Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies

# Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies

François Nadal Commissariat à l’Energie Atomique, 33114 Le Barp, France    Eric Lauga Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Center for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWA, United Kingdom
###### Abstract

Recent experiments showed that standing acoustic waves could be exploited to induce self-propulsion of rigid metallic particles in the direction perpendicular to the acoustic wave. We propose in this paper a physical mechanism for these observations based on the interplay between inertial forces in the fluid and the geometrical asymmetry of the particle shape. We consider an axisymmetric rigid near-sphere oscillating in a quiescent fluid along a direction perpendicular to its symmetry axis. The kinematics of oscillations can be either prescribed or can result dynamically from the presence of an external oscillating velocity field. Steady streaming in the fluid, the inertial rectification of the time-periodic oscillating flow, generates steady stresses on the particle which, in general, do not average to zero, resulting in a finite propulsion speed along the axis of the symmetry of the particle and perpendicular to the oscillation direction. Our derivation of the propulsion speed is obtained at leading order in the Reynolds number and the deviation of the shape from that of a sphere. The results of our model are consistent with the experimental measurements, and more generally explains how time periodic forcing from an acoustic field can be harnessed to generate autonomous motion.

## I Introduction

The transport of synthetic micro- and nano-scale particles is a well-studied field of research, starting with the first studies on the effect of electric fields on colloidal suspensions in the 1920s. The topic has recently seen a revival of activity, due in part to the possible biomedical and environmental use of these devices Nelson2010 (). Indeed, small controlled bodies could be employed to achieve transport of cargo and drug delivery Sundararajan2008 (); Burdick2008 (), analytical sensing in biological media Campuzano2011b (); Wu2010 (). Furthermore, their fast motion could also be efficiently used to perform wastewater treatment Soler2013 ().

While deformable synthetic micro-swimmers Dreyfus2005 () are of fundamental interest to mimic the locomotion of real cellular organisms Bray2000 (); Lighthill1975 (); Lighthill1976 (); Brennen1977 (); Lauga2009 (), rigid synthetic micro- and nano-swimmers appear to provide a more practical alternative. A number of different mechanisms have been proposed to achieve propulsion of small rigid objects, as recently reviewed by Ebbens & Howse Ebbens2010 () and Wang et al. Wang2013 (). The propulsion mechanisms can be sorted into two generic categories: external mechanisms, in which a directional field is used to drive the object, and autonomous mechanisms, where the object performs a local conversion of the energy from an exterior source field. In the latter case, symmetry breaking of the particle itself (shape, composition) is usually required to achieve propulsion.

External strategies typically lead to a global motion of the assembly of micro particles. For instance, applying an electric field on a suspension of charged spherical colloids in an electrolyte leads to a collective motion of the assembly parallel to the field lines, a phenomenon known as electrophoresis Smoluchowsky1921 (). Applying a non uniform electric field on dielectric uncharged spherical particles in an electyrolyte leads as well to an ensemble motion of the colloids parallel to the field lines (dielectrophoresis Pohl1978 ()). Rigid particles can also be propelled by the mean of magnetic fields. For example, a time-varying magnetic field can be used to actuate in rotation an helical (chiral) body Ghosh2009 (); Zhang2009 (); Zhang2010 ().

Whereas external control is convenient for targeting and navigation, autonomous strategies are more suitable for swarming and cleaning tasks. In this case, particles show independent trajectories able to cover a given region of fluid in a limited amount of time than unidirectional similar trajectories resulting from external driving. Autonomous motion can be achieved by methods which typically require a breaking of the symmetry of the particle (not a requirement in the case of external forcing). Catalytic bimetallic microrods can propel at high velocities (10 ms) in a liquid medium by self-generating local electric fields maintained by a local gradient of charged species (self-electrophoresis) Paxton2004 (); Ibele2007 (); Ebbens2011 (). If the generated species is uncharged, the concentration gradient can also trigger a net motion of the particle through self-diffusiophoresis Pavlick2011 (); Pavlick2013 (); Cordova2008 (). Similarly, autonomous propulsion can be achieved by taking advantage of self-thermophoresis effects Jiang2010 (); Baraban2012 (); Qian2013 (). Self-electrophoresis and self-diffusiophoresis have the important drawbacks to be incompatible with biological media such as blood, for these processes rely on the use of toxic fuels – e.g. hydrogene peroxide Paxton2004 (); Ebbens2011 (), hydrazine Ibele2007 () in the case of self-electrophoresis or norborene in the case of self-diffusiophoresis Pavlick2011 () – and are inefficient in high-ionic strength media. Self-thermophoresis requires temperature differences of a few Kelvins which makes it difficult to use for medical applications.

