Oct 21st 2012

In this paper we study the self-propulsion of a dumbbell micro-robot submerged in a viscous fluid. The micro-robot consists of two rigid spherical beads connected by a rod or a spring; the rod’s/spring’s length is changing periodically. The constant density of each sphere differs from the density of a fluid, while the whole micro-robot has neutral buoyancy. An effective oscillating gravity field is created via rigid-body oscillations of the fluid. Our calculations show that the micro-robot undertakes both translational and rotational motion. Using an asymptotic procedure containing a two-timing method and a distinguished limit, we obtain analytic expressions for the averaged self-propulsion velocity and averaged angular velocity. The important special case of zero angular velocity represents rectilinear self-propulsion with constant velocity.

Dumbbell micro-robot driven by flow oscillations] Dumbbell micro-robot driven
by flow oscillations

V. A. Vladimirov] V.A.Vladimirov

1 Introduction

The study of self-propelling micro-robots is a flourishing modern research topic, striving to create a fundamental base for modern applications in medicine and technology, see e.g. ???????????????. We define self-propulsion as the motion of a micro-robot which is subjected to zero external total force. The simplicity of micro-robots’ geometry represents a major advantage in contrast to the extreme complexity of self-swimming microorganisms. This advantage allows us to describe the motions of micro-robots in greater depth.

Figure 1: Two spheres, linked by a rod of periodically changing length.

The major problem in the designing of a micro-robot is the need for an external source of energy to provide for its oscillatory behaviour. Proposed sources include an oscillating (or rotating) magnetic field (see ???), an electric field (see ?), and even molecular Brownian forces (see ?). At the same time, the major oscillatory forces available in fluid have not been exploited; these are the forces caused by fluid oscillations which are imposed by periodically varying boundary conditions, waves, or turbulence. The ratio of characteristic spatial scales (several microns for a micro-robot vs. millimeters, centimeters, or greater scales for flow oscillations) makes it clear that the first problem to study is the behaviour of a micro-robot in a fluid that oscillates as a rigid body.

In this paper, we consider the self-propulsion of a two-sphere Buoyancy-driven Dumbbell micro-robot (we call it BD-robot, see the figure). The whole micro-robot is neutrally buoyant (in order to avoid sedimentation); one of its spherical beads is positively buoyant and the other is negatively buoyant. We study two versions of BD-robots. In the first one the beads are connected by a rod of prescribed oscillating length, in the second one the beads are linked by an elastic spring. First, we study the case of a rod and, next, we consider the changes that appear after replacing the rod with a spring. A mathematical formulation of the problem leads us to the study of creeping motions with time-periodic forces. The problem is solved by employing a version of the two-timing method and distinguished limit arguments, developed in ???. The approach allows any motion of BD-robot to be described analytically. Our calculations show that, generally, the BD-robot participates in both translational and rotational motion. Rectilinear translational self-propulsion with constant velocity represents a special case of this solution. We have calculated the velocity of rectilinear self-propulsion and the ranges of governing parameters that correspond to translational motions.

2 Problem formulation

The BD-robot represents a dumbbell configuration, which consists of two homogeneous rigid spherical beads of different radii , connected by a rod of length (see the figure). We study two-dimensional motions of a tree-dimensional dumbbell in cartesian coordinates . The centers of the spheres are described as


where is the radius-vector of a center of reaction. The axis of symmetry of a dumbbell is given by the vector , . The unit vectors , and the angle are given by


where the subscript stands for . The length is changing periodically


where is a constant averaged value and is a -periodic function of with zero average value (throughout the paper a ‘tilde’ above a function of time denotes that this function is oscillating and has zero mean value). The spheres experience external friction forces while the rod is so thin (in comparison with either ) that its interaction with the fluid can be considered negligible.

