2 Problem set-up and general theory

We consider the flow of a viscous fluid (density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D707}$ ) in the vicinity of a large obstacle of characteristic radius of curvature $R$ , which is described by a velocity field $\boldsymbol{u}(\boldsymbol{x})$ that is assumed to be fully known. The flow has a characteristic velocity scale $u_{0}$ and a characteristic length scale $\ell$ that, in general, may or may not be related to $R$ , cf. figure 1. As an example, gradients of flow past a large curved obstacle depend on the size of the obstacle (i.e. $\ell \sim R$ ). By contrast, in pressure-driven flow through a channel with plane walls ( $R\rightarrow \infty$ ), flow gradients are determined by the width of the channel $(\ell \ll R)$ . Similarly, hydrodynamic interactions in dilute suspensions are associated with flow gradients on the scale of the interparticle separation $\ell$ and are generally unrelated to the curvature of nearby walls (Swan & Brady Reference Swan and Brady2007).

At small Reynolds numbers ( $Re_{u}\equiv \unicode[STIX]{x1D70C}u_{0}\ell /\unicode[STIX]{x1D707}\ll 1$ ), the flow is described by the Stokes equations

(2.1a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{u}=\mathbf{0}\quad \text{and}\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D748}_{u}=-p_{u}\unicode[STIX]{x1D644}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}})$ is the stress tensor and $p_{u}$ is the pressure. Although the precise shape of the obstacle is not ultimately important for the formalism developed here, we consider two classes of obstacles widely encountered in physical systems. The first corresponds to solid obstacles on whose surface the flow must satisfy the no-slip boundary condition (denoted BC I), and the second corresponds to obstacles with free surfaces, e.g. a clean gas bubble, on which tangential components of stress vanish (denoted BC II). The boundary conditions satisfied by the flow ( $\boldsymbol{u},\unicode[STIX]{x1D748}_{u}$ ) on the surface $S_{w}$ of the obstacle are accordingly either

(2.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \text{BC I (no-slip)}:\quad \boldsymbol{u}=\mathbf{0},\quad \text{or}\\ \displaystyle \text{BC II (no-stress)}:\quad \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{m}=0,\quad \text{and}\quad (\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{u})\times \boldsymbol{m}=\mathbf{0}\end{array}\right\}\quad \text{on }S_{w},\end{eqnarray}$$

where $\boldsymbol{m}$ denotes the unit normal to the surface of the obstacle, directed into the fluid (figure 1). The flow ( $\boldsymbol{u},\,\unicode[STIX]{x1D748}_{u}$ ), which we refer to as the ‘background flow’, may satisfy some generally inhomogeneous boundary conditions at infinity.

We now introduce to the flow a rigid spherical particle of radius $a$ with a centre at a distance $h$ from the surface of the obstacle. The particle, whose centre is at a position $\boldsymbol{x}_{p}(t)$ translates with velocity $\boldsymbol{v}_{p}$ and rotates with angular velocity $\unicode[STIX]{x1D734}_{p}$ , as indicated in figure 1. Due to the motion of the particle, the velocity field is no longer $\boldsymbol{u}(\boldsymbol{x})$ , but rather

(2.3) $$\begin{eqnarray}\boldsymbol{U}(\boldsymbol{x})\equiv \boldsymbol{u}(\boldsymbol{x})+\boldsymbol{v}(\boldsymbol{x}),\end{eqnarray}$$

where $\boldsymbol{v}(\boldsymbol{x})$ is the disturbance flow due to the moving particle. Assuming that the disturbance velocity scale $v_{0}$ is at most of the order of $u_{0}$ , the net flow $\boldsymbol{U}$ also satisfies the Stokes equations, the boundary conditions at the wall (either BC I or BC II as appropriate) and the conditions at infinity satisfied by $\boldsymbol{u}$ . In addition, to observe the no-slip condition on the surface $S_{p}$ of the rigid particle, $\boldsymbol{U}$ must satisfy

(2.4) $$\begin{eqnarray}\boldsymbol{U}=\boldsymbol{v}_{p}+\unicode[STIX]{x1D734}_{p}\times \boldsymbol{r}\quad \text{on }S_{p},\end{eqnarray}$$

where $\boldsymbol{r}\equiv \boldsymbol{x}-\boldsymbol{x}_{p}$ is the position of a point in space relative to the particle centre.

The disturbance flow $\boldsymbol{v}$ is therefore governed by

(2.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}=\mathbf{0}\quad \text{and}\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0,\end{eqnarray}$$

(2.5c ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}=\boldsymbol{w}(\boldsymbol{x})\equiv \boldsymbol{v}_{p}-\boldsymbol{u}(\boldsymbol{x})+\unicode[STIX]{x1D734}_{p}\times \boldsymbol{r}\quad \text{for }\boldsymbol{x}\in S_{p},\end{eqnarray}$$

(2.5d ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}\rightarrow \mathbf{0}\quad \text{as }|\boldsymbol{r}|\rightarrow \infty \quad \text{and either}\end{eqnarray}$$

(2.5e ) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \text{BC I}:\quad \boldsymbol{v}=\mathbf{0},\quad \text{or}\\ \displaystyle \text{BC II}:\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{m}=0,\quad (\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{u})\times \boldsymbol{m}=\mathbf{0}\end{array}\right\}\quad \text{on }S_{w}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D748}_{v}=-p_{v}\unicode[STIX]{x1D644}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}\boldsymbol{v}+\unicode[STIX]{x1D735}\boldsymbol{v}^{\text{T}})$ is the stress tensor associated with the disturbance flow $\boldsymbol{v}$ (with $p_{v}$ being the pressure). Also, $\boldsymbol{w}(\boldsymbol{x})$ is the disturbance flow surface velocity distribution on the sphere required for the net flow $\boldsymbol{U}$ to satisfy the no-slip condition on the particle.

Since disturbance flow quantities depend on the local background flow properties through (2.5c ), $\boldsymbol{v}(\boldsymbol{x})$ is generally also time dependent along the trajectory of the particle. Inertial effects associated with this time dependence are small for $\mathit{Re}_{u}\ll 1$ and are neglected here, making the disturbance flow $\boldsymbol{v}(\boldsymbol{x})$ quasi-steady and the time dependence of the flow implicit; see Lovalenti & Brady (Reference Lovalenti and Brady1993, Reference Lovalenti and Brady1995) for a detailed discussion. We also note that the boundary conditions (2.5e ) assume that the introduction of the particle does not produce stresses that deform the surface of the obstacle. For a free surface in particular, this assumption only holds if the disturbance pressure is small relative to the Laplace–Young pressure jump across the interface. The deformation of the interface is much smaller than the particle size if the capillary number $\mathit{Ca}\equiv (\ell /a)^{2}\unicode[STIX]{x1D707}u_{0}/\unicode[STIX]{x1D6FE}\ll 1$ , $\unicode[STIX]{x1D6FE}$ being the surface tension of the interface $S_{w}$ . This condition is satisfied in typical micron-scale applications.

