Hydrodynamics of a Rotating Torus

Eur. Phys. J. B 60, 325–336 (2007) DOI: 10.1140/epjb/e2007-00358-1 THE EUROPEAN PHYSICAL JOURNAL B

Hydrodynamics of a rotating torus

R.M. Thaokar1,a, H. Schiessel2, and I.M. Kulic3 1 Department of chemical engineering, Indian Institute of Technology, Bombay, Mumbai 400076, India 2 Instituut-Lorentz, Universiteit Leiden, Leiden, Postbus, 9506, 2300 RA, The Netherlands 3 School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Received 23 July 2007 / Received in ﬁnal form 2 November 2007 Published online 22 December 2007 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007

Abstract. The hydrodynamics of a torus is important on two counts: firstly, most stiff or semiflexible ring polymers, e.g. DNA miniplasmids are modeled as a torus and secondly, it has the simplest geometry which can describe self propelled organisms (particles). In the present work, the hydrodynamics of a torus rotating about its centerline is studied. Analytical expression for the velocity of a force free rotating torus is derived. It is found that a rotating torus translates with a velocity which is proportional to its internal velocity and to the square of the slenderness ratio, , similar to most low Reynolds number swimmers. The motion of a torus along a cylindrical track is studied numerically and it is observed that a force free torus changes its direction of motion (from a propelled state (weak wall effects) to a rolling state (strong wall effects)) as the diameter of the inner circular cylinder is increased. The rolling velocity is found to depend only on when the inner cylinder diameter approaches that of the torus.

PACS. 47.85.Dh Hydrodynamics, hydraulics, hydrostatics

1 Introduction

The list of models suggested to explain swimming at low Reynolds number is ever increasing, now spanning almost five decades [1–6]. The peculiarity of low Reynolds number flow lies in its quasi-static, time reversible nature, which renders impossible any reciprocal motion to propel the organism. This means that only certain types of shape changes that are non-reciprocal and break the time- reversal symmetry can make the microorganism move, and most suggested models in the literature take this into ac- count [4–6]. Purcell [3] in his famous article on life at low Reynolds number, suggests a rotating torus which can translate as a possible mechanism of motility of liv- Fig. 1. Hydrodynamics of a torus: (depicted hereas a mini- ing organisms. Similar recommendation was discussed by plasmid) (a) Translation of a single torus, (b) A torus moving Taylor [7], although no calculations to date exist on the along a cylindrical rail. propulsion velocity of such an object and its dependence on the angular velocity about the centerline. The other instance where a torusisencounteredisthe the longitudinal axis (a spinning torus), on edge rotation description of stiff and semiflexible polymer rings (e.g. and flow around an expanding torus. However, they did DNA miniplasmids), which can be best modeled as a not address the problem of rotation around the center- torus. Recently, we [8] proposed a nanomachine, consist- line (Fig. 1a). There are very few studies that discuss this ing of a DNA mini-plasmid, which is set into rotations by problem [10], although the self-propelled characteristics rectifying thermal fluctuations, using the ratchet effect. are not addressed in any of these and this forms the basis In an extensive work on the hydrodynamics of a torus, of the present study. Johnson [9] studied the flow around a torus for five dif- Interestingly, while suggesting a torus as a possible ferent situations: Translation along the longitudinal axis, geometry to explain a low Reynolds number swimmer, translation along the two transverse axes, rotation along Taylor [7] qualitatively argues about a situation in which a cylindrical rod threads through the hole in the torus a e-mail: [email protected] (Fig. 1b). Such a torus would reverse its direction and 326 The European Physical Journal B can possibly go from a propulsion state to a rolling kind z z motion. In our recent work on a DNA miniplasmid as a Stokeslet nanomachine, we [8] argue that although a DNA mini- f Rotlet plasmid can self-propel itself, its effect can be nullified by R m the substantial out of plane rotational diffusion prevalent ζ a φ r in these machines owing to their small sizes [11]. This can X x θ be prevented though, by putting the DNA mini-plasmid y on a rail, for example a DNA threading through the centerline. This is identical to a torus rotating about its cen- ω terline and moving along an infinitely long cylinder. An- alytical solutions were obtained for the case of a slender y torus moving inside a tube and the force acting on such a torus [12,13]. However, there are no calculations to date which address the motion of a rotating torus along a cir- Fig. 2. Coordinate system for the torus: Local representation cular cylinder, and a possible reversal of direction, which in cylindrical coordinates. is exactly the proposed mechanism for the nanomachine discussed by us [8]. In the present work, the hydrodynamics of a torus, Navier Stokes equations are given by rotating about its center-line in the zero Reynolds number limit is addressed, which is the relevant flow regime ∇·u =0 (3) 2 for less than micron sized objects. We construct the so- Re [∂tu + u ·∇u ]=−∇p + ∇ u, (4) lution to this problem to an order higher in slenderness b/a 2 2 ratio (O( ) ), than that considered earlier in the litera- where the Reynolds number Re = a ρf /(µω). For micron ture (O(b/a)), [10]. The full velocity profile for a rotating sized particles with speeds of few microns per second, the torus which is not allowed to translate is derived next and Reynolds number is indeed very low. In this work, we con- the force on such a torus is determined. Analytical results sider the limit of low Reynolds number, so that the Navier are presented in the limit of a slender torus, a limit which Stokes equations reduce to the familiar Stokes equations is relevant in for e.g. DNA mini-plasmids [8]. Although slender torus results are useful, especially in the case of ∇·u =0 (5) polymer rings, there might be instances in which the torus 2 −∇p + ∇ u =0. (6) thickness is of the same order as the internal radius, and the the boundary integral method is used to numerically Here we use two coordinate systems (Fig. 2), the Carte- solve the Stokes equation. We verify our analytical results sian (ex, ey, ez), and the cylindrical coordinate system using the numerical scheme and then extend the calcula- (ex, er, eθ). The unit vectors of the two coordinate sys- tions to the non-slender limit. The problem of a force free tems are related by torus moving along a cylindrical rod is addressed next. The force free velocity is calculated in several regimes and ey = er cos θ + eθ sin θ (7) comparisons made with existing literature. The calcula- ez = er sin θ − eθ cos θ. (8) tions are then extended to the lubrication limit. The solution for velocity can be expressed in terms of fundamental solutions of Stokes flow like the rotlet, the 2 Analytical solution of a rotating torus Stokeslet, the stresslet and the potential dipole [15,16]. For a rotating torus about its centerline, we assume a uni- form distribution of rotlets of strength mex × er acting Consider a torus with smaller diameter b and larger diam- along the center line of the torus, m is a scalar and ex is eter a, rotating about its centerline with an angular veloc- the unit vector in the direction of the axis of symmetry of ity ω in a Newtonian fluid, in the zero Reynolds number the torus. er is the unit vector joining the center line with limit (Fig. 2). In the analysis to follow, we indicate dimen- any point on the surface of torus and ex × er is tangent to sional quantities by a tilde. The governing equations for thecircleatpointζ = aer [10]. The dominant singularity the fluid are the Navier Stokes equations (continuity and representing the velocity field of a “self propelled rotat- momentum), ing torus” is expected to be the rotlet. The effect of the Stokeslet (force singularity) and the potential dipole (a ∇·˜ u˜ =0 (1) higher order singularity) is considered later. The velocity 2 in terms of the rotlet is then given by ρf ∂t˜u˜ + u˜ · ∇˜ u˜ = −∇˜ p˜ + µ∇˜ u˜ (2) e × ζ × R u m( x ) . where µ is the viscosity of the fluid. The physical quan- = R3 (9) tities are non-dimensionalized as follows: The lengths are scaled by a, velocity with ωa,timeby1/ω, the stresses and Any reference point in the fluid, can be repre- c the pressure by µω. With this non-dimensionalization the sented in cartesian coordinates (denoted with () )by R.M. Thaokar et al.: Hydrodynamics of a rotating torus 327

