arXiv subjects

B. U. Felderhof

Publications and source records attributed to B. U. Felderhof.

At least 19 recordsLinked to original sources

arXiv2024

Effect of retarded friction and added mass on the swimming speed of a vibrating two-sphere

A theoretical expression is derived for the mechanical contribution to the mean swimming speed of a vibrating two-sphere immersed in a viscous incompressible fluid. The two spheres are connected by an elastic spring which provides a harmonic potential for oscillations about a mean distance between centers. The system is made to oscillate at a chosen frequency by activating forces which sum to zero. The mechanical contribution to the resulting mean swimming velocity is calculated from the mechanical equations of motion and the corresponding impedance matrix of linear response. The frequency-dependent pair friction coefficients are found from approximate expressions derived earlier. The mechanical contribution is calculated to second order in the amplitude of stroke as a function of the scaling number, a dimensionless combination of size, frequency, and kinematic viscosity. Retarded friction and added mass determine the functional behavior.

physics.flu-dyn↗

arXiv2023

Transition in steady streaming and pumping caused by a sphere oscillating in a viscous incompressible fluid

The steady streaming flow pattern caused by a no-slip sphere oscillating in an unbounded viscous incompressible fluid is calculated exactly to second order in the amplitude. The pattern depends on a dimensionless scale number, determined by sphere radius, frequency of oscillation, and kinematic viscosity of the fluid. At a particular value of the scale number there is a transition with a reversal of flow. The analytical solution of the flow equations is based on a set of antenna theorems. The flow pattern consists of a boundary layer and an adjacent far-field of long range, falling off with the inverse square distance from the center of the sphere. The boundary layer becomes thin in the limit where inertia dominates over viscosity. The system acts as a pump operating in two directions, depending on the scale number. The efficiency of the pump is estimated from a comparison of the rate of flow with the rate of dissipation.

physics.flu-dyn↗

arXiv2022

Optimizing the Mean Swimming Velocity of a Model Two-sphere Swimmer

The swimming of a two-sphere system oscillating in a viscous fluidis studied on the basis of simplified equations of motion which take account of both friction and inertial effects. In the model the friction follows from an Oseen approximation to the mobility matrix, and the inertial effects follow from a dipole approximation to the added mass matrix. The resulting mean swimming velocity is evaluated analytically in a first harmonics approximation. For specific choices of the parameters this is compared with the exact result following from a numerical calculation including higher harmonics. The Oseen-Dipole model is compared with the simpler Oseen* model, in which the added mass effects are approximated by just the effective mass of the single spheres and dipole interactions are neglected. The expression for the mean swimming velocity can be reduced to a dimensionless scaling form. For given viscosity and mass density of the fluid the frequency of the stroke and the ratio of radii can be chosen such that the swimming velocity is optimized.

cond-mat.soft↗

arXiv2021

Comment on: arXiv:2008.08305 "Scallop Theorem and Swimming at the Mesoscale" by M. Hubert, O. Trosman, Y. Collard, A. Sukhov, J. Harting, N. Vandewalle, and A.-S. Smith, Phys. Rev. Letters \vc{126}, 224501 (2021)

The expression for the mean swimming velocity of the Oseen two-sphere model derived in the Letter [arXiv:2008.08305] is compared with an exact one derived earlier, as well as one derived in first harmonic approximation. The latter is found to be a very accurate approximation to the exact result.

physics.flu-dyn↗

arXiv2020

Swimming of a uniform deformable sphere in a viscous incompressible fluid with inertia

The swimming of a deformable uniform sphere is studied in second order perturbation theory in the amplitude of the stroke. The effect of the first order reaction force on the first order center of mass velocity is calculated in linear response theory by use of Newton's equation of motion. The response is characterized by a dipolar admittance, which is shown to be proportional to the translational admittance. As a consequence the mean swimming velocity, calculated in second order perturbation theory, depends on the added mass of the sphere. The mean swimming velocity and the mean rate of dissipation are calculated for several selected strokes.