The flow $\boldsymbol{u}(\boldsymbol{x})$ and the geometry of the obstacle’s surface $S_{w}$ have so far not been specified. To make analytical progress, we restrict our attention to particles that are small relative to the ambient flow scale, i.e. $a/\ell \ll 1$ . In addition, we only consider obstacles $S_{w}$ with spherical boundaries (this includes planar surfaces), and thereby draw upon the well-developed literature for two-sphere hydrodynamic interactions (Stimson & Jeffery Reference Stimson and Jeffery1926; Brenner Reference Brenner1961; Maude Reference Maude1961). Henceforth, ‘sphere’ and ‘particle’ will both refer to the spherical particle $S_{p}$ .

With no loss of generality, we take $\boldsymbol{x}=\mathbf{0}$ to represent the point on $S_{w}$ closest to the surface of the particle $S_{p}$ and $\boldsymbol{e}_{\bot }$ to represent the unit normal to $S_{w}$ at this point, cf. figure 1. By construction, $\boldsymbol{e}_{\bot }$ points toward the centre of the particle, so that $\boldsymbol{x}_{p}=h\boldsymbol{e}_{\bot }$ . It is also convenient to introduce cylindrical coordinates $(r,\unicode[STIX]{x1D703},z)$ with origin at the centre of the particle as indicated in figure 1. The position of a point relative to the particle centre is then $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}_{p}=r\boldsymbol{e}_{r}+z\boldsymbol{e}_{z}$ ; note that $\boldsymbol{e}_{z}=\boldsymbol{e}_{\bot }$ . Although our primary focus in on the case where the particle is small relative to the obstacle ( $a/R\ll 1$ ), the formalism that we develop here is valid for arbitrary $a/R$ and can therefore be applied more generally to sphere–sphere interactions.

Gradients of $\boldsymbol{u}(\boldsymbol{x})$ are set by $\ell$ , while those of $\boldsymbol{v}(\boldsymbol{x})$ are determined by the radius of the particle $a$ , owing to the no-slip condition (2.5c ) on its surface. The background velocity field $\boldsymbol{u}(\boldsymbol{x})$ in the vicinity of a small particle ( $a/\ell \ll 1$ ) can be approximated by a Taylor expansion about its centre. To quadratic order in distance from the particle centre, we can write

(2.6) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x})=\boldsymbol{u}|_{\boldsymbol{x}_{p}}+\boldsymbol{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}|_{\boldsymbol{x}_{p}}+\frac{1}{2}\boldsymbol{r}\boldsymbol{r}\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{u}|_{\boldsymbol{x}_{p}}+O\left(u_{0}\frac{a^{3}}{\ell ^{3}}\right). & & \displaystyle\end{eqnarray}$$

Then, the surface velocity distribution $\boldsymbol{w}(\boldsymbol{x})$ in (2.5c ) is

(2.7) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x})=\boldsymbol{w}^{(0)}+\boldsymbol{w}^{(1)}+\boldsymbol{w}^{(2)}+\cdots \,,\end{eqnarray}$$

with

(2.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{w}^{(0)}\equiv \boldsymbol{v}_{p}-\boldsymbol{u}|_{\boldsymbol{x}_{p}}, & \displaystyle\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{w}^{(1)}\equiv -\boldsymbol{r}\boldsymbol{\cdot }\unicode[STIX]{x1D640}+\left(\unicode[STIX]{x1D734}_{p}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D74E}\right)\times \boldsymbol{r}, & \displaystyle\end{eqnarray}$$

(2.8c ) $$\begin{eqnarray}\displaystyle & \boldsymbol{w}^{(2)}\equiv -\boldsymbol{r}\boldsymbol{r}\boldsymbol{ : }\unicode[STIX]{x1D642}, & \displaystyle\end{eqnarray}$$

(2.9a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D640}\equiv {\textstyle \frac{1}{2}}(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}})|_{\boldsymbol{x}_{p}},\quad \unicode[STIX]{x1D74E}\equiv \unicode[STIX]{x1D735}\times \boldsymbol{u}|_{\boldsymbol{x}_{p}}\quad \text{and}\quad \unicode[STIX]{x1D642}\equiv {\textstyle \frac{1}{2}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{u}|_{\boldsymbol{x}_{p}}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D640}$ , $\unicode[STIX]{x1D74E}$ and $\unicode[STIX]{x1D642}$ are background flow properties evaluated at the particle centre $\boldsymbol{x}_{p}$ , viz. the rate-of-strain tensor $(\unicode[STIX]{x1D640})$ , the vorticity ( $\unicode[STIX]{x1D74E}$ ) and a rank-3 tensor ( $\unicode[STIX]{x1D642}$ ) that quantifies the curvature of the background flow (Nadim & Stone Reference Nadim and Stone1991).

The hydrodynamic force on the particle under the net flow $\boldsymbol{U}=\boldsymbol{u}+\boldsymbol{v}$ is given by $\boldsymbol{F}=\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}\,\text{d}S,$ where $\boldsymbol{n}$ is the unit normal on the particle surface directed into the fluid. Due to linearity, each $\boldsymbol{w}^{(i)}$ in (2.8) can be associated with a flow ( $\boldsymbol{v}^{(i)},\unicode[STIX]{x1D748}_{v}^{(i)}$ ). The force on the particle is then expressible as

(2.10) $$\begin{eqnarray}\boldsymbol{F}=\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}\,\text{d}S=\boldsymbol{F}^{(0)}+\boldsymbol{F}^{(1)}+\boldsymbol{F}^{(2)}+\cdots \,,\end{eqnarray}$$

where $\boldsymbol{F}^{(i)}=\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}^{(i)}\,\text{d}S$ is the force associated with the surface velocity distribution $\boldsymbol{w}^{(i)}$ in (2.8). Thus, $\boldsymbol{F}^{(0)}$ is the hydrodynamic force due to the translation of the sphere relative to the mean local background velocity, $\boldsymbol{F}^{(1)}$ is due to the background velocity gradient and the rotation of the sphere and $\boldsymbol{F}^{(2)}$ is the contribution of the curvature of the background velocity field in the vicinity of the particle. In the absence of the boundary $S_{w}$ (i.e. in an infinite medium) $\boldsymbol{F}^{(0)}$ reduces to the Stokes drag, while $\boldsymbol{F}^{(2)}$ becomes the Faxén contribution to the force, proportional to $\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}$ . By contrast, the $\boldsymbol{F}^{(1)}$ contribution, which depends on gradient of the background velocity, is identically zero for a sphere in an unbounded fluid (Happel & Brenner Reference Happel and Brenner1965; Batchelor Reference Batchelor1970). However, the introduction of the boundary $S_{w}$ breaks the symmetry of the disturbance flow $\boldsymbol{v}$ around the sphere, making $\boldsymbol{F}^{(1)}$ generally non-zero, as we show in subsequent sections.

