2.1 Governing equations and boundary conditions

Consider the motion of a swimmer near a weakly deforming fluid–fluid interface, with the swimmer fully immersed in the lower fluid (fluid 2). The flow field in both fluids is governed by the Stokes equation and the incompressibility condition

where the superscript $k=1$ or 2 indicates fluid 1 $(z>0)$ or 2 $(z<0)$ and $p^{(k)}$ , $\boldsymbol{u}^{(k)}$ and $\unicode[STIX]{x1D707}^{(k)}$ denote the pressure, velocity and the dynamic viscosity of the kth fluid. In this work, we consider swimmers that propel themselves by means of a slip velocity $\boldsymbol{u}_{s}$ on their surface. Because of this slip velocity, the swimmer moves with a translation velocity $(\boldsymbol{U})$ and rotational velocity $(\unicode[STIX]{x1D734})$ , so the boundary condition on its surface is given by

In addition, there are hydrodynamic force-free and torque-free constraints on the swimmer, given by

Here, $\boldsymbol{x}^{s}$ is a position vector from the centre of the swimmer to its surface, $\unicode[STIX]{x1D64F}^{(2)}$ denotes the stress tensor in fluid 2, $S_{p}$ denotes the surface of the swimmer, $\boldsymbol{n}$ is the unit normal vector to the surface of the swimmer that is pointing into the fluid and dS is an infinitesimal surface area on the surface of the swimmer. For describing the surface deformation, we use the traditional ‘squirmer’ model (Lighthill Reference Lighthill1952; Blake Reference Blake1971) which assigns a tangential velocity on the surface of the swimmer as a manifestation of the metachronal beating of cilia on its surface. A periodic stroke on the surface of a swimmer can be described using two different approaches: (i) in a reference frame fixed to the swimmer, we prescribe at each instant, a periodic surface velocity. This description is referred to as the Eulerian periodic stroke in Michelin & Lauga (Reference Michelin and Lauga2013). (ii) Otherwise, one can prescribe periodic trajectories of material surface points. This description is referred to as the Lagrangian periodic stroke in Michelin & Lauga (Reference Michelin and Lauga2013). Because of its simplicity and its widespread usage in modelling swimmers with steady surface and flow velocities (Ishikawa, Simmonds & Pedley Reference Ishikawa, Simmonds and Pedley2006; Short et al. Reference Short, Solari, Ganguly, Powers, Kessler and Goldstein2006; Doostmohammadi, Stocker & Ardekani Reference Doostmohammadi, Stocker and Ardekani2012), we choose Eulerian periodic strokes to model a swimmer. We truncate the infinite series in the slip velocity to first two terms (Ishikawa et al. Reference Ishikawa, Simmonds and Pedley2006; Li & Ardekani Reference Li and Ardekani2014; Matas-Navarro et al. Reference Matas-Navarro, Golestanian, Liverpool and Fielding2014). This slip velocity is given by

Here, $B_{n}(t)$ is the swimming gait for the $n$ th squirming mode, while $P_{n}^{\prime }$ denotes the derivative of the Legendre polynomial with respect to its argument, $\boldsymbol{e}$ denotes the direction of the head of the swimmer. If the squirming gait, $B_{n}$ , is proportional to $\sin (\unicode[STIX]{x1D714}t)$ , such a swimmer cannot swim on an average according to Purcell’s scallop theorem (Purcell Reference Purcell1977). However, the nonlinearities due to the interface deformation break the scallop theorem leading to the motion of a swimmer performing time-reversible surface deformation. The slip velocity of a squirmer can be written in terms of the spherical coordinates whose origin is at the centre of the swimmer as

Figure 1. Schematic of the problem. The interface shown is weakly deforming. ‘ $O$ ’ is the origin of the Cartesian and cylindrical coordinate systems, both fixed at the undeformed interface. $O^{\prime }$ is the origin of the spherical coordinate system, fixed at the centre of the sphere. The swimmer is at a distance of $l$ from the undeformed interface.

Here, $\unicode[STIX]{x1D703}_{z}$ is the angle made by the head of the swimmer ( $\boldsymbol{e}$ ) with the interface-fixed $z$ -axis (pointing from fluid 2 to fluid 1), while $\unicode[STIX]{x1D719}_{x}$ is the angle made by the projection of $\boldsymbol{e}$ in the $xy$ plane (equivalently $XY$ plane) with the $x$ -axis (equivalently $X$ -axis). The origin of $xyz$ coordinate system is at the undeformed interface, while the origin of $XYZ$ coordinate system is at the centre of the swimmer. Without loss of generality, we consider $\unicode[STIX]{x1D719}_{x}=0$ . Also, $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ are the coordinate variables in spherical coordinates fixed at the centre of the swimmer, $\boldsymbol{i}_{\unicode[STIX]{x1D703}}$ and $\boldsymbol{i}_{\unicode[STIX]{x1D719}}$ are the unit vectors in the polar and azimuthal directions as shown in figure 1. As the swimmer is moving in a quiescent fluid, the flow field far from the swimmer is given by

