2.1. General framework

We adopt a continuum description of diffusio-osmotic transport, following established theoretical frameworks (Golestanian et al. Reference Golestanian, Liverpool and Ajdari2007; Jülicher & Prost Reference Jülicher and Prost2009; Sabass & Seifert Reference Sabass and Seifert2012), to analyse the two-dimensional flow field generated within a wedge formed by two semi-infinite lines starting from one point. The fluid within the wedge is characterised by dynamic viscosity $\eta$ and density $\rho _0$ , and contains solute of local concentration $\tilde {C}$ (number of particles per unit volume), with diffusivity $\kappa$ . The chemical activity $\mathcal{A}$ of a surface $\mathcal{S}$ quantifies the fixed rate of solute release ( $\mathcal{A}\gt 0$ ) or absorption ( $\mathcal{A}\lt 0$ ) on the surface by

(2.1) \begin{equation} \kappa \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla \!}{\tilde {C}} =-\mathcal{A}\quad \textrm { on } \quad \mathcal{S}, \end{equation}

where $\boldsymbol{n}$ is the normal unit vector on $\mathcal{S}$ . Due to the short-range interaction of solute molecules with the cavity boundary, local concentration gradients result in the motion of the solute, and consequently drive the motion of the fluid (Anderson Reference Anderson1989). Assuming that the thickness of the interaction layer is small compared to the dimensions of the cavity, the classical slip-velocity formulation can be used (Michelin & Lauga Reference Michelin and Lauga2014), and local solute gradients induce an effective slip velocity on the boundaries,

(2.2) \begin{equation} {\tilde {\boldsymbol{u}}} =\mathcal{M}(\boldsymbol{1}-\boldsymbol{n}\boldsymbol{n})\boldsymbol{\cdot }\boldsymbol{\nabla \!}{\tilde {C}}\quad \textrm { on } \quad \mathcal{S}, \end{equation}

which drives the bulk motion of the fluid. Here, $\mathcal{M}$ is the local phoretic mobility at the surface $\mathcal{S}$ . It is related to the surface–solute interaction potential (Anderson Reference Anderson1989).

Introducing $\mathcal{R}$ as the characteristic length scale, and $\mathcal{U}=|\mathcal{AM}|/\kappa$ as the characteristic phoretic velocity generated along $\mathcal{S}$ , we can define the Péclet and Reynolds numbers

(2.3) \begin{equation} \textit{Pe}=\dfrac {\mathcal{UR}}{\kappa }, \quad \textit{Re}=\dfrac { {\rho _0}\, \mathcal{U} \mathcal{R}}{\eta }, \end{equation}

which quantify, respectively, the relative importance of solute advection and diffusion as transport mechanisms and effects of inertia and viscosity in forces that shape the flow. When $\textit{Pe}$ is small enough, the solute dynamics is purely diffusive and is governed by Laplace’s equation

(2.4) \begin{equation} {\nabla} ^2 {\tilde {C}}= 0, \end{equation}

in the whole domain. Provided that inertial effects are negligible (i.e. $\textit{Re}\ll 1$ ), the flow and pressure fields satisfy the incompressible Stokes equations:

(2.5) \begin{align} \eta {\nabla} ^2 {\tilde {\boldsymbol{u}}} &=\boldsymbol{\nabla \!}{\tilde {P}}, \end{align}

(2.6) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }{\tilde {\boldsymbol{u}}} &= 0. \end{align}

The diffusive Laplace’s problem for the solute concentration $\tilde {C}$ effectively decouples from the hydrodynamic problem and may be solved independently. The concentration $\tilde {C}$ can be used to compute the slip flow along $\mathcal{S}$ via (2.2), which serves as the boundary condition for the flow field in (2.5) within the cavity.

We note here that while the fluid is assumed incompressible, in phoretic problems involving the transport of particles, effective particle velocity fields can appear compressible due to concentration-dependent drift. For instance, Raynal et al. (Reference Raynal, Bourgoin, Cottin-Bizonne, Ybert and Volk2018) showed that diffusio-phoretic drift can induce an effectively compressible colloid flow, even in an incompressible solvent, leading to transient particle focusing. Similarly, Chu et al. (Reference Chu, Garoff, Tilton and Khair2020) studied colloid transport under transient solute gradients where spatially varying drift produces apparent compression or expansion, though the underlying fluid remains incompressible.

Given the boundary conditions that we assume in this problem, and the governing Laplace equation, if $\tilde {C}_0$ is a constant and the concentration field $\tilde {C}$ is a solution, then the excess concentration $\tilde {c} = \tilde {C} - \tilde {C_0}$ is a solution too. The boundary conditions (2.1) and (2.2) do not depend on the absolute value of the local concentration. For all $\tilde {C}_0$ , the resulting flow will be the same. This also implies that $\tilde {c}$ can be negative. After shifting by a positive  $\tilde {C}_0$ , we thus will still obtain a physical solution. In the following subsections, we will therefore replace concentration $\tilde {C}$ with field $\tilde {c}$ .

