Method Of Multipole Expansions Research Articles

Slow Rotation of Coaxial Slip Colloidal Spheres about Their Axis

The flow field around a straight chain of multiple slip spherical particles rotating steadily in an incompressible Newtonian fluid about their line of centers is analyzed at low Reynolds numbers. The particles may vary in radius, slip coefficient, and angular velocity, and they are permitted to be unevenly spaced. Through the use of a boundary collocation method, the Stokes equation governing the fluid flow is solved semi-analytically. The interaction effects among the particles are found to be noteworthy under appropriate conditions. For the rotation of two spheres, our collocation results for their hydrodynamic torques are in good agreement with the analytical asymptotic solution in the literature obtained by using a method of twin multipole expansions. For the rotation of three spheres, the particle interaction effect indicates that the existence of the third particle can influence the torques exerted on the other two particles noticeably. The interaction effect is stronger on the smaller or less slippery particles than on the larger or more slippery ones. Torque results for the rotation of chains of many particles visibly show the shielding effect among the particles.

Hydrodynamic performance and energy absorption of multiple spherical absorbers along a straight coast

The development and utilization of wave energy have great potentiality to alleviate the urgent problem of global energy shortage. Spherical bodies can be used as point absorbers to extract wave energy, and much attention has been paid to the performance of spherical absorbers in an open water domain. This study focuses on the hydrodynamic performance and energy absorption of multiple spherical absorbers in front of a straight coast. The coast is assumed to be a fully reflecting vertical wall, and all the absorbers are restricted to only heave motion. An analytical solution based on linear potential flow theory is developed for the problem of wave diffraction and radiation by multiple absorbers. In the solution procedure, the hydrodynamic problem is transformed into an equivalent problem in an open water domain by applying the image principle. The velocity potential of the fluid motion is solved using the method of multipole expansions combined with the shift of local spherical coordinate systems. Then, the wave excitation force, added mass coefficient, radiation damping coefficient, and energy extraction performance of the absorbers are calculated. Case studies are presented to analyze the effects of the coastal reflection and hydrodynamic interaction among absorbers on the energy extraction performance of the wave energy converter (WEC) system. The effects of wave frequency, incident angle, spacing between the absorber and coast, submergence depth, absorber number, and plane layout are also clarified. The results suggest that the energy extraction performance of an isolated absorber is significantly improved when the motions of the waves and absorber are in resonance, and the coastal reflection can enhance the overall energy extraction performance for a WEC system with multiple absorbers. In addition, when the number of absorbers increases, the effects of the coastal reflection and hydrodynamic interaction become more complicated.

Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays.

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.

Water wave scattering by a submerged horizontal semi-circular barrier in a two-layer fluid

This paper studies water wave scattering by a submerged horizontal bottom-mounted semi-circular barrier in a two-layer fluid based on the linear potential flow theory. Two cases, i.e., the barrier completely located in the lower fluid and the barrier intersecting the internal interface of two fluids, are considered independently. In the former case, an analytical solution for the wave scattering problem is derived by using the method of multipole expansions, whereas in the latter case, a numerical solution is developed by employing the multi-domain boundary element method (BEM). The conservation relations of wave energy for incident waves in surface and internal modes are derived, and the reflection and transmission energy coefficients as well as the wave forces acting on the barrier are calculated. Comparisons of calculated results between the analytical solution and the BEM numerical solution are conducted. Calculation examples are presented to examine the effects of various physical parameters (angular frequency, interface position, density ratio of two fluids and incident angle) on wave forces acting on the barrier and the energy transformation between two wave modes. It is found that there is a noticeable difference in the energy transformation between the barrier intersecting the fluid interface or not.

Water wave diffraction and radiation by a submerged sphere in front of a vertical wall

This paper studies water wave diffraction and radiation by a submerged sphere in front of a vertical wall. Based on the image principle, the physical problem is transformed into an imaginary problem of two spheres, which are symmetric with respect to the original vertical wall, in a horizontally open water domain. Then, an analytical solution for the imaginary problem is developed in the context of linear wave theory. The velocity potential of fluid motion is obtained by adopting the method of multipole expansions combined with the shift of local spherical coordinate systems. The wave exciting forces in surge, sway and heave directions, and added mass and radiation damping due to the sphere oscillation are calculated. The convergence of the series solution is very rapid, and the results of analytical solution and numerical solution based on boundary element method are highly consistent. Calculation examples are presented to examine the effects of wave/oscillation frequency, spacing between sphere and wall, submergence depth and wave incident angle on various hydrodynamic quantities of the sphere. Results show that because the submerged sphere is subjected to not only incident/radiated waves but also reflected waves from the wall, the hydrodynamic quantities noticeably differ from those in an open water domain.

Effects of flexible bottom on radiation of water waves by a sphere submerged beneath an ice-cover

A particular hydro-elastic model is considered to examine a radiation problem involving an immersed sphere in an infinitely extended ice-covered sea, where the lower surface is enveloped by a flexible base surface. Both the flexible base surface and floating ice-plate are modelled as thin elastic plates with different configurations and are based on the Euler–Bernoulli beam equation. The appearance of surface tension at the surface below the floating ice-plate is ignored. Under such circumstance, two different modes of propagating waves appear in the fluid for any particular frequency. One of the modes with lower wavenumber propagates along the surface beneath the ice-plate and the other with higher wavenumber propagates along the elastic base surface. The method of multipole expansions is used to calculate the solutions of the heave and sway radiation problems involving a submerged sphere in an ice-covered fluid. Furthermore, this procedure gives rise to an infinite system of linear equations, which can be solved computationally by any regular method. The added-mass as well as damping coefficients in case of heave as well as sway motions are calculated, and displayed graphically in various submergence depths of the oscillating sphere and elastic specifications of both the flexible base surface as well as the floating ice-plate.

Antiplane Shear of an Elastic Body with Elliptic Inclusions Under the Conditions of Imperfect Contact on the Interfaces

We study the problem of аntiplane shear of an elastic body containing a finite array of arbitrarily located and oriented elliptic inclusions under the conditions of imperfect mechanical contact on the interfaces. The analytic solution of the problem is obtained by the method of multipole expansions with the use of the technique of complex potentials. By expanding the disturbances of the field of displacements caused by inclusions in a series in the system of elliptic harmonics and using the formulas for their reexpansion and exact validity of all contact conditions, we reduce the boundary-value problem of the theory of elasticity to an infinite system of linear algebraic equations. It is also proved that the reduction method is applicable to the indicated system, the rate of convergence of the solution is investigated, and the accumulated results are compared with the data available from the literature. The presented numerical results of parametric investigations reveal the presence of a strong dependence of stress concentration on the conditions of contact on the interfaces, as well as on the sizes, shapes, and relative positions of the inclusions.

Free-surface hydrodynamics of a submerged prolate spheroid in finite water depth based on the method of multipole expansions

Free-surface hydrodynamics of a submerged elongated (prolate) spheroid in water of finite depth is considered by employing spheroidal harmonics and the method of multipole expansions. In particular, we present semi-analytic solutions for both the wave making resistance and wave diffraction fundamental problems. Although these two cases are generally treated separately, it is demonstrated here that by using the present general methodology, the analytic procedures of these two problems are quite similar and thus the corresponding solutions can be obtained by using almost a single effort. The motion of the prolate spheroid is rectilinear and steady in a direction parallel to the undisturbed free surface and rigid planar bottom. The ambient wave field is assumed to be monochromatic with an arbitrary inclination angle