X =(x, r cos θ, r sin θ)c, while the centerline of the torus a Stokeslet is given by ζ˜ ,a φ, a φ c θ is given by =(0 cos sin ) . Here, is the az- R · f imuthal angle in the cylindrical coordinate system for p 2 . = 3 (15) any reference point X in the fluid, while φ defines the R azimuthal angle subtended by the point rotlet ζ,acting a An axially symmetric situation is considered here, i.e. along the centerline. Choosing the unit-length , the vec- a Stokeslet acting in the flow direction, x, and of con- tor joining the reference point and the centerline becomes stant magnitude f so that f = fxex. The velocity is then R X − ζ x, r2 − r φ − θ ,φ− θ cyl = =( +1 2 cos ( ) ) with given by (...)cyl denoting the cylindrical coordinates with magni- e Rx u x f . tude R given by r2 + x2 +1− 2rcos(φ − θ). The veloc- = R + R3 x (16) ity at point X generated by the distribution of the point R X − ζ rotlets acting along the centerline can be obtained by in- Similarly as in the rotlet case , using = with mag- 2 2 2 tegrating over φ. The velocity components are nitude R = x +(r +1) − 4r cos (φ/2), and integrat- φ ing over the distribution of the ring of point Stokeslets, 2π a − r φ − θ we obtain u dφ m cos( ) x = 3 (10) 0 R 2π x2 I I u f dφ 1 f 1 x2 3 2π φ − θ x = R + R3 = x C + C3 (17) u dφ mxcos( ) 0 r = 3 (11) 0 R 2π r φ − θ xf rI − I u fx dφ − cos( ) x( 3 4) 2π φ − θ r = R3 R3 = C3 u dφ mxsin( ) . 0 θ = 3 (12) 0 R (18) 2π cos(φ − θ) 2xfxI3 Since the rotlet is a special case of the fundamental so- p fx dφ . =2 R3 = C3 (19) lution which is identically annihilated by the Laplacian 0 ∇2u = 0, the pressure due to the rotlet, equation (6), is a constant. The flow due to a rotlet is vorticity free It is known that the Stokeslet and its higher derivatives are solutions to Stokes equations [16]. Thus if G represents and it turns out to be a solution to both the Stokes and ∇2G potential flow equations. This explains the similarity of a Stokeslet then is also a solution to the Stokes equa- results obtained in our work with the propulsion of vortex tion and is called as the potential dipole. The velocity due rings [17,18]. to a potential dipole can be easily derived as

