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the swimming of fish and the flying of birds continue to pose challenging theoretical problems . the physics of bird flight was first studied in detail by otto lilienthal in the nineteenth century @xcite . since then , significant progress has been made in many years of dedicated research @xcite-@xcite . the goal of theory is to calculate the time - averaged speed and power for given periodic shape variations of the body , at least for a simple model system . it is assumed that the motion of the fluid is well described by the navier - stokes equations for an incompressible viscous fluid . on average over a period the force exerted by the body on the fluid vanishes , so that thrust and drag cancel . in early work by lighthill @xcite and wu @xcite the thrust and power were calculated approximately as functions of the speed on the basis of potential flow theory for a planar strip . this work and subsequent developments have been reviewed by childress @xcite , by wu @xcite,@xcite , and by sparenberg @xcite . however , an independent calculation of the mean speed for given periodic shape variations is still lacking . measurement of the power consumption has led to a surprisingly small friction coefficient , much smaller than that of an inert body , as was first observed by gray @xcite . it was first shown by taylor @xcite that in the slow swimming of a microorganism the calculation of thrust can be circumvented . in this limiting case one can use the time - independent stokes equations . the mean swimming velocity and mean rate of dissipation then follow from a purely kinematic calculation @xcite,@xcite . for small amplitude swimming both quantities are quadratic in the amplitude of the stroke to lowest order . for a simple system , where the body is modeled as an assembly of rigid spheres held together by direct interaction forces and subject to periodic actuating forces which sum to zero , we have shown that in the high viscosity limit the swimming velocity and power can be calculated for any amplitude of stroke from kinematics alone @xcite,@xcite .    in the following we investigate questions of thrust , velocity , and power for swimming or flying in a fluid of any viscosity , including the limit of low viscosity , for the same mechanical model as before . we assume for simplicity that the spheres experience stokes friction . in addition we incorporate hydrodynamic interactions via virtual mass effects , as found from potential flow theory . we use hamilton s equations of motion with added damping terms . in the limit of high viscosity , where resistive forces dominate , the earlier results are recovered . the model provides valuable insight also in the limit of low viscosity , where reactive forces dominate . in that regime the motion is dominated by virtual mass effects . bernoulli forces and modified linear friction should be taken into account in a more realistic model . nonetheless , the principle of the calculation , which exploits elimination of the fluid degrees of freedom , remains valid . the flow is assumed to be laminar at all times . it is now realized that the boundary layer of swimming fish is laminar even at high reynolds number @xcite . virtual mass effects were discussed earlier by lighthill @xcite . the numerical modeling of animal swimming and flight was reviewed by deng et al . @xcite .    as an example a system of three equal spheres moving along a common axis is studied . for this simple system the mean swimming speed and mean power to second order in the amplitude of stroke can be evaluated analytically . the solution to a corresponding eigenvalue problem provides the optimal stroke to this order , as we found elsewhere in the resistive regime @xcite . in our model the mean thrust , i.e. the frictional force exerted on the fluid averaged over a period in periodic swimming , vanishes identically . we find that the velocity of the geometric center of the assembly is driven by a different force , which we call the impetus . it has both a reactive and a resistive component . the impetus determines the center velocity with retardation . the mean impetus does not vanish . it is known for small amplitude swimming in the resistive regime that the mean power is directly proportional to the mean velocity . we find for our example that the relation between mean power and mean velocity is nearly linear also for large amplitude swimming . presumably the near linearity