The shape of the interface is represented by $F\equiv z-f(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},t;l)=0$ , the boundary conditions at the interface are given by (Leal Reference Leal2007)

where $\unicode[STIX]{x1D70E}$ is the interfacial tension between the two fluids, $g$ is the acceleration due to gravity, $t$ is time, $\unicode[STIX]{x1D70C}^{(k)}$ denotes the density of the $k$ th fluid, $\boldsymbol{n}=\unicode[STIX]{x1D735}F/|\unicode[STIX]{x1D735}F|$ denotes the unit vector normal to the interface, $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},z)$ are the coordinate variables of the cylindrical coordinate system with the origin fixed at the undeformed interface and $l$ denotes the distance of the centre of the swimmer from the undeformed interface.

We hereby non-dimensionalize the governing equations and boundary conditions. For this purpose, we use $u_{ref}=b_{1}$ (assuming $B_{n}=b_{n}\unicode[STIX]{x1D6FD}_{n}(t)$ ), $l_{ref}=R$ (radius of the squirmer), $p_{ref}^{(k)}=T_{ref}^{(k)}=\unicode[STIX]{x1D707}^{(k)}u_{ref}/R$ and $t_{ref}=R/u_{ref}$ as the characteristic scales for velocity, length, stress field and time, respectively. Additionally, any forces and torques (especially encountered in the complementary Stokes problem of the reciprocal theorem) are non-dimensionalized by $\unicode[STIX]{x1D707}^{(2)}u_{ref}R$ and $\unicode[STIX]{x1D707}^{(2)}u_{ref}R^{2}$ , respectively. We hereby denote the dimensionless variables by the same symbols as used for the dimensional ones. All the variables appearing hereafter are dimensionless variables. The dimensionless governing equations are given as

The boundary conditions on the surface of the swimmer, the force-free and torque-free conditions and the slip velocity on the surface of the swimmer are given by

where $\unicode[STIX]{x1D6EC}=b_{2}/b_{1}$ . The flow field far from the swimmer is still given by

The boundary conditions at the interface are given by (Lee et al. Reference Lee, Chadwick and Leal1979; Lee & Leal Reference Lee and Leal1980; Berdan & Leal Reference Berdan and Leal1982; Leal Reference Leal2007)

where $\unicode[STIX]{x1D6FD}=Ca^{-1}=\unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D707}^{(2)}u_{ref});\unicode[STIX]{x1D6FC}=Bo/Ca=(gR^{2}(\unicode[STIX]{x1D70C}^{(2)}-\unicode[STIX]{x1D70C}^{(1)}))/(\unicode[STIX]{x1D707}^{(2)}u_{ref});\unicode[STIX]{x1D706}=\unicode[STIX]{x1D707}^{(1)}/\unicode[STIX]{x1D707}^{(2)}$ .

For $\unicode[STIX]{x1D6FD}\gg 1$ or $\unicode[STIX]{x1D6FC}\gg 1$ , the deformation of the interface is very small, so we define $\unicode[STIX]{x1D716}=1/(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})$ to consider the influence of either $\unicode[STIX]{x1D6FC}$ or $\unicode[STIX]{x1D6FD}$ . For $\unicode[STIX]{x1D716}\ll 1$ , we can write $f(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},t;l)=\unicode[STIX]{x1D716}f_{1}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719};t,l)+\unicode[STIX]{x1D716}^{2}f_{2}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719};t,l)+\cdots \,.$ Expanding all the flow field variables and the swimmer velocities as a regular perturbation series in $\unicode[STIX]{x1D716}$ , we get (Berdan & Leal Reference Berdan and Leal1982)

Note that the subscript denotes the order of perturbation, whereas the superscript denotes fluid 1 or 2. For small interface deformations, we write the boundary conditions at the interface, equations (2.15)–(2.17) as a series of boundary conditions applied at an undeformed interface using Taylor series expansion about $z=0$ . We then substitute the regular perturbation expansion of the flow field variables, equation (2.18) in the governing equations and the boundary conditions (applied at the undeformed interface) and we eventually collect the terms at each order of $\unicode[STIX]{x1D716}$ (the usual domain perturbation methodology). At $O(1)$ , the boundary condition at the surface of the swimmer and at the interface are given as (Berdan & Leal Reference Berdan and Leal1982)

At the undeformed interface,

where $\unicode[STIX]{x1D716}=1/(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})$ and $\unicode[STIX]{x1D6FB}_{2}^{2}$ is a 2-D Laplacian in cylindrical coordinates for a constant $z$ . The normal component of (2.22) is used for finding the leading-order interface deformation. Assuming that the slip velocity on the surface of the swimmer is $O(1)$ , the boundary conditions at $O(\unicode[STIX]{x1D716})$ on the surface of the swimmer and at the interface are given as (Berdan & Leal Reference Berdan and Leal1982)