As an alternative, acoustic fields are good candidates to enable autonomous propulsion in biocompatible media, as recently demonstrated experimentally by Wang et al. Wang2012 (). In that work, it was shown that micron size metallic and bimetallic rods located in the pressure nodal plane of a standing acoustic wave could undergo planar autonomous motion with speeds of up to 200 ms. In this paper, we provide a model for these experimental results. Specifically, we propose asymmetric steady fluid streaming as a generic physical mechanism inducing the propulsion of rigid particles in a standing acoustic wave. This mechanism requires a shape asymmetry of the particle, does not involve any other physical process than pure Newtonian hydrodynamics (in particular, no chemical reaction), and takes its origin in the non-zero net forces induced in the fluid by inertia under time-periodic forcing.

After drifting towards the pressure nodal plane under the effect of the radiation pressure Doinikov1994a (); Mitri2009 (), a rigid particle can be viewed as a body oscillating in a uniform oscillating velocity field - note that this is does not hold in the general case where the particle is located at an arbitrary -position in the resonator (see section V). If and refer respectively to the wavenumber of the acoustic radiation and the typical size of the particle, this assumption of local uniformity of the field is justified provided that , a limit true in the experiments in Ref. Wang2012 (). The motion of the particle relative to the surrounding fluid leads then to an oscillating perturbative flow which can be computed in the framework of unsteady Stokes flows. Such a viscous flow, when coupled with itself through the convective term of the Navier-Stokes equation, forces a steady flow (so-called steady streaming), together with a flow at twice the original pulsation. If the particle has a non-spherical shape, the force coming from the integration of the corresponding steady streaming stress over the surface of the particle will generically non cancel out, leading to propulsion. Critically, in the absence of inertia, no propulsion would be possible since the initial transverse oscillatory motion is time-reversible. The breaking of symmetry in the geometry is also indispensable and, as originally shown by Riley Riley1966 (), the net force coming from the integration of steady inertial stress (steady streaming stress) over the surface of an oscillating sphere is zero.

In order to mathematically model this physical mechanism, we first consider the problem of an axisymmetric near-sphere oscillating in a prescribed fashion in the transverse direction in a quiescent fluid. The particle is assumed to be force-free in the direction of its axis of symmetry. We start by a near sphere of harmonic polar equation (i.e. one whose shape differs from the sphere by a cosine of small amplitude) before considering an arbitrary axisymmetric shape. The case of a free particle in an oscillating uniform velocity field is then addressed as it corresponds to the experimental situation in which the particle is trapped at the pressure node of a standing acoustic radiation. The problem is governed by two dimensionless parameters: a shape parameter,quantifying the distance to a perfect sphere, and the Reynolds number. Our calculations will present the derivation of the propulsion speed at leading order in both, giving rise to a propulsive force on the order of shape parameterReynolds number. To perform the perturbation analysis, we expand the fields in Reynolds number and to introduce the shape parameter at each separated order in Reynolds.