physics.flu-dyn↗

arXiv2020

Collinear velocity relaxation of two spheres in a viscous incompressible fluid

Collinear velocity relaxation of two spheres immersed in a viscous incompressible fluid is studied on the basis of an approximate expression for the retarded hydrodynamic interaction. After a sudden impulse applied to one sphere, the other one instantaneously starts to move as well, with amplitude determined by the added mass effect. The velocities of both spheres eventually decay with a t^{-3/2} long-time tail, but the relative velocity decays with a t^{-5/2} long-time tail. The three relaxation functions are approximated by simple expressions involving only a small number of poles in the complex square root of frequency plane.

physics.flu-dyn↗

arXiv2019

Dynamics of cruising swimming by a deformable sphere for two simple models

The dynamics of periodic swimming is studied for two models of a deformable sphere, the dipole-quadrupole model and the quadrupole-octupole model. For the two models the solution of the Navier-Stokes equations can be found exactly to second order in the amplitude of stroke. Hence the oscillating force exerted by the fluid on the body is calculated. This allows calculation of the periodic time-dependent center of mass velocity.

physics.flu-dyn↗

arXiv2019

Retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid

Retarded or frequency-dependent hydrodynamic interactions are relevant for velocity relaxation of colloidal particles immersed in a fluid, sufficiently close that their flow patterns interfere. The interactions are also important for periodic motions, such as occur in swimming. Analytic expressions are derived for the set of scalar mobility functions of a pair of spheres. Mutual hydrodynamic interactions are evaluated in one-propagator approximation, characterized by a single Green function acting between the two spheres. Self-mobility functions are evaluated in a two-propagator approximation, characterized by a single reflection between the two spheres. The approximations should yield accurate results for intermediate and long distances between the spheres. Both translations and rotations are considered. For motions perpendicular to the line of centers there is translation-rotation coupling. Extensive use is made of Faxén theorems which yield the hydrodynamic force and torque acting on a sphere in an incident oscillating flow.

physics.flu-dyn↗

arXiv2019

Second harmonic generation and vortex shedding by a dipole-quadrupole and a quadrupole-octupole swimmer in a viscous incompressible fluid

Vortex shedding by a swimming sphere in a viscous incompressible fluid is studied for surface modulation characterized by a superposition of dipolar and quadrupolar, as well as for quadrupolar and octupolar displacements, varying harmonically in time. The time-dependent swimming velocity and the flow velocity are calculated to second order in the amplitude of surface modulation for both models. The models are also useful for the discussion of bird flight.

physics.flu-dyn↗

arXiv2018

Effect of fluid inertia on swimming of a sphere in a viscous incompressible fluid

Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function of kinematic viscosity for axisymmetric strokes of the swimming sphere. The calculation covers the full range of scale number, from the friction-dominated Stokes regime in the limit of vanishing scale number to the inertia-dominated regime at large scale number. The model therefore provides paradigmatic insight into the fluid dynamics of swimming or flying of a wide range of organisms.

physics.flu-dyn↗

arXiv2018

Vanishing mean volume velocity in isothermal isobaric diffusion of a binary fluid mixture

It is shown that in isothermal isobaric diffusion of a binary fluid mixture the mean volume velocity vanishes in the linear regime, independent of the equation of state. The partial specific volumes of the two components are uniform and constant in the process of mutual diffusion. The properties lead to a simple derivation of the de Groot-Mazur thermodynamic factor in the diffusion coefficient. The properties also imply that the diffusive volume flux defined by Brenner is proportional to the mass current density, and is therefore not a quantity of independent interest.

physics.chem-ph↗

arXiv2017

Generalized Einstein relation for the mutual diffusion coefficient of a binary fluid mixture

The method employed by Einstein to derive his famous relation between the diffusion coefficient and the friction coefficient of a Brownian particle is used to derive a generalized Einstein relation for the mutual diffusion coefficient of a binary fluid mixture. The expression is compared with the one derived by de Groot and Mazur from irreversible thermodynamics, and later by Batchelor for a Brownian suspension. A different result was derived by several other workers in irreversible thermodynamics. For a nearly incompressible solution the generalized Einstein relation agrees with the expression derived by de Groot and Mazur. The two expressions also agree to first order in solute density. For a Brownian suspension the result derived from the generalized Smoluchowski equation agrees with both expressions.