holds also for other systems in the whole regime of viscosity . if true , this would resolve the so - called gray paradox @xcite , which is based on the mistaken notion that the power is quadratic in the velocity , as in stokes friction . we consider a set of @xmath0 rigid spheres of radii @xmath1 and masses @xmath2 , centered at positions @xmath3 , and immersed in an incompressible viscous fluid of shear viscosity @xmath4 and mass density @xmath5 . the fluid is of infinite extent in all directions . the flow velocity @xmath6 and pressure @xmath7 of the fluid are assumed to satisfy the navier - stokes equations @xmath8=\eta\nabla^2{\mbox{\boldmath $ v$}}-\nabla p,\qquad\nabla\cdot{\mbox{\boldmath $ v$}}=0.\ ] ] the flow velocity @xmath9 is assumed to satisfy the no - slip boundary condition on the surface of the spheres . the fluid is set in motion by time - dependent motions of the spheres . at each time @xmath10 the velocity field @xmath6 tends to zero at infinity , and the pressure @xmath7 tends to the constant ambient pressure @xmath11 .    as the spheres move in the fluid they experience a frictional force . in addition there may be applied forces @xmath12 and direct interaction forces which depend on the relative positions @xmath13 of sphere centers . we shall assume that the sum of applied forces vanishes , so that @xmath14 the sum of direct interaction forces vanishes owing to newton s third law . we assume that the frictional forces are linear in the sphere velocities , as given by low reynolds number hydrodynamics on the slow timescale @xcite . the spheres are freely rotating so that there are no frictional torques . the forces exerted by pressure gradients resist instantaneous acceleration and give rise to virtual mass effects @xcite . for a single sphere @xmath15 immersed in infinite fluid the virtual mass would be @xmath16 , where @xmath17 is the mass of fluid displaced by the sphere . the virtual mass effect for a collection of @xmath0 spheres is embodied in a @xmath18-dimensional mass matrix @xmath19 . this can be derived from potential flow theory by considering the irrotational flow pattern generated instantaneously by a set of sudden impulses @xmath20 from a state of rest . the total corresponding kinetic energy is @xmath21 where @xmath22 is the set of sphere velocities . the kinetic energy is a sum of two contributions , @xmath23 , with @xmath24 where the integration is over the part of space occupied by fluid and @xmath25 is the velocity potential corresponding to the set of instantaneous velocities @xmath26 . the potential @xmath25 is linear in the sphere velocities @xmath22 , so that the kinetic energy @xmath27 is a quadratic form in @xmath26 . the contribution @xmath27 to the total kinetic energy defines the virtual mass . this depends parametrically on the positions @xmath28 , leading to corresponding hydrodynamic interactions .    besides the mass matrix @xmath19 it will be useful to define also the mobility matrix @xmath29 , the friction matrix @xmath30 , and the inverse mass matrix @xmath31 , @xmath32 the four matrices are symmetric and depend on the relative positions of the sphere centers . the sphere momenta , including the virtual mass contribution , are given by @xmath33 correspondingly we postulate the equations of motion @xmath34 where @xmath35 is given by @xmath36 and @xmath37 is the potential of direct interaction forces . the partial derivative @xmath38 is taken at constant momenta @xmath39 . it is clear that the equations of motion ( 2.7 ) have only limited validity . the frictional forces are assumed to be linear in the sphere velocities , and basset memory forces are neglected . nonetheless it is of interest to explore the consequences of the equations as they stand . we note that it follows from eq . ( 2.7 ) that total dressed sphere momentum @xmath40 is not conserved but changes due to friction with the fluid . if an impulse @xmath41 is imparted at time @xmath42 to the spheres in a state of rest due to applied forces @xmath43 , then part of the momentum is transferred instantaneously to the fluid , reducing the sphere velocities to @xmath44 , as can be seen by integration over an infinitesimal time interval about @xmath42 and use of eq . the remaining momentum of the spheres is transferred to the fluid in the course of time by friction . the total momentum of spheres and fluid is conserved at all times . we are looking for a solution of the equations of motion where the center of the