At the undeformed interface, we have

It can be seen that the $O(1)$ problem is the motion of the swimmer near a plane interface. This is a linear problem so if $\boldsymbol{u}^{s}$ is time reversible, we expect the swimmer velocities at this order $(\boldsymbol{U}_{0},\unicode[STIX]{x1D734}_{0})$ to also be time reversible, i.e., the time-averaged swimmer velocities to be zero. Consequently, on an average, the swimmer will not move, consistent with the predictions of the scallop theorem at $O(1)$ . Besides this, we need to solve the $O(\unicode[STIX]{x1D716})$ problem to find the leading-order effect of the interface deformation on the swimmer velocities, $(\boldsymbol{U}_{1},\unicode[STIX]{x1D734}_{1})$ . This, however, requires the solution of the $O(1)$ problem as the boundary conditions at $O(\unicode[STIX]{x1D716})$ involve flow variables at $O(1)$ . Before explaining the solution methodology to solve the $O(1)$ problem, we first describe the reciprocal theorem used for obtaining the $O(\unicode[STIX]{x1D716})$ swimmer velocities directly without solving for the $O(\unicode[STIX]{x1D716})$ flow fields. This reciprocal theorem is similar to that obtained for the motion of rigid sphere near a weakly deforming interface (Berdan & Leal Reference Berdan and Leal1982).

2.2 Reciprocal theorem for a swimmer near a deforming interface

We begin with the generalized reciprocal theorem between two Stokes flows $(\boldsymbol{u}^{\prime },\unicode[STIX]{x1D64F}^{\prime }),(\boldsymbol{u}^{\prime \prime },\unicode[STIX]{x1D64F}^{\prime \prime })$ in the same flow geometry given by

As we would like to find the velocity of a squirmer at $O(\unicode[STIX]{x1D716})$ near a weakly deforming interface using the reciprocal theorem, we define $(\boldsymbol{u}^{\prime },\unicode[STIX]{x1D64F}^{\prime })\equiv (\boldsymbol{u}_{1}^{(k)},\unicode[STIX]{x1D64F}_{1}^{(k)})$ . We also consider the auxiliary problem (or the complementary Stokes problem) as the motion of a rigid sphere near a flat interface, we denote this problem by variables with a hat over them, $(\boldsymbol{u}^{\prime \prime },\unicode[STIX]{x1D64F}^{\prime \prime })=(\hat{\boldsymbol{u}}^{(k)},\hat{\unicode[STIX]{x1D64F}}^{(k)})$ . Applying the reciprocal theorem in fluid 1, we have

whereas the application of reciprocal theorem in fluid 2 gives

where $A_{F}$ is the area of the flat interface, $A_{2}$ includes the area of the flat interface and the area of the sphere $(A_{2}=A_{F}+S_{p})$ . Multiplying (2.28) with $\unicode[STIX]{x1D706}$ and subtracting from (2.29), we have

Using the boundary condition on the surface of the swimmer, $\boldsymbol{u}_{1}^{(2)}=\boldsymbol{U}_{1}+\unicode[STIX]{x1D734}_{1}\times \boldsymbol{x}$ , on the surface of the rigid sphere in the complementary Stokes problem, $\hat{\boldsymbol{u}}^{(2)}=\hat{\boldsymbol{U}}+\hat{\unicode[STIX]{x1D734}}\times \boldsymbol{x}$ and the boundary conditions at the interface in the complementary Stokes problem, equation (2.30) can be simplified to

where $\hat{\boldsymbol{u}}=\hat{\boldsymbol{u}}^{(1)}=\hat{\boldsymbol{u}}^{(2)}$ (on the interface), $\hat{\boldsymbol{F}},\hat{\boldsymbol{L}}$ are the force and torque acting on the rigid sphere in the complementary Stokes problem. Using the boundary conditions at the interface for $O(\unicode[STIX]{x1D716})$ problem, equations (2.24)–(2.26), along with (2.31), we see that we only need to know the solution of $O(1)$ problem and complementary Stokes flow problem to find the swimmer velocity at $O(\unicode[STIX]{x1D716})$ .

Below, we outline the methodology used for solving the $O(1)$ problem corresponding to the motion of a swimmer near a plane interface which is governed by (2.10), (2.12)–(2.14), (2.19)–(2.22). We use two approaches: (i) the bipolar coordinate approach and (ii) the method of reflections. Corresponding approaches are used for solving the complementary Stokes problem of the motion of a rigid sphere near a plane interface. These complementary Stokes problems were already solved using the bipolar coordinate approach (Lee & Leal Reference Lee and Leal1980), and also using Lorentz reciprocal theorem approach (Lee et al. Reference Lee, Chadwick and Leal1979). So, we will not repeat the solution of the complementary Stokes problem here.