If we use the representative scale $v_{p}$ for the translational speed of the particle and take length scales $\ell$ and $a$ , respectively, to measure gradients of $\boldsymbol{u}$ and $\boldsymbol{v}$ , we obtain from (2.8) and (2.10) that the different force contributions scale as

(2.11a-c ) $$\begin{eqnarray}\boldsymbol{F}^{(0)}=O(\unicode[STIX]{x1D707}a(v_{p}-u_{0})),\quad \boldsymbol{F}^{(1)}=O\left(\unicode[STIX]{x1D707}au_{0}\frac{a}{\ell }\right)\quad \text{and}\quad \boldsymbol{F}^{(2)}=O\left(\unicode[STIX]{x1D707}au_{0}\left(\frac{a}{\ell }\right)^{2}\right).\end{eqnarray}$$

The force on a sphere moving in free space in the Stokes flow limit is described exactly by a combination of the Stokes drag $\boldsymbol{F}^{(0)}$ and the Faxén contribution $\boldsymbol{F}^{(2)}$ , i.e. Faxén’s law. In the presence of a nearby boundary, however, $\boldsymbol{F}^{(1)}$ is non-zero and can, in particular, exceed $\boldsymbol{F}^{(2)}$ since it is nominally $O(\ell /a)$ greater than the Faxén force contribution.

Here, we seek to develop analytical expressions for the force on the particle for arbitrary separations $h$ (relative to $a$ ) between the particle and the obstacle. We further focus our attention on the force normal to the obstacle, which, as we will show, can be obtained by exploiting the geometric symmetry. The computations for the force parallel to the wall are beyond the focus of this paper, but can be obtained using the techniques presented in the following sections.

3 A reciprocal relation for the force normal to the obstacle

In this paper, we focus on the hydrodynamic force normal to the surface $S_{w}$ of the obstacle, $F_{z}=\boldsymbol{F}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ , for which we use the reciprocal theorem for Stokes flows. To this effect, we introduce as a model problem the Stokes flow due to a spherical particle translating normal to the surface of a stationary spherical obstacle (either no slip or no stress) at a velocity $V^{\prime }$ in a quiescent background fluid of viscosity $\unicode[STIX]{x1D707}$ . As in the main problem introduced in § 2, we take the radius of the spherical particle to be $a$ , the radius of the obstacle as $R$ and the perpendicular distance of the particle’s centre to the surface of the obstacle as $h$ .

It is convenient to describe the model problem in dimensionless form using the characteristic scales $a$ , $V^{\prime }$ and $\unicode[STIX]{x1D707}V^{\prime }/a$ , respectively, to scale distances, velocities and stresses. Defining $\tilde{\boldsymbol{r}}=\boldsymbol{r}/a$ , and dimensionless model velocity and stress fields by $\boldsymbol{v}^{\prime }$ and $\unicode[STIX]{x1D748}^{\prime }$ , respectively, the model problem satisfies

(3.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=\mathbf{0}\quad \text{and}\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}^{\prime }=0,\end{eqnarray}$$

(3.1c ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}^{\prime }=\boldsymbol{e}_{z}\quad \text{on }S_{p},\end{eqnarray}$$

(3.1d ) $$\begin{eqnarray}\displaystyle |\boldsymbol{v}^{\prime }|\rightarrow 0\quad \text{as }|\tilde{\boldsymbol{r}}|\rightarrow \infty \quad \text{and either}\end{eqnarray}$$

(3.1e ) $$\begin{eqnarray}\displaystyle \left.\!\!\!\!\begin{array}{@{}l@{}}\displaystyle \text{BC I}:\quad \boldsymbol{v}^{\prime }=\mathbf{0}\quad \text{or}\\ \displaystyle \text{BC II}:\quad \boldsymbol{v}^{\prime }\boldsymbol{\cdot }\boldsymbol{m}=0,\quad (\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime })\times \boldsymbol{m}=\mathbf{0}\end{array}\right\}\quad \text{on }S_{w}.\end{eqnarray}$$

Solutions to the model problem (both for the no-slip and stress-free cases) depend on the dimensionless geometric ratios (i) $h/a$ , which quantifies the separation distance of the particle from the surface $S_{w}$ of obstacle and (ii) the curvature ratio $a/R$ . Exact solutions to the model problem can be constructed for arbitrary $h/a$ and $a/R$ using known results (Brenner Reference Brenner1961; Maude Reference Maude1961; Adamczyk, Adamczyk & Van de Ven Reference Adamczyk, Adamczyk and Van de Ven1983) and are summarized in appendix A. An approach similar to the one used in this paper was adopted to evaluate wall effects on the motion of a sphere in a viscoelastic fluid (Becker, McKinley & Stone Reference Becker, McKinley and Stone1996).

Note that while the model flow ( $\boldsymbol{v}^{\prime },\unicode[STIX]{x1D748}^{\prime }$ ) is expressed in dimensionless form, we continue to express the main disturbance flow ( $\boldsymbol{v},\unicode[STIX]{x1D748}_{v}$ ) in dimensional terms due to its dependence on the arbitrary background flow $\boldsymbol{u}(\boldsymbol{x})$ . For the two Stokes flows $\boldsymbol{v}$ and $\boldsymbol{v}^{\prime }$ the reciprocal theorem has the form

(3.2) $$\begin{eqnarray}\int _{S_{p}+S_{w}+S_{\infty }}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}\boldsymbol{\cdot }\boldsymbol{v}^{\prime }\,\text{d}S=\frac{\unicode[STIX]{x1D707}}{a}\int _{S_{p}+S_{w}+S_{\infty }}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{v}\,\text{d}S,\end{eqnarray}$$

where $S_{\infty }$ refers to a bounding surface at infinity and $\boldsymbol{n}$ is the unit normal directed into the fluid. The factor $\unicode[STIX]{x1D707}/a$ in (3.2) is due to the definition of the dimensionless stress tensor $\unicode[STIX]{x1D748}^{\prime }$ in the model problem. The integral over $S_{\infty }$ vanishes since $\boldsymbol{v}$ and $\boldsymbol{v}^{\prime }$ decay to zero as $|\boldsymbol{r}|\rightarrow \infty$ . Additionally, both $(\boldsymbol{v},\unicode[STIX]{x1D748}_{v})$ and $(\boldsymbol{v}^{\prime },\unicode[STIX]{x1D748}^{\prime })$ either have zero velocities on a rigid wall, or vanishing tangential stress and normal velocity components at a plane interface, due to which the surface integrals on $S_{w}$ in (3.2) are identically zero. On applying the boundary conditions for $\boldsymbol{v}$ and $\boldsymbol{v}^{\prime }$ on $S_{p}$ , the reciprocal relationship gives the normal component of the hydrodynamic force on the sphere as

(3.3) $$\begin{eqnarray}F_{z}=\boldsymbol{e}_{z}\boldsymbol{\cdot }\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{v}\,\text{d}S=\unicode[STIX]{x1D707}a\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{w}(\boldsymbol{x})\,\text{d}\tilde{S},\end{eqnarray}$$

where $\text{d}\tilde{S}\equiv a^{-2}\,\text{d}S$ is a dimensionless differential area element and we recall that $\boldsymbol{e}_{z}=\boldsymbol{e}_{\bot }$ .

