Generalized squirming motion of a sphere
A number of swimming microorganisms such as ciliates (Opalina) and multicellular colonies of flagellates (Volvox) are approximately spherical in shape and swim using beating arrays of cilia or short flagella covering their surfaces. Their physical actuation on the fluid may be mathematically modeled as the generation of surface velocities on a continuous spherical surface – a model known in the literature as squirming, which has been used to address various aspects of the biological physics of locomotion. Previous analyses of squirming assumed axisymmetric fluid motion and hence restricted all swimming kinematics to take place along a line. In this paper we generalize squirming to three spatial dimensions. We derive analytically the flow field surrounding a spherical squirmer with arbitrary surface motion, and use it to derive its three-dimensional translational and rotational swimming kinematics. We then use our results to physically interpret the flow field induced by the swimmer in terms of fundamental flow singularities up to terms decaying spatially as . Our results will enable to develop new models in biological physics, in particular in the area of hydrodynamic interactions and collective locomotion.
Due to their small sizes, microorganisms inhabit a world where viscous forces dominate and inertial effects are negligible. The Reynolds number, which characterizes the relative importance of inertial to viscous forces, ranges typically from for the smallest bacteria up to for spermatozoa . Fluid-based locomotion of microorganisms is vital in a number of biological processes, including reproduction, locating nutrient sources, preying, and escaping from predation [2, 3, 4]. Physically, locomotion at low Reynolds numbers suffers from the constraints due to the absence of inertia, mathematically manifested by the linearity and time-independence of the governing equation – the Stokes equations . Purcell illustrated the difficulties encountered in small-scale locomotion by introducing his scallop theorem , which states that any reciprocal motion (body deformation possessing a time-reversal symmetry) cannot lead to any net propulsion at zero Reynolds number .
Nature showcases a variety of mechanisms able to overcome the constraints of the theorem and achieve micro-propulsion. Many cells use one or more appendages, called a flagellum (plural, flagella), for propulsion. Traveling waves are then propagated along the flagellum either by internal bending (seen, for example, in the spermatozoon of eukaryotic cells) or passive rotation of a rigid helical flagellum (the case of swimming bacteria), in both cases allowing to break the time-reversal symmetry and hence escape from the constraints of the scallop theorem . A number of microorganisms possess multiple flagella. Escherichia coli is a bacterium with a few helical flagella that can wrap into a bundle to move the cell forward when the motor turns in a specific direction. Chlamydomonas reinhardtii is an alga (eukaryotic cell) with two flagella. Ciliates such as Opalina and Paramecium (illustrated in Fig. 1a) and colonies of flagellates such as Volvox (shown in Fig. 1b) have their surface covered by arrays of cilia (or short flagella) beating in a coordinated fashion .
Over the past 60 years, theoretical and experimental studies on locomotion of microorganisms have improved our understanding of life under the microscope [1, 10, 5]. Topics of recent active interest include the locomotion of cells in environments with complex geometries [11, 12, 13] and complex fluids [14, 15, 16, 17], the role of motility in the formation of biofilms [18, 19, 20, 21], collective dynamics of active particles [22, 23], and the effect of Brownian noise on swimming [24, 25, 26]. Considerable attention has also been given on the design of artificial microscopic swimmers  for potential biomedical applications such as microsurgery and targeted drug delivery .
Historically, Taylor  pioneered the theoretical modeling of flagellar hydrodynamics by analyzing the motion of a waving sheet in Stokes flows. The propulsion speed of the sheet was solved asymptotically in the limit of small waving amplitude compared to its wavelength. Subsequently, most of the theoretical studies in the field have derived their results asymptotically, meaning they are physically valid only in specific mathematical limits: small amplitudes [29, 30, 14, 15], long-wavelength [31, 32], slender filaments [33, 34, 35, 36, 37, 38], or in the far-field [39, 21]. As a result, very few exact solutions for swimming in Stokes flows exist.