2.3 Swimmer near a plane interface: bipolar coordinate approach

Following the traditional approach (Lauga & Powers Reference Lauga and Powers2009), we divide the $O(1)$ problem into two subproblems. The first subproblem, also known as the thrust problem, consists of a swimmer fixed in space. As the swimmer is fixed in space, it will experience a hydrodynamic force and torque $(\boldsymbol{F}_{T},\boldsymbol{L}_{T})$ . The second subproblem, also known as the drag problem, consists of the motion of a rigid sphere with unknown translational and rotational velocities of the swimmer, thereby experiencing a drag force and torque $(\boldsymbol{F}_{D},\boldsymbol{L}_{D})$ . As the swimmer itself is force free and torque free, we have

Knowing the force and torque experienced by the swimmer in the drag problem, the unknown swimmer's translational and rotational velocities can be found using the linear relations between forces and velocities given by

where $\unicode[STIX]{x1D646}_{T},\unicode[STIX]{x1D646}_{C},\unicode[STIX]{x1D646}_{R}$ are second-order tensors known as the resistance tensors. The appropriate form of these tensors for the motion of a rigid sphere near a plane interface are given by Lee & Leal (Reference Lee and Leal1980) using the bipolar coordinate approach. Thus, we do not repeat the derivation of these resistance tensors. We rather focus on deriving the force experienced by the swimmer in the thrust problem using the bipolar coordinate approach.

For the motion of a sphere near a plane interface, we represent the velocity field in cylindrical coordinates, which in turn is expressed in terms of bispherical eigensolutions. The transformation between cylindrical and bipolar coordinates is given by (Happel & Brenner Reference Happel and Brenner1981)

where $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D719})$ are the bispherical coordinate variables. In these coordinates, the surface of the sphere is at $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{0}=-\text{cosh}^{-1}(l)$ , the plane interface is at $\unicode[STIX]{x1D702}=0$ and $c=\sqrt{l^{2}-1}$ . A general solution of the Stokes equation for fluid 2 for the motion of a sphere near a plane interface is given by (even though flow field variables in $O(1)$ problem are denoted by $\boldsymbol{u}_{0}^{(k)}$ and $p_{0}^{(k)}$ , in this section, for the purposes of brevity, we denote the $O(1)$ flow variables in fluid 2 by $\boldsymbol{u}$ and $p$ )

where $(u,v,w)$ are the $\unicode[STIX]{x1D70C}$ , $\unicode[STIX]{x1D719}$ , and $z$ components of the velocity field, $\unicode[STIX]{x1D701}=\cos (\unicode[STIX]{x1D709})$ , $P_{n}^{m}$ is the associated Legendre polynomial of degree $n$ and order $m$ , while $\unicode[STIX]{x1D6FC}_{m}$ are the constants and

A similar set of equations can also be obtained for the motion of fluid above the interface, fluid 1. Satisfying the boundary conditions, continuity equation and boundedness of the flow field provides us with an infinite set of linear algebraic equations among the constants $A_{n}^{m}$ , $B_{n}^{m}$ , $C_{n}^{m}$ , $E_{n}^{m}$ , $F_{n}^{m}$ , $G_{n}^{m}$ and $H_{n}^{m}$ . As these equations were already derived by Lee & Leal (Reference Lee and Leal1980), we hereby do not repeat them. Since the order of magnitude of these constants for large values of $n$ is very small, we, therefore, truncate these infinite series to some finite number $N$ such that the error in evaluating these constants is $O(10^{-6})$ . As the major difference in this work is due to the boundary conditions at the surface of the swimmer, we, therefore, proceed to describe the methodology in applying the boundary conditions at the surface of the swimmer in the thrust problem, given by

The surface velocity can be written in terms of bispherical eigenfunctions as

Hence, for $m=0$ , we expand $u_{s}^{0}$ and $v_{s}^{0}$ as

whereas for $m\geqslant 1$ , we have

while for all $m$ , we have

The boundary condition on the surface of the swimmer determines the values of constants $X_{n}^{m}$ , $Y_{n}^{m}$ and $Z_{n}^{m}$ , which in turn determine the constants $A_{n}^{m}$ , $B_{n}^{m}$ , $C_{n}^{m}$ , $E_{n}^{m}$ , $F_{n}^{m}$ , $G_{n}^{m}$ and $H_{n}^{m}$ . We, therefore, describe in detail the derivation of constants $X_{n}^{m}$ , $Y_{n}^{m}$ and $Z_{n}^{m}$ for a squirming boundary condition at the surface of the sphere in appendix A.