2.2. Non-dimensionalisation of the model

We have already defined $\mathcal{U}$ and $\mathcal{R}$ as the characteristic velocity and length. The choice of the characteristic length depends on the problem at hand – we discuss this in detail when defining specific problems, but in general it may be set e.g. by the size of an active patch on the wall. Since $\tilde {c}$ is the excess concentration, its variations scale with the magnitude of the gradients produced at confining boundaries, which are induced by surface chemistry. Its natural scale is therefore ${\mathscr{C}}=|\mathcal{A}|\,\mathcal{R}/\kappa$ , where $|\mathcal{A}|$ is the typical magnitude of the chemical surface activity. The characteristic pressure is constructed from the typical velocity scale, given by the mobility and the magnitude of concentration, and reads ${\mathscr{P}=}\eta\, |\mathcal{AM}|/\mathcal{R}\kappa$ . The dimensionless pressure, concentration and velocity fields are thus given by $P={\tilde {P}}/{\mathscr{P}}$ , $c={\tilde {c}}/{\mathscr{C}}$ and $\boldsymbol{u} = {\tilde {\boldsymbol{u}}}/{\mathcal{U}}$ . Similarly, the dimensionless activity and mobility are given by $A = \mathcal{A}/|\mathcal{A}|$ and $M = \mathcal{M}/|\mathcal{M}|$ . Since activity can vary over the surface $\mathcal{S}$ , the choice of the characteristic activity is not always obvious. Because $\textit{Pe},\textit{Re} \propto |\mathcal{A}|$ , choosing the maximal value of $|\mathcal{A}|$ on the surface $\mathcal{S}$ leads to the strongest conditions on $ \textit{Pe}$ and $\textit{Re}$ . For the diffusion of solute, the non-dimensional governing equations are

(2.7a) \begin{align}{\nabla} ^2 c &= 0, \end{align}

(2.7b) \begin{align} \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla \!}c &=-A\quad \textrm { on } \quad \mathcal{S}, \end{align}

while the flow problem becomes

(2.7c) \begin{align}{\nabla} ^2 \boldsymbol{u} &=\boldsymbol{\nabla \!}P, \end{align}

(2.7d) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} &= 0, \end{align}

(2.7e) \begin{align} \boldsymbol{u} &=M(\boldsymbol{1}-\boldsymbol{n}\boldsymbol{n})\boldsymbol{\cdot }\boldsymbol{\nabla \!}c\quad \textrm { on } \quad \mathcal{S}. \end{align}

In the following, we apply this general framework to the specific geometry of a narrowing corner to explore the ways in which fluid motion can be actuated within such a cavity by purely chemical means, with no moving mechanical parts.

3.1. Solute concentration

We first focus on the problem of diffusion of a solute in the corner domain. Upon introducing cylindrical polar coordinates $(\rho ,\theta )$ (see figure 1), the stationary Laplace’s equation for the concentration field $c$ takes the form

(3.1) \begin{equation} {\rho }\frac {\partial }{\partial \rho }\!\left (\rho \frac {\partial c}{\partial \rho }\right ) + \frac {\partial ^2 c}{\partial \theta ^2} = 0. \end{equation}

Suppose that the wedge extends in polar coordinates from $\theta =0$ to $\theta =\alpha$ . The boundary conditions are imposed on the normal gradient of concentration, $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla \!}c$ , and are thus given by the chemical activity distribution on the walls, $A_0(\rho )$ and $A_\alpha (\rho )$ , as

(3.2) \begin{equation} \frac {1}{\rho }\frac {\partial c}{\partial \theta } = -A_0(\rho ), \end{equation}

if the condition is imposed on the wall $\theta =0$ , and

(3.3) \begin{equation} \frac {1}{\rho }\frac {\partial c}{\partial \theta } = A_\alpha (\rho ), \end{equation}

if the condition is imposed on the wall $\theta =\alpha$ . For the two-dimensional diffusion equation to be well-posed, we require the integrals of both activity functions over the walls to add to zero, as we discuss later in the paper.

Figure 1. Geometry of the diffusio-osmotic corner flow set-up in polar coordinates $(\rho ,\theta )$ . In a wedge of opening angle $\alpha$ , an active patch on the $\theta =0$ wall covering the radial section $\rho \in [a,b]$ releases solute (grey arrows) and generates an inhomogeneous concentration field that drives circulatory flow indicated by schematic streamlines.