assembly moves on average with uniform translational velocity and the individual spheres perform periodic motions about the moving center . the velocity @xmath45 of the geometric center is defined by @xmath46 where the symbol @xmath47 denotes a @xmath48-dimensional vector with @xmath49 on the @xmath50 positions , @xmath51 on the @xmath52 positions , and cyclic . the @xmath48-dimensional displacement vector @xmath53 is defined by @xmath54 with the property @xmath55 and with @xmath56 a set of equilibrium positions for which the direct interaction forces vanish . correspondingly the velocity vector is decomposed as @xmath57 summation over repeated greek indices is implied . by substitution into eqs . ( 2.6 ) and ( 2.7 ) we find @xmath58+\frac{\partial\mathcal{k}}{\partial{{\bf\sf r}}}+{\mbox{\boldmath $ \zeta$}}\cdot(u_\beta{{\bf\sf u}}_\beta+\dot{{{\bf\sf d}}})+\frac{\partial v_{int}}{\partial{{\bf\sf r}}}={{\bf\sf e}}(t).\ ] ] multiplying from the left with the vector @xmath59 we obtain @xmath60 with mass tensor @xmath61 and friction tensor @xmath62 defined by @xmath63 we have used the fact that @xmath19 depends only on relative coordinates , so that @xmath64 . also we have used newton s third law and eq . ( 2.2 ) . we note that in eq . ( 3.6 ) the center velocity @xmath65 occurs only in the two places explicitly shown . we rewrite eq . ( 3.6 ) as @xmath66 with time - dependent impetus @xmath67 given by @xmath68 the mean impetus , averaged over a period @xmath69 , @xmath70 does not vanish , even though the total mean force exerted by the fluid on the assembly does vanish . the drag exerted on the assembly by the fluid is @xmath71 the thrust @xmath72 is equal and opposite to the drag , @xmath73 . there are frictional forces on the spheres , but on time average the total drag vanishes , @xmath74 , as follows from eqs . ( 2.2 ) , ( 2.7 ) and newton s third law for the interaction forces . we use the fact that the integral of @xmath75 over a period vanishes by periodicity , as well as the property @xmath64 , mentioned below eq . ( 3.7 ) . from @xmath76 it follows that the mean impetus can also be expressed as @xmath77    we may regard eq . ( 3.8 ) as a balance between central and internal motion . the equation must be solved for the center velocity @xmath45 for given impetus @xmath67 , which can be calculated from the displacement vector @xmath78 . in the resistive limit the total drag vanishes at any time @xcite and the reactive forces vanish , so that then @xmath79 . the displacement vector @xmath78 may be calculated from displacements in relative space by using a transformation to center and relative coordinates . the geometric center of the assembly is given by @xmath80 with cartesian unit vectors @xmath81 . we define relative coordinates @xmath82 with @xmath83 as @xmath84 and denote the corresponding @xmath85-vector @xmath86 . the @xmath48-vector @xmath87 is related to the vector @xmath28 by a transformation matrix @xmath88 according to @xmath89 with explicit form given by eqs . ( 3.13 ) and ( 3.14 ) . the displacement vector @xmath90 is derived from the displacement @xmath91 in relative space as @xmath92 therefore the impetus is determined by displacements in relative space . the time - dependent rate of dissipation can be expressed in the same matrix formalism . the rate of dissipation is given by @xmath93 once the center velocity @xmath45 has been calculated for known displacements we can also calculate the time - dependent rate of dissipation @xmath94 by use of eq . ( however , we derive an alternative expression which will be useful in the following . we can solve for the product @xmath95 from the equation of motion @xmath96 where @xmath97 is the vector of mechanical forces acting on the spheres , @xmath98 which has the property @xmath99 . using this property we find for the rate of dissipation @xmath100 using here eq . ( 3.18 ) again we can rewrite this as @xmath101 with friction vector @xmath102 . this generalizes an expression derived earlier in the resistive limit @xcite . for vanishing displacements eq . ( 3.8 ) has the solution @xmath103 . by formal series expansion in powers of the displacement vector @xmath90 we obtain a corresponding expansion of the center velocity @xmath104 the first order velocity @xmath105 satisfies the equation @xmath106 where the superscript @xmath51 indicates that the quantity is calculated for the configuration @xmath107 . in particular for oscillating displacement , in complex notation @xmath108,\ ] ] we find correspondingly @xmath109^{-1}_{\alpha\beta}{{\bf\sf u}}_\beta\cdot(\omega^2{{\bf\sf m}}^0+i\omega{\mbox{\boldmath $ \zeta$}}^0)\cdot{{\bf\sf d}}_\omega.