The paper is organized as follows. Section II is devoted to the presentation of the problem. Geometry, governing equations, and boundary conditions are detailed. Section III is dedicated to the derivation of an integral expression of the first-order (in Reynolds) propulsion speed. Zeroth and first-order (in Reynolds) problems are successively addressed. The full solution to the zeroth-order – transverse oscillation of a near-sphere in a purely viscous fluid – is presented first. As we are interested in the first-order (in Reynolds) propulsion speed rather than in the full first-order flow field, the latter is not derived explicitly and instead, we use a suitable form of Lorenz’s reciprocal theorem to establish an integral expression of the propulsion speed Ho1974 (). Results provided by the numerical integration of the integral expression of the propulsion speed are presented in section IV. We then use section V to address the dynamics of an axisymmetric near-sphere free to move in an uniform oscillating exterior velocity field. We show in particular that the zeroth-order (in Reynolds) rotational oscillation of the near-sphere is of second-order (in shape perturbation number), so that the propulsion speed computed in the case of a non rotating particle (section III) can be used as is. We conclude the paper by a discuss of the numbers predicted by the model in relation to the original experiment Wang2012 (). In Appendix A, we demonstrate that the calculated propulsion speed does not depend on the choice of the origin of the coordinate system, a technical but important detail. As the integral form of the propulsion speed involves the expression of the flow field induced by an oscillating sphere in a purely viscous fluid, we recall its expression in Appendix B. Some further technical details concerning the zeroth-order problem are given in Appendix C. The use of the reciprocal theorem requires an auxiliary flow field. The characteristic of such a flow (axial translation of an axisymmetric solid body at constant speed in a purely viscous fluid) are given in Appendix D. Finally, in Appendix E we discuss the dipolar forces appearing when the rigid particle is not located at a pressure node of the acoustic field.

## Ii Problem formulation

### ii.1 Geometry and kinematics

The setup of our calculation in shown in figure 1. Both cartesian and spherical coordinate systems are used. Unit vectors of the cartesian (resp. spherical) coordinate system are referred to as , , and (resp. , , and ). The position is denoted by , and the spherical coordinates by , and . We use capital letters to refer to dimensional quantities, force, position, and velocity variables. Corresponding dimensionless quantities are denoted by small letters (this rule obviously does not apply for constants).

We first consider an axisymmetric homogeneous solid body the axis of which is in the -direction (in section V, the particle will be free to rotate). The body oscillates in a Newtonian fluid (density , viscosity ) along the transverse -direction at frequency . The amplitude of its oscillations relative to the quiescent fluid is denoted such that the relative velocity of any point of the body is , where .

In order to allow an analytical solution, the solid body is assumed to take the form of a slightly deformed, axisymmetric sphere. We thus write its shape as

 R=R0[1+ϵξ(θ)], (1)

where is the radius of the reference sphere, is the dimensionless small shape parameter and a dimensionless function of order one. The surface of the axisymmetric near-sphere is referred to as and its volume is denoted by . In our calculations we first assume that is of the form , with (). The value is not considered since the corresponding body is equivalent to a sphere at order (see Appendix A) and odd values of would lead to no propulsion by symmetry. The case of an arbitrary (axisymmetric) shape is dealt with in section III.4, but we first perform the analysis for one of the terms of the Fourier expansion of the shape function susceptible to provide a finite propulsion speed of the body along the direction of its axis of symmetry (). Note that the function with satisfies the condition

 ∫S0ξdS=0, (2)

where is the surface of sphere of radius . Consequently, the sphere of radius is the equivalent-volume sphere and . Note also that the origin of the spherical coordinate system used in the paper is in general not the center of mass of the body (except in section V), and we have thus that the equality

 ∫S0ξ\boldmathndS=\boldmath0 (3)

is not satisfied. This fact will be important when we address the translation/rotation coupled problem of the dynamics of a near-sphere in a uniform exterior oscillating velocity field (section V).

### ii.2 Governing equations and boundary conditions

The solid particle is moving in the laboratory reference frame and we choose to work in the frame of reference of the body. The dimensional velocity and pressure fields satisfy the incompressible Navier-Stokes equations

 ∂\boldmathU∂T+(\boldmathU⋅\boldmath∇)\boldmathU=−1ρ% \boldmath∇P+ν\boldmath∇2\boldmathU, (4) \boldmath∇⋅\boldmathU=0, (5)

where is the kinematic viscosity of the fluid. In equation (4), the additional inertial force field due to the acceleration of the origin of the (non Galilean) reference frame has been incorporated in the pressure term (since this force field is the gradient of the linear pressure field ). These equations can be made dimensionless by choosing , , , and as typical length, velocity, time, and pressure scales, and one gets

 λ2∂\boldmathu∂t+{% Re}(\boldmathu⋅\boldmath∇)\boldmathu=−\boldmath∇p+\boldmath∇2\boldmathu, (6) \boldmath∇⋅\boldmathu=0. (7)