3.2. Stokes flow

Once the concentration distribution $c(\rho ,\theta )$ is known, the slip flow velocity $\boldsymbol{u}_s = u_{\!s} \boldsymbol{e}_{\!\rho}$ on the boundary is determined as

(3.4) \begin{equation} u_{\!s} = M\!\left .\frac {\partial c}{\partial \rho }\right |_{\mathcal{S}}, \end{equation}

which becomes the boundary condition for a two-dimensional flow in the fluid domain. In this problem, it is convenient to introduce the two-dimensional stream function $\varPsi (\rho ,\theta )$ (Batchelor Reference Batchelor2000; Deville Reference Deville2022) such that in polar coordinates,

(3.5) \begin{equation} u_{\!\rho} = \frac {1}{\rho }\frac {\partial \varPsi }{\partial \theta }, \quad u_\theta = -\frac {\partial \varPsi }{\partial \rho }. \end{equation}

Since $\boldsymbol{u}=(u_{\!\rho} ,u_\theta )$ satisfies the incompressible Stokes equations (2.7c )–(2.7d ), $\varPsi$ obeys the biharmonic equation

(3.6) \begin{equation}{\nabla} ^4 \varPsi = 0, \end{equation}

with the boundary conditions on the two planes $\theta = \{0 , \alpha \}$ being

(3.7) \begin{equation} u_{\!\rho} = \frac {1}{\rho }\frac {\partial \varPsi }{\partial \theta } = u_{\!s}, \quad u_\theta = -\frac {\partial \varPsi }{\partial \rho } = 0, \quad \text{for} \quad \theta = \{0 , \alpha \}. \end{equation}

3.3. Solution of Laplace and biharmonic equations in the Mellin space

We now introduce the formalism of Mellin transforms (Butzer & Jansche Reference Butzer and Jansche1997; Debnath & Bhatta Reference Debnath and Bhatta2016) to solve the more general case of a single active sector. Indeed, Mellin transforms have become the traditional method for solving boundary-value problems in wedge-shaped regions (Tranter Reference Tranter1948; Moffatt Reference Moffatt1964a , Reference Moffattb ; Martin Reference Martin2017). In cylindrical coordinates, Laplace’s equation takes the form (3.1). We denote by $\bar {c}(p,\theta )$ the Mellin transform $\mathscr{M}_{\!p}$ of concentration $c(\rho ,\theta )$ , where $p$ is the transform variable, and

(3.8) \begin{equation} \mathscr{M}_{\!p}\{c(\rho ,\theta )\}:= \int _{0}^{\infty }\! \rho ^{p-1}\, c(\rho ,\theta ) \,\mathrm{d}\rho =: \bar {c}(p,\theta ). \end{equation}

In addition, $\mathscr{M}^{-1}$ is the inverse Mellin transform, which reads

(3.9) \begin{equation} c(\rho ,\theta )=\mathscr{M}^{\kern1pt -1}\{\bar {c}(p,\theta )\}=\frac {1}{2\pi\! i}\int _{\gamma -\text{i}\infty }^{\gamma +\text{i}\infty } \bar {c}(p,\theta )\, \rho ^{-p}\, \mathrm{d} p. \end{equation}

The conditions for the existence of the inverse transform are that for a chosen parameter  $\gamma$ , the integral $\int _{0}^{\infty }\rho ^{\gamma -1}\, c(\rho ,\theta ) \,{\rm d}\!\rho$ exists, and the complex integration line $(\gamma - \text{i} \infty , \gamma + \text{i}\infty )$ lies within the strip of analyticity of $\bar {c}(p,\theta )$ . When these conditions are satisfied, the inverse is independent of $\gamma$ . Unless stated otherwise, we take $\gamma =0$ . Then, for appropriate $c(\rho ,\theta )$ , we have

(3.10) \begin{equation} \mathscr{M}_{\!p}\{\rho\, \partial _\rho c(\rho ,\theta )\}=\mathscr{M}_{\! p+1}\{ \partial _\rho c(\rho ,\theta )\}=-p \bar {c}(p,\theta ), \end{equation}

where we introduced an abbreviation for derivatives, $\partial _\rho \equiv \partial /\partial \rho$ , etc. The Laplace equation for concentration, (3.1), implies that $p^2\bar {c}+{\partial ^2_\theta } \bar {c}=0$ , and for some $C(p)$ , $D(p)$ , we have

(3.11) \begin{equation} \bar {c}(p,\theta )= C(p)\, \text{e}^{\text{i}p\theta }+ D(p)\,\text{e}^{-\text{i}p\theta }. \end{equation}

We now turn to the flow problem. Let $\varPsi$ be the stream function. The biharmonic equation ${\nabla} ^4 \varPsi =0$ in cylindrical coordinates and after applying the Mellin transform $\mathscr{M}_{\! p+4}$ becomes

(3.12) \begin{equation} \left \{\partial _\theta ^4 + [(p+2)^2 +p^2]\,\partial _\theta ^2 + p^2(p+2)^2\right \} \bar {\varPsi }(p,\theta )=0, \end{equation}

where we write $\bar {\varPsi }(p,\theta ):=\mathscr{M}_{\! p}\{ \varPsi (\rho , \theta )\}$ . The general solution then reads

(3.13) \begin{equation} \bar {\varPsi }= F_1(p) \cos (p\theta )+ F_2(p)\sin (p\theta )+G_1(p)\cos ((p+2)\theta )+ G_2(p)\sin ((p+2)\theta ). \end{equation}