\ ] ] in this situation the first order velocity oscillates in time and vanishes on time average .    multiplying eq . ( 3.8 ) by the mobility tensor @xmath110 and expanding to second order we obtain a more complicated equation for the second order velocity @xmath111 . it suffices to derive an expression for the time - average over a period @xmath112 , @xmath113 using periodicity and the first order equation eq . ( 4.2 ) we obtain the expression @xmath114 alternatively the expression can be derived directly from eq . ( 3.12 ) by use of @xmath115 . here we can use @xmath116 where @xmath117 indicates the gradient operator in configuration space , and the notation @xmath118 indicates that the value of the gradient is taken at @xmath107 . the time average in the first term in eq . ( 4.6 ) can be expressed as @xmath119,\ ] ] with derivative friction matrix @xmath120 as introduced earlier @xcite .    in the second term in eq . ( 4.6 ) we use the identity @xmath121 to show that @xmath122 with gradient vectors @xmath123 the first order velocity @xmath124 is eliminated by use of eq . then the second order mean swimming velocity can be expressed as @xmath125,\ ] ] with frequency - dependent matrix @xmath126 given by @xmath127 with reduced derivative friction matrix @xmath128 with admittance tensor @xmath129^{-1},\ ] ] and impedance vector @xmath130    the matrix @xmath131 has the property @xmath132 since @xmath30 depends only on relative coordinates @xmath133 , and hence @xmath134 as a consequence @xmath135 these properties generalize those derived earlier at zero frequency @xcite . to second order in the displacements the last term in eq . ( 3.21 ) can be written as @xmath136 in the average over a period the first term on the right does not contribute . hence for periodic motion the mean second order rate of dissipation can be expressed as @xmath137\;dt.\ ] ] substituting eq . ( 4.4 ) we find @xmath138,\ ] ] with the complex matrix @xmath139 the matrix is symmetric and has the properties @xmath140={{\bf\sf 0}},\qquad\mathrm{re}[{{\bf\sf p}}(\omega)]\cdot{{\bf\sf u}}_\alpha={{\bf\sf 0}}.\ ] ] the properties eq . ( 4.20 ) and ( 4.25 ) allow us to reduce the dimension of the matrix description by three by introducing center and relative coordinates . the transformation given by eqs . ( 3.13 ) and ( 3.14 ) can be used to reduce the calculation of the second order swimming velocity and rate of dissipation to one in relative space . the matrices @xmath126 and @xmath141 are transformed to @xmath142 the first three rows of @xmath88 consist of @xmath143 and the first three columns of @xmath144 consist of @xmath59 . it follows from the properties eq . ( 4.20 ) and ( 4.25 ) that the first three rows and columns of the transformed matrices @xmath145 and @xmath146 vanish identically . hence in this representation we can drop the center coordinates and truncate the matrices by erasing the first three rows and columns . we denote the truncated @xmath147-matrices as @xmath148 and @xmath149 and define displacements @xmath150 in relative space by @xmath151 with this notation the mean second order swimming velocity and rate of dissipation are given by @xmath152 with the matrix @xmath153\;{\mbox{\boldmath $ \hat{}$}}.\ ] ] this @xmath147 dimensional matrix consists of numerical coefficients and is obtained from the corresponding @xmath154 matrix by truncation , as indicated by the final hat symbol . we rewrite the expressions in eq . ( 5.3 ) in a more convenient form with vectors and matrices consisting of dimensionless numbers . we introduce the complex dimensionless vector @xmath155 where @xmath156 is a typical length scale , and define @xmath157,\nonumber\\ { { \bf\sf a}}&=&\frac{1}{b\eta}\;[\big({{\bf\sf c}}_t\cdot\hat{{{\bf\sf p}}}_t(\omega)\big|_0\big)^{s\prime}+i\big({{\bf\sf c}}_t\cdot\hat{{{\bf\sf p}}}_t(\omega)\big|_0\big)^{a\prime\prime}\big],\end{aligned}\ ] ] where the superscript @xmath158 indicates the symmetric part , the superscript @xmath159 the antisymmetric part , the single prime the real part , and the double prime the imaginary part . with the scalar product @xmath160 the mean swimming velocity and mean rate of dissipation can then be expressed as @xmath161 the matrices @xmath162 and @xmath163 are hermitian . we call @xmath162 the velocity matrix and @xmath163 