In equation (6), quantifies the dimensionless distance over which the vorticity diffuses, and is the Reynolds number. In the following, will be referred to as the viscous parameter. Due to the assumption , the Reynolds number is smaller than by a factor . Considering our choice of nondimensionalization, the polar equation of the surface is now written as

 r=1+ϵξ(θ). (8)

Equations (6) and (7) have to be supplemented by a suitable set of boundary conditions. We assume that, due to inertial effects, the force-free body will propel in the -direction (the only one allowed by symmetry) and the corresponding dimensionless propulsion speed is denoted . As the analysis is performed in the reference frame of the particle, the boundary condition for the velocity field then takes the following form

 \boldmathu=\boldmath0on S, (9) \boldmathu→−\boldmathv∥−\boldmathv⊥=−v∥\boldmathez−e−% \scriptsize it\boldmathex,for|\boldmathr|→∞. (10)

With the aim of applying the reciprocal theorem (§III.3), we write the difference , transforming equations (6) and (7) into new equations as

 λ2∂\boldmathu′∂t+{Re}[(\boldmathu′⋅\boldmath∇)%\boldmath$u$′−(\boldmathv∥⋅\boldmath∇)\boldmathu′]=−\boldmath∇p′+\boldmath∇2\boldmathu′, (11) \boldmath∇⋅\boldmathu′=0, (12)

where since no additional pressure (stress) is associated with the uniform field . The new set of boundary condition is

 \boldmathu′=\boldmathv∥on S, (13) \boldmathu′→−\boldmathv⊥=−e−\scriptsize it\boldmath% exfor|\boldmathr|→∞. (14)

For notation convenience, we drop the primes in the rest of the paper.

## Iii Inertial propulsion speed

In the section, we consider the effects of inertia in the case of a near-sphere oscillating in the transverse direction in a prescribed way. We first expand the velocity and pressure fields in powers of the Reynolds number. The perturbation in shape is introduced once the governing equations are obtained at each order in Reynolds. We first consider in §III.1 the Stokes problem of an oscillating near-sphere (zeroth-order in Reynolds). In §III.2 we introduce the first-order (in Reynolds) problem. We then use a suitable form of the reciprocal theorem in §III.3 in order to obtain the axial propulsion speed at leading order in an integral form, thereby bypassing the calculation of the full flow at first order in Reynolds. The case of an arbitrary axisymmetric shape is finally presented in III.4.

We first expand the velocity, pressure, and stress fields in powers of the Reynolds number as follows

 u = \boldmathu(0)+{Re}% \boldmathu(1)+O({Re}2), (15) p = p(0)+{Re}p(1)+O({Re}2), (16) σ = \boldmathσ(0)+{Re}\boldmathσ(1)+O({Re}2). (17)

The stress expansion is a consequence of the velocity and pressure expansions since at each order,

 \boldmathσ(i)=−p(i)\boldmathδ+[\boldmath∇\boldmathu(i)+\boldmath∇\boldmathu(i)†], (18)

where the superscript refers to the transposed tensor and is the unit tensor. Introducing equations (15) –(17) in the Navier-Stokes equations, equations (11) and (12), leads to the two sets of similar equations satisfied by the zeroth and first order velocity/pressure fields. We consider them successively below.

### iii.1 Zeroth-order solution in Reynolds

The zeroth-order flow field satifies the Stokes equations

 λ2∂\boldmathu(0)∂t=−\boldmath∇p(0)+\boldmath∇2\boldmathu(0), (19) \boldmath∇⋅\boldmathu(0)=0, (20)

with the boundary conditions

 \boldmathu(0) =\boldmath0on S, (21) \boldmathu(0) →−\boldmathv⊥=−e−\scriptsize it\boldmathexfor|\boldmathr|→∞. (22)

Note that the oscillating transverse velocity is entirely taken into account in the zeroth-order boundary conditions. Note also that no axial propulsion speed is expected at that order since the kinematics corresponding to the transverse oscillation of the body can not lead to any net force in the axial direction (reversibility). From here, we make the additional assumption that

 λ2≪1. (23)

This condition means that the viscous penetration scale is much larger than the typical size of the body. The flow is therefore approximately Stokesian in the entire space, enabling us to use Lorenz’s reciprocal theorem. In the opposite limit (), the viscous flow would be confined to a thin layer of thickness , and the flow would be irrotational outside the viscous layer Riley1966 ().