4.1. A single active wall

In the first simple example, we consider a single active wall, with constant activity $A$ on the surface. To ensure the existence of a steady-state solution, we assume that the other boundary is absorbing, thus we can write the boundary conditions for the concentration problem as

(4.1) \begin{align} \frac {\partial c}{\partial \theta }= -A\rho \quad &\text{for } \quad \theta = 0, \end{align}

(4.2) \begin{align} c = 0 \quad &\text{for } \quad \theta = \alpha. \end{align}

Using the separated form $c=R(\rho )\,\varTheta (\theta )$ as an ansatz, we find the general solution

(4.3) \begin{equation} R(\rho ) = {r_1} \rho ^\lambda + {r_2} \rho ^{-\lambda }, \quad \varTheta (\theta ) = {q_1}\cos \lambda \theta + {q_2} \sin \lambda \theta , \end{equation}

where $r_1$ , $r_2$ , $q_1$ , $q_2$ , $\lambda$ are constants that need to be determined. Applying the boundary conditions, we obtain the concentration field as

(4.4) \begin{equation} c(\rho ,\theta ) =- \frac {A \rho \sin (\theta - \alpha )}{\cos \alpha }. \end{equation}

The fact that the concentration increases linearly with the distance from the tip is a consequence of the fixed flux boundary condition and the fact that the active patch extends to infinity. In reality, the finite size of the active patch limits the concentration field, as we discuss later. As a result, the tangential gradient of the concentration profile yields a constant slip velocity distribution on the surface,

(4.5) \begin{align} \boldsymbol{u}= \textit{MA} \tan \alpha \ \boldsymbol{e}_{\!\rho} \quad &\text{for } \quad \theta = 0, \end{align}

(4.6) \begin{align} \boldsymbol{u}=\boldsymbol{0} \quad &\text{for } \quad \theta = \alpha. \end{align}

With this velocity profile, the problem becomes identical to the well-known Taylor’s scraper (Taylor Reference Taylor1962), with the imposed slip velocity $U=MA\tan \alpha$ . The stream function then has the form

(4.7) \begin{equation} \varPsi = U\! \rho {F}(\theta ), \end{equation}

where ${F}(\theta )$ satisfies an ordinary differential equation

(4.8) \begin{equation} F{''''} + 2 F{''} + F = 0, \end{equation}

with the boundary conditions $F(0) = 0$ , $F'(0)=1$ , $F(\alpha )=0$ , $F'(\alpha )=0$ . We note that by (3.7), the boundary conditions for $F$ correspond to the normal velocity at the surface, while those for $F'$ pertain to the induced slip (tangential) velocity component. This is a particular form of the general solution to planar elasticity problems for the biharmonic Airy stress function due to Michell (Reference Michell1899). The resulting stream function reads

(4.9) \begin{equation} \varPsi = U\! \rho {F}(\theta ) = \frac {U\! \rho }{\alpha ^2 - \sin ^2 \alpha }\left [ \alpha (\alpha -\theta )\sin \theta -\theta \sin \alpha \sin (\alpha -\theta ) \right ]. \end{equation}

The resulting flow fields are drawn in figure 2 for three values of the wedge opening angle $\alpha =\{\pi /4,\pi /2,3\pi /4\}$ . The flow is asymmetric, reflecting the boundary conditions. We observe the strongest flow along the active boundary, and a gradual decay of the velocity field closer to the other, no-slip wall. We note that the magnitude of the bulk velocity field remains a large fraction of the surface slip velocity, confirming the pronounced effect of surface actuation.

Figure 2. Diffusio-osmotic flow induced by the activity of one phoretic wall in a wedge of angle $\alpha$ for (a–c) $\alpha =\{\pi /4,\pi /2,3\pi /4\}$ , respectively. Flow streamlines are marked in white. The colour map indicates the total velocity magnitude. The emergent bulk flow remains comparable in magnitude to the driving slip flow on the active boundary, and decays rapidly close to the inert, no-slip wall.

4.2. Two phoretic walls

A simple generalisation of the problem of one wall involves covering both walls of the wedge with catalyst, thereby imposing a slip velocity symmetrically for both $\theta = 0$ and $\theta =\alpha$ . In this case, we expect the flow to be symmetrical about the wedge bisector.

The concentration profile satisfying the fixed-flux boundary conditions at both walls, namely,

(4.10a) \begin{align} \frac {\partial c}{\partial \theta } =-A\rho \quad &\textrm {for} \quad \theta =0, \end{align}

(4.10b) \begin{align} \frac {\partial c}{\partial \theta } =A\rho \quad &\textrm {for} \quad \theta =\alpha , \end{align}

follows directly as

(4.11) \begin{equation} c(\rho ,\theta ) = A \rho \dfrac {\sin (\theta -\alpha )- \sin \theta }{1-\cos \alpha }. \end{equation}

Equation (4.8) remains valid with the new boundary conditions $F(0)=F(\alpha )=0$ (no normal velocity at the surface) and $F'(0)=F'(\alpha )=1$ (two active walls with non-zero slip velocity). The stream function in this case takes on the form

(4.12) \begin{equation} \varPsi = \frac {U\! \rho }{\alpha -\sin \alpha }\left [ (\alpha - \theta )\sin \theta -\theta \sin (\alpha -\theta ) \right ]. \end{equation}

Figure 3. Diffusio-osmotic flow induced by the activity of two phoretic walls in a wedge of angle $\alpha =\{\pi /4,\pi /2,3\pi /4\}$ . The colour map indicates the total velocity magnitude. We note a strong flow close to the driving active boundaries, and a counterflow along the wedge bisector.