2.4 Swimmer near a plane interface: method of reflections

In this section we solve the $O(1)$ problem for the motion of the squirmer near a plane interface up to an accuracy of $O(1/l^{3})$ , using the Lorentz reciprocal theorem. Such a method was used by Lee et al. (Reference Lee, Chadwick and Leal1979) for solving the motion of a rigid sphere near a plane interface. The authors proposed a lemma using which one can find the solution of creeping flow equations for the motion of an arbitrarily shaped particle near a plane interface if one already knows the solution of creeping flow equations for the motion of the same particle in an unbounded medium. If $\boldsymbol{u}$ and $p$ are the solutions of creeping flow equations for the motion of a particle in an infinite fluid 2, then the solutions near a plane interface are given by

For the purposes of brevity, we again denote the flow field variables in the $O(1)$ problem of motion of a swimmer near a plane interface as $\boldsymbol{u}^{(k)}$ and $p^{(k)}$ . Here $(\tilde{\boldsymbol{u}},\tilde{p})$ is the associated solution of $(\boldsymbol{u},p)$ and is defined as (Lee et al. Reference Lee, Chadwick and Leal1979)

and $(\boldsymbol{u}^{\ast },p^{\ast })$ is the reflected image solution of $(\boldsymbol{u},p)$ given by

where $(\tilde{\boldsymbol{u}}^{\ast },\tilde{p}^{\ast })$ is the associated solution of $(\boldsymbol{u}^{\ast },p^{\ast })$ . The operator $\boldsymbol{J}$ is defined as $J_{ij}\equiv \unicode[STIX]{x1D6FF}_{ij}-2\unicode[STIX]{x1D6FF}_{i3}\unicode[STIX]{x1D6FF}_{j3}$ , where $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta. These solutions satisfy the continuity of velocity, shear stress and zero normal velocity at the plane interface.

The creeping flow equations for the motion of the squirmer in an unbounded fluid were already solved in the literature (Blake Reference Blake1971; Ishikawa et al. Reference Ishikawa, Simmonds and Pedley2006; Pak & Lauga Reference Pak and Lauga2014). For two squirming modes $(B_{1},B_{2})$ , this solution can be readily written in terms of point force singularities located at the centre of the squirmer as

where $\boldsymbol{x}_{+}=(x,y,z+l)$ and $\boldsymbol{u}_{SS},\boldsymbol{u}_{D},\boldsymbol{u}_{D4}$ are the flow fields due to stresslet, source dipole and source quadrupole, respectively. Their expressions are provided in appendix B. Here, the subscript denotes the level of approximation in the context of normal reflection calculation procedure. For instance, $\boldsymbol{u}_{0}^{(k)}$ denotes the flow field in an unbounded domain, $\boldsymbol{u}_{1}^{(k)}$ denotes the (first) reflection from the interface, $\boldsymbol{u}_{2}^{(k)}$ denotes the (second) reflection from the sphere. Even though these velocity fields are dimensionless, i.e.,  $B_{1}=\unicode[STIX]{x1D6FD}_{1}(t)$ and $B_{2}=\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6FD}_{2}(t)$ , we write them in terms of $B_{1}$ and $B_{2}$ to see the correspondence with the dimensional velocities. Using the lemma and the above solution, one can derive the solution of creeping flow equations for the motion of a squirmer near a plane interface. After subtracting the flow field in an unbounded fluid, equations (2.60)–(2.62) from the solution calculated using the lemma, we obtain the flow field due to the interface (denoted as $\boldsymbol{u}_{1}^{(k)},p_{1}^{(k)}$ ). Using the Faxen’s laws with the flow field due to the interface in fluid 2 $(\boldsymbol{u}_{1}^{(2)},p_{1}^{(2)})$ and the force-free, torque-free condition on the squirmer, we obtain the corrections to the swimmer velocity because of the presence of the interface which are given as

Figure 2. Velocities of a swimmer moving near a plane interface, (a) $U_{z}$ and (b) $\unicode[STIX]{x1D6FA}_{y}$ for $B_{2}/B_{1}=4$ , $\unicode[STIX]{x1D703}_{z}=3\unicode[STIX]{x03C0}/8$ , $\unicode[STIX]{x1D706}=10^{-3},1$ and 10. Lines indicate bipolar coordinate results while symbols denote the method of reflections results.