The most popular exact solution is originally due to Lighthill  and Blake  and was developed to address the propulsion of ciliates. In their model, sometimes referred as the envelope model, the motion of closely packed cilia tips are modeled as a continuously deforming surface (envelope) over the body of the organism, taken to be of spherical shape [40, 41]. The deformation of the envelope can then be expanded about the surface of the spherical cell body and to leading order, the action of cilia is represented by distributions of radial and tangential velocities on the spherical surface. Lighthill  first derived the exact solution to the Stokes equation due to such a squirming motion on a sphere, with subsequent corrections and generalizations by Blake . Since then, the squirmer model has been all but adopted as the hydrogen atom of low-Reynolds number swimming. Originally developed to specifically model the swimming of ciliates, the squirmer model can also be useful in studying other types of swimming microorganisms, broadly categorized as “pushers” and “pullers” . Pushers obtain their thrust from the rear part of their body, such as the swimming of all peritrichous bacteria (like Escherichia coli). In contrast, for pullers the thrust comes from their front part, such as the breaststroke swimming of algae genus Chlamydomonas. The squirmer model can represent pushers and pullers by correspondingly changing the surface actuation (squirming profile), rendering it a general model to investigate the locomotion of microorganisms. As such it was used to study many problems including hydrodynamic interactions of swimmers [42, 43], suspension dynamics [44, 45], nutrient transport and uptake by microorganisms [46, 47, 48], optimal locomotion , as well as non-Newtonian [50, 51] and inertial effects .
Most studies on squirming motion follow the notation of Lighthill  and Blake  and assume that the surface distortion is axisymmetric. This simplifies the analysis significantly and results in swimmers undergoing swimming along their axis of symmetry only. Real microorganisms, however, actuate the fluid in a non-axisymmetric fashion. The protozoan Paramecium, for instance, rotates as it swims and has a helical distribution of cilia (Fig. 1a). Stone & Samuel  derived formulae which relate the translational and rotational velocities of a squirmer to the arbitrary squirming profiles on the sphere via the reciprocal theorem. In experiments with V. carteri, Drescher et al.  measured non-axisymmetric squirming profiles and utilized these reciprocal relations to analyze the swimming kinematics of the cell. However, the reciprocal relations do not give any information on the flow surrounding the squirmer.
In this paper, we generalize the classical squirming results to non-axisymmetric actuation. Using Lamb’s general solution in Stokes flow , we derive analytically the exact solution for the flow field surrounding the swimmer, together with the swimming kinematics, for a general non-axisymmetric squirmer. Lamb’s general solution is ideally suited for problems with spherical or nearly spherical  geometries, and a detailed description of the solution and its applications can be found in classical textbooks [57, 58]. Our results will be useful for addressing the role of non-axisymmetric actuation in a variety of problems in the biological physics of locomotion, including feeding and sensing, and the rheology of active suspensions. Furthermore, from a fundamental fluids perspective, our study allows to make the link between arbitrary surface motion and the appearance of non-axisymmetric flow singularities.
The structure of the paper is the following. The problem is mathematically formulated in Sec. 2, followed by a summary of the axisymmetric case in Sec. 3, where the swimming kinematics (Sec. 3.1) and flow structure (Sec. 3.2) are presented. We then generalize the analysis to a non-axisymmetric squirmer in Sec. 4, where we present the swimming kinematics (Sec. 4.1 & Sec. 4.2), the three-dimensional flow structure (Sec. 4.3), the rate of work due to swimming (Sec. 4.4), and the decomposition of arbitrary surface velocities in the form of Lamb’s general solution (Sec. 4.5). We conclude in Sec. 5. In Appendix A, we include detailed expressions of flow singularities used throughout the paper. While in the main text we present the results of a squirmer with purely tangential deformation, we include the more general case of non-axisymmetric squirmer with radial deformation in Appendices B–D.
We model theoretically the motion of a spherical ciliate of radius in an incompressible fluid at zero Reynolds number using spherical coordinates (Fig. 2), with , , and as the basis vectors. Following the envelope model, the action of the cilia is represented by a general squirming profile (tangential and radial surface velocities) over the spherical surface at . The fluid around the squirmer is governed by the Stokes equation
and the continuity equation for incompressible flows
where and represent the velocity and pressure fields respectively, and denotes the dynamic viscosity of the fluid. In the main text, we present only the results in the case of purely tangential squirming motion (i.e. no radial surface velocities), , as it is the most widely used squirming model in the literature. The more general case which includes the radial deformation is presented in Appendices B–C for completeness.
where is the position vector, and and are both harmonic functions
One can expand the functions and in series of solid spherical harmonics, and , where and denote spherical harmonics of order as
By taking the divergence of Eq. 1 and utilizing the continuity equation, Eq. 2, we obtain that the pressure satisfies the Laplace equation . The pressure is therefore also harmonic, and we can again expand it in a series of solid spherical harmonics, , where
One can then use these results to construct a particular solution to the Stokes equation as
where denotes the magnitude of the position vector.