The angular component of the velocity goes to zero by I 3RR u − · d. symmetry. The velocities in equations (10) and (11) can = R3 R5 (20) be integrated and easily expressed in terms of complete E k K k k elliptic integrals ( )and ( ) (Appendix A), where Consider the distribution of potential dipoles along a ring, is deﬁned as, k2 =4r/ x2 +(r +1)2 . acting in the x direction, such that d = dpxex,thex and The velocities in terms of these elliptic functions are the r directional velocities are given by, given by (Appendix A), 2π x2 I x2I m I − rI u d dφ 1 − 3 f 3 − 3 6 ( 3 4) x = px 3 5 = x 3 5 (21) ux = (13) R R C C C3 0 2π r φ − θ mxI4 u xd dφ − cos( ) ur = r =3 px 5 + 5 (22) C3 0 R R 3dpxx(rI6 − aI7) 2 2 − . with C = x +(r +1) and I3 and I4 elliptic integrals = C5 given in Appendix A. The latter is exactly the result of [10]. Note that there is no pressure contribution of the potential In our analysis we consider external forces acting on dipole [16]. The net velocity resulting from the contribu- the torus, and analyze two additional singularities, the po- tion of the rotlets, Stokeslets and potential dipoles can be tential dipole of strength dpx and the Stokeslet of strength written as fx in the x direction. We now discuss the velocity due to m I I a ring of point forces and point dipoles, distributed along u I − rI d 3 − x2 6 x = C3 ( 3 4)+ px C3 3 C5 the centerline. Consider the velocity due to a Stokeslet of strength f which is given by I I f 1 x2 3 + x C + C3 (23) I RR u = + · f. (14) mxI d x xf R R3 u 4 − 3 px rI − I x rI − I r = C3 C5 ( 6 7)+ C3 ( 3 4) RR I 2fxxI3 Here denotes the dyadic product, is the identity p = (24) matrix and f is a point force vector. The pressure due to C3 328 The European Physical Journal B

z 2.1 Calculation of the velocity of a force free rotating torus b ψ The velocity of a force free torus is obtained by substitut- ing fxo = 0 in equations (25) and (26). The strength of the 1 dipole can be obtained from equation (26) as dpxo = − 8 . x a Equation (25) indicates that the velocity is given by

2 8 1 Ux = log − . (27) x 2 2 This is the nondimensional translational velocity of a freely rotating torus. The dimensional velocity of a rotating torus with an angular velocity ω can therefore be y given by ωb2 8a 1 U˜x = log − . (28) 2a b 2 which is exactly identical to the well known expression for Fig. 3. Local parameterization of the torus. velocity due to an ideal vortex ring as predicted by the potential flow theory [17,18]. This is expected because for- u where p is the pressure and has contribution only from mally a rotlet rot is also the expression for the velocity the Stokeslet. For a torus rotating with angular velocity ω field of an ideal line vortex. However, despite the kinematic analogy between the twirling torus and an ideal vortex and translating with velocity Ux the boundary conditions ring, the physics behind is quite different. The propaga- areu ˜x = ω(a − r˜)+U˜x andu ˜r = ωx˜ at the surface of the torus. The non-dimensional boundary conditions thus tion of an ideal vortex ring does not require any external forces/torques and is governed by conservation of ki- become ux =(1− r)+Ux and ur = x. In the special netic energy and momentum. In sharp contrast the low case of an immobile torus, we set Ux = 0. We further consider here the case of a slender torus (the ratio of the two Reynolds-number flow is governed by the dissipation in radii of the torus, = b/a is small), and the surface is the fluid, which is manifested as a local force and a lo- locally parameterized by an angle ψ, such that x = cos ψ cal torque acting on the torus. The rotating torus at low and r =1+ sin ψ (Fig. 3). This implies the expansion Reynolds numbers has to constantly dissipate energy to 2 maintain its rotation as opposed to the inertia dominated k2 − O 3 =1 4 + ( ) in the slender torus limit. The asymp- ideal vortex ring [17,18]. k totic expressions for the elliptic integrals in the =1 If we consider a rotating torus as a model of a self limit (Appendix A) can therefore be used to simplify the propelled organism [3], the translational velocity is pro- expressions. The boundary conditions demand a scaling , 2 4 2 portional to the “internal velocity” ω and quadratic in m = m0 , dpx = dpx0 and fx = fx0. The bound- b/a x r the ratio ( ) in agreement with the previous results for ary conditions for the and directional no slip velocity the models of swimming of microorganism [5,4]. conditions read as follows:

2 (mo +4dpxo −2fxo) Ux − sin ψ = −2mo sin ψ − cos 2ψ 2.2 Calculation of force on a non-translating torus 2 2 8 1 8 1 + 2mo(log − )+4fxo(log + ) (25) 2 2 2 To calculate the velocity field due to a rotating torus, prevented from translating by an external force, con- 2 m 1 (mo +4dpxo − 2fxo) sider equations (25) and (26). Here o = 2 satisfies cos ψ =2mo cos ψ − sin 2ψ. O O 2 2 the ( )equation.At ( ), the quantities dependent (26) on the angular parts can be balanced appropriately by O m 1 2fxo−mo At ( ), we get the strength of the rotlet as o = 2 which dpxo = 4 . The Stokeslet strength fxo can be calcu- is the central result of [10]. lated by equating the translational velocity to zero, and Equations (25) and (26) also justify the use of only the the strength of the Stokeslet can now be easily determined rotlet, the Stokeslet and the potential dipole in the expan- as sion since the boundary conditions can be satisfied using 2 ln(8/) − 1/2 fx = − . (29) these singularities. The higher order terms from these sin- 4 ln(8/)+1/2 gularities can be balanced by additional singularities like the r directional stokeslet, but we restrict the analysis here The strength of the potential dipole is then given by O 2 to ( ). 8 Equations (25) and (26) can be solved in two situations 1 log dpxo = − 8 1 . (30) which we now discuss. 4 log + 2 R.M. Thaokar et al.: Hydrodynamics of a rotating torus 329