Using the decomposition of $\boldsymbol{w}(\boldsymbol{x})$ in (2.7) we can write

(3.4a ) $$\begin{eqnarray}\displaystyle F_{z}^{(0)} & = & \displaystyle \unicode[STIX]{x1D707}a\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{w}^{(0)}\,\text{d}\tilde{S}=\unicode[STIX]{x1D707}a\boldsymbol{w}^{(0)}\boldsymbol{\cdot }\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\,\text{d}\tilde{S},\end{eqnarray}$$

(3.4b ) $$\begin{eqnarray}\displaystyle F_{z}^{(1)} & = & \displaystyle \unicode[STIX]{x1D707}a\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{w}^{(1)}\,\text{d}\tilde{S}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D707}a^{2}\!\left\{-\unicode[STIX]{x1D640}\boldsymbol{ : }\!\int _{S_{p}}\tilde{\boldsymbol{r}}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime })\,\text{d}\tilde{S}+\left(\unicode[STIX]{x1D734}_{p}-\frac{1}{2}\unicode[STIX]{x1D74E}\right)\!\boldsymbol{\cdot }\!\int _{S_{p}}\tilde{\boldsymbol{r}}\times (\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime })\,\text{d}\tilde{S}\right\}\quad \text{and}\quad\end{eqnarray}$$

(3.4c ) $$\begin{eqnarray}\displaystyle F_{z}^{(2)} & = & \displaystyle \unicode[STIX]{x1D707}a\int _{S_{p}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{w}^{(2)}\,\text{d}\tilde{S}=-\unicode[STIX]{x1D707}a^{3}\unicode[STIX]{x1D642}\,\boldsymbol{\vdots }\int _{S_{p}}\tilde{\boldsymbol{r}}\tilde{\boldsymbol{r}}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime })\,\text{d}\tilde{S}.\end{eqnarray}$$

$F_{z}^{(i)}$

$i$

$\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }$

The above expressions involve only integrals on the particle surface $S_{p}$ . Thus, they are generally valid for an arbitrary particle suspended in a quadratic flow as long as the model flow $(\boldsymbol{v}^{\prime },\unicode[STIX]{x1D748}^{\prime })$ satisfies (3.1), although we note that the solution to the corresponding model problem may not be known for arbitrary geometries. We also remark that a more general application of the reciprocal theorem, with the appropriate choice of model problem, can be used to compute the full mobility matrix for a sphere near a boundary; see e.g. Becker et al. (Reference Becker, McKinley and Stone1996), Haber & Brenner (Reference Haber and Brenner1999). While solutions to the relevant model problems (wall-parallel translation and rotation of a sphere) are known (Dean & O’Neill Reference Dean and O’Neill1963; O’Neill Reference O’Neill1964), we do not attempt these more involved calculations here. Rather, we focus on the normal force component, which can be computed explicitly by utilizing the symmetry in the model problem (3.1).

3.1 Axisymmetry of the model problem

To exploit the spherical symmetry, we parametrize the surface of the particle in the $rz$ section through using intrinsic coordinates, see e.g. Stimson & Jeffery (Reference Stimson and Jeffery1926), Happel & Brenner (Reference Happel and Brenner1965). As shown in figure 2, we define $s$ as the dimensionless arc length in the $rz$ plane (non-dimensionalized using the particle radius $a$ ), such that $s=0$ and $s=\unicode[STIX]{x03C0}$ correspond, respectively, to the points on the sphere farthest and closest to $S_{w}$ , i.e.

(3.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle s=0\quad \text{at }(\tilde{r}=0,\,\tilde{z}=+1)\quad \text{and} & \displaystyle\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle s=\unicode[STIX]{x03C0}\quad \text{at }(\tilde{r}=0,\,\tilde{z}=-1). & \displaystyle\end{eqnarray}$$

The surface of the particle is then described by rotating the curve defined by $0<s<\unicode[STIX]{x03C0}$ about the $z$ -axis through an azimuthal angle $\unicode[STIX]{x1D703}$ of $2\unicode[STIX]{x03C0}$ . Under the change of coordinates, surface integrals transform as

(3.6) $$\begin{eqnarray}\int _{S_{p}}f\,\text{d}\tilde{S}=\int _{0}^{\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}f\,\tilde{r}\,\text{d}\unicode[STIX]{x1D703}\,\text{d}s\end{eqnarray}$$

for an arbitrary scalar field $f$ defined on the spherical particle surface $S_{p}$ .

Figure 2. Parametrization of the particle surface in polar coordinates ( $\tilde{r},\tilde{z}$ ), showing intrinsic coordinates ( $s$ , $n$ ). The evaluation of the force ultimately involves bipolar coordinates ( $\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}$ ), which are introduced in § 4. Here, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are dimensionless quantities relevant to the bipolar coordinate system that are related to $h/a$ and $R/a$ and are defined in § 4.

Due to the boundary conditions and the geometry (spherical particle and obstacle surfaces), the model problem is axisymmetric about the $z$ -axis. The $\unicode[STIX]{x1D703}$ integration in the surface integrals of (3.4), when written using (3.6), can then be immediately evaluated without explicit knowledge of $\unicode[STIX]{x1D748}^{\prime }$ . Defining

(3.7) $$\begin{eqnarray}\overline{\boldsymbol{w}}^{(i)}\equiv \frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}\left\{(\boldsymbol{e}_{r}\boldsymbol{\cdot }\boldsymbol{w}^{(i)})\boldsymbol{e}_{r}+(\boldsymbol{e}_{z}\boldsymbol{\cdot }\boldsymbol{w}^{(i)})\boldsymbol{e}_{z}\right\}\text{d}\unicode[STIX]{x1D703},\quad \text{with }i\in \{0,1,2\},\end{eqnarray}$$

one can write

(3.8) $$\begin{eqnarray}F_{z}^{(i)}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a\int _{0}^{\unicode[STIX]{x03C0}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\overline{\boldsymbol{w}}^{(i)}\;\tilde{r}\,\text{d}s\quad \text{on }S_{p}.\end{eqnarray}$$

