Method Of Multipole Expansions Research Articles

Simulation of diffraction of ocean waves by a submerged sphere in finite depth

Based on the linear diffraction theory, an investigation is made on the interaction of water waves with a completely submerged sphere in water of finite depth in this paper. The method of multipole expansions is used to obtain the fluid velocity potential in the form of double series of the associated Legendre functions with the unknown coefficients of the infinite set of infinite matrix equations. The truncation property of the matrices and the convergence of the multipole series coefficients are investigated for various wavelengths and depths. The systematic numerical simulation, based on our analytical solution, is carried out and the fields of the hydrodynamic diffraction pressure and fluid velocity around the sphere, the three-dimensional free surface elevation, and total exciting forces acting on the sphere are graphically presented for a wide range of the body submergences, ocean depths and wavelengths.

Rate of diffusion-limited reactions in a cluster of spherical sinks

The reaction rate is calculated for a cluster of perfectly absorbing, stationary spherical sinks in a medium containing a mobile reactant. The diffusive interactions are accurately taken into account by employing the first-passage technique. The configurations of the clusters include three spheres, four spheres, regular polygons, linear chains, squares, and finite cubic arrays. For a given number of sinks, the reaction rate is decreased with increasing the compactness of the structure due to the screening effect. For a specified configurations, the asymptotic expressions for the reaction rate varying with the number of sinks are confirmed. By comparing results with these “exact” data, the method of multipole expansions up to the dipole level, which is suitable for a finite system of many sinks, proved to be an excellent approximation.

Low-Reynolds-number hydrodynamic interactions in a suspension of spherical particles with slip surfaces

The motion of two rigid spherical particles in an arbitrary configuration in an infinite viscous fluid at low Reynolds numbers is considered. The fluid is allowed to slip at the surfaces of the spheres and the particles may differ in radius. The resistance and mobility functions that completely characterize the linear relations between the forces and torques and the translational and rotational velocities of the particles are analytically calculated in the quasi-steady limit using a method of twin multipole expansions. For each function, an expression of power series in r −1 is obtained, where r is the distance between the particle centers. The agreement between these expressions and the relevant results in the literature is quite good. Based on a microscopic model, the analytical results for two-sphere hydrodynamic interactions are used to find the effect of the volume fraction of particles of each type on the average settling velocities in a bounded suspension of slip spheres. Our results, presented in simple closed forms, agree very well with the existing solutions for the limiting cases of no slip and perfect slip at the particles surfaces. In general, the particle-interaction effects are found to be more significant when the slip coefficients at the particle surfaces become smaller. Also, the influence of the interactions on the smaller particles is stronger than on the larger ones.

Nusselt number for flow perpendicular to arrays of cylinders in the limit of small Reynolds and large Peclet numbers

The problem of determining the Nusselt number N, the nondimensional rate of heat or mass transfer, from an array of cylindrical particles to the surrounding fluid is examined in the limit of small Reynolds number Re and large Peclet number Pe. N in this limit can be determined from the details of flow in the immediate vicinity of the particles. These are determined accurately using a method of multipole expansions for both ordered and random arrays of cylinders. The results for N/Pe1/3 are presented for the complete range of the area fraction of cylinders. The results of numerical simulations for random arrays are compared with those predicted using effective-medium approximations, and a good agreement between the two is found. A simple formula is given for relating the Nusselt number and the Darcy permeability of the arrays. Although the formula is obtained by fitting the results of numerical simulations for arrays of cylindrical particles, it is shown to yield a surprisingly accurate relationship between the two even for the arrays of spherical particles for which several known results exist in the literature suggesting thereby that this relationship may be relatively insensitive to the shape of the particles.

The energy of singular effects in an elastic anisotropic medium

The problem of determining the accumulated energy function, the strain energy and the energy concentration coefficient corresponding to singularities in the form of concentrated forces and moments in an infinite homogeneous linearly elastic anisotropic medium is considered. The method of multipole expansions is used. Examples are presented.

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.

Simple shear flow of suspensions of liquid drops

The shearing motion of monodisperse suspensions of two-dimensional deformable liquid drops with uniform interfacial tension is studied by means of numerical simulations. In the theoretical model, the drops are distributed randomly within a square that is repeated periodically in two directions yielding a doubly periodic flow. Under the assumption that inertial effects are negligible and the viscosity of the drops is equal to that of the suspending fluid, the motion is investigated as a function of the area fraction of the suspended drops and of the capillary number. The evolution of the suspension from an initial configuration with randomly distributed circular drops is computed using an improved implementation of the method of interfacial dynamics which is based on the standard boundary integral formulation for Stokes flow. The numerical procedure incorporates the method of multipole expansions to account for far-drop interactions, and interpolation through tables for computing the doubly periodic Green's function ; the latter allows considerable savings in the cost of the computations. Dynamic simulations are carried out for suspensions with up to 49 drops within each periodic cell, for an extended period of time up to kt = 60, where k is the shear rate. Comparisons with previous numerical results for solid particles reveal that particle deformability and interfacial mobility play an important role in the character of the motion. The effects of particle area fraction and capillary number on the effective rheological properties of the suspension are discussed, and the statistics of the drop motion is analysed with reference to the drop-centre pair distribution function and probability density functions of drop aspect ratio and inclination. It is found that the effective rheological properties may be predicted with remarkable accuracy from a knowledge of the instantaneous mean drop deformation and orientation alone, even at high area fractions. Cluster formation is not as important as in suspension of solid particles. The apparent random motion of the individual drops, when viewed at a sequence of time intervals that are large compared to the inverse shear rate, is described in terms of an effective non-isotropic long-time diffusivity tensor, and the transverse component of this tensor is computed from the results of the simulations with some uncertainty.

Periodic solutions for three sedimenting spheres

Exact periodic solutions are found for the relative motion of three spheres sedimenting in a Stokes fluid. Nearby solutions are found to be nearly periodic for a long time. Existence of the exact periodic solutions is proved using the point-particle approximation and symmetry properties of Stokes equations. Numerical simulations for finite-sized particles are performed using a method of multipole expansions.