The local stress tensor can be expressed as would, however, be interesting to extend the results to the T =[(σxx,σxr), (σrx,σrr)], where different elements of the case in which the torus thickness is of the same order as stress tensor have following definitions: the internal radius. For the motion of such a non-slender torus it is necessary to revert to some kind of numerical dux method. Here we use the boundary integral method which σxx = −p +2 (31) dx is a singularity method and is best suited to solve Stokes dux dur equations. The drag calculation (resistance problem) in- σxr = σrx = + (32) dr dx volves solving integral equation of the first kind, which are du known to generate ill conditioned matrices [15], although σ −p r . rr = +2 dr (33) converged non-oscillatory solutions are reported in certain specific cases [19]. For force calculation in the torus, The traction vector (force per unit area) is t = Tn,andits we do get well behaved converged solutions which are in value in the x direction is given by exTn,wheren is the good agreement with the analytical solution in the slender unit normal given by (cos ψ, sin ψ). The net force on the torus limit. The calculations are then extended to the non- torus is then calculated by integrating the traction over slender limit. In fact, the analytical expressions reported the area of the torus given by dψ(1 + sin ψ)2π.Thus in Section 2 are multipole expansions of the complete in- the net x directional non-dimensional force is given by tegral equation, for force distribution, which can be solved

2π numerically using the boundary integral method [16]. Note 2 2 ln(8/) − 1/2 that multipole expansions are different moments of the Fx = dψ(1 + sin ψ)2πtx =4π 0 ln(8/)+1/2 Stokeslet about the centerline of the torus [16]. (34) The representation of the velocity by the boundary and the dimensional force by integral equation can be written as 2 2 ln(8/) − 1/2 F˜x =4π µb . (35) 1 / / ui x0 − dS x Gij x, x0 fj x ln(8 )+1 2 ( )= πµ ( ) ( ) ( ) 4 The main contribution to the force therefore comes from 1 + dS(x)uj(x)Tijk (x,x0)nk(x) (38) the Stokeslet and has two parts: the skin drag due to vis- 4π cous stresses and the form drag due to pressure variation around the torus. Unlike the case of a sphere where the δij xixj where Gij (x, x0)= + 3 and Tijk(x, x0)= form drag is half the skin drag, it is found that for the r r −6 xixj xk torus the two are of the same magnitude and given by r5 where r = |x − x0| , S(x) is the surface area 2 2 ln(8/)−1/2 n 2π ln(8/)+1/2 . of the body over which integration is carried out and k The net torque on the torus about the centerline is is the local unit outward normal to the surface. However, given by the moment of the tangential force which is given the second term in this equation can be modified [15] to by tTn about the center (lever arm ) avoid singular kernels and equation (38) can be solved numerically if suitably modified (Eq. (73), discussed in Ap- pendix C). We make use of the axisymmetry in the prob- Nc = dψ(1 + sin ψ)2π(tTn ) lem, akin to the analytical solution discussed in the earlier section. The Greens functions are modified after integra- / − / 2 2 2 5 1 ln(8 ) 1 2 tion in the azimuthal direction and are provided in several =8π 1 − + eθ (36) 16 8 ln(8/)+1/2 references [15,16]. These are expressed in terms of elliptic integrals of the first and second kind. In the above equa- and hence the net torque integrates out to zero. The di- tion ui(x)isgivenby(Ux +(1−r),x). The unknowns fj(x) mensional, locally felt,torqueisgivenby are then solved for by numerically evaluating the integrals

/ − / by discretizing the arc length into numerous elements and N˜ π2µb2a − 2 5 1 ln(8 ) 1 2 e interpolating using cubic splines. The additional unknown c =8 1 + / / θ 16 8 ln(8 )+1 2 Ux is obtained by imposing the no-force in the x direction (37) condition which is dS(x)fx(x) = 0. The Greens function One should note though, that, to maintain the local torque exhibit − log r singularity as r → 0thatisx → x0.The some form of internal driving mechanism of the torus is singularity is handled in the usual way [15] by subtracting needed, which is possibly provided by the metabolic ac- the singular part from the Greens functions and carefully tivity of the swimmer,or rectiﬁcation of thermal motion integrating it analytically. The force, arc length and all as in [8]. other variables are also expressed as cubic splines. The condition of no net x directional force (force free torus) is enforced to get the unknown longitudinal translational 3 The boundary integral method (BIM) velocity. The comparison of the asymptotic and numerical solutions are discussed in the following section. In all The results in the slender limit are practically useful, es- the simulations reported in this work, we have used 80 el- pecially, in the case of polymer rings and miniplasmids. It ements for discretization. The simulations were repeated 330 The European Physical Journal B

2.5 4

2 3 1.5 2 1

0.5 1 r r

, f 0 , f

x 0 x f f −0.5 −1 −1

−1.5 −2

−2 −3

−2.5 0 1 2 3 4 5 6 7 −4 0 1 2 3 4 5 6 7 ψ ψ x r Fig. 4. Comparison of the analytical and numerical and Fig. 5. Comparison of the analytical and numerical x and r di- . Fx directional forces for a slender ( =0025) torus ( , (Num), rectional forces for a non-slender ( =0.5) torus ( , Fx(Num), ··· Fx · · Fr Fr , (Anal), , (Num), , (Anal)). Note that ···, Fx(Anal), · ·, Fr(Num), , Fr(Anal)) the analytical and numerical results are identical and almost indistinguishable. 0.8