These expressions in the limit $\unicode[STIX]{x1D706}\rightarrow \infty$ or $\unicode[STIX]{x1D706}=0$ are identical to the expressions derived by Spagnolie & Lauga (Reference Spagnolie and Lauga2012) for the motion of a swimmer (represented by force dipole, source dipole and source quadrupole) near a plane wall and plane interface, respectively. In the case of a stresslet near a plane interface, these expressions reduce to those derived by Lopez & Lauga (Reference Lopez and Lauga2014), for arbitrary values of viscosity ratios. Additionally, in figure 2, we compare these expressions of swimmer velocities with the bipolar coordinate results, for various values of viscosity ratios. The flow field due to the interface in fluid 2 $(\boldsymbol{u}_{1}^{(2)},p_{1}^{(2)})$ , however, does not satisfy the boundary conditions on the surface of the squirmer given by $\boldsymbol{U}_{1}+\unicode[STIX]{x1D734}_{1}\times \boldsymbol{x}^{s}$ . Consequently, we need to put additional singularities at the centre of the squirmer to satisfy the boundary conditions on its surface $(\boldsymbol{U}_{1}+\unicode[STIX]{x1D734}_{1}\times \boldsymbol{x}^{s}-\boldsymbol{u}_{1}^{(2)}(\boldsymbol{x}^{s}))$ . We expand this boundary condition in terms of $1/l$ , and we choose singularities located at the centre of the squirmer only to cancel the first term of this boundary condition expressed in terms of $1/l$ . The non-zero velocity on the surface of the squirmer expanded in terms of $1/l$ is given as

The flow field satisfying these boundary conditions on the surface of the sphere and the Stokes flow equations can be written as a linear combination of shear flow and extensional flow past a sphere. A shear flow past a sphere can be represented by a stresslet, rotlet and a potential quadrupole located at the centre of the sphere, while an extensional flow past a sphere can be generated by a stresslet and a potential quadrupole kept at the centre of the sphere (Chwang & Wu Reference Chwang and Wu1975). Consequently, we find that this boundary condition can be satisfied by a suitable combination of stresslet and a potential quadrupole located at the centre of the squirmer, given by

where $\unicode[STIX]{x1D71E}_{1}=\boldsymbol{i}_{x}-\boldsymbol{i}_{z};\unicode[STIX]{x1D71E}_{2}=\frac{5}{6}(\boldsymbol{i}_{x}+\boldsymbol{i}_{z})$ and

In summary, the singularities required to represent the flow field due to a squirmer near a plane interface, accurate to $O(1/l^{3})$ , are given by

For simplicity, we will derive the velocity of the squirmer near a weakly deforming interface, accurate to $O(1/l)$ . For this purpose, we will need the flow field due to the motion of a squirmer near a plane interface, accurate to $O(1/l)$ . From the above expressions, we can clearly see that such a flow field, accurate up to $O(1/l)$ , is given by $-(B_{2}/2)\boldsymbol{u}_{SS}(\boldsymbol{x}_{+};\boldsymbol{e},\boldsymbol{e})+(B_{1}/3)\boldsymbol{u}_{D}(\boldsymbol{x}_{+};\boldsymbol{e},\boldsymbol{e})$ .

3.1 Interface deformation

The leading-order interface shape is given by the solution of differential equation which is obtained from the $O(1)$ normal stress balance at the interface, given by

The expressions for $g_{0}$ , $g_{1}$ and $g_{2}$ using the method of reflections are given by

The solution of equation (3.1) for any arbitrary values of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ is given by

where, $m=0$ , 1 or 2, $\unicode[STIX]{x1D70C}_{ref}=\sqrt{\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC}}$ , $\text{I}_{m}$ and $\text{K}_{m}$ are the modified Bessel functions of first and second kind of order $m$ . The interface shape for the limiting values of $\unicode[STIX]{x1D6FC}\ll \unicode[STIX]{x1D6FD}$ is given by

and for $\unicode[STIX]{x1D6FC}\gg \unicode[STIX]{x1D6FD}$ is given by

For the motion of a rigid sphere normal to the undeformed interface, Aderogba & Blake (Reference Aderogba and Blake1978) remarked that if we set $\unicode[STIX]{x1D6FC}$ identically to zero in the differential equation for the interface deformation, then the interface deformation has a log singularity and it has no solution which is both finite at the origin and zero as $\unicode[STIX]{x1D70C}$ goes to infinity. Note that when $\unicode[STIX]{x1D6FC}$ is identically zero, the flow field is dominated by interfacial tension. Berdan & Leal (Reference Berdan and Leal1982) suggested that the interfacial tension can only restrict the surface curvature but not the surface deformation, thereby allowing the occurrence of a log singularity as predicted by Aderogba & Blake (Reference Aderogba and Blake1978). We note that there is no such log singularity in the interface deformation when $\unicode[STIX]{x1D6FC}$ is set identically to zero in (3.1), for the motion of a squirmer near a deforming interface. To generalize, one cannot find an interface deformation satisfying all the boundary conditions for a Stokeslet (far-field approximation of a rigid sphere) oriented normal to an undeformed interface. However, this is not the case for a Stokes dipole (far-field approximation of a swimmer) oriented normal to an undeformed interface. Such an ill-posed problem for rigid bodies versus a well-posed problem for swimmers can also be found for the motion of two-dimensional bodies. For instance, the translational motion of an infinitely long cylinder in an unbounded fluid is an ill-posed problem (Happel & Brenner Reference Happel and Brenner1981; Guazzelli, Morris & Pic Reference Guazzelli, Morris and Pic2011). On the contrary, the motion of a cylindrical swimmer in an unbounded fluid is a well-posed problem which can also be solved using a reciprocal theorem framework (Elfring Reference Elfring2015).