The resulting flow fields are drawn in figure 3 for three values of the wedge opening angle $\alpha =\{\pi /4,\pi /2,3\pi /4\}$ . The flow is symmetrical, with the fluid dragged along the walls towards the wedge vertex, and expelled along the bisector.

5.1. The case $a\neq 0$ and $b\lt \infty$

The solution in Mellin space is

(5.2) \begin{equation} \bar {c}(p,\theta )=A\frac {b^{p+1}-a^{p+1}}{p(p+1)} \frac {\sin (p(\alpha -\theta ))}{\cos p\alpha }. \end{equation}

It is assumed that $0\lt \alpha \lt 2\pi$ . The solution has singularities at $p'_k$ such that $p'_k\alpha ={\pi }/{2}+k\pi$ for $k\in \mathbb{Z}$ , and also at $p_0=-1$ , and a removable singularity at $p=0$ .

To evaluate the concentration field, we must now invert the Mellin transform. The concentration in the real space can be written as $c=I_b-I_a$ , where

(5.3) \begin{align} f_{\!a} &=A\frac {a}{p(p+1)} \frac {\sin (p(\alpha -\theta ))}{\cos p\alpha } \left (\frac {a}{\rho }\right )^p \!, \end{align}

(5.4) \begin{align} I_a &=\frac {1}{2\pi\! i}\int _{\gamma -\text{i}\infty }^{\gamma +\text{i}\infty } f_{\!a}\, {\rm d}\!p. \end{align}

The complex integral above can be evaluated by applying Cauchy’s residue theorem to a square integration contour sketched in figure 4. For $a \leqslant \rho$ , the integration contour lies in the half-plane $\textit{Re}(p) \gt \gamma$ . For $a\gt \rho$ , the appropriate contour lies in the $\textit{Re}(p) \lt \gamma$ half-plane. Such a choice ensures that the integral over three sides of the square contour vanishes when the contour is extended to infinity.

Figure 4. Contours of integration for the evaluation of the inverse Mellin transform. The green contour is used for $a\gt \rho$ , and the blue contour for $a\leqslant \rho$ . Both integration contours are shifted by $\gamma$ along the real axis, and the poles of the integrand are denoted by $p_k$ , where $k\in \mathbb{Z}$ . To evaluate the integral, one takes the limit of the square side length approaching infinity. In the limit, contributions from the three dashed sides of each square contour vanish, and the desired integral along the imaginary axis can be evaluated using the method of residues.

In particular, for the integral in (5.4), we select $\gamma =0$ . The poles are listed as

(5.5) \begin{equation} p_k= \begin{cases} \displaystyle {\alpha ^{-1}\!\left (\frac {\pi }{2}+(k-1)\pi \right )} & \text{for } \quad k\in \mathbb{Z}_+,\\ -1 & \text{for } \quad k=0,\\ \displaystyle \alpha ^{-1}\!{\left (\!-\frac {\pi }{2}+(k+1)\pi \right )} & \text{for } \quad k\in \mathbb{Z}_-. \end{cases} \end{equation}

Under the assumption that for all $k$ , $(({ {\pi }/{2})+k\pi })/{\alpha }\neq -1$ (i.e. $\alpha \neq {\pi }/{2},\pi , {3\pi }/{2}$ ), the residues are

(5.6a) \begin{align} \textrm {for }& \quad k\neq 0, \quad \mathop {\mathrm{Res}}_{p = p_k} f_{\!a}= -\frac {Aa}{\alpha p_k(p_k +1)}\cos (p_k\theta ) \left (\frac {a}{\rho }\right ) ^{p_k}\!, \end{align}

(5.6b) \begin{align} \textrm {for }& \quad k=0, \quad \mathop {\mathrm{Res}}_{p = p_k} f_{\!a}=A \frac {\sin (\alpha -\theta )}{\cos \alpha } \rho . \end{align}

Depending on the ratio ${a}/{\rho }$ , different residues are contained in the integration contour, and the orientation of the contour is different, hence

(5.7) \begin{equation} I_a(\rho ,\theta )= \begin{cases} \phantom {-} \sum _{k\leqslant 0} {\mathrm{Res}}_{p = p_k} f_{\!a} & \text{for } \quad a\gt \rho,\\[6pt] - \sum _{k\geqslant 1} {\mathrm{Res}}_{p = p_k} f_{\!a} & \text{otherwise,} \end{cases} \end{equation}

with the full solution being

(5.8) \begin{equation} c(\rho ,\theta )=I_b(\rho ,\theta )-I_a(\rho ,\theta ). \end{equation}

Note that the residue at $p=-1$ is responsible for satisfying the condition (5.1a ). In the special cases $\alpha = {\pi }/{2},\pi , {3\pi }/{2}$ , solutions can be obtained by taking, for example, the limit of $\alpha \to {\pi }/{2}$ . The solutions for $\alpha ={\pi }/{2}-\epsilon$ and $\alpha ={\pi }/{2}+\epsilon$ can then be compared to check if the desired accuracy has been achieved for a chosen small parameter $\epsilon$ .