We consider the motions of a BD-robot in a viscous incompressible fluid which, in the absence of the BD-robot, oscillates as a rigid body. These rigid-body oscillations are prescribed as a two-dimensional translational spatial displacement of fluid particles (at infinity in space); the related acceleration is , where the subscripts stand for related derivatives. The problem can be studied in an oscillating (non-inertial) system of reference, in which a fluid at infinity is in a state of rest. In this frame, according to Einstein’s principle of equivalence, or according to a related transformation of a Lagrangian function, the equations of fluid motion are standard, however, they contain an additional oscillating gravity force


which causes buoyancy forces , where the coefficient is equal to the difference in the mass of a sphere and the mass of a displaced fluid; can be either positive or negative. The potential energy of a sphere is . We consider a BD-robot of neutral total buoyancy, with total potential energy


The problem formulation contains three characteristic lengths: the length of the rod , the radius of the spheres , and the amplitude of the rod’s oscillation . In addition we have the characteristic time-scale , excess mass , gravity , and viscous force . We have chosen these scales as , , , , , where is the fluid viscosity. The dimensionless variables (marked with asterisks) are , , . Three independent small parameters of the problem are


Below we use only dimensionless variables, however all asterisks are omitted. Note, that in the chosen dimensionless units, .

We choose the generalised coordinates of BD-robot to be . The motion of the BD-robot, with a given (2.0), is described by the Lagrangian function , which includes the constraint (2.0) with lagrangian multiplier


where , and are kinetic energy and potential energy (2.0) of BD-robot; represents an additional unknown function of time. The Lagrange equations are


where is the generalized external viscous force, exerted by a fluid on the BD-robot. As one can see, we use latin subscripts () for cartesian components of vectors and tensors, subscript for generalised coordinates, and subscripts (or superscripts) to identify the spheres.

We accept, that the fluid flow past BD-robot is described by the Stokes equations, where all inertial terms are neglected. In the consistent approximation the mass of a rod and masses of the spheres are negligible, hence . Therefore (2.0),(2.0),(2) give use to the following system of equations


which is supplemented by constraint (2.0). The great advantage of a Lagrangian formalism is its self-sufficiency. In particular, the conditions of zero force (2.0) and the balance of torques (2.0) appear automatically, while eqn.(2.0) allows us to find the reaction of constraint . In (2.0),(2.0) we have accepted that with constant ; this is our physical assumption, which states that two small parameters and (out of three parameters in the list (2.0)) are of the same order. Physically, it means that the difference between the densities (of each sphere and the fluid) or the amplitude of oscillations of the fluid is small (or both these parameters are small). The explicit expressions for are


Each force represents the first approximation (with the error ) for the Stokes friction force exerted on a sphere moving in a flow field generated by another sphere. To construct (2.0) we use a classical explicit formula for the fluid velocity past a moving sphere, see ???. Eqns. (2.0)-(2.0) represent a system of four equations for four unknown functions of time: , , , and . For the prescribed (2.0), the equation (2.0) need not to be considered if we are interested only in the motion of the micro-robot and are not calculating of reaction force . For future use, we rewrite (2.0),(2.0) as


where .

3 Two-timing method and asymptotic procedure

3.1 Functions and notations

The following dimensionless notations and definitions are in use:

(i) and denote slow and fast times; subscripts and stand for related partial derivatives.

(ii) A dimensionless function, say , belongs to the class if and all partial -, and -derivatives of (required for our consideration) are also . In this paper all functions belong to class , while all small parameters appear as explicit multipliers.

(iii) We consider only functions periodic in , where -dependence is not specified. Hence, all functions considered below belong to .

(iv) For arbitrary the averaging operation is


(v) The oscillating part of an integral is:

(vi) The tilde-function (or purely oscillating function) represents a special case of -function with zero average . The bar-function (or mean-function) does not depend on . For any periodic function a unique decomposition is valid.