Substituting (2.8) into (3.7) and using the symmetries of the tensors $\unicode[STIX]{x1D640}$ and $\unicode[STIX]{x1D642}$ , we obtain

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\boldsymbol{w}}^{(0)}=w_{z}^{(0)}\boldsymbol{e}_{z}, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\boldsymbol{w}}^{(1)}=-a\unicode[STIX]{x1D60C}_{zz}\left(-\frac{\tilde{r}}{2}\boldsymbol{e}_{r}+\tilde{z}\boldsymbol{e}_{z}\right)\quad \text{and} & \displaystyle\end{eqnarray}$$

(3.9c ) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\boldsymbol{w}}^{(2)}=-a^{2}\unicode[STIX]{x1D60E}_{zzz}\left\{-\tilde{r}\tilde{z}\boldsymbol{e}_{r}+\left(\tilde{z}^{2}-\frac{\tilde{r}^{2}}{2}\right)\boldsymbol{e}_{z}\right\}-a^{2}\unicode[STIX]{x1D60E}_{jjz}\frac{\tilde{r}^{2}}{2}\boldsymbol{e}_{z}\,\quad \text{with }j\in \{1,2,3\}, & \displaystyle\end{eqnarray}$$

where summation over $j$ is implied in $\unicode[STIX]{x1D60E}_{jjz}$ . We observe that the normal force only depends on certain components of the tensors $\unicode[STIX]{x1D640}$ and $\unicode[STIX]{x1D642}$ , in particular $\unicode[STIX]{x1D60C}_{zz}=\boldsymbol{e}_{z}\boldsymbol{e}_{z}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}|_{\boldsymbol{x}_{p}}$ , $\unicode[STIX]{x1D60E}_{zzz}=(1/2)\boldsymbol{e}_{z}\boldsymbol{e}_{z}\boldsymbol{e}_{z}\,\boldsymbol{\vdots }\,\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{u}|_{\boldsymbol{x}_{p}}$ and $\unicode[STIX]{x1D60E}_{jjz}=(1/2)\boldsymbol{e}_{z}\boldsymbol{\cdot }\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}|_{\boldsymbol{x}_{p}}$ . We note also that $F_{z}^{(1)}$ is independent of the angular velocity $\unicode[STIX]{x1D734}_{p}$ of the sphere and the vorticity $\unicode[STIX]{x1D74E}$ of the background flow as a consequence of the rotational symmetry of the particle about the $z$ axis.

The unit tangent and outward normal vectors $\boldsymbol{s}$ and $\boldsymbol{n}$ , respectively, to the surface of the sphere are by definition

(3.10a,b ) $$\begin{eqnarray}\boldsymbol{s}\equiv \frac{\text{d}\tilde{\boldsymbol{r}}}{\text{d}s}=\frac{\text{d}\tilde{r}}{\text{d}s}\boldsymbol{e}_{r}+\frac{\text{d}\tilde{z}}{\text{d}s}\boldsymbol{e}_{z}\quad \text{and}\quad \boldsymbol{n}\equiv \boldsymbol{s}\times \boldsymbol{e}_{\unicode[STIX]{x1D703}}=-\frac{\text{d}\tilde{z}}{\text{d}s}\boldsymbol{e}_{r}+\frac{\text{d}\tilde{r}}{\text{d}s}\boldsymbol{e}_{z}.\end{eqnarray}$$

Under the axisymmetry of the model flow, we can then write

(3.11) $$\begin{eqnarray}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=-p^{\prime }\boldsymbol{n}+\unicode[STIX]{x1D714}^{\prime }\boldsymbol{s}+\left(\frac{v_{r}^{\prime }}{\tilde{r}}\boldsymbol{n}-2\frac{\unicode[STIX]{x2202}\boldsymbol{v}^{\prime }}{\unicode[STIX]{x2202}s}\times \boldsymbol{e}_{\unicode[STIX]{x1D703}}\right)\quad \text{on }S_{p},\end{eqnarray}$$

where $p^{\prime }$ is the dimensionless pressure (in units of $\unicode[STIX]{x1D707}V^{\prime }/a$ ) and $\unicode[STIX]{x1D714}^{\prime }=\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{v}^{\prime })$ is the dimensionless vorticity field (in units of $V^{\prime }/a$ ) of the model problem (Happel & Brenner Reference Happel and Brenner1965). The no-slip condition on the sphere in the model problem ( $\boldsymbol{v}^{\prime }=\boldsymbol{e}_{z}$ on $S_{p}$ , cf. equation 3.1c ), ensures that the term in parentheses in (3.11) is identically zero. Substituting (3.10) and (3.11) into (3.8) then results in

(3.12a ) $$\begin{eqnarray}\displaystyle F_{z}^{(0)} & = & \displaystyle 2\unicode[STIX]{x03C0}aw_{z}^{(0)}\int _{0}^{\unicode[STIX]{x03C0}}\left\{-p^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{\tilde{r}^{2}}{2}\right)+\unicode[STIX]{x1D714}^{\prime }\tilde{r}\frac{\unicode[STIX]{x2202}\tilde{z}}{\unicode[STIX]{x2202}s}\right\}\text{d}s,\end{eqnarray}$$

(3.12b ) $$\begin{eqnarray}\displaystyle F_{z}^{(1)} & = & \displaystyle -2\unicode[STIX]{x03C0}a^{2}\unicode[STIX]{x1D60C}_{zz}\int _{0}^{\unicode[STIX]{x03C0}}\left\{-p^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{\tilde{r}^{2}\tilde{z}}{2}\right)+\unicode[STIX]{x1D714}^{\prime }\tilde{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{\tilde{z}^{2}}{2}-\frac{\tilde{r}^{2}}{4}\right)\right\}\text{d}s,\end{eqnarray}$$

(3.12c ) $$\begin{eqnarray}\displaystyle F_{z}^{(2)} & = & \displaystyle -2\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D60E}_{zzz}\int _{0}^{\unicode[STIX]{x03C0}}\left\{-p^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{\tilde{r}^{2}\tilde{z}^{2}}{2}-\frac{\tilde{r}^{4}}{8}\right)+\unicode[STIX]{x1D714}^{\prime }\left(\frac{\tilde{r}\tilde{z}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{\tilde{z}^{2}-\tilde{r}^{2}}{2}\right)-\frac{\tilde{r}^{3}}{2}\frac{\unicode[STIX]{x2202}\tilde{z}}{\unicode[STIX]{x2202}s}\right)\right\}\text{d}s\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D60E}_{jjz}\int _{0}^{\unicode[STIX]{x03C0}}\left\{-p^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\!\!\left(\frac{\tilde{r}^{4}}{8}\right)+\unicode[STIX]{x1D714}^{\prime }\frac{\tilde{r}^{3}}{2}\frac{\unicode[STIX]{x2202}\tilde{z}}{\unicode[STIX]{x2202}s}\right\}\text{d}s,\end{eqnarray}$$

where it is understood that the kernels of the integrals are evaluated on $S_{p}$ .