Superimposing the homogenous and particular solutions gives the general solution
usually named after Lamb .
Here we require that the solution decays at infinity () and hence all harmonics of positive order are discarded. In addition, following Brenner , we replace by in Eq. 10 to obtain the form of Lamb’s solution convenient for exterior problems
and the pressure field is given by . Notice that the solutions of the case have also been discarded since they correspond to sources and sinks, which are unphysical in problems related to rigid particles [40, 41, 58].
After performing all the differential operations in Eq. 11, Lamb’s general solution in spherical coordinates takes the form
Here we have employed a recursion expression of associated Legendre polynomials
From the radial velocity component, Eq. 15, the requirement of purely tangential deformation leads to the relations
Enforcing the conditions in Eq. 19, the general flow field due to purely tangential squirming motion becomes
To reiterate, the above flow fields decay at infinity in the laboratory frame and correspond to purely tangential velocities at the body surface; the case with radial velocities is detailed in Appendices B–C.
Note that for simplicity in this paper we consider a neutrally buoyant squirmer, where the buoyancy force from the fluid balances the gravitational force on the squirmer. Hence, there is no net force and torque acting on the fluid (the force- and torque-free conditions). Should there be a density offset between the squirmer and the fluid it would result in a net force  and thus would add a Stokeslet component (Appendix A.1) to the flow field around the squirmer, which can be superimposed to the results of the current work.
3 Axisymmetric Squirming Motion
In this section, we use Lamb’s general solution in order to reproduce the axisymmetric results first derived by Lighthill  and Blake . The analysis also identifies new axisymmetric modes. With the general solution, Eqs. 20–22, the axisymmetric flow field () reduces to
for purely tangential squirming motion, where we have denoted the Legendre polynomials of degree . The surface velocities on the sphere have the form
Upon setting and using a simple rescaling
Here we have identified new axisymmetric modes, denoted , acting in the azimuthal direction (right hand side of Eq. 24) and the corresponding flow fields (last sum in Eq. 23), which were not accounted for in previous works. While Stone & Samuel  and Drescher et al.  employed the reciprocal theorem to discuss the swimming kinematics of a squirmer subject to arbitrary squirming profiles, the current results complete the analysis of axisymmetric squirming motion by providing the whole flow field. The physical interpretation of these new axisymmetric modes is discussed below in Sec. 3.2.
3.1 Swimming of an axisymmetric squirmer
When studying the swimming of a squirmer, it is best to think about the problem in two separate steps. In the first step, we consider the above solution, Eq. 23, with boundary conditions from Eq. 24 so that the squirmer is fixed in space (by an external force) and not allowed to move. This is sometimes referred to as the “pumping problem” in the literature. In the second step, we allow the squirmer to move freely and compute the induced translational () and rotational () velocities, given the boundary actuation in the pumping problem, Eq. 24. This allows the separation of the surface velocities due to the boundary actuation of the squirmer from the contribution due to the induced translation and rotation. To obtain the overall flow field, , of a swimming squirmer, we superimpose the solution of the pumping problem, , to the flow fields due to the induced translation, , and rotation, , and thus write
This first step (computing ) was accomplished in Eq. 23. We now determine the unknown swimming kinematics, , when the squirmer is free to move. This involves computing all the forces and torques acting on the swimming squirmer: the fluid force, , and torque, , due to the boundary actuation in the pumping problem, and the drag, , and torque, , due to the induced translation and rotation of the squirmer. We then enforce the overall force-free and torque-free conditions in swimming problems of Stokes flows as
The drag and torque due to the translation and rotation of a spherical squirmer are simply and respectively. For the contributions from the pumping problem, the net force and torque due to the boundary actuation can be conveniently computed in Lamb’s general solution as and respectively [57, 58]. The force and torque balances, Eqs. 27–28, therefore become
where we have used Eq. 19. Note that the propulsion and rotational velocities may also be obtained using a reciprocal theorem approach  as is discussed in Sec. 4.2. The translational swimming velocity agrees with that given by Lighthill  and Blake  provided the rescaling from Eq. 25 is used, giving .
For an axisymmetric squirmer, propulsion and rotation can only occur in the same direction (here, the -direction), and the squirmer hence can only follow a straight swimming trajectory. Also notice that among all the modes in the squirming profile, Eq. 24, just one mode contributes to propulsion, namely mode . Similarly, among all the new azimuthal modes in the boundary condition, only mode contributes to the rotation of the squirmer.