In order to obtain the right order in the final propulsion speed, we have to expand the zeroth-order (in Reynolds) velocity and pressure fields to the first order in shape parameter, . We thus write

 \boldmathu(0) = \boldmathu0+ϵ% \boldmathuϵ+O(ϵ2), (24) p(0) = p0+ϵpϵ+O(ϵ2), (25)

where and are the velocity and pressure fields corresponding to the oscillations of the equivalent-volume sphere in a purely viscous fluid, and and are the first corrections due to the difference in shape between the particle and the equivalent-volume sphere.

Working in Fourier space and denoting Fourier transforms with a hat, the Fourier components of the velocity and pressure fields, and , satisfy the Stokes equations

 −iλ2^\boldmathu0=−\boldmath∇^p0+\boldmath∇2^\boldmathu0, (26) \boldmath∇⋅^\boldmathu0=0, (27)

together with the boundary conditions

 ^\boldmathu0 =\boldmath0on S0, (28) ^\boldmathu0 →−^\boldmathv⊥=−\boldmathexfor|\boldmathr|→∞. (29)

The Stokes flow induced by the oscillation of a sphere in a viscous fluid has been derived by Lamb Lamb () – see also Riley1966 (); Kim&Karrila (). This is a classical result and we recall its characteristics in Appendix B.

Due to the linear nature of the problem at zeroth-order in Reynolds, the corrective quantities and also satisfies the unsteady Stokes equations

 −iλ2^\boldmathuϵ=−% \boldmath∇^pϵ+\boldmath∇2^% \boldmathuϵ, (30) \boldmath∇⋅^\boldmathuϵ=0. (31)

The fist boundary condition satisfied by the corrective flow is found by Taylor expanding the boundary condition (21) to the first order in . Using equations (8) and (24), one then obtains the expression of the correction in shape on the spherical surface (i.e. at ) as

 ^\boldmathuϵ|r=1=−ξ(θ)∂^\boldmathu0∂r∣∣ ∣∣r=1. (32)

As is recalled in Appendix B, the radial derivative of the velocity field, , is given at the spherical surface by

 ∂^\boldmathu0∂r∣∣ ∣∣r=1=−32^% \boldmathv⊥⋅(1+e−% \scriptsize iπ/4λ)(\boldmathδ−\boldmathn%\boldmath$n$), (33)

where is the unit vector normal to which points towards the fluid (here ). Given that , the explicit form of the first boundary condition, expressed in the basis of the spherical coordinate system is thus

 ^\boldmathuϵ|r=1=Kcos(nθ)⎛⎜⎝0cosϕcosθ−sinϕ⎞⎟⎠, (34)

with . As the corrective velocity flow (due to the difference in shape from that of the sphere) must vanish at large distances from the particle, the following condition takes place

 ^\boldmathuϵ→\boldmath0% for|\boldmathr|→∞. (35)

The general form of the solution to the system formed by equations (30-31) has been derived by Chandrasekhar Chandrasekhar () as a sum of spherical harmonics. Taking the curl of equation (30) leads to the equation which governs the vorticity. After projecting the latter on the radial direction, one gets

 (iλ2+\boldmath∇2)(r^χϵ)=0, (36)

where is the radial component of the vorticity. Similarly, an equation for the radial component of the velocity is obtained by taking the radial component of the curl of the vorticity equation and we obtain

 \boldmath∇2(iλ2+\boldmath∇2)(r^uϵr)=0. (37)

The objective is now to derive explicit expressions for the radial components and of the velocity and vorticity fields. In principle, the three components of the velocity must satisfy the boundary condition (34). Unfortunatly, only the radial components of velocity and vorticity are involved in the governing equations (36) and (37). As is classically done in such situations Miller1968 (), we keep the condition of continuity of the radial component of the velocity

 ^uϵr=0atr=1, (38)

and build two alternative boundary conditions involving the surface divergence and the radial component of the surface curl of the velocity, by recombining the velocity components (and their derivative) given by (34). These new boundary conditions are then used below instead of the continuity conditions on the polar and azimuthal velocity components. The advantage of such an approach is that and are the only quantities involved in the new set of boundary conditions. The surface divergence and the radial component of the surface curl of the velocity at are given by Kim&Karrila ()