5.2. The case $a=0$ , $b\lt \infty$ , i.e. sector $[0,b]$

The solution is obtained similarly to the previous case as

(5.9) \begin{equation} c(\rho ,\theta )= \begin{cases} \phantom {-} \sum _{k\leqslant 0} {\mathrm{Res}}_{p = p_k} f_b & \text{for } \quad b\gt \rho ,\\[6pt] - \sum _ {k\geqslant 1} {\mathrm{Res}}_{p = p_k} f_b & \text{otherwise.} \end{cases} \end{equation}

5.3. The case $a\neq 0$ and $b=\infty$ , i.e. sector $[a,\infty ]$

In this case, existence of the Mellin transform requires that ${\gamma } \lt -1$ .

We thus take any $\gamma$ such that $-1 \gt \gamma \gt -{{\pi }/{2\alpha }}$ , and write the solution as

(5.10) \begin{equation} c(\rho ,\theta )= \begin{cases} - \sum _{k\leqslant -1} {\mathrm{Res}}_{p = p_k} f_{\!a} & \text{for } \quad a\gt \rho,\\[6pt] \phantom {-} \sum _{k\geqslant 0} {\mathrm{Res}}_{p = p_k} f_{\!a} & \text{otherwise.} \end{cases} \end{equation}

A simple computation shows that in all the cases above, $c(\rho , \theta )$ with an appropriate choice of $\gamma$ satisfies the condition for the existence of the inverse transform.

In figure 5, we sketch the solutions for the concentration field for a chosen wedge opening angle $\theta =\pi /6$ in three representative cases. The catalytic sector is marked in red, and the colours indicate the absolute value of solute concentration. In all cases, the resulting concentration field is heterogeneous, which offers the possibility to drive slip flow along the active boundary.

Figure 5. (a–c) Solute concentration fields, and (d–f) isolines of the stream function $\psi$ , for an ideally absorptive wall at $\theta =\pi /6$ and a catalytic (active) wall at $\theta =0$ , with a catalytic sector at $(a,b)= \{ (1,3), (0,3), ({1},\infty )\}$ marked in red. Here, we assume $A=1$ , and the plotted radius of the wedge is $\rho \lt 4$ . The scale bar for the absolute concentration field $|c|$ is common for plots (a,b) and different for (c).

6.1. Solution for a single catalytic sector with reflective walls

Consider a setting in which the wall at $\theta = 0$ contains a catalytic sector with activity $A$ and the wall at $\theta =\alpha$ is not active, thus it is a reflective wall, with no-flux boundary condition. We denote the corresponding solute concentration field by $c_0$ . The boundary conditions are then

(6.1a) \begin{align} \frac {\partial c_0}{\partial \theta }=-A\rho\, \unicode{x1D7D9}_{{[a,b]}} \quad &\textrm {for} \quad \theta =0, \end{align}

(6.1b) \begin{align} \frac {\partial c_0}{\partial \theta }=0 \quad &\textrm {for} \quad \theta =\alpha . \end{align}

We again assume that $0\lt \alpha \lt 2\pi$ . A solution satisfying these boundary conditions in the Mellin space reads

(6.2) \begin{equation} \bar {c}_0(p,\theta )=-A\frac {{b^{p+1}-a^{p+1}}}{p(p+1)} \frac {\cos (p(\alpha -\theta ))}{\sin (p\alpha )}. \end{equation}

If $c_\alpha$ is a solution for the active sector with activity $A$ on the wall $\theta =\alpha$ , i.e. for the boundary conditions

(6.3a) \begin{align} \frac {\partial c_\alpha }{\partial \theta }=0 \quad &\textrm {for} \quad \theta =0, \end{align}

(6.3b) \begin{align} \frac {\partial c_\alpha }{\partial \theta }=A\rho\, \unicode{x1D7D9}_{{[a,b]}} \quad &\textrm {for} \quad \theta =\alpha , \end{align}

then the corresponding solution takes the form

(6.4) \begin{equation} \quad \bar {c}_\alpha (p,\theta )=\bar {c}_0 (p,\alpha -\theta )=-A\frac {{b^{p+1}-a^{p+1}}}{p(p+1)} \frac {\cos (p\theta )}{\sin (p\alpha )}. \end{equation}