Finally, by superimposing the solution of the pumping problem, Eq. 26, and the flow fields due to the swimming kinematics, Eqs. 31 and 32, we obtain the overall flow field of an axisymmetric swimming squirmer, , in the laboratory frame as
Note that throughout the paper we will refer to the flow fields in the pumping and swimming problems as and respectively.
3.2 The axisymmetric flow structure
In this section, we identity the flow structure generated by a swimming squirmer as due to a superposition of flow singularities. This allows a physical interpretation of the flows caused by different modes of ciliary action in terms of combinations of point forces and torques and their spatial derivatives . Such an understanding is useful for constructing approximations for swimmers in theoretical modeling and computer simulations, where one can retain only modes relevant to the aspects of physics of interest. For instance, in the axisymmetric case, it is common to retain only the mode contributing to swimming (the source dipole mode) and the mode due to two point forces (Stokes dipoles) [42, 44, 45, 46], where the arrangement of the two point forces (the sign of the Stokes dipole) can represent different types of swimmers (“pushers” vs. “pullers”, see Sec. 1). We first revisit below the known correspondences between the flow field around an axisymmetric squirmer and different flow singularities. We then proceed to discuss the new axisymmetric modes and the interpretation of their corresponding flow singularities.
3.2.1 The mode
The primary fundamental singularity in Stokes flows is the flow due to a point force of magnitude and direction at the origin, where is the dirac delta function. The solution is given by , where
and is called a Stokeslet. That flow is long-ranged and decays as . A Stokeslet acting in the -direction has the explicit form in spherical coordinates
In the pumping problem, Eq. 23, we can then identify that the mode
To obtain the overall flow field, , surrounding a swimming squirmer, Eq. 26, the solution to the pumping problem, , needs to be superimposed with that due to a translating sphere at a velocity , Eq. 31, leading to
which also contains a Stokeslet and a source dipole. Unsurprisingly, the Stokeslet components cancel each other exactly and satisfy the overall force-free condition of a free swimming squirmer. Therefore, a Stokeslet component does not appear in the swimming flow field from Eqs. 33–35. Notice however that the cancellation of the source dipole components is incomplete, leaving a residual source dipole in the swimming flow field as
3.2.2 The and modes
The part decaying as can be be interpreted as the contribution of a Stokes dipole, which is a higher order singularity of Stokes flows and obtained by taking a derivative of a Stokeslet (directed in the direction) along the direction
The symmetric part of a Stokes dipole is termed a stresslet, first defined by Batchelor , and given by
which physically represents straining motion of the fluid. The antisymmetric part is termed a rotlet
where represents the strength (magnitude and direction) of the flow due to a singular point torque. The Stokes dipole with corresponds to only a stresslet
In the mode of the flow field, Eq. 43, we can readily identify a Stokes dipole (stresslet) (see Appendices A.2 & A.3), while the other part decaying as corresponds to a source quadrupole, a higher order singularity.
The first azimuthal mode in the pumping problem, Eq. 23, is given by the mode
and represents a rotlet in the -direction, . Similar to the translation case, the pumping problem solution, , is superimposed with the flow field due to the induced rotation at the rate (Eq. 32)
leading to the total flow field surrounding a swimming squirmer, (Eq. 26). Again, unsurprisingly, the rotlet components exerting torques on the fluid cancel out completely thus satisfying the overall torque-free condition. The mode is hence absent from the resulting swimming flow field, Eq. 33–35, leaving no trace of the rotational motion of the squirmer. The important difference between rotation and translation is that the rotational mode is due to velocities which are all in the direction of the rotation, so a complete cancellation of the flow field satisfying the torque-free condition is possible by simply rotating in the opposite direction at the same rate. In contrast, for translational swimming, such exact cancellation is not possible because the surface velocity has a distribution of directions all along the sphere relative to its swimming direction. In other words, one can construct the ultimate stealth rotating sphere using purely tangential modes but not a similarly stealth translating sphere.
In summary, the and modes together contain the representation of a Stokes dipole (stresslet plus rotlet) with the direction and gradient taken both in the -direction ().