3.2 Asymptotic procedure and successive approximations

The introduction of a fast time variable and a slow time variable represents a crucial step in our asymptotic procedure. We choose and . This choice can be justified by the same distinguished limit arguments as in ?. Here we present this choice without proof, however, its most important part (that this choice leads to a valid asymptotic procedure) is exposed and exploited below. We use the chain rule


and then we accept (temporarily) that and represent two independent variables. Further more we consider series expansions in the small parameter and restrict our attention to terms which are at most . Simultaneously, we keep at most linear in terms. It does not mean that in our setting , since in all expressions appears not separately but as products with various degrees of . Hence, we do not specify the dependence of unknown functions on ; such dependence reveals itself naturally during the calculations. The unknown functions are taken as regular series in


with a similar expression for . We accept that

which express a basic property of our solutions that long distances of self-swimming and large angles of rotation are caused by small oscillations. The application of (3.0) to (3.0) gives


and a similar expression for . In the calculations below all bar-functions belong to the zero approximation, while all tilde-functions belong to the first approximation; therefore we omit the related subscripts.

The successive approximations of equations (2.0),(2.0) yield:

Terms give the identities .

Terms lead to


The use of (2.0) transforms the second equation to the form . Then the integration of (3.0) in the class of periodic functions yields


Terms : These terms do not vanish and can be easily calculated. However, as one can see below, they do not participate in the leading terms of the average motion that appear in the orders and .

Terms : Eqns.(2.0)),(3.0) give


where we have used the integration by parts in the average operation (3.0). The use of the definition of matrix (2.0) and (2.0) yield . Then (3.0) takes the form


Similarly, from (2.0), we can derive the equation for and obtain the system of equations


where we have used the equality , which is valid by virtue of (2),(3.0), and . One can see, that the dynamics of a dumbbell is determined by the values of two parameters , and by three correlations , , and . We can make a general conclusion, based on (3.0), that the mean translational velocity is always perpendicular to the mean symmetry axes of a dumbbell ( is directed along the normal vector ).

4 Prescribed oscillations of the Bd-robot

4.1 Unidirectional oscillations of a fluid are not effective

The simplest motion takes place when . Physically, it means that fluid oscillations are unidirectional, see (2.0). In this case equations (3.0) produce the integral


with a vectorial constant of integration . This equality shows that changes along a circular path (or along an arc of a circle) of small radius . The equation for (3.0) can be integrated exactly. For unidirectional oscillations along the -axis, when (2.0), the second equation (3.0) takes the form . It can be integrated, having an initial value , as


which shows that for we have ; it means that the axis of symmetry of a dumbbell is turning monotonically towards the direction of oscillations. Equation (4.0) describes the simultaneous change of along the arc of a small circle. It is clear, that in the general case of unidirectional oscillations along any direction (different from ), the result is the same: the axis of the dumbbell asymptotically approaches the direction parallel to the oscillations, and changes along the arc of a small circle. Therefore, we can conclude that any unidirectional oscillations of the fluid do not result in the self-propulsion of the BD-robot.

4.2 Rectilinear self-propulsion without rotation

For we first consider motions without rotation . In these cases the angular part of (3.0) gives , which immediately leads to


Physically, the restriction (4.0) means that the BD-robot can move without rotation if the oscillations are ‘strong enough’. In this case the first eqn. in (3.0) gives , which shows that the BD-robot moves with constant speed in the fixed direction , which is given by the angle (4.0), where the sign is determined by the correlation . Striving for more general results, one can show, that the system (3.0) can be integrated analytically in the general case , with the conclusion, that if the parameters satisfy (4.0) then a trajectory with any initial data asymptotically (when ) approaches the same straight paths (4.0) as described above. Exact integration outside of the range of parameters (4.0) is also accessible analytically; it produces motions with rotation , which we do not consider in this paper.

Let us consider a particular example


For the BD-robot propels itself with constant speed along a straight path . It is remarkable, that the self-propulsion speed does not depend on the amplitude . However, one should keep in mind that such solutions are available only for ‘strong’ oscillations, when ; for ‘very strong’ oscillations, when , we have .