 −r\boldmath∇s⋅^\boldmathuϵ=r∂^uϵr∂r=−2^uϵr−1sinθ∂∂θ(^uϵθsinθ)−1sinθ∂^uϵϕ∂ϕ, (39) r\boldmather⋅\boldmath∇s×^\boldmathuϵ=r^χϵ=1sinθ∂∂θ(^uϵϕsinθ)−1sinθ∂^uϵθ∂ϕ, (40)

where is the surface gradient operator. After introducing the polar and azimutal components of at the surface given by equations (34) in equations (39) and (40), we obtain, for

 −\boldmath∇s⋅^\boldmathuϵ=Kcosϕ[nsin(nθ)cosθ+2cos(nθ)sinθ], (41) \boldmather⋅\boldmath∇s×^\boldmathuϵ=Ksinϕnsin(nθ). (42)

We further show in Appendix C that expressions (41) and (42) of the surface divergence and curl can be rewritten as sums of associated Legendre functions of order 1. Thus, the previous equations can be put in the form

 −\boldmath∇s⋅^\boldmathuϵ=Kcosϕk∑q=0B2(q+1)P12(q+1)(cosθ), (43) \boldmather⋅\boldmath∇s×^\boldmathuϵ=Ksinϕk∑q=0B2q+1P12q+1(cosθ), (44)

where the constants and are also given in Appendix C. Consequently, we can search for the radial components of velocity and vorticity in the form

 r^uϵr=k∑q=0r^uϵ2(q+1),withr^uϵ2(q+1)=KU2(q+1)(r)P12(q+1)(cosθ)cosϕ, (45) r^χϵ=k∑q=0r^χϵ2q+1,withr^χϵ2q+1=KX2q+1(r)P12q+1(cosθ)sinϕ. (46)

Introducing equations (45) and (46) into (36) and (37), and solving the resulting equations in , one obtains the general forms of and as

 U2(q+1) (r)=α02(q+1)r2(q+1)+β02(q+1)⎛⎜⎝πe−\scriptsize iπ/42λr⎞⎟⎠1/2J2q+52(e\scriptsize iπ/4λr) +α∞2(q+1)r−(2q+3)+β∞2(q+1)⎛⎜⎝πe−\scriptsize iπ/42λr⎞⎟⎠1/2H(1)2q+52(e\scriptsize iπ/4λr), (47) X2q+1 (r)=γ02q+1⎛⎜⎝πe−\scriptsize iπ/42λr⎞⎟⎠1/2J2q+32(e\scriptsize iπ/4λr) +γ∞2q+1⎛⎜⎝πe−\scriptsize iπ/42λr⎞⎟⎠1/2H(1)2q+32(e\scriptsize iπ/4λr). (48)

In the previous expressions and are Bessel functions and Hanckel functions of the first kind respectively. Boundary condition (35) imposes allowing us to drop the superscript in the following. The coefficients , and are then to be determined using the boundary conditions at the surface. After using equations (47) and (48) in the continuity conditions for the radial components of the velocity, equation (38), surface divergence, equation (43), and surface curl, equation (44), we obtain

 α2(q+1)+β2(q+1)⎛⎜⎝πe−\scriptsize iπ/42λ⎞⎟⎠1/2H(1)2q+52(e\scriptsize iπ/4λ)=0, (49) −α2(q+1)(2q+4)+β2(q+1)⎛⎜⎝πe−\scriptsize iπ/42λ⎞⎟⎠1/2[(2q+1)H(1)2q+52(e\scriptsize iπ/4λ) −(e% \scriptsize iπ/4λ)H(1)2q+72(e\scriptsize iπ/4λ)]=B2(q+1), (50) γ2q+1⎛⎜⎝πe−% \scriptsize iπ/42λ⎞⎟⎠1/2H(1)2q+32(e\scriptsize iπ/4λ)=B2q+1. (51)

When solved, the system of equations (49)–(50) gives the values of and , while the last equation gives directly the value of