3.2.3 The and modes
Higher order flow singularities can be obtained by repeatedly taking derivatives of the lower order singularities. For example, a Stokes quadrupole can be obtained by taking a derivative of a Stokes dipole, , along the direction , leading to
The flow field due to such a Stokes quadrupole decays as . In particular, a Stokes quadrupole with takes the simple form
which is useful in interpreting the mode in Lamb’s solution as we will see below.
Several components of the Stokes quadrupole have particularly clear physical meanings, such as the potential (source) dipole
where denotes its direction. A potential dipole in the -direction is given by
which is the residual component of the mode in the overall flow field of a swimming squirmer (see Eq. 41 or Eqs. 33–35). This potential dipole component contained in the mode, together with the part decaying as in the mode and given by
contain the representation of the Stokes quadrupole, , expressed in Eq. 51. The other component decaying as in the mode corresponds to a source octupole.
The first azimuthal component appearing in the flow field around a swimming squirmer is given by the mode
which represents another well-known component of the Stokes quadrupole, named a rotlet dipole. A rotlet dipole can be obtained by taking a derivative along the direction of a rotlet with the direction
In particular, a rotlet dipole with takes the simple form . This corresponds to the mode in Lamb’s general solution, Eq. 55, and provides the leading-order mode in the azimuthal direction. In Fig. 3, we plot the slowest decaying flow field, given by the mode (a stresslet in the far field, Fig. 3a), as well as the slowest decaying flow in the azimuthal direction, given by the mode (a rotlet dipole, Fig. 3b).
To summarize, the and modes contain physically the Stokes quadrupole, ; the mode corresponds to a rotlet dipole, , which is part of a Stokes quadrupole different than (see also Sec. 4.3 for further details).
4 Non-axisymmetric Squirming Motion
Recall that the more general analysis which includes nonzero radial surface velocities is given in Appendices B–C and we focus below on the swimming problem of a non-axisymmetric squirmer with purely tangential squirming profiles.
4.1 Swimming of a non-axisymmetric squirmer
We follow closely the analysis presented in the axisymmetric case (Sec. 3) to investigate the situation where the surface motion is non-axisymmetric. As discussed in Sec. 3.1, we consider the swimming problem as a superposition of a pumping problem with the boundary actuation in Eqs. 57–59 and the flow field due to the induced translation and rotation of the squirmer (given in Eq. 26). Applying the force and torque balances in this non-axisymmetric case, the results from Eqs. 29 and 30 continue to hold, but with solutions for and which are more involved, resulting in
For an axisymmetric squirmer, only the and modes contribute to propulsion and rotation respectively. In contrast, for a general non-axisymmetric squirmer, Eqs. 60-61 identify all the modes contributing to its three-dimensional locomotion. Specifically, three modes, , , and , contribute to the translational swimming in the , , and -directions respectively; similarly, three modes, , , and , lead to rotation in the , , and -directions respectively. By superimposing the flow fields due to the induced translation and rotation according to the velocities determined, Eqs. 60–61, with the solution to the non-axisymmetric pumping problem, Eqs. 20–22, we obtain the flow field around a non-axisymmetric swimming squirmer as
4.2 Swimming kinematics by integral theorems
The swimming kinematics of a squirmer can also be arrived using the reciprocal theorem approach taken by Stone & Samuel  without having to solve for the whole flow field. The theorem relates the swimming velocity to the surface distortion, , via a surface integral on the sphere S
which in spherical coordinates reads
By transforming the basis vectors in spherical coordinates to those in Cartesian coordinates, the integral simplifies due to the orthogonality of sinusoidal functions in the azimuthal angle , and we obtain
The integrals associated with and vanish upon integration by parts. The remaining integrals can be evaluated using a general expression derived below (see Eq. 78) in the special cases of and 1. We then obtain the results identical to those given by Lamb’s solution, Eq. 60. Similarly, one can employ the reciprocal theorem to derive the rotational rotational velocity 
and obtain the same result as above (Eq. 61). However, while the reciprocal theorem is a useful tool for determining swimming kinematics, it provides no information about the flow around the swimmer, which is a main result of our work.
4.3 The non-axisymmetric flow structure
In the axisymmetric case, we have interpreted the flow fields due to different modes of the squirming profile as fundamental flow singularities. We extend the idea here to physically interpret the flow induced by the non-axisymmetric terms. Note that, in each of the modes discussed below, the corresponding flow field is not a far-field approximation of the flow induced by the squirmer but an exact solution valid in the entire space, and it is an appropriate superposition of these modes which satisfies arbitrary boundary conditions on the spherical surface.