Two-sphere System Research Articles

Pair trajectories of uncharged conducting spheres in an electric field

In this paper, we study the role of electrostatic forces on pair trajectories of two uncharged conducting spheres subject to an external electric field. We consider the hydrodynamic interactions between the spheres as they move relative to one another. Previous studies have shown that electric-field-induced forces on a two-sphere system are always attractive, except for the configuration when the line joining the centers is perpendicular to the external electric field. In the current study, we derive the asymptotic form of the interparticle force induced by the electric field in the lubrication limit for arbitrary size ratios. The attractive electric force diverges as the separation approaches zero. Thus, our calculation shows that the electric-field-induced forces can overcome the continuum lubrication resistance and allow finite time contact between the surfaces of two spherical conductors. We calculate the asymptotic variation of interparticle separation using the near-field asymptotic expressions for the electric-field-induced forces, exploring the role of hydrodynamic interactions in interparticle motion parallel and perpendicular to the electric field.

Optimizing the mean swimming velocity of a model two-sphere swimmer

The swimming of a two-sphere system oscillating in a viscous fluid is 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.

On the Motion of Two Microspheres in a Stokes Flow Driven by an External Oscillator Field

Hydrodynamic interactions of a two-solid microspheres system in a viscous incompressible fluid at low Reynolds number is investigated analytically. One of the spheres is conducting and assumed to be actively in motion under the action of an external oscillator field, and as the result, the other nonconducting sphere moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the two-sphere system is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of each sphere is formulated. The trajectory of the sphere was found to be inversely proportional to the frequency of the external field. The results demonstrated that no collisions occur between the spheres as the system moves in a circular motion with a fixed separation distance.

Effect of spacing on laminar natural convection flow and heat transfer from two spheres in vertical arrangement

Laminar natural convection heat transfer from two vertically arranged spheres has been investigated numerically in air (Pr=0.72) for large Rayleigh number (Ra=105). The effect of sphere spacing (S) on fluid flow and heat transfer is comprehensively analyzed within the ratio of spacing to radius (S/R) ranging from 4 to 48, i.e., 2⩽S/D⩽24. Extensive results are obtained for complete descriptions on natural convection of two-sphere system, such as the streamline patterns and isotherm contours, dimensionless velocities and temperature profiles, drag coefficient, local and average Nusselt numbers. Comparisons with the isothermal bisphere have also been reported with the previous studies. Due to the interaction of preheating effect and flow mixing, the lower sphere has a great influence on the fluid flow and heat transfer around the upper sphere with the change of spacing between the two spheres. The variation of dimensionless temperature with the dimensionless distance between the two spheres has been also concluded to explain the impact of preheating effect. Finally, correlations have been developed for accurate prediction of natural convection heat transfer for each single sphere and the whole two-sphere system, which can be applied to relevant engineering calculations, and provide values for academic research.

Effect of fluid inertia on the motion of a collinear swimmer.

The swimming of a two-sphere system and of a three-sphere chain in an incompressible viscous fluid is studied on the basis of simplified equations of motion which take account of both Stokes friction and added mass effects. The analysis is based on an explicit expression for the asymptotic periodic swimming velocity and a corresponding evaluation of the mean rate of dissipation. The mean swimming velocity of the two-sphere system is found to be nonvanishing provided that the two spheres are not identical. The swimming of a comparable chain of three identical spheres is much more efficient.

A regularised singularity approach to phoretic problems

An efficient, accurate, and flexible numerical method is proposed for the solution of the swimming problem of one or more autophoretic particles in the purely diffusive limit. The method relies on successive boundary element solutions of the Laplacian and the Stokes flow equations using regularised Green's functions for swift, simple implementations, an extension of the well-known method of "regularised stokeslets" for Stokes flow problems. The boundary element method is particularly suitable for phoretic problems, since no quantities in the domain bulk are required to compute the swimming velocity. For time-dependent problems, the method requires no re-meshing and simple boundaries such as a plane wall may be added at no increase to the size of the linear system through the method of images. The method is validated against two classical examples for which an analytical or semi-analytical solution is known, a two-sphere system and a Janus particle, and provides a rigorous computational pipeline to address further problems with complex geometry and multiple bodies.