the power matrix . we ask for the stroke with maximum swimming velocity in a class of strokes with equal rate of dissipation for fixed values of the geometric parameters , fixed frequency @xmath164 , and given values of viscosity @xmath4 and mass density @xmath5 . maximizing the quadratic form @xmath165 with lagrange multiplier @xmath166 we obtain the generalized eigenvalue problem @xmath167 the eigenvalues @xmath168 are real . the maximum efficiency @xmath169 for motion in direction @xmath170 is given by the maximum eigenvalue as @xmath171 the set @xmath172 depends on the choice of cartesian coordinate system . further optimization may be possible by a rotation of axes . in particular cases a natural choice of axes will suggest itself . we view the swimmer or flyer as a dynamical system in periodic motion , driven by actuating forces @xmath173 satisfying @xmath174 and @xmath175 . in an expansion in powers of the actuating forces the first order displacements are given by eq . ( 4.3 ) . to second order the mean swimming velocity and mean rate of dissipation are given by eq . ( 5.8 ) . in this section we relate the mean power supplied by the actuating forces to the mean rate of dissipation . the instantaneous power supplied by the actuating forces is given by @xmath176 from eq . ( 2.7 ) @xmath177 hence we find for periodic motion @xmath178 this shows that on time average over a period the power is fully dissipated by friction . since the mean thrust vanishes the usual definition of energy wastage @xcite makes no sense here . instead we define the energy wastage @xmath179 as the difference @xmath180 in the theory of fish swimming and bird flight the energy wastage has been associated with energy being lost to the vortical wake @xcite . we define the corresponding froude efficiency as @xmath181 this concept may be useful for the comparison of different strokes . in order to perform explicit calculations we must specify the form of the hydrodynamic interactions appearing in the friction matrix and the mass matrix . in practice one uses approximate expressions which are presumed to be reasonably accurate in the range of distances considered . the friction matrix can be calculated from an approximation to the mobility matrix based on oseen s pair interaction @xcite . in this approximation the pair mobility tensor for the pair @xmath182 is given by @xmath183(1-\delta_{jk}).\ ] ] the mobility matrix @xmath29 is composed of pair tensors , and the friction matrix @xmath30 is its inverse . the calculation of the mass matrix @xmath19 is based on potential flow theory . a dipole approximation to the mass matrix can be evaluated on the basis of an expression for the force on a sphere in a uniform flow in potential flow theory , as given by landau and lifshitz @xcite and by batchelor @xcite .    in potential flow theory the flow velocity is expressed as @xmath184 with a scalar potential @xmath185 , which satisfies laplace s equation @xmath186 by incompressibility . a sphere of radius @xmath159 , centered at the origin and moving with velocity @xmath187 in a fluid at rest generates a potential @xmath188 corresponding to the dipole moment @xmath189 if the sphere is placed in a uniform flow @xmath190 this is modified to @xcite @xmath191 for a collection of @xmath0 spheres in a fluid at rest at infinity the velocities and dipole moments are related in dipole approximation as @xmath192 with dipole interaction tensor @xmath193 where @xmath194 . we abbreviate eq . ( 7.5 ) as @xmath195    the velocity of sphere @xmath15 after a sudden impulse @xmath196 from a state of rest is given by @xcite,@xcite @xmath197 substituting from eq . ( 7.7 ) and solving for the velocities we find by use of eq . ( 7.5 ) and @xmath17 @xmath198 with matrix @xmath199^{-1},\ ] ] where the matrices @xmath200 and @xmath201 are diagonal with elements @xmath202 and @xmath203 . the approximate effective mass matrix is @xmath204 if the spheres are neutrally buoyant one has simply @xmath205 in our application we shall consider this case . the simplest application of the theory is to a three - sphere swimmer or flyer with the three spheres aligned on the @xmath50 axis , as studied by golestanian and ajdari @xcite in the resistive limit . the spheres move along the @xmath50 axis , and the @xmath206 and @xmath207 coordinates can be ignored @xcite . there are only two relative coordinates @xmath208 and @xmath209 , and the relevant parts of the matrices @xmath210 and @xmath163 are two - dimensional . the relevant parts are denoted as @xmath211 and @xmath212 . in the bilinear theory we consider a point @xmath213 in @xmath214-space with coordinates @xmath215 , corresponding to the configuration @xmath107 of the rest system . as an example we consider the case of equal - sized spheres with @xmath216 , equal masses @xmath217 , and equal distances between centers @xmath218 . for this case analytic expressions for the matrices @xmath211 and @xmath212 can be derived , but are too complicated to be presented . in the high viscosity limit the expressions reduce to those derived previously @xcite . the matrices , defined with @xmath219 in eq . ( 5.6 ) , depend only on the ratio @xmath220 and the dimensionless viscosity @xcite @xmath221 the two eigenvalues @xmath222 , as well as the corresponding eigenvectors @xmath223 with @xmath224 , also depend only on these two variables . the dependence on the viscosity @xmath225 is surprisingly slight over the whole range of @xmath225 values . the absolute value @xmath226 is close to unity over the whole range . in fig . 1 we show the variation of @xmath227 and @xmath226 with @xmath225 for @xmath228 . the argument of @xmath229 increases slightly from @xmath230 at @xmath231 to @xmath232 at @xmath233 .    in the bilinear theory the optimal orbit @xmath234 in relative space is given by @xmath235 with @xmath236 and @xmath237,\ ] ] with amplitude factor @xmath238 . the corresponding displacement vector in configuration space is given by @xmath239 in fig . 2 we show a snapshot of the spheres and their velocities in the instantaneous rest frame at @xmath42 for @xmath240 and @xmath241 . in fig . 1 of ref . 15 we showed the elliptical orbit in relative space for @xmath228 and @xmath242 in the limit of high viscosity . in fig . 3 we compare this with the corresponding orbit in the limit of zero viscosity , corresponding to small @xmath225 . the two plots are indistinguishable on the scale of the figure .    from the periodic displacement @xmath243 we can calculate the instantaneous swimming velocity @xmath244 as a series of harmonics from eq . the zeroth harmonic yields the mean swimming velocity @xmath245 . from @xmath244 and @xmath243 we can calculate the time - dependent rate of dissipation @xmath94 , and hence the mean @xmath246 , by use of eq . ( 3.17 ) .    in fig . 4 we show the reduced mean swimming velocity @xmath247 as a function of @xmath238 for @xmath228 and @xmath241 . in fig . 5 we show the reduced mean power @xmath248 vs. the reduced mean swimming velocity @xmath247 in the range @xmath249 . in fig . 6 we show the efficiency @xmath250 as a function of @xmath238 . the efficiency increases monotonically with the amplitude factor . in fig . 7 we show the time - dependence of the impetus @xmath251 and the center velocity @xmath244 as functions of time during a period for @xmath252 . we also plot separately the resistive contribution to the impetus . at this value of @xmath225 this is much smaller than the reactive part . it is noteworthy that the variations in time of the center velocity are much smaller than those of the impetus . the center velocity follows the impetus with an after - effect . in fig . 8 we show the absolute value of the fourier coefficients @xmath253 of the velocity @xmath244 , normalized to @xmath254 , for @xmath252 . this shows that only a small number of harmonics contribute appreciably .    the equality of mean power and mean rate of dissipation given by eq . ( 6.3 ) can be checked numerically . the froude efficiency @xmath255 , defined in eq . ( 6.5 ) , for these three values of @xmath238 is @xmath256 , respectively . it is of interest to compare the above results with values obtained from the numerical solution of the equations of motion eq . ( 2.7 ) with hydrodynamic interactions given by eqs . ( 7.1 ) and ( 7.12 ) and with prescribed oscillating actuating forces . in order to stabilize the system we consider microswimmers with internal harmonic interactions @xcite . in matrix form the forces may be expressed as @xmath257 with a real symmetric matrix @xmath258 with the property @xmath259 . the actuating forces @xmath260 can be chosen to correspond to the eigenvector with maximum eigenvalue in the problem eq . the first term in eq . ( 8.4 ) represents a harmonic approximation to the direct interactions . we use harmonic interactions given by the @xmath261-matrix @xmath262 with elastic constant @xmath263 . this corresponds to nearest neighbor interactions