Let us introduce the notation

(6.5) \begin{align} g_{{a}}(p,\theta )&=-A\frac {a}{p(p+1)} \frac {\cos (p(\alpha -\theta ))}{\sin (p\alpha )}\left (\frac {a}{\rho }\right )^p \!, \end{align}

(6.6) \begin{align} I_{{a}}&=\frac {1}{2\pi\! i}\int _{\gamma -\text{i}\infty }^{\gamma +\text{i}\infty } g_{{a}}\, {\rm d}\!p. \end{align}

Applying the inverse Mellin transform to the solution (6.2), we can write

(6.7) \begin{equation} c_0=I_{{b}}-I_{{a}}=\frac {1}{2\pi i}\int _{\gamma -\text{i}\infty }^{\gamma +\text{i}\infty } (g_{{b}}-g_{{a}})\, {\rm d}\!p. \end{equation}

We now note that $g_{{a}}$ has poles at $p'_k={k\pi }/{\alpha }$ , with $k\in \mathbb{Z}$ , and at $p=-1$ . Under the assumption that for all $k$ , $p'_k\neq -1$ (i.e. $\alpha \neq \pi$ ), the residues are

(6.8a) \begin{align} \textrm {for }& \quad k\neq 0, \quad \mathop {\mathrm{Res}}_{p = p_k} g_{a}= -\frac {Aa}{\alpha } \frac {\cos (p_k(\alpha -\theta ))}{p_k(p_k +1)}\left (\frac {a}{\rho }\right ) ^{p_k} \!, \end{align}

(6.8b) \begin{align} \textrm {for }& \quad k=0, \quad \mathop {\mathrm{Res}}_{p = {0}} g_a= -\frac {Aa}{\alpha }\left [\log \left (\frac {{a}}{\rho }\right )-1\right ]\!, \end{align}

(6.8c) \begin{align} \textrm {for }& \quad p=-1, \quad \mathop {\mathrm{Res}}_{p = -1}g_a= -A\rho \frac {\cos (\alpha -\theta )}{\sin \alpha }. \end{align}

The choice of $\gamma$ such that $\textrm {max}\{p'_k: p'_k\lt 0\}=-{\pi }/{\alpha }\lt \gamma \lt 0$ , $\gamma \gt -1$ , guarantees that the Mellin transform of $c_0$ exists. Integrating over a square contour in one of the half-planes, we obtain

(6.9) \begin{equation} I_{{a}}(\rho ,\theta )= \begin{cases} \sum _{k\leqslant {-1}} {\mathrm{Res}}_{p = p'_k} g_a +{{\mathrm{Res}}_{p = -1}g_a} & \text{for} \quad {a\gt \rho},\\ -\sum _{k\geqslant {0}} {\mathrm{Res}}_{p = p'_k}g_a & \text{otherwise.} \end{cases} \end{equation}

Using (6.7), $c_0$ can now be computed straightforwardly. The total activity of the walls not being zero does not cause a contradiction, since there are no constraints on activity at $\rho =\infty$ , and the solute can be absorbed or emitted at infinity. The case $\alpha = \pi$ can be solved in Cartesian coordinates or by taking the limit $\alpha \to \pi$ in the above solution.

6.2. Diffusion for multiple catalytic sectors

By linearity, the solution can be obtained by superposition of solutions for single sectors. If sectors $[a_i, b_i]$ have activity $A_i$ , then the condition for equilibrium is

(6.10) \begin{equation} \mathop{\sum} \nolimits_ i A_i (b_i-a_i)=0, \end{equation}

where the sum is taken over all active sectors. This is also the condition for the solution to be physical (solute conservation), i.e. unless we allow the solute to be released/absorbed at $\rho =\infty$ . It can be seen that the resulting solutions are indeed continuous. Note that the sectors can overlap, so step activity profiles can be created to mimic arbitrary activity functions $A_0 (\rho )$ and $A_\alpha (\rho )$ of the walls.

An example of a setting involving multiple sectors is presented in figure 6 for two active patches of opposite activity on each of the walls, and for different wedge opening angles. The effect of activity of patches closer to the tip of the wedge is diminished by their mutual influence, and the concentration fields produced by patches further away from the confinement are more pronounced. This demonstrates the intricate interaction between the geometry of the wedge and the chemical activity of its walls.

Figure 6. (a,b) Solute concentration fields, and (c,d) corresponding stream functions $\psi$ , for wedges with multiple active sectors. The wedge angles are $\alpha = \{4\pi /7,3\pi /7 \}$ . There are two active sectors of opposite activity $|A|=1$ on each wall. They span the sections $\rho \in (0.5,1.5)$ and $\rho \in (2.5,3.5)$ . The plotted radius of the wedge is $\rho \lt 4$ . Patches of positive activity are marked in red, while those of negative activity are marked in blue. Panels (a,b) share the concentration scale bar placed in the middle of the figure, while the stream function scale bar for panels (c,d) is in the top right corner.