0.7 with 160 elements and no signiﬁcant diﬀerence was obtained up to the third decimal place. 0.6

0.5 a ω

4 Comparison of analytical and numerical / x 0.4 results U=U 0.3 The results derived for a freely rotating torus require a force and torque free motion. However, locally at each 0.2 point on the torus, there is a ﬁnite force and torque acting and analytical expressions can be easily obtained as t·T·n, 0.1 where T is given by equations described in section 2.2. 0 2 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (16 + 6 − 3 (cos 2ψ + log 64 − 2log)) sin ψ ε=b/a fx = 8 Fig. 6. Comparison of the numerical ( ) and analytical − 2 cos 2ψ ( ) translational velocity (39)

2 cos ψ(16 − 6 + (−3 (cos 2ψ − log 64 + 2 log ) . fr = − for slenderness ratio as high as 0 5. This can be attributed 8 to the O(2) analysis carried out in the present work. For − 4sinψ cos ψ. a really “fat” torus, the translational velocities are lower (40) than those predicted by the analytical slender limit re- Note that both the forces when integrated over the sur- sults. A “fatter” torus seems to be slower than that pre- face of the torus equate to zero, indicating a globally force dicted analytically, which could be due to the hydrody- free, torque free torus. Figure 4 compares the numeri- namic interaction of the flow fields of various sections of cal solution with the analytical expressions for the forces the moving torus. (Eqs. (39, 40)) and shows a good agreement in the slender torus limit. At higher values of , a deviation between the numerical and analytical results is clearly seen (Fig. 5). 5 Mobility and resistance matrices The main analytical results from the calculations is equation (27) which gives the translational propulsion ve- Mobility and resistance matrices are commonly used in locity as a function of the angular velocity and the aspect describing the collective hydrodynamics of a group of par- ratio of the torus. ticles, where the forces and the torques are related to the Figure 6 shows the comparison of the analytical translational and rotational velocities of the particles. The (Eq. (27)) and numerical results for the translational ve- resistance or the mobility matrices are dense, although locities. Interestingly, the analytical expressions are valid symmetric. However, for single particles, the matrices are R.M. Thaokar et al.: Hydrodynamics of a rotating torus 331 often diagonal, indicating that the forces typically cause 2 b translation and torques cause rotation. A torus is one of δ d=b δ r d=b the few bodies with a regular geometry in which a resistance matrix can be written even for a single particle R=a−b−d 2 a (torus) due to the coupling of the torques and forces. Here x we derive a resistance matrix (Mkl) in terms of dimen- 2 a 2 R sional quantities, relating the angular velocity ωc (about 2 b the circular axis) and velocity ux (in the x direction) with the corresponding external torque N˜c and force F˜x,sothat Fig. 7. Schematics of a torus moving along a rod. Fx M11 M12 ux =4π2µ . Nc M21 M22 ωc the proposed mechanism by [8] of a DNA mini-plasmid Combining the previous expressions obtained for Fx and onatrack(Fig.1b).Thesystem can be best described Nc (Eqs. (34) and (37)), together with the results for a in cylindrical coordinates. We work here with dimensional translating torus (Appendix B, [9]) for the drag on a rigid quantities and follow the same notation as that of [12]. slender torus, we obtain the entries for the mobility matrix Consider a torus (Fig. 7) translating in the x direction M at the leading order as: with velocity U and rotating about its centerline with an- ω M M gular velocities . The torus has smaller and larger radius, M 11 12 . b a = M M and , and moves along an infinitely long inner cylinder 21 22 of radius R. We introduce two small parameters here, δ Such that and , such that the distance between the torus surface and the cylinder d = a − b − R = δb and b = a,which −1 M11 =2a(log 8/ +1/2) (41) correspond to lubrication limit of a slender torus. In addi- β R/a − − δ M M b2 / − / / / −1 tion, we describe the parameter = =1 in 12 = 21 = (log 8 1 2)(log 8 +1 2) conjunction with the notations used in [12] and [13] such 2 M22 =2b a. that β>1 indicates a torus translating inside a cylinder and β<1 denotes a torus moving outside an infinitely Note that the symmetry of the resistance matrix is a gen- long cylinder. For β<1, β → 0(R a)indicates limit eral feature of swimmers in Stokes flow and provides a of weak wall effect and β → 1(R ≈ a) indicates presence good check for the consistency of the involved calculations. of strong wall effect. Suitable non-dimensional variables The numerical results for propulsion velocity in the described here are relevant only to this section. The equa- paper are in agreement with [14]. tions of motion at low Reynolds number (taking advantage of the azimuthal symmetry) can be written as:

6 Motion of a torus along a cylindrical rod 1 ∂(rur) ∂ux + = 0 (42) r ∂r ∂x It is interesting to study the motion of a torus along a ∂p ∂2u ∂u u ∂2u − µ r 1 r − r r cylindrical rod, since the direction of motion can possi- ∂r + ∂r2 + r ∂r r2 + ∂x2 = 0 (43) bly reverse as the diameter of the cylindrical rod is made 2 2 larger [7]. Such a setup has been suggested [8] in the case ∂p ∂ ux 1 ∂ux ∂ ux − + µ + + =0. (44) of a nanomachine, essentially a DNA mini-plasmid, suit- ∂x ∂r2 r ∂r ∂x2 ably selected, so that it can be converted into a nanoswim- mer. It was argued that although the mini-plasmid could, The boundary conditions are the following: at r = R = under certain circumstances, self-propel, its effect can be a − b − d,thevelocityux,ur = 0, and on the surface of nullified by the substantial out of plane rotational dif- the torus ux = ω(a − r)+U, ur = ωx. fusion prevalent in these machines owing to their small One of the motivations for earlier works on this con- sizes [11]. It was therefore proposed to thread a DNA figuration [12,13] of a torus sedimenting inside a cylinder through the center of the DNA miniplasmid thereby al- was to explain the experimental results of a translating lowing only unidirectional motion. This situation was first torus [20]. The geometry was selected because the exper- addressed by Taylor [7], albeit qualitatively and the cal- imental results showed deviation from the predictions of culations were not provided. In another study Cox and a sedimenting torus in an infinite medium and the dis- O’Neil separately [12,13] obtained analytical solutions for crepancy was attributed to the presence of the walls of the case of a slender torus moving inside a tube, in two dif- the cylindrical vessel in which the experiments were car- ferent parameter regimes: non-lubrication and lubrication. ried out. In these studies, analysis was performed using They obtained the force on the torus when it is sliding singularity method for two different regimes, β → 1and both inside as well as outside of an infinitely long cylinder. β → 0, which correspond to strong and weak wall effects However, there are no calculations to date which address respectively. The work was also extended to the case of the motion of a force free, rotating torus along a circu- a torus sedimenting on the outside and along the axis of lar cylinder, and its propulsion velocity, which is exactly an infinitely long cylinder (β<1) [12], which is also the 332 The European Physical Journal B

22 140

20 120

18 100

16 80 F (Force ) F (Force) 14 60

12 40

10 20 −0.2 −0.1 −0.3 −0.2 −0.1 10 10 10 10 10 β β F Fig. 8. Comparison of the variation of the total force (F ) Fig. 9. Comparison of the variation of the total force ( ) β R/a ◦ with β = R/a of [12] (◦) and the present BIM study (*) for a with = of [12] ( ) and the present BIM study (*) for a U . ,a . ,µ . . sedimenting torus U =1.0,a=1.0,µ=1.0, =0.0005. sedimenting torus =10 =10 =10, =005. case in the present study, and again, both β → 1and torus along a cylinder or the force on a rotating torus is β → 0 regimes were considered. The analytical expression zero. This is identical to the result obtained by [21] for a for the total force acting on the torus in the β → 1isgiven rotating and translating cylinder near a wall exerting zero by [12], force and torque respectively. π2aµU F 8 , The numerical results however show that the force on = 2(a−R) (45) ln a rotating or translating torus about a cylinder is non- b zero [12,13]. The reason for this can be attributed to the for the parameter regime <δ<1. We use the boundary fact that unlike the case of a cylinder, the torus has a integral method, described in the earlier section, to calcu- finite length l =2πa with a curvature of the order of 1/a late the drag force on the torus sedimenting along a cylin- and has an internal degree of freedom. der. The only difference in this case is that the integration We now extend the analysis to the case of a force is performed over both the torus and the cylinder areas, free torus, rotating at a constant angular velocity ω and axi-symmetry being still valid. The results of [12] are ver- moving along the cylinder, using the boundary element ified first, and comparisons made with the analytical ex- method. To solve for the force free torus, we adopt a pression (Eq. (45)). Typically, the infinitely long cylinder strategy similar to that for a single self-propelled torus. was kept 80 times the larger torus diameter (a =1)and The integral equation is solved for the unknown stokeslet 160 elements were equally distributed over the cylinder strength fi with an additional constraint that the net and rod. The results were verified by doubling the length force vanishes dS(x)fi(x) = 0. This extra equation al- of the cylinder and the number of elements. lows the calculation of the unknown propulsion velocity Figure 8 shows the comparison of the results obtained Us. To describe the results in a compact way, we nondi- from the present BIM study with the expression given mensionalise the lengths as earlier with the outer torus by [12] (Eq. (45)). For =0.0005, the results show a diameter, a and the velocities are with aω.Wereportthe very good agreement, however, for =0.05(Fig.9,it force free translational (propulsion) velocity scaled by the is found that the deviation is large especially for β → 1. non dimensional single torus (in an infinite fluid) velocity 2 ∗ This is expected since the results of [12] are valid in the Us = /(log(8/) − 1/2) so that U = U/Us.Figure10 limit <δ<1, whereas for a given , δ → 0asβ → 1, indicates that the torus undergoes reversal of direction the assumption, <δin [12] is violated. Note that β = as the thickness of the inner cylinder increases. At small 1 − (1 − δ). In the present work, we are interested in the thicknesses of the cylinder (β → 0), the torus is propelled regime δ< <1 for a torus rotating about its centerline by the theory discussed earlier for a single rotating torus and translating along the cylinder axis. Thus, the limit indicated by U ∗ = 1. However even for very small thick- considered is one in which, the slender torus moves along nesses of the cylinder, substantial deviations are observed the cylinder at gap widths δ much smaller than the smaller from that of a freely rotating torus which is typical of diameter of the torus. Stokes flow. As the thickness of the inner cylinder is in- At the leading order a torus rotating about its center- creased the translational motion reverses and the torus line with a cylinder passing through its center is identical now starts moving in the opposite direction (β → 1). to a rotating cylinder near an infinite wall in the lubrica- This is the rolling motion observed for force free rotat- tion limit. Therefore, a naive lubrication theory analysis ing objects near a solid wall (Fig. 12). To get the correct of this problem indicates that the torque on a translating scaling of rotation in the lubrication limit, we carried out R.M. Thaokar et al.: Hydrodynamics of a rotating torus 333