of equal strength @xmath263 between the three spheres . the stiffness of the assembly is characterized by the dimensionless number @xmath264 defined by @xmath265    we use the first order equations of motion , @xmath266 to calculate the actuating forces @xmath267 corresponding to the optimal linear motion given by eqs . ( 8.3 ) and ( 4.4 ) . these have the property @xmath268 , so that the sum of actuating forces vanishes . we choose initial conditions corresponding to the rest configuration @xmath269    the kinetic energy term in eq . ( 2.7 ) makes the direct numerical solution of the equations of motion time - consuming . instead we use an iterative procedure , neglecting the kinetic energy term in first approximation . then the equations can be solved by a fast procedure . from the solution the kinetic forces , defined as @xmath270 , can be calculated as a function of time . in the next step we include the kinetic forces and repeat the procedure . the kinetic forces are small compared to the actuating forces and the solution converges after a few iterations .    in fig . 9 we show the numerical solution of the equations of motion eqs . ( 2.6 ) and ( 2.7 ) with forces given by eq . ( 8.4 ) for @xmath228 , viscosity @xmath241 , stiffness @xmath271 , and amplitude factor @xmath272 for the first fifty periods . in fig . 10 we show the mean value of the kinetic energy for successive periods . the orbit for the last period hardly differs from the ellipse given by eq . ( 8.3 ) , as shown in fig . 11 . the mean swimming velocity and the mean rate of dissipation can be calculated as time - averages over the last period . the efficiency is @xmath273 , equal to the value calculated from the periodic orbit by use of eq . ( 3.8 ) for displacements @xmath243 with @xmath272 , shown in fig . in general the performance of a swimmer or flyer can be measured in terms of the dimensionless efficiency @xmath274 , defined as the ratio @xmath275 where @xmath159 is a conveniently chosen length scale , @xmath245 is the mean speed and @xmath276 is the mean power , averaged over a period @xmath112 . we note that the lower bound of the efficiency @xmath274 vanishes , since for a periodic motion in relative phase space which is time - reversible the mean swimming velocity vanishes . a striking result of the present analysis is that for the simple three - sphere model , for which analytic calculations can be performed , the maximum efficiency @xmath277 is nearly independent of the dimensionless viscosity @xmath278 , as shown in fig . 1 , ( 5.10 ) . as a consequence the mean speed is nearly inversely proportional to the shear viscosity @xmath4 for given power . we expect that this is a general feature of the mechanical system defined by equations of motion of the type eq . ( 2.7 ) with hydrodynamic interactions as detailed in sec . this explains the great advantage that birds in air have over fish in water . a second result of the analysis is that the mean power is equal to the mean rate of dissipation , as shown in sec . there is no additional energy loss related to the rate of change of the virtual mass . the mean speed and mean power can be evaluated for more sophisticated model swimmers or flyers by similar analysis . elsewhere we studied longer chains with both longitudinal and transverse excitation in the resistive limit @xcite . that analysis can be extended to the full range of viscosity , based on the equations of motion eqs . ( 2.6 ) and ( 2.7 ) .    for assumed periodic displacements the velocity of the assembly can be derived from the equation of motion eq . ( 3.8 ) by decomposition in harmonics , as demonstrated in the three - sphere model . the mean power is found as a collateral of the calculation . the full range of viscosity is covered , so that the analysis provides an interesting link between the flying of birds , the swimming of fish , and the swimming of bacteria . 99    o. lilienthal , _ der vogelflug als grundlage der fliegekunst _ ( r. gaertner s verlagsbuchhandlung , berlin , 1889 ) . m. j. lighthill , _ mathematical biofluiddynamics _ ( siam , philadelphia , 1975 ) . s. childress , _ mechanics of swimming and flying _ ( cambridge university press , cambridge , 1981 ) .    c. j. pennycuick , _ modelling the flying bird _ ( academic press , burlington ma , 2008 ) . h. tennekes , _ the simple science of flight _ ( mit press , cambridge ma , 2009 ) . m. j. lighthill , `` note on the swimming of slender fish '' , j. fluid mech . * 9 * , 305 ( 1960 ) . t. y. wu , `` swimming of a waving plate '' , j. fluid mech . * 10 * , 321 ( 1961 ) . t. y. wu , `` on theoretical modeling of aquatic and aerial animal locomotion '' , adv . 