Figure 3. Interface deformation for various orientations of swimmer at $l=2$ , $B_{2}/B_{1}=4$ , $\unicode[STIX]{x1D706}=1$ , $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=1$ , $\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=20$ . The orientations of the swimmer are (a) $\unicode[STIX]{x1D703}_{z}=0$ , (b) $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/3$ , (c) $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/2$ and (d) $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}$ . Interface shape is axisymmetric when $\unicode[STIX]{x1D703}_{z}=0$ or $\unicode[STIX]{x03C0}$ .

In this section, we study the interface deformation for steady squirming modes. Figure 3 shows the interface deformation for the squirmer orientations of $\unicode[STIX]{x1D703}_{z}=0,\unicode[STIX]{x03C0}/2,\unicode[STIX]{x03C0}$ and $\unicode[STIX]{x03C0}/3$ . When $\unicode[STIX]{x1D703}_{z}$ is 0 or $\unicode[STIX]{x03C0}$ , the problem is axisymmetric, hence it can be seen from (3.3) that $f_{1}=h_{0}$ (as $g_{1}$ and $g_{2}$ are zero, so $h_{1}$ and $h_{2}$ are zero). Hence, the interface shape is the same for these two orientations but the magnitude of the interface deformation is larger for $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}$ due to a corresponding larger magnitude of $A_{1}$ when $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}$ (note that $h_{0}\propto g_{0}\propto A_{1}$ ).

Figure 4. The effect of $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}$ on the interface deformation for the motion of a squirmer near an interface. The values of parameters are taken as: $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/6$ , $\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=20$ , $B_{2}/B_{1}=4$ , $\unicode[STIX]{x1D706}=1$ and $l=3$ . Lines denote the bipolar coordinate results while symbols denote the MOR results.

We hereby study the effect of the relative importance of density difference and the interfacial tension on the interfacial deformation, i.e., the effect of $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}$ on $f_{1}$ . For this purpose, we fix the sum $\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}$ as 20 (equivalent to fixing $\unicode[STIX]{x1D716}=0.05$ ) and study the interfacial deformation as $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}$ is varied from 0.01 to $\infty$ . We note that $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\ll 1$ implies that the physics is determined by the interfacial tension forces while $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\gg 1$ means that the density difference is the dominating factor. Figure 4 shows the variation of interfacial deformation with radial coordinate (in cylindrical coordinates) for $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=0.01,0.1,1,10$ and $\infty$ , $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/6$ , $\unicode[STIX]{x1D706}=1$ , $l=3$ and $B_{2}/B_{1}=4$ . As deformation itself is two-dimensional as shown in figure 3, for simplicity, we study the behaviour of deformation as observed from the equatorial plane $(\unicode[STIX]{x1D719}=0,\unicode[STIX]{x03C0})$ . It is to be noted that the behaviour of the interface deformation with $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}$ is qualitatively the same for all other combination of parameters not reported here. It can be seen from the figure that the maximum deformation is obtained when the forces due to the interfacial tension are dominant, i.e.,  $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\ll 1$ while the minimum deformation is obtained when the forces due to both interfacial tension and density difference are of the same order of magnitude, i.e.,  $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=O(1)$ . This result needs to be contrasted with the interface deformation for the motion of a rigid sphere near an interface. In the latter case, the maximum deformation occurs for $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\ll 1$ but the minimum deformation is obtained for $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\gg 1$ . Berdan & Leal (Reference Berdan and Leal1982) gave a reason for such an observation by saying that the forces due to the density difference act directly in restricting the interface deformation (thereby giving a minimum deformation when $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\gg 1$ ) but the interfacial tension forces act to restrict the surface curvature but not the deformation (thereby giving a maximum deformation when $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}\ll 1$ ). Even though the physical nature of forces due to the interfacial tension and density difference are still the same in both problems, the minimum interface deformation occurs when $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=1$ for the motion of a squirmer near an interface. As shown in figure 4, the MOR results predict the interface deformation quite accurately for all values of $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}$ , for the given parameters of interest.

Figure 5. (a) The role of the distance of the squirmer from the undeformed interface on the interface deformation. The values of the parameters are taken as $\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=20$ , $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=1$ , $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/6$ , $B_{2}/B_{1}=4$ , $\unicode[STIX]{x1D706}=1$ , $l=2$ , 3 and 4. (b) Effect of viscosity ratio on the interface deformation. The values of most of the parameters are the same as those of panel (a) except $l$ is fixed at 2 and $\unicode[STIX]{x1D706}$ is varied from $10^{-3}$ to $10^{4}$ . Again the lines indicate the bipolar coordinate solutions while symbols indicate the MOR solutions.