Through an integration by parts, integrals involving $p^{\prime }$ can be recast in terms of integrals involving $\unicode[STIX]{x2202}p^{\prime }/\unicode[STIX]{x2202}s$ ; boundary terms are identically zero since $\tilde{r}=0$ for both $s=0$ and $s=\unicode[STIX]{x03C0}$ . Writing $\unicode[STIX]{x2202}p^{\prime }/\unicode[STIX]{x2202}s=\boldsymbol{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}p^{\prime }$ and noting that the momentum equation for the axisymmetric dimensionless model problem can be expressed as $\unicode[STIX]{x1D735}p^{\prime }=-\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D714}^{\prime }\boldsymbol{e}_{\unicode[STIX]{x1D703}})$ , we write

(3.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}s}=\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}(\tilde{r}\unicode[STIX]{x1D714}^{\prime })}{\unicode[STIX]{x2202}n},\end{eqnarray}$$

where $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}n=\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the derivative normal to the surface of the sphere. Thus, on eliminating the pressure $p^{\prime }$ from (3.12) in favour of the vorticity $\unicode[STIX]{x1D714}^{\prime }$ using (3.13), the expressions for $F_{z}^{(0)}$ , $F_{z}^{(1)}$ and $F_{z}^{(2)}$ become

(3.14a ) $$\begin{eqnarray}\displaystyle F_{z}^{(0)} & = & \displaystyle \unicode[STIX]{x03C0}\unicode[STIX]{x1D707}aw_{z}^{(0)}\int _{0}^{\unicode[STIX]{x03C0}}\tilde{r}^{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}\left(\frac{\unicode[STIX]{x1D714}^{\prime }}{\tilde{r}}\right)\text{d}s,\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle F_{z}^{(1)} & = & \displaystyle -\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{2}\unicode[STIX]{x1D60C}_{zz}\int _{0}^{\unicode[STIX]{x03C0}}\tilde{r}^{3}\tilde{z}^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}\left(\frac{1}{\tilde{z}}\frac{\unicode[STIX]{x1D714}^{\prime }}{\tilde{r}}\right)\text{d}s\quad \text{and}\end{eqnarray}$$

(3.14c ) $$\begin{eqnarray}\displaystyle F_{z}^{(2)} & = & \displaystyle -\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{3}\left[\unicode[STIX]{x1D60E}_{zzz}\int _{0}^{\unicode[STIX]{x03C0}}\left\{\tilde{r}^{3}\tilde{z}^{4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}\left(\frac{1}{\tilde{z}^{2}}\frac{\unicode[STIX]{x1D714}^{\prime }}{r}\right)-\frac{r^{7}}{4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}\left(\frac{\unicode[STIX]{x1D714}^{\prime }}{\tilde{r}^{3}}\right)\right\}\text{d}s\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\unicode[STIX]{x1D60E}_{jjz}\int _{0}^{\unicode[STIX]{x03C0}}\frac{\tilde{r}^{7}}{4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}\left(\frac{\unicode[STIX]{x1D714}^{\prime }}{\tilde{r}^{3}}\right)\text{d}s\,\right].\end{eqnarray}$$

The force normal to the surface of the obstacle can therefore be computed using only the vorticity field of the model problem. The dependence on $\unicode[STIX]{x1D707}$ , $a$ and the background flow has been scaled out explicitly in (3.14), so that the integrals themselves are dimensionless and only depend on the dimensionless ratios $a/h$ and $a/R$ .

Note that since $F_{z}^{(0)}$ depends on the translational velocity of the sphere relative to the fluid, it is related to the force on the particle in the model problem $F_{z}^{\prime }$ as $F_{z}^{(0)}=F_{z}^{\prime }(w_{z}^{(0)}/V^{\prime })$ . This contribution to the force is already well known (Brenner Reference Brenner1961; Cox & Brenner Reference Cox and Brenner1967) for a translating sphere in a quiescent fluid, but is obtained here as the leading term of a more general expansion of the background flow. The expressions for $F_{z}^{(1)}$ and $F_{z}^{(2)}$ have a similar structure to $F_{z}^{(0)}$ and are calculated in the next section.

4 Evaluation of the normal force on the sphere

4.1 Vorticity field in the model problem

The normal force on the sphere, equation (3.14), can be computed given the vorticity field in the model problem, which corresponds to the translation of a spherical particle normal to a stationary spherical obstacle (on which the flow satisfies either no-slip or stress-free conditions) in a quiescent fluid. For arbitrary $a/h$ and $a/R$ , the solution to the model problem can be expressed using bipolar coordinates $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ , in which the particle surface $S_{p}$ and the obstacle surface $S_{w}$ are delineated by the coordinate lines $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FD}$ , respectively (see figure 2), where (Adamczyk et al. Reference Adamczyk, Adamczyk and Van de Ven1983)

(4.1a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\equiv \cosh ^{-1}\left({\displaystyle \frac{\left({\displaystyle \frac{h}{a}}+{\displaystyle \frac{R}{a}}\right)^{2}-\left({\displaystyle \frac{R}{a}}\right)^{2}+1}{2\left({\displaystyle \frac{h}{a}}+{\displaystyle \frac{R}{a}}\right)}}\right)\quad \text{and}\quad {\displaystyle \frac{\sinh \unicode[STIX]{x1D6FD}}{\sinh \unicode[STIX]{x1D6FC}}}\equiv -\frac{a}{R}.\end{eqnarray}$$

Taken together, the values of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ specify both $a/h$ and $a/R$ and therefore uniquely determine the geometry of the problem. The coordinates $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ are related to the cylindrical coordinates $(r,z)$ as (Stimson & Jeffery Reference Stimson and Jeffery1926)

(4.2a,b ) $$\begin{eqnarray}\tilde{r}=\frac{\sinh \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D702}}{\cosh \unicode[STIX]{x1D709}-\cos \unicode[STIX]{x1D702}}\quad \text{and}\quad \tilde{z}=\frac{\sinh \unicode[STIX]{x1D6FC}\sinh \unicode[STIX]{x1D709}}{\cosh \unicode[STIX]{x1D709}-\cos \unicode[STIX]{x1D702}}-\cosh \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Thus, $\unicode[STIX]{x1D709}$ takes values in the range $[\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}]$ , where we take $\unicode[STIX]{x1D6FD}<0$ and $\unicode[STIX]{x1D6FC}>0$ without loss of generality, cf. (4.1). The coordinate $\unicode[STIX]{x1D702}$ lies in $[0,\unicode[STIX]{x03C0}]$ ; on the particle $S_{p}$ , $\unicode[STIX]{x1D702}=0\;\Longleftrightarrow \;s=0$ and $\unicode[STIX]{x1D702}=\unicode[STIX]{x03C0}\;\Longleftrightarrow \;s=\unicode[STIX]{x03C0}$ , cf. figure 2. We note that for a fixed curvature ratio $a/R$ , the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ corresponds to the case where the surfaces are touching $(h\rightarrow a)$ and $\unicode[STIX]{x1D6FC}\rightarrow \infty$ to large separations ( $h/a\rightarrow \infty$ ).