physics.flu-dyn↗

arXiv2017

Coaxial collisions of a vortex ring and a sphere in an inviscid incompressible fluid

The dynamics of a circular thin vortex ring and a sphere moving along the symmetry axis of the ring in an inviscid incompressible fluid is studied on the basis of Euler's equations of motion. The equations of motion for position and radius of the vortex ring and those for position and velocity of the sphere are coupled by hydrodynamic interactions. The equations are cast in Hamiltonian form, from which it is seen that total energy and momentum are conserved. The four Hamiltonian equations of motion are solved numerically for a variety of initial conditions.

physics.flu-dyn↗

arXiv2016

Swimming at small Reynolds number of an elliptical disk propelled by an elliptically polarized surface wave

The swimming of an elliptical disk at small Reynolds number is studied on the basis of a perturbative solution of the Navier-Stokes equations for fluid flow near a deformable infinite sheet. A stroke involving an elliptically polarized plane surface wave is studied, in extension of work by Taylor and Tuck. In general the elliptic polarization of the stroke leads to an asymmetry of the flow in the upper and lower half-space. On the basis of results for an infinite sheet expressions for the mean translational and rotational swimming velocity of an elliptical disk of size much larger than the wavelength of the stroke are deduced. In addition expressions are derived for the mean power and the efficiency of swimming.

physics.flu-dyn↗

arXiv2016

Swimming of a sphere in a viscous incompressible fluid with inertia

The swimming of a sphere immersed in a viscous incompressible fluid with inertia is studied for surface modulations of small amplitude on the basis of the Navier-Stokes equations. The mean swimming velocity and the mean rate of dissipation are expressed as quadratic forms in term of the surface displacements. With a choice of a basis set of modes the quadratic forms correspond to two hermitian matrices. Optimization of the mean swimming velocity for given rate of dissipation requires the solution of a generalized eigenvalue problem involving the two matrices. It is found for surface modulations of low multipole order that the optimal swimming efficiency depends in intricate fashion on a dimensionless scale number involving the radius of the sphere, the period of the cycle, and the kinematic viscosity of the fluid.

physics.flu-dyn↗

arXiv2016

Swimming of a circular disk at low Reynolds number

The swimming of a circular disk at low Reynolds number is studied for distortion waves along its two planar surfaces with wavelength much smaller than the size of the disk. The calculation is based on an extension of Taylor's work for a planar sheet. It is shown that in general the disk performs both translational and rotational swimming, resulting in a circular orbit.

physics.flu-dyn↗

arXiv2016

Swimming of a deformable slab in a viscous incompressible fluid with inertia

The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a polarization angle, allows optimization of the swimming efficiency with respect to polarization. The mean swimming velocity and mean rate of dissipation are calculated to second order in the amplitude of the stroke. The optimum efficiency depends on the ratio of viscosity and mass density of the fluid. For high viscosity a stroke is found with significantly higher efficiency than Taylor's solution for a swimming sheet. For low viscosity the efficiency is optimal for a nearly irrotational flow pattern.

physics.flu-dyn↗

arXiv2016

Swimming at small Reynolds number of a planar assembly of spheres in an incompressible viscous fluid with inertia

Translational and rotational swimming at small Reynolds number of a planar assembly of identical spheres immersed in an incompressible viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The translational and rotational swimming velocities of the assembly are deduced from momentum and angular momentum balance equations. The mean power required during a period is calculated from an instantaneous power equation. Expressions are derived for the mean swimming velocities and the power, valid to second order in the amplitude of displacements from the relative equilibrium positions. Hence these quantities can be evaluated for prescribed periodic displacements. Explicit calculations are performed for three spheres interacting such that they form an equilateral triangle in the rest configuration.

physics.flu-dyn↗