, black) correspond to $\theta _{c}={\rm \pi} /2$ and $\theta _{c}={\rm \pi} /4$ or $3{\rm \pi} /4$, respectively. Inset showing the variation in non-dimensional viscosity $\mu$ with $C\!u_{\lambda }$ at $y=0$ and $\theta =\theta _{c}={\rm \pi} /2$. (c) Variation of non-Newtonian contribution to the swimming velocity (arising from the bulk stresses) with $C\!u_{B}$. The convergence was obtained by accounting for first 15 modes. (d) Comparison between the two non-Newtonian components of swimming velocity for $C\!u_{\lambda }=10^{2}$. Other parameters: $\chi =0.1$, $n=0.25$, $\zeta =16$.

3.3. Swimming velocity

We now evaluate the swimming velocity using the reciprocal theorem (Stone & Samuel Reference Stone and Samuel1996; Khair et al. Reference Khair, Posluszny and Walker2012)

(3.13)\begin{equation} U=\frac{-1}{4{\rm \pi}} \int_{S_{p}} \left.\left( \boldsymbol{u}_{0} + \chi \,\boldsymbol{u}_{1} \right)\right\vert_{r=1} \boldsymbol{\cdot} \boldsymbol{e}_{z} \, \textrm{d}S \; - \; \frac{1}{6{\rm \pi}} \chi \int_{V_{f}} \mu_{1} \left( \gamma_{0} \right) \boldsymbol{\mathsf{A}}_{0} \boldsymbol{:} \boldsymbol{\nabla}{\boldsymbol{u}^{t}}\,\textrm{d}V. \end{equation}

As in § 2.4, the first integral contains the Newtonian component $U_{\lambda _0}$ (given by (2.31a,b)) and non-Newtonian component $U_{\lambda _1}$

(3.14)\begin{equation} U_{\lambda_{1}} = \frac{\chi}{2} \int_{0}^{{\rm \pi}} M_{1}(\theta) c_{\theta}(1,\theta) \sin^{2} \theta\, \textrm{d}\theta. \end{equation}

Substituting (3.12) in (3.13) and integrating numerically, we find that the contribution from $U_{\lambda _1}$ modestly affects the swimming of a Janus particle. This is shown in figure 12(b) for three different surface coverages. The positive sign of the ratio shows that $U_{\lambda _1}$ adds to the contribution arising from the Newtonian slip ($U_{\lambda _{0}}$). As $C\!u_{\lambda }$ increases, the viscosity in the interaction layer decreases, which results in a faster diffusio-osmotic flow. However, this enhancement reaches a plateau as the viscosity reduction stagnates at high $C\!u_{\lambda }$ (see inset in figure 12b). For repulsive interactions, this enhancement in swimming velocity ($U_{\lambda _{1}}$) is qualitatively similar (see the Appendix); shear thinning enhances the swimming velocity, irrespective of the interaction between solute molecules and the particle.

The second integral ($U_{B}$) accounts for the bulk stresses arising from viscosity variations. The modification due to bulk non-Newtonian stresses is quantified in figure 12(c) for three different surface coverages. (We follow the approach of Blake (Reference Blake1971) and Datt etal. (Reference Datt, Natale, Hatzikiriakos and Elfring2017) to find $U_{B}$ for smooth activity. The details of implementation and reproduced results (for step activity) are provided in the supplementary material.) The negative sign of $U_{B}/U_{\lambda _{0}}$ denotes that the bulk stresses always reduce the swimming velocity, for all surface coverages (also reported by Datt et al. (Reference Datt, Natale, Hatzikiriakos and Elfring2017) for step activity). The non-monotonic behaviour is consistent with the fact that a viscosity, following Carreau model, reduces to a Newtonian fluid of a lower viscosity ($\mu _{\infty }^{*}$) as $C\!u_{B} \rightarrow \infty$. As a result, the bulk non-Newtonian stresses are maximum at intermediate $C\!u_{B}$ and vanish at high $C\!u_{B}$ (Khair et al. Reference Khair, Posluszny and Walker2012; Datt et al. Reference Datt, Zhu, Elfring and Pak2015, Reference Datt, Natale, Hatzikiriakos and Elfring2017).

Figure 12(d) compares the contribution from the two non-Newtonian components to swimming velocity, i.e. $U_{B}$ and $U_{\lambda _{1}}$. We take $C\!u_{\lambda }=10^{2}$, as shear rate in the interaction layer is generally high. This yields $C\!u_{B} = (\epsilon ) 10^{2}$. Thus, in figure 12(d), we compare the two components, keeping $\epsilon$ as an independent variable. Since the current analysis is valid for asymptotically thin interaction layers, we vary $\epsilon$ from $10^{-4}$ to $10^{-2}$. Physically, $C\!u_{\lambda }/C\!u_{B} = \epsilon ^{-1}$ corresponds to the disparity in shear rates in the inner and outer regions. For $\epsilon \sim 10^{-4}$, the shear rate inside the interaction layer is very high and thus the contribution from $U_{\lambda _{1}}$ dominates that from $U_{B}$. As $\epsilon$ increases (i.e. thickness of interaction layer), the shear in the interaction layer decreases, which increases the magnitude of $U_{B}/U_{\lambda _{1}}$. This continues until $U_{B}$ reaches a plateau. (The above results are for $\zeta =16$. The Appendix shows the effect $\zeta$ on the swimming velocity.)

The contribution from both non-Newtonian components is modest in magnitude. In context of electrophoresis, Khair et al. (Reference Khair, Posluszny and Walker2012) also reported a similar enhancement and trends in a shear-thinning medium. Analogous to their study, our result (3.10) also reveals that the shear-thinning modification of the slip does not add a size dependency to the motion induced due to self-diffusiophoresis. On the other hand, the modification due to bulk non-Newtonian stresses adds a size dependency (it enters through the bulk Carreau number).