38 * , 291 ( 2001 ) . t. y. wu , `` fish swimming and bird / insect flight '' , annu . rev . fluid mech . * 43 * , 25 ( 2011 ) . j. a. sparenberg , `` survey of the mathematical theory of fish locomotion '' , j. eng . math . * 44 * , 395 ( 2002 ) . j. gray , `` studies of animal locomotion '' , j. expl . * 13 * , 192 ( 1936 ) .    g. i. taylor , `` analysis of the swimming of microscopic organisms '' , proc . london a * 209 * , 447 ( 1951 ) . j. lighthill , `` flagellar hydrodynamics : the john von neumann lecture , 1975 '' , siam review * 18 * , 161 ( 1976 ) . e. lauga and t. r. powers , `` the hydodynamics of swimming microorganisms '' , rep . prog . phys . * 72 * , 09660 ( 2009 ) . b. u. felderhof , `` swimming of an assembly of rigid spheres at low reynolds number '' , eur . j. e * 37 * , 110 ( 2014 ) . b. u. felderhof , `` efficient swimming of an assembly of rigid spheres at low reynolds number '' , arxiv1504.05794[physics.flu - dyn ] . j. lighthill , `` large - amplitude elongated body theory of fish locomotion '' , proc . b * 179 * , 125 ( 1971 ) . deng , y .- q . chen , h. dai , j. wian , and f .- b . tian , `` on numerical modeling of animal swimming and flight '' , comput . mech . * 52 * , 1221 ( 2013 ) .    j. happel and h. brenner , _ low reynolds number hydrodynamics _ ( noordhoff , leyden , 1973 ) . j. lighthill , _ an informal introduction to theoretical fluid mechanics _ ( clarendon press , oxford , 1986 ) . b. u. felderhof , `` comment on : ' ' circular motion of asymmetric self - propelling particles `` '' , phys . rev . lett . * 113 * , 029801 ( 2014 ) . l. d. landau and e. m. lifshitz , _ fluid mchanics _ ( pergamon , oxford , 1987 ) .    g. k. batchelor , _ an introduction to fluid dynamics _ ( cambridge university press , cambridge , 1967 ) . b. u. felderhof , `` virtual mass and drag in two - phase flow '' , j. fluid mech . * 225 * , 177 ( 1991 ) . r. golestanian and a. j. ajdari , `` analytic results for the three - sphere swimmer at low reynolds number '' , phys . e * 77*. 036308 ( 2008 ) . b. u. felderhof , `` optimization of flagellar swimming by a model sperm '' , arxiv:1412.3937 .    a. s. sangani and a. prosperetti , `` numerical simulation of the motion of particles at large reynolds numbers '' in _ particulate two - phase flow _ ed . m. c. roco ( butterworth - heinemann , boston , 1993 ) . b. u. felderhof , `` the swimming of animalcules '' , phys . fluids * 18 * , 063101 ( 2006 ) . plot of the eigenvalue @xmath227 and the absolute value @xmath226 for the corresponding eigenvector @xmath279 of the three - sphere model with @xmath228 as functions of the dimensionless viscosity @xmath225 . plot of the elliptical orbit in the @xmath280 plane corresponding to eq . ( 8.2 ) with @xmath281 and @xmath241 ( solid curve ) . we also plot the elliptical orbit for the high viscosity limit ( dashed curve ) . the two curves can not be distinguished on the scale of the figure . plot of the impetus @xmath251 ( solid curve ) , the resistive contribution to the impetus ( short dashes ) and of the center velocity @xmath244 ( long dashes ) as functions of time during a period for the three - sphere swimmer for @xmath228 , @xmath241 and amplitude factor @xmath252 . plot of the absolute values of the fourier coefficients @xmath253 of harmonics of frequency @xmath285 , normalized to @xmath254 , of the center velocity @xmath244 of the three - sphere swimmer for @xmath228 and @xmath241 with amplitude factor @xmath252 .
a mechanical model of swimming and flying in an incompressible viscous fluid in the absence of gravity is studied on the basis of assumed equations of motion . 
 the system is modeled as an assembly of rigid spheres subject to elastic direct interactions and to periodic actuating forces which sum to zero . 
 hydrodynamic interactions are taken into account in the virtual mass matrix and in the friction matrix of the assembly . 
 an equation of motion is derived for the velocity of the geometric center of the assembly . the mean power 
 is calculated as the mean rate of dissipation . 
 the full range of viscosity is covered , so that the theory can be applied to the flying of birds , as well as to the swimming of fish or bacteria . as an example 
 a system of three equal spheres moving along a common axis is studied .