5 Elastic Bd-robot

The above results correspond to the prescribed periodic function (2.0), which can be chosen arbitrary and represents a given time-dependent constraint. However, in practice, the oscillations produced by the forces exerted from an oscillating fluid on the beads, are more interesting. A simple way to consider such oscillations is to replace the rod with a spring of stiffness . In this case the dimensionless Lagrangian function (2.0) and potential energy (2.0) become


One can check that for this Lagrangian function the equations for total force and torque (2.0),(2.0) remain the same, while the equation for the reaction of constraint (2.0) must be replaced with


The relation between the problem for an elastic -robot and the previous problem for a -robot with the arbitrary oscillation of a rod is evident: the latter considers all possible solutions, while the former corresponds to a special subclass of only, that appears as the result of spring oscillations. Hence the ability for self-propulsion can only ‘worsen’ after the introduction of a spring. The order equation (5.0) produces a linear equation for


It gives us which must be substituted into and in (3.0) instead of an arbitrary chosen function . The rest of the problem remains unchanged. The general solution of (5.0) can be obtained analytically in an integral form, or in the form of a Fourier series. Both forms are rather cumbersome and are not considered in this paper. Instead, we present an example for a gravity field , that coincides with (4.0). The related solution of (5.0) is

where an exponentially decreasing complementary solution has been dropped. It leads to an explicit formula for (3.0)


which determines the system of equations (3.0). In this case we obtain the following equations for the motion without rotation ()


We can see, that any direction of a rectilinear self-propulsion can be arranged by an appropriate choice of and . It is also interesting that the speed of self-propulsion, , does not depend on the spring stiffness ; however, the required condition shows that both a small and high stiffness do not lead to rectilinear motion. Another interesting conclusion is: for rectilinear motion to exist, the values of vibrational amplitudes and can not be chosen close to each other. It means that imposed vibrations (2.0) must be anisotropic (the circular vibrations with and close to them are excluded). Again, for , the system of equations (3.0),(5.0) can be integrated analytically. The integration shows that any trajectory asymptotically (for ) approaches (5.0). In the case the full system (3.0),(5.0) also allows explicit analytical integration; it leads to the motions with rotation , which we do not consider in this paper.

6 Discussion

(i) Our choice of slow time (3.0) agrees with classical studies of self-propulsion for low Reynolds numbers, see ???, as well as the geometric studies of ?.

(ii) The scale of slow time implies that in order to obtain physical dimensionless velocities we have to multiply and (3.0) by . Accordingly, the mean translational velocity , while mean angular velocity .

(iii) We have built an asymptotic procedure with two small parameters: and . Such a setting usually requires the consideration of different asymptotic paths on the plane when, say . In our case we can avoid this additional analysis, since does not appear separately, but only in combinations like .

(iv) In this paper we consider only plane motions of a three-dimensional dumbbell. This class of motions corresponds to two-dimensional oscillations/gravity (2.0). At the same time, for experimental realization, it could be necessary to solve a full three-dimensional problem.

(v) For the first experimental studies of self-propulsion of the BD-robot one can consider: rigid-body oscillations of a fluid, enclosed within a vibrating container; the viscous flows, caused by oscillatory boundary conditions; or the oscillations of a fluid due to an external acoustic wave.

(vi) The calculated velocities of self-propulsion are much smaller than flow oscillations and can be even smaller than some secondary flows (like acoustic streaming). In this case, one may look for situations when a slow (but permanent) self-propulsion leading to new physical effects or providing advantages in applications.

(vii) It is well known, that an oscillating dumbbell is able to self-swim when an oscillating external torque, exerted on a dumbbell, is present; the related discussion can be found in ???. The self-swimming of a magnetically driven oscillating dumbbell has been studied by ?.

(viii) Our approach is technically different from all previous methods employed in the studies of micro-robots. The possibility to describe any motion of a BD-robot explicitly, shows the strength and analytical simplicity of our method. The studies of different micro-robots by the same method (as in this paper) can be found in ???. In ? the same method resulted in a new asymptotic model and a new equation (the acoustic-drift equation) for the averaged flows generated by acoustic waves.

The author is grateful to Profs. M.A. Bees, C.J. Fewster, P.H. Gaskell, A.D. Gilbert, K.I. Ilin, H.K. Moffatt, T.J. Pedley, and J.W. Pitchford for useful discussions.

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