 α2(q+1) = B2(q+1)⎡⎢ ⎢⎣e\scriptsize i% π/4λH(1)2q+72(e\scriptsize iπ/4λ)H(1)2q+52(e\scriptsize iπ/4λ)−(4q+5)⎤⎥ ⎥⎦−1, (52) β2(q+1) = −√2πB2(q+1)[(e% \scriptsize iπ/4λ)1/2H(1)2q+72(e\scriptsize iπ/4λ)−(4q+5)(e\scriptsize iπ/4λ)−1/2H(1)2q+52(e% \scriptsize iπ/4λ)]−1, (53) γ2q+1 = B2q+1⎡⎢ ⎢⎣⎛⎜⎝πe−\scriptsize iπ/42λ⎞⎟⎠1/2H(1)2q+32(e\scriptsize iπ/4λ)⎤⎥ ⎥⎦−1. (54)

As shown in Ref. Sani1963 (), the complete velocity field can be reconstructed from the radial velocity and vorticity components as

 ^\boldmathuϵ=^uϵr\boldmather+r22(q+1)[k∑q=0\boldmath∇sD^uϵ2(q+1)(2q+3)−\boldmather×\boldmath∇s^χϵ2q+1(2q+1)], (55)

where the operator is defined as

 D[...]=1r2∂∂r[r2...]. (56)

The expressions we then obtain for the components , , of the flow field are given in Appendix C.

### iii.2 First-order solution

We now consider the derivation for the first-order solution (in Reynolds) . That flow field contains terms of different frequencies, but we are here only interested in the steady part of the flow. For the sake of simplicity, we use to denote to the steady component of this first-order flow. The latter sastifies the following set of equations

 \boldmath∇⋅\boldmathσ(1)=14[(^\boldmathu–––––––––––––(0)⋅\boldmath∇)^\boldmathu(0)+(^\boldmathu(0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)], (57) \boldmath∇⋅\boldmathu(1)=0, (58)

where complex conjugate quantities are underlined. In the first-order governing equations, the term has been dropped since this term is time-dependent (dimensionless frequency 1) and we are only interested in steady flows. Equations (57) and (58) have to be completed by the boundary conditions

 \boldmathu(1)=\boldmathv(1)on S, (59) \boldmathu(1)→% \boldmath0at infinity. (60)

where the unknown quantity is linked to by the relationship

 \boldmathv∥={Re}% \boldmathv(1). (61)

In order to obtain the first-order translation speed, we could try to derive the full velocity and stress fields and , and integrate the stress over the particle surface to obtain the propulsive force. However, it is more convenient to use a suitable version of the reciprocal theorem, as suggested by Ho & Leal Ho1974 () (the standard version of the Lorentz reciprocal theorem can be found in Ref. Kim&Karrila ()).

### iii.3 Reciprocal theorem and propulsion speed

For the same geometry, we consider now an auxiliary Stokes velocity and stress fields satisfying

 \boldmath∇⋅¯\boldmathσ=%\boldmath$0$, (62) \boldmath∇⋅¯\boldmathu=0, (63)

with suitable boundary conditions to be specified below. Subtracting the inner product of equation (57) with and the inner product of equation (62) with , and integrating over the volume of fluid leads to the equality of virtual powers as

 ∫V[¯\boldmathu⋅(% \boldmath∇⋅\boldmathσ(1))− \boldmathu(1)⋅(% \boldmath∇⋅¯\boldmathσ)]dV= 14∫V¯\boldmathu⋅[(^\boldmathu–––––––––––––(0)⋅% \boldmath∇)^\boldmathu(0)+(^\boldmathu(0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (64)

Then, using the general vector identity

 ¯\boldmathu⋅(\boldmath∇⋅%\boldmath$σ$(1))− \boldmathu(1)⋅(% \boldmath∇⋅¯\boldmathσ)= \boldmath∇⋅(¯\boldmathu⋅%\boldmath$σ$(1)−\boldmathu(1)⋅¯\boldmathσ)+(\boldmath∇\boldmathu(1):¯\boldmathσ−\boldmath∇¯\boldmathu:\boldmathσ(1)), (65)

and realizing that the second term in the right-hand side of equation (65) vanishes for a Newtonian fluid, we can rewrite equation (64) as