1 0.05

Propulsion 0.049 0

0.048

−1 0.047 )−0.5) ε −2 0.046 U

/(log(8/ 0.045 2 Rolling ε −3

0.044 =2 U/ *

U −4 0.043

−5 0.042

0.041 −6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β Separaton (d/b) Fig. 10. Variation of the scaled translation velocity with the Fig. 11. Comparison of dimensional translational (Rolling) the scaled cylinder thickness for slenderness ratio =0.05, velocity in the lubrication limit for two different inner diame- b . a a ······ =0.25. ters of a torus ( =005, =1, =2, Best fit U =0.05 b(1–3/2 d/b)). simulations in low gap limit in dimensional quantities. Fig- ure 11 shows the variation of the dimensional translational velocity as a function of the gap distance for two different tori diameter a =1anda = 2. The velocity is seen to vary as 3 d U = bω 1 − (46) 2 b d bδ where = is the dimensional separation. The result ab c indicates that the rolling velocity is independent of the Fig. 12. Schematic of transition from propulsion to rolling torus diameter. This can be compared with the result for a regime, (a) propulsion of a free torus, (b) propulsion of a torus the rolling of a force free sphere of radius for which moving along a cylinder, (c) rolling of a torus along a cylinder. the translational velocity goes as U = aω/4 [22] and the deviation from aω by a factor of 4 is described as the slip, a result discussed in details in the literature. The force free cylinder on the other hand does not show any translation [21]. The numerical results for the torus show local forces and torques are also calculated. Finally the a clear scaling of U ∼ bω for the translational velocity in mobility matrix for the rotating-translating torus is pro- the limit of separation d → 0. vided. The results would be of immense use in calculating the friction and writing the correct Langevin equations for DNA mini-plasmids of different slenderness ratio which is 7Conclusion the subject of our future publication. The results for the motion of a slender torus along a cylinder show that the The paper presents analytical and numerical calculations torus can undergo reversal of translational motion from for a rotating torus. The analytical results are valid up a propulsion type to rolling when the internal diameter to O( 2) in the slenderness ratio. The calculations reveal of the cylinder is changed. Interestingly, the translational that a rotating torus can translate. The reason for this can velocity is found to be independent of the size of the torus be attributed to the presence of non-zero non-diagonal in the lubrication limit. terms in the reduced mobility and resistance matrices. The expression for the translational velocity is identical to that of translating smoke rings in inviscid flow. Such a We thank one of the referees for bringing to our notice a similar translating torus is force free and torque free. The force work [14]. The authors in that paper have addressed propuslion and torque required to prevent the translation of a rotat- of a Purcell’s swimmer using toroidal coordinates. The final re- ing torus is also calculated. The results of slender torus sults are obtained in a numerical fashion and are in agreement are extended numerically to finite aspect ratio using the with the results of our present work. The emphasis in the afore- boundary integral method. It is found that the analyti- said work though is on the efficiency of such a swimmer and cal expression provided is correct to a slenderness ratio as comparison of a toroidal swimmer with a similar object made high as 0.2 which is attributed to the O(2) analysis. The from a collection of rotating spheres. 334 The European Physical Journal B Appendix A: Elliptic Integrals 2π cos2 θ 4 I8 = dθ = 5 2 2 0 1 − k cos θ/2 3k (1 − k)

2(2−k)(2−2k−k2) × (8−8k−k2)F (k)− E(k) The complete elliptic integrals are deﬁned as [23]: 1−k 2π cos3 θ I9 = dθ π/2 5 2 dθ 0 1 − k cos θ/2 F (k)= (47) 2 2 0 1 − k sin θ 4 2 k4 k3 − k2 k− E k = 3 ( + 33 +64 32) ( ) π/2 3k (1 − k) 2 − k E k dθ − k2 2 θ. ( )= 1 sin (48) 4 2 0 + −(2−k)(k +32k−32)F (k) . (54) 3k3(1 − k) The elliptic integrals have the following asymptotic ex- k pansion around =1, Appendix B: Analytical solution for a translating, non-rotating torus 1 16 F (k)= ln (49) 2 1 − k2 The force and torque acting on a translating torus was cal- 1 − k2 16 1 culated by Johnson [9], apart from several other results. E(k)=1+ ln − (50) 2 1 − k2 2 We rederive these equations using the method outlined here and arrive at the same results. The method described is general enough and can be easily extended to the calcu- and around k = 0, the asymptotic expression reads lation of hydrodynamic interaction between two tori which is a subject of our future publication. π π 9πk4 F (k)= + k2 + (51) 2 8 128 Constructing velocity proﬁles for a translating torus π π 3πk4 E(k)= − k2 − . (52) 2 8 128 For a torus translating in the x direction with velocity U, we scale the lengths by a,thevelocitywithU,time by a/U, the stresses and the pressure by µa/U.Theθ