The model vorticity field $\unicode[STIX]{x1D714}^{\prime }$ can be described in these coordinates, for arbitrary $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , by first defining the functions

(4.3a ) $$\begin{eqnarray}\displaystyle U_{n}(\unicode[STIX]{x1D709}) & = & \displaystyle a_{n}\cosh (n-{\textstyle \frac{1}{2}})\unicode[STIX]{x1D709}+b_{n}\sinh (n-{\textstyle \frac{1}{2}})\unicode[STIX]{x1D709}\nonumber\\ \displaystyle & & \displaystyle +\,c_{n}\cosh (n+{\textstyle \frac{3}{2}})\unicode[STIX]{x1D709}+d_{n}\sinh (n+{\textstyle \frac{3}{2}})\unicode[STIX]{x1D709}\quad \text{and}\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle V_{n}(\unicode[STIX]{x1D70F}) & = & \displaystyle \frac{P_{n-1}(\unicode[STIX]{x1D70F})-P_{n+1}(\unicode[STIX]{x1D70F})}{2n+1},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}\equiv \cos \unicode[STIX]{x1D702}$ and $P_{n}$ denotes a Legendre polynomial of order $n$ . The dimensionless coefficients $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ are determined by the boundary conditions on $S_{p}$ and $S_{w}$ and depend on $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ ; explicit formulas are given in appendix A. The dimensionless quantity $\unicode[STIX]{x1D714}^{\prime }/\tilde{r}$ appearing in the expressions for $F_{z}$ can then be written as (Stimson & Jeffery Reference Stimson and Jeffery1926; Brenner Reference Brenner1961)

(4.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{\prime }}{\tilde{r}}=\frac{g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})}{(1-\unicode[STIX]{x1D70F}^{2})\sinh ^{4}\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where

(4.5) $$\begin{eqnarray}\displaystyle g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F}) & = & \displaystyle (\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F})^{5/2}\mathop{\sum }_{n=0}^{\infty }\bigg[V_{n}(\unicode[STIX]{x1D70F})\bigg\{\frac{\text{d}^{2}U_{n}(\unicode[STIX]{x1D709})}{\text{d}\unicode[STIX]{x1D709}^{2}}-\frac{2\sinh \unicode[STIX]{x1D709}}{\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F}}\frac{\text{d}U_{n}(\unicode[STIX]{x1D709})}{\text{d}\unicode[STIX]{x1D709}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{3}{4}\frac{\cosh \unicode[STIX]{x1D709}+3\unicode[STIX]{x1D70F}}{\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F}}U_{n}(\unicode[STIX]{x1D709})\bigg\}\nonumber\\ \displaystyle & & \displaystyle +\,(1-\unicode[STIX]{x1D70F}^{2})\,U_{n}(\unicode[STIX]{x1D709})\left\{\frac{\text{d}^{2}V_{n}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}^{2}}+\frac{2}{\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F}}\frac{\text{d}V_{n}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\right\}\bigg].\end{eqnarray}$$

This completes the description of the model vorticity field $\unicode[STIX]{x1D714}^{\prime }$ required to calculate the normal force on the particle using (3.14). We remark that the curvature ratio $a/R$ and boundary conditions on both surfaces enter the problem through the coefficients $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ , which depend on both $a/h$ and $a/R$ (via $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ ). In the limit $a/R\rightarrow 0$ (planar $S_{w}$ ), we find from (4.1) that $\unicode[STIX]{x1D6FC}\rightarrow \cosh ^{-1}(h/a)$ and $\unicode[STIX]{x1D6FD}\rightarrow 0$ . The solutions to the model problems (both no-slip and stress-free cases) in this limit are given by Brenner (Reference Brenner1961) for arbitrary $h/a$ (i.e. arbitrary $\unicode[STIX]{x1D6FC}$ ). For the general case of a curved obstacle ( $\unicode[STIX]{x1D6FD}\neq 0$ ) and no-slip surfaces (BC I), the appropriate expressions for the coefficients $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ can be constructed using two-sphere solutions (Maude Reference Maude1961; Adamczyk et al. Reference Adamczyk, Adamczyk and Van de Ven1983) and depend on both $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ (see appendix A). The coefficients corresponding to a stress-free curved obstacle $S_{w}$ ( $\unicode[STIX]{x1D6FD}\neq 0$ ; BC II) have been computed in appendix A.

4.2 Expressions for the force

To utilize the expression (4.4) for $\unicode[STIX]{x1D714}^{\prime }$ in the integrals of (3.14), it is necessary to transform $\text{d}s$ and $\text{d}n$ to bipolar coordinates. Writing $\text{d}n=\boldsymbol{n}\boldsymbol{\cdot }\text{d}\boldsymbol{r}$ and $\text{d}s=\boldsymbol{s}\boldsymbol{\cdot }\text{d}\boldsymbol{r}$ in (3.10) and using the transformation (4.2), we obtain

(4.6) $$\begin{eqnarray}\displaystyle \left(\text{d}n,\,\text{d}s\right)=\frac{\sinh \unicode[STIX]{x1D6FC}}{\cosh \unicode[STIX]{x1D6FC}-\cos \unicode[STIX]{x1D702}}\left(-\text{d}\unicode[STIX]{x1D709},\,\text{d}\unicode[STIX]{x1D702}\right)\quad \text{on }\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}, & & \displaystyle\end{eqnarray}$$

so that

(4.7) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}n}\,\text{d}s=-\int _{0}^{\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\,\text{d}\unicode[STIX]{x1D702}=-\int _{-1}^{1}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\frac{\text{d}\unicode[STIX]{x1D70F}}{\sqrt{1-\unicode[STIX]{x1D70F}^{2}}}\quad \text{on }S_{p},\end{eqnarray}$$

for any function $f$ . Substituting the expressions (4.2)–(4.7) for the coordinate transformation $(r,z)\mapsto (\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ and the model vorticity field $\unicode[STIX]{x1D714}^{\prime }$ into (3.14), we obtain