Forces and torque on a pair of uncharged conducting spheres in an external electric field

Exact results are given for the forces acting on two conducting spheres in an applied electric field E. The torque acting on the two-sphere system is proportional to the difference between the longitudinal and transverse polarizabilities: τ=(αℓ−αt)E2 sinθ cosθ (θ is the angle between the applied field and the line-of-centers of the spheres). The forces acting on the two spheres are equal and opposite, and given by the derivatives of αℓ and αt with respect to the sphere separation. Simple analytic forms for the torque and forces are found at small and at large separations. At all separations, the torque always acts to align the line of centers of the spheres with the external field. Possible applications are to colloidal suspensions of spherical conducting particles.

Ultrahigh-contrast and large-bandwidth thermal rectification in near-field electromagnetic thermal transfer between nanoparticles

We exploit the unique properties of optical waves in nanophotonic structures to enhance the capabilities for active control of electromagnetic thermal transfer at nanoscale. We show that the near-field thermal transfer between two nanospheres can exhibit a thermal rectification effect with very high contrast and large operating bandwidth. In this system, the scale invariance properties of the resonance modes result in a large difference in the coupling constants between relevant modes in the forward and reverse scenarios. Such a difference provides a mechanism for thermal rectification. The two-sphere system can also exhibit negative differential thermal conductance.

Coupling Effects of Depletion Interactions in Colloidal Suspensions

The depletion interactions of the three-sphere system in which the three spheres are on one line are studied by Monte Carlo simulations. The depletion interactions are determined by ARM, and the coupling effect was proved by the numerical result that the depletion interactions in the three-sphere system are larger than that of the corresponding two-sphere system. Furthermore, we find that the mechanisms of the coupling effect and the effect on depletion force from the geometry factor are the same. In addition, the numerical results also show that this coupling effect will be affected by both the volume fraction and separation of three-sphere system.

Mechanisms of Slower Nitric Oxide Uptake by Red Blood Cells and Other Hemoglobin-containing Vesicles

Nitric oxide (NO) acts as a smooth muscle relaxation factor and plays a crucial role in maintaining vascular homeostasis. NO is scavenged rapidly by hemoglobin (Hb). However, under normal physiological conditions, the encapsulation of Hb inside red blood cells (RBCs) significantly retards NO scavenging, permitting NO to reach the smooth muscle. The rate-limiting factors (diffusion of NO to the RBC surface, through the RBC membrane or inside of the RBC) responsible for this retardation have been the subject of much debate. Knowing the relative contribution of each of these factors is important for several reasons including optimization of the development of blood substitutes where Hb is contained within phospholipid vesicles. We have thus performed experiments of NO uptake by erythrocytes and microparticles derived from erythrocytes and conducted simulations of these data as well as that of others. We have included extracellular diffusion (that is, diffusion of the NO to the membrane) and membrane permeability, in addition to intracellular diffusion of NO, in our computational models. We find that all these mechanisms may modulate NO uptake by membrane-encapsulated Hb and that extracellular diffusion is the main rate-limiting factor for phospholipid vesicles and erythrocytes. In the case of red cell microparticles, we find a major role for membrane permeability. These results are consistent with prior studies indicating that extracellular diffusion of several gas ligands is also rate-limiting for erythrocytes, with some contribution of a low membrane permeability.