Now, we study the effect of distance of the centre of the squirmer from the undeformed interface on the interfacial deformation as shown in figure 5(a) for $\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=20$ , $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=1$ , $\unicode[STIX]{x1D703}_{z}=\unicode[STIX]{x03C0}/6$ , $B_{2}/B_{1}=4$ , $\unicode[STIX]{x1D706}=1$ , $l=2$ , 3 and 4. As expected, the interface deformation decreases with the distance of the squirmer from the undeformed interface. It can also be seen in figure 5(a) that the error in predictions of interface deformation using MOR decreases with the distance of the squirmer from the interface. This is expected as the MOR results are accurate when the squirmer is far away from the interface. In figure 5(b), we study the effect of viscosity ratio on the interface deformation when $\unicode[STIX]{x1D706}$ is varied from $10^{-3}$ to $10^{4}$ . We note that $\unicode[STIX]{x1D706}\ll 1$ corresponds to an air–water interface, whereas $\unicode[STIX]{x1D706}\gg 1$ corresponds to an oil–water interface (e.g. for a crude oil–water interface, $\unicode[STIX]{x1D706}=12$ ). The interface deformation predicted using bipolar coordinates varies negligibly with the viscosity ratio. On the other hand, the MOR results predict that the interface deformation is independent of the viscosity ratio. This can also be seen from the expression for the interface deformation, equation (3.3) which does not depend on the viscosity ratio. However, the MOR results for the motion of a rigid sphere near an interface predict that the interface deformation does vary, even though slowly with the viscosity ratio. These different behaviours of interface deformation with the viscosity ratio predicted by the MOR in the two cases can be understood by looking at the singularities that occur in the $O(1)$ problem. The singularities that enter into the solution due to the reflections from the surface of the squirmer (i.e. to satisfy the boundary condition in (2.66)–(2.68)) are $O(1/l^{3})$ or lower. Thus, these singularities do not enter into the $O(\unicode[STIX]{x1D716})$ problem, because of which the interfacial deformation does not depend on the viscosity ratio as predicted by the MOR. Such a phenomenon does not happen for the motion of a rigid sphere near an interface.

3.2 Swimming with steady squirming modes

Once the leading-order interface deformation is determined, we can derive the $O(\unicode[STIX]{x1D716})$ swimming velocities using the reciprocal theorem, equation (2.31). As the swimming dynamics (attraction/repulsion to the interface and reorientation) near an interface is governed by $U_{z}$ and $\unicode[STIX]{x1D6FA}_{y}$ , we report only these two velocities. In general, the swimming velocity is of the form

where the functions $\boldsymbol{M}_{1}$ , $\boldsymbol{M}_{2}$ , $\boldsymbol{N}_{1}$ , $\boldsymbol{N}_{2}$ , $\boldsymbol{K}_{11}$ , $\boldsymbol{K}_{12}$ and $\boldsymbol{K}_{22}$ are obtained either from bipolar coordinate approach or from the method of reflections. Now if $B_{1},B_{2}\propto j(t)$ , then the swimmer velocity is given by $\boldsymbol{U}(z,\unicode[STIX]{x1D703}_{z},t)=\boldsymbol{M}(z,\unicode[STIX]{x1D703}_{z})j(t)+\unicode[STIX]{x1D716}[\boldsymbol{N}(z,\unicode[STIX]{x1D703}_{z})(\unicode[STIX]{x2202}j/\unicode[STIX]{x2202}t)+\boldsymbol{K}(z,\unicode[STIX]{x1D703}_{z})j^{2}]$ , where $\boldsymbol{M}=\boldsymbol{M}_{1}+\unicode[STIX]{x1D6EC}\boldsymbol{M}_{2}$ , $\boldsymbol{N}=\boldsymbol{N}_{1}+\unicode[STIX]{x1D6EC}\boldsymbol{N}_{2}$ and $\boldsymbol{K}=\boldsymbol{K}_{11}+\unicode[STIX]{x1D6EC}\boldsymbol{K}_{12}+\unicode[STIX]{x1D6EC}^{2}\boldsymbol{K}_{22}$ . Also, if $B_{1}$ and $B_{2}$ are constants (independent of time), then the swimmer velocity is given by $\boldsymbol{U}(z,\unicode[STIX]{x1D703}_{z},t)=\boldsymbol{M}(z,\unicode[STIX]{x1D703}_{z})+\unicode[STIX]{x1D716}\boldsymbol{K}(z,\unicode[STIX]{x1D703}_{z})$ . We note that the velocity of the swimmer located far away from a plane interface is $O(1)$ while the velocity due to the interface deformations is $O(\unicode[STIX]{x1D716})$ . Hence the dynamics of a swimmer with steady squirming gait near a weakly deforming interface is similar to that near a plane interface except if it is very close to the interface.

3.3 Swimming with time-reversible squirming modes