(4.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(0)}=-6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}aw_{z}^{(0)}{\mathcal{I}}_{0}, & \displaystyle\end{eqnarray}$$

(4.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(1)}=-6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{2}\unicode[STIX]{x1D60C}_{zz}\left({\mathcal{I}}_{0}\cosh \unicode[STIX]{x1D6FC}-{\mathcal{I}}_{1}\right)\quad \text{and} & \displaystyle\end{eqnarray}$$

(4.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(2)}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{3}\Big\{\unicode[STIX]{x1D60E}_{zzz}\left({\mathcal{I}}_{0}\cosh ^{2}\unicode[STIX]{x1D6FC}-2{\mathcal{I}}_{1}\cosh \unicode[STIX]{x1D6FC}+{\mathcal{I}}_{2}-{\mathcal{J}}\right)+\unicode[STIX]{x1D60E}_{jjz}{\mathcal{J}}\Big\}, & \displaystyle\end{eqnarray}$$

where

(4.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{I}}_{k}\equiv \frac{1}{6}\int _{-1}^{1}\frac{(\sinh \unicode[STIX]{x1D709})^{3k-1}}{\,(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F})^{2k+3}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left\{\left(\frac{\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F}}{\sinh \unicode[STIX]{x1D709}}\right)^{k}g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\right\}\,\text{d}\unicode[STIX]{x1D70F}\bigg|_{\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}}\quad \text{and} & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{J}}\equiv \frac{1}{24}\int _{-1}^{1}\frac{(1-\unicode[STIX]{x1D70F}^{2})\,\sinh \unicode[STIX]{x1D709}}{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F})^{7}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\bigg\{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D70F})^{2}g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\bigg\}\,\text{d}\unicode[STIX]{x1D70F}\,\bigg|_{\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}}. & \displaystyle\end{eqnarray}$$

The integrals ${\mathcal{I}}_{k}$ and ${\mathcal{J}}$ are functions of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ and involve the coefficients $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ , which enter through $g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ in (4.5).

The explicit evaluation of ${\mathcal{I}}_{k}$ and ${\mathcal{J}}$ utilizes some properties of the Legendre polynomials that appear in $g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ ; the details are given in appendix B. The integration ultimately results in expressions for the force in terms of the coefficients $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ of the model flow, which we express as

(4.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(0)}=-6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}aw_{z}^{(0)}\,{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(1)}=-6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{2}\unicode[STIX]{x1D60C}_{zz}\,{\mathcal{B}}, & \displaystyle\end{eqnarray}$$

(4.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{z}^{(3)}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a^{3}\left(\unicode[STIX]{x1D60E}_{zzz}\,{\mathcal{C}}+\unicode[STIX]{x1D60E}_{jjz}\,{\mathcal{D}}\right), & \displaystyle\end{eqnarray}$$

where ${\mathcal{A}}$ , ${\mathcal{B}}$ , ${\mathcal{C}}$ and ${\mathcal{D}}$ are dimensionless hydrodynamic resistances given by

(4.11a ) $$\begin{eqnarray}\displaystyle {\mathcal{A}} & = & \displaystyle -\frac{\sqrt{2}}{3\sinh \unicode[STIX]{x1D6FC}}\mathop{\sum }_{n=0}^{\infty }\{a_{n}+b_{n}+c_{n}+d_{n}\},\end{eqnarray}$$

(4.11b ) $$\begin{eqnarray}\displaystyle {\mathcal{B}} & = & \displaystyle {\mathcal{A}}\cosh \unicode[STIX]{x1D6FC}+\frac{\sqrt{2}}{3}\mathop{\sum }_{n=0}^{\infty }\{(2n-1)(a_{n}+b_{n})+(2n+3)(c_{n}+d_{n})\},\end{eqnarray}$$

(4.11c ) $$\begin{eqnarray}\displaystyle {\mathcal{C}} & = & \displaystyle 2{\mathcal{B}}\cosh \unicode[STIX]{x1D6FC}-{\mathcal{A}}\cosh ^{2}\unicode[STIX]{x1D6FC}-\frac{\sqrt{2}\sinh \unicode[STIX]{x1D6FC}}{9}\mathop{\sum }_{n=0}^{\infty }\big[n(n+1)(a_{n}+b_{n}+c_{n}+d_{n})\nonumber\\ \displaystyle & & \displaystyle +\,\{(2n-1)^{2}(a_{n}+b_{n})+(2n+3)^{2}(c_{n}+d_{n})\}\big],\end{eqnarray}$$

(4.11d ) $$\begin{eqnarray}\displaystyle {\mathcal{D}} & = & \displaystyle \frac{{\mathcal{A}}}{3}-\frac{{\mathcal{C}}}{5}+\frac{2\sqrt{2}}{45}\mathop{\sum }_{n=0}^{\infty }\big[(2(2n+1)-\coth \unicode[STIX]{x1D6FC})\{(a_{n}-b_{n})\text{e}^{\unicode[STIX]{x1D6FC}}+(c_{n}-d_{n})\text{e}^{-\unicode[STIX]{x1D6FC}}\}\nonumber\\ \displaystyle & & \displaystyle -\,5\{(a_{n}-b_{n})\text{e}^{\unicode[STIX]{x1D6FC}}-(c_{n}-d_{n})\text{e}^{-\unicode[STIX]{x1D6FC}}\}\big]\text{e}^{-(2n+1)\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

The dimensionless resistances ${\mathcal{A}}$ , ${\mathcal{B}}$ , ${\mathcal{C}}$ and ${\mathcal{D}}$ are positive and can be written as functions of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ by substituting the analytical expressions for $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ (see appendix A).

Thus, we have obtained general expressions for the hydrodynamic resistances of a spherical particle near a stationary spherical obstacle suspended in a locally quadratic background flow. The component of the hydrodynamic force on the sphere normal to the obstacle, accurate up to the quadratic moment of the flow, is therefore

(4.12) $$\begin{eqnarray}\displaystyle F_{\bot }\equiv \boldsymbol{F}\boldsymbol{\cdot }\boldsymbol{e}_{\bot } & = & \displaystyle 6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a\bigg[\bigg\{-{\mathcal{A}}\,(\boldsymbol{v}_{p}-\boldsymbol{u})-a\,{\mathcal{B}}\,(\boldsymbol{e}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})\nonumber\\ \displaystyle & & \displaystyle +\frac{a^{2}}{2}{\mathcal{C}}\,(\boldsymbol{e}_{\bot }\boldsymbol{e}_{\bot }\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{u})+\frac{a^{2}}{2}{\mathcal{D}}\,\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\bigg\}\boldsymbol{\cdot }\boldsymbol{e}_{\bot }\bigg]_{\boldsymbol{x}_{p}}.\end{eqnarray}$$