Analytical expression for the electric field enhancement between two closely-spaced conducting spheres

An external electric field applied to two conducting spheres in close approach is enhanced (by charge separation on the spheres) in the region between the spheres. For spheres of equal size, this enhancement is a universal function of the ratio of the separation of the spheres to their radius, and increases without limit as this ratio decreases. We calculate the enhancement factor analytically for perfectly conducting spheres, providing a simple formula valid when the spheres are close together, that is when the enhancement is large and the known series solution is difficult to evaluate. The same methods allow us to find the close-approach forms of the longitudinal and transverse polarizabilities of the two-sphere system.

A Third-Order Dispersive Flow for Closed Curves into Kähler Manifolds

This article is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into Kahler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method.

Micro-Rotation by Flow-Induced Torque in an Optical Trap

We report a new, simplified mechanism for performing micro-rotation. A sample chamber filled with aqueous solution and mounted on a piezoelectric transducer (PZT)-driven stage was used to produce flow-induced torque for rotating micro-objects. We demonstrated the scheme by rotating ellipsoidal yeast particles and a two-sphere system with 20 mW of laser power. Clockwise or counterclockwise rotation could be controlled by changing the initial angular position of the object trapped and by reversing the direction of the PZT-driven stage. This mechanism has potential applications in microfabrication, biotechnology and fluidic technology.

Hydrodynamic interactions of spherical particles in quadratic Stokes flows

The quasi-steady hydrodynamic Stokes drag force and torque exerted on each of N non-identical particles immersed in a general quadratic undisturbed flow field at infinity is analytically investigated. Explicit results are given for the case of two spherical particles. In particular, expressions are obtained for these forces and torques in terms of: (1) the linear and angular velocities of each of the particles together with the local shear rate of the quadratic flow, all evaluated at the center of volume of the two-sphere system; and (2) the position-independent triadic shear-rate gradient characterizing the undisturbed quadratic flow field. These tensorial (matrix) formulas, obtained by the method of reflections for the intrinsic Stokes resistance coefficients for each of the spheres, are expressed nondimensionally in terms of the respective sphere radii, the center-to-center distance 2 H between spheres, a unit vector e lying along the line-of-centers, and a characteristic length-scale R 0 of the undisturbed quadratic flow field — typically the radius of a tube bounding an axisymmetric Poiseuille flow in which the two spheres are suspended. The reflection scheme is accurate within an error of O r/H α r/R 0 β , wherein α+ β≤6 and in which r is a characteristic sphere radius. The respective translational velocities of the two spheres are derived for the particular case where each is freely suspended in an unbounded Poiseuille flow (i.e., wall effects are neglected). It is shown in this case that a net radial motion of the pair of neutrally buoyant spheres ensues across the undisturbed streamlines as a consequence of the quadratic nature of the flow field (coupled with the interparticle hydrodynamic interaction). The direction of net migration calculated for the two spheres is from low to high shear rates. This contrasts with shear-induced migration phenomena observed in concentrated suspensions subjected to inhomogeneous shearing flows, where the direction of particle migration is from high to low shear rates. The two opposing effects acting in concert may help explain how a steady-state radial particle concentration distribution can be achieved for suspensions of non-Brownian particles.

Influence of Agglomerates of Conducting Spheres on the Response of a Phase-Doppler Anemometer

The influence of agglomerates of solid conducting spheres on the response of a phase-Doppler anemometer (PDA) is described for a two-sphere system by using a ray theory model. First- and second-order reflection and diffraction are considered for far-field calculations of the PDA phase difference. The numerical simulations are accompanied and supported by experimental results. Two-sphere systems of Sn63Pb37 alloy particles were captured inside an electrodynamic trap and investigated with a standard phase-Doppler system.

Qualitative light-scattering angular correlations of conglomerate particles

The scattering phase functions of micrometer-sized glycerol droplets containing spherical latex inclusions undergo random fluctuations with time. We measure scattering intensities in the near-forward and near-backward scattering directions and find them to have strong positive correlations during some time periods and strong negative correlations during other time periods. The characteristic time constants of these correlations are of the order of seconds. We calculate scattering correlations from two types of scattering system. Correlations from a two-sphere system generally are positive, whereas correlations from a sphere containing a single spherical inclusion may be both positive and negative. Calculations of correlations from our experimental data are consistent with diffusion of inclusions within the host droplet, rather than interference effects between the inclusions.