2π component of the velocity goes to zero by symmetry. The dθ velocities are easily expressed in terms of complete elliptic I1 F k = 2 =4 ( ) (53) 0 1 − k cos θ/2 integrals with a ring of Stokeslet, dipoles and rotlets as 2π cos θ 4 m I I I I I2 = dθ = ((2 − k)F (k) − 2E(k)) 3 2 6 1 2 3 2 ux = (I3 −rI4)+dpx −3x +fx +x 0 1 − k cos θ/2 k C3 C3 C5 C C3 2π dθ 4 (55) I3 = = E(k) 3 2 mxI d x xf 0 1 − k cos θ/2 1 − k 4 3 px x ur = − (rI6 − aI7)+ (rI3 − I4) 2π C3 C5 C3 cos θ 4 2 − k I4 = dθ = E(k) − 2F (k) 2fxxI3 3 2 k − k p . 0 1 − k cos θ/2 1 = C3 (56) 2π cos2 θ I5 = dθ p 3 2 where is the pressure due to the Stokeslet. 0 1 − k cos θ/2 s The boundary conditions for the problem areu ˜x = 2 s 4 k − 8k +8 U andu ˜r = 0 at the surface of the torus. The non- = E(k) − 4(2 − k)F (k) s k2 − k dimensional boundary conditions become ux =1and 1 s 2π ur = 0. We consider here the case of a slender torus (the dθ b/a I6 = ratio of the two radii of the torus, = is small), and 5 2 0 1 − k cos θ/2 ψ the surface can be locally parameterized by an angle , 4 2 such that x =cosψ and r =1+ sin ψ. The boundary − k E k − F k 2 2 = (2 ) ( ) ( ) conditions demand a scaling, m = m0 , dpx = dpo,and 3(1 − k) 1 − k dpo fx/ fx O 2π = 2, with = (1). The boundary conditions are cos θ given by, I7 = dθ 5 2 s 0 1 − k cos θ/2 ui = ui(S) (57)

4 2 2 2π 2π = (1−k+k )E(k)−(2−k)F (k) s 3k(1 − k) 1 − k dθ(x × ui )= dθ(x × ui(S)), (58) 0 0 R.M. Thaokar et al.: Hydrodynamics of a rotating torus 335

s where ui is the velocity on the surface of a torus Uex. Appendix C: Derivation of boundary integral Integrating equation (57) leads to, equation

8 U fo , The basic equation for velocity at a point in flow over a = 1+2log (59) particle, when the point lies outside the particle is given by such that, U 1 f . ui(x0)=− dS(x)Gij (x, x0)fj(x) o = 8 (60) 8πµ (1 + 2 log ) 1 dS x u x T x,x n x . Similarly, the strength of the rotlet is obtained by inte- + π ( ) j( ) ijk ( 0) k( ) (69) grating equation (58) over the surface of the torus. The 8 method is similar to deriving Faxen’s second law for a The first integral is called as single layer potential, whereas sphere [24] and requiring that the velocity on the surface the second is called the double layer potential. When the us Ue is i = x gives, singular point is moved on to the surface, the equation is 8 − modified as log 1 m0 = −f0 . (61) 2 u x − 1 dS x G x, x f x i( 0)= πµ ( ) ij ( 0) j( ) 4 1 Calculation of force and torque on a translating torus + dS(x)uj(x)Tijk (x,x0)nk(x). (70) 4π The stress tensor is expressed as T = [(σxx,σxr), (σrx,σrr)], where the different elements The singularity in the double layer potential can be elim- of the stress tensor have following definitions: inated by re-writing the equation as du x 1 σxx = −p +2 (62) u x − dS x G x, x f x dx i( 0)= πµ ( ) ij ( 0) j( ) 4 dux dur σxr = σrx = + (63) 1 dS x u x − u x T x,x n x dr dx + π ( )( j( ) j( 0)) ijk( 0) k( ) 4 dur σrr = −p +2 . (64) 1 dr + uj(x0) dS(x)Tijk (x,x0)nk(x). (71) 4π The traction vector (force per unit area) is t = Tn,and Using the property of double layer potential, the traction in the x direction is given by ex · Tn,where n is the unit normal given by (cos ψ, sin ψ). The net force on the torus is then calculated by integrating the traction dS(x)Tijkx,x0)nk(x)=−4πδij (72) over the area of the torus given by dψ(1 + sin ψ)2π. Thus the non-dimensional net x directional force per unit the governing equation can be re-written as π center line length (2 )isgivenby 2π 2 1 16π ui(x0)=− dS(x)Gij (x, x0)fj(x) Fx = dψ(1 + sin ψ)2πtx = − (65) 8πµ 0 1+2log8/ 1 + dS(x)(uj (x) − uj(x0))Tijk (x,x0)nk(x). (73) and the dimensional force is given by 8π π2aµU ˜ 8 In the present work, it was found that the contribution of Fx = − 8 1 . (66) log + 2 the second integral increases as the slenderness ratio increases. This is understandable as the assumption of rigid The net torque on the torus about the centerline is given body approximation becomes more invalid with .How- by the moment of the tangential force which is given by ever, the absolute contribution of the second integral was tTn about the center (lever arm ) found to be minimal and results were affected by around (1 + sin ψ)2π 2 2 1% (if the second integral was neglected) even in the case Nc = dψ (tTn)=4π (f0 − 4m0) . 2π of a torus of thickness =08. The change was less than 0.1% for slender tori (<0.05) log 8/ − 1/2 =4π2 2 (67) log 8/ +1/2 References and the total dimensional torque is given by 1. G.I. Taylor, Proc. R. Soc. London, Ser. A. 209, 447 (1951) 2 2 log 8/ − 1/2 2. M. Lighthill, Commun. Pure Appl. Math. 5, 109 (1952) N˜c =4π µb U . (68) log 8/+1/2 3. E.M. Purcell, Amer. J. Phys. 45, 3 (1977) 4. J. Avron, O. Gat, O. Kenneth, Phys. Rev. Lett. 93, 186001 Expressions (66) and (68) are the central results of [9]. (2004) 336 The European Physical Journal B

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