6.3. The diffusion problem for an arbitrary catalytic wall

Let us now consider a setting where the wall at $\theta = 0$ has arbitrary activity $A=A(\rho )$ , and the wall at $\theta =\alpha$ is reflective. We will approach this problem using the Green’s function method (Arfken, Weber & Harris Reference Arfken, Weber and Harris2013), because the Stokes equation is linear. Let $c^G_0 (\rho , \rho ', \theta )$ be the Green’s function, which is a formal solution to the Laplace equation with a point activity $\delta (\rho - \rho ')$ at $\rho = \rho '$ , $\theta =0$ . Boundary conditions for $c^G_0 (\rho , \rho ', \theta )$ take the form

(6.11a) \begin{align} \frac {\partial c^G_0}{\partial \theta }(\rho , \rho ', \theta )=-\rho \delta (\rho -\rho ') \quad \textrm {for} \quad \theta =0, \end{align}

(6.11b) \begin{align} \frac {\partial c^G_0}{\partial \theta }(\rho , \rho ', \theta )=0 \quad \textrm {for} \quad \theta =\alpha . \\[9pt] \nonumber \end{align}

The solution in the Mellin space reads

(6.12) \begin{equation} \bar {c}^G_0(p,\rho ',\theta )=-\frac {\cos (p(\alpha -\theta ))}{p \sin (p\alpha )} \rho '^p. \end{equation}

Here, $\bar {c}_0 (p,\theta )$ is obtained by accounting for all point activities $A(\rho ')\delta (\rho {-}\rho ')$ along the boundary:

(6.13) \begin{equation} \bar {c}_0 (p,\theta )=\int _0^\infty A(\rho ')\, \bar {c}^G_0(p,\rho ',\theta )\, {\rm d}\!\rho '. \end{equation}

Residues of $\bar {c}^G_0$ are $p_k={k\pi }/{\alpha }$ . We now change the order of integration in the back-transformed concentration field to arrive at

(6.14) \begin{equation} c_0 (\rho ,\theta )=\frac {1}{2\pi\! i}\int _0^\infty {\rm d}\!\rho ' A(\rho ') \int _{\gamma -\text{i}\infty }^{\gamma +\text{i}\infty } -\frac {\cos (p(\alpha -\theta ))}{p \sin (p\alpha )} \left ( \frac {\rho '}{\rho } \right )^p {\rm d}\!p, \end{equation}

where $\gamma$ has to satisfy the same condition as in the previous section. Defining

(6.15) \begin{equation} {h}(p,\rho ,\rho ',\theta )= -\frac {\cos (p(\alpha -\theta ))}{p \sin (p\alpha )} \left ( \frac {\rho '}{\rho } \right )^p \!, \end{equation}

we evaluate the following residues:

(6.16) \begin{equation} \mathrm{Res}_{p=p_k} {h} = \begin{cases} -\frac {1}{\alpha }\frac {\cos (p_k(\alpha -\theta ))}{p_k}\left ( \frac {\rho '}{\rho } \right )^{{p_k}} & \text{for} \quad k\neq 0,\\ -\frac {1}{\alpha } \log \left ( \frac {\rho '}{\rho } \right ) & \text{for} \quad {k=0}. \end{cases} \end{equation}

Choosing the contours of integration the same way as in the previous section, we finally obtain

(6.17) \begin{align} c_0 (\rho ,\theta )= \int _{{\rho }}^{{\infty }} {\rm d}\!\rho ' A(\rho ') \sum _{k\leqslant {-1}} {\mathrm{Res}}_{p_k}\, {h}(p,\rho ,\rho ',\theta ) - \int _{{0}}^{{\rho }} {\rm d}\!\rho ' A(\rho ') \sum _{k {\geqslant }0} {\mathrm{Res}}_{p_k}\,{h}(p,\rho ,\rho ',\theta ). \end{align}

We note that for two arbitrary catalytic walls, superposition of solutions can be used. The condition to satisfy solute conservation is

(6.18) \begin{equation} \int _0^\infty (A_0(\rho ) + A_\alpha (\rho ))\,{\rm d}\!\rho =0, \end{equation}

where $A_0 (\rho )$ and $A_\alpha (\rho )$ are the activities of the walls $\theta =0$ and $\theta =\alpha$ , respectively. It is worth noting that for practical purposes, approximating the solution for a given function $A(\rho )$ by making a superposition of solutions for constant activity sectors could potentially be more efficient than truncating the above series and evaluating the integrals.

Figure 7. (a) Isolines of the stream function $\psi$ of diffusio-osmotic corner flow for $\theta =\pi /3$ , with active sectors at $\rho \in (0,1)$ , emitting at $\theta = 0$ and absorbing at $\theta = \alpha$ . The flow in the corner eddy (yellow) is anticlockwise and drives clockwise rotation of another eddy further away from the corner. Colours code the magnitude of $\psi$ . (b) Transversal velocity profile $v_\theta (\rho )$ on the bisector angle of the wedge. Roots of the velocity indicate the centres of vortices.