RELATIONSHIP BETWEEN THE BROWNIAN MOTION OF THE SUSPENDED PARTICLES IN ERF AND THEIR AGGREGATION

It is known that the Brownian motion is concerned with the aggregation of particle- chains in electrorheological fluids (ERF) under external field. In this paper we suggest a two-sphere model in which we assume that there is internal motion between two spheres but they keep separation from each other for simplifying particle-chain in ERF. Analysing the properties of the Brownian motion of the two-sphere system the result of its trace fractal dimension less than 2 is obtained. In view of the dimension of motion-trace a qualitative explanation of the aggregation of particle-chains in ERF subjected to external electric field, i.e. ,phase separation,is given.

The viscosity of suspensions — an effective medium approach that incorporates Brownian motion

By an extension of the hydrodynamic equivalent to the Clausius-Mossotti (CM) and Bruggeman (BM) formulas in dielectrics, new theories for the viscosity of a fluid suspension are constructed that include the effects due to Brownian motion hitherto ignored in previous works using the effective medium (EM) approach. This entails knowledge of the averaged dipole polarisability due to the hydrodynamic interaction of (at least) a two-sphere system. When incorporated into the theories, the resultant viscosity of the suspension is clearly temperature dependent. A requirement, desired in this work, is that in the limits of high and low Péclet numbers (Pe = E ij a 2 / D 0), the theories reproduce the well-known (exact) and (semi-exact) results for the low density expansions due to Einstein (1905, 1911) and Batchelor (1977) respectively. We demonstrate this explicitly for the CM case. Our numerical estimates also indicate that this requirement can only be met in the BM case, provided that two-body corrections are included symmetrically in the theory. In the latter, a coupling of the two-particle distribution function and the EM equation results, with the natural consequence that the renormalised Péclet number (Pe φ = E ij a 2 / D( φ)) now becomes the relevant parameter (whose value must be determined self-consistently). The near coincidence of the second order coefficient at low Péclet number with the low density expansion of the BM formula (without Brownian motion) is further discussed, together with a comparison with experiments, computer simulations and more recent studies.

Nonequilibrium molecular dynamics simulation of free-molecule gas flows in complex geometries with application to Brownian motion of aggregate aerosols.

A nonequilibrium molecular dynamics method is presented for the study of free-molecule, rarefied gas flows in complex geometries. Simulation results for a system of N rigid spheres in a rarefied gas are given, including particle friction tensor and force autocorrelation function. For single spheres, simulation results are shown to agree with known analytical results. For a two-sphere system, computational results are presented that indicate the two-sphere friction tensor is nonsymmetric, as opposed to the symmetric behavior in continuum and near continuum fluids. The nonsymmetric behavior can be traced to geometric shielding effects and leads to an effective attractive force between two spheres in free-molecule flows. It is possible to use the nonequilibrium molecular dynamics method given here to determine the many-sphere friction behavior for combined use in Langevin dynamics. Such a procedure, in principle, could allow for a rigorous, long-term dynamical analysis of particles in complex fluid systems.

Interactive dipole model for two-sphere system

Approximate “dipole” equations are derived for the interaction of two uncharged dielectric spheres immersed in a uniform electric field. Each sphere is treated as a point polarizable particle in a locally uniform field whose strength is that which would be present at the sphere's center if the sphere were removed. The response of each sphere is a dipole moment. In this way each sphere sees the uniform external field and the dipole field of the other sphere. These interactive dipole moments and forces are calculated for the two orientations of the spheres relative to the external field. Data are presented which show the range of validity of the results: the exclusion of all higher order multipoles is acceptable only as long as the spheres are separated by at least one radius.