 ∫V\boldmath∇⋅(¯\boldmathu⋅\boldmathσ(1)−\boldmathu% (1)⋅¯\boldmathσ)dV=14∫V¯\boldmathu⋅[(^\boldmathu––––––––––––––(0)⋅\boldmath∇)^\boldmathu(0)+(^\boldmathu% (0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (66)

Using the divergence theorem allows to simplify the left-hand side term and obtain

 ∫S\boldmathn⋅(¯\boldmathu⋅\boldmathσ(1)−\boldmathu(1)⋅¯\boldmathσ)dS=−14∫V¯\boldmathu⋅[(^\boldmathu––––––––––––––(0)⋅\boldmath∇)^\boldmathu(0)+(^\boldmathu% (0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (67)

We now define the boundary conditions for the auxiliary problem, . We assume that it represents a solid-body motion with translational and angular velocities and , so that the auxiliary velocity at the surface of the body is given by

 ¯\boldmathu=¯\boldmathv+¯\boldmathω×\boldmathr, (68)

where is the position vector. Since is the first-order propulsion speed, we can introduce equations (59) and (68) into equation (67), leading to the equality

 ¯\boldmathv⋅∫S% \boldmathn⋅\boldmathσ(1)dS+ ¯\boldmathω⋅∫S%\boldmath$r$×(\boldmathn⋅\boldmathσ(1))dS−\boldmathv(1)⋅∫S\boldmathn⋅¯\boldmathσdS =−14∫V¯\boldmathu⋅[(^\boldmathu–––––––––––––(0)⋅% \boldmath∇)^\boldmathu(0)+(^\boldmathu(0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (69)

In equation (69), the first term on the left-hand side is nothing but the inner product of the auxiliary translational velocity of the solid body with the hydrodynamic force, , in the main problem. The second term is the inner product of the auxiliary angular velocity with the torque, , applied on the solid body by the main flow. The third term is of similar nature to the first one with the role of the flows reversed. Denoting by the force applied by the auxiliary flow on the solid body, we obtain a convenient form of equation (69) as

 ¯\boldmathv⋅\boldmathf(1)+¯\boldmathω⋅\boldmatht(1)−\boldmathv(1)⋅¯% \boldmathf=−14∫V¯\boldmathu⋅[(^\boldmathu–––––––––––––(0)⋅% \boldmath∇)^\boldmathu(0)+(^\boldmathu(0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (70)

For the problem of oscillation of the near sphere considered in this paper, only two quantities are important to compute in equation (70). Either the particle is free to move and we want to calculate or the particle is tethered and we wish to compute the hydrodynamics force applied by the fluid, balancing the external force tethering it. In both cases, we can therefore pick and arbitrary. The flow with these boundary conditations has been calculated in the very general case of an arbitrary near-sphere Happel&Brenner (). The particular case of an axisymmetric near-sphere in axial translation is presented in Appendix D. As , equation (70) becomes

 \boldmathv(1)⋅¯% \boldmathf=14∫V¯\boldmathu⋅[(^\boldmathu–––––––––––––(0)⋅% \boldmath∇)^\boldmathu(0)+(^\boldmathu(0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV. (71)

in the case of the force-free near-sphere (), and

 ¯\boldmathv⋅\boldmathf(1)=−14∫V¯\boldmathu⋅[(^\boldmathu––––––––––––––(0)⋅\boldmath∇)^\boldmathu(0)+(^\boldmathu% (0)⋅\boldmath∇)^\boldmathu–––––––––––––(0)]dV (72)

in the case of a tethered oscillating near-sphere (). Note that in these two equations, the magnitude of the solid body motion in the auxiliary problem is arbitrary, since the hydrodynamic force scales linearly with the magnitude of the imposed velocity.

Focusing on the force-free swimming case, we now consider the expansion of the right-hand side of (71) to order . We first write the auxilliary flow as , where is the field generated by an equivalent-volume sphere translating at a velocity and the perturbative field due to the non sphericity of the particle. We also expand the steady drag as , where is the steady drag of the sphere (of magnitude ) and the corrective drag due to the non sphericity of the particle. The expressions of and are both given in Appendix D. Noticing that

 V=V0+∑iVi++∑iVi−, (73)

where is the volume of fluid outside the equivalent-volume sphere and and are defined in figure 1, and recalling that can also be written as , equation (71) can be expanded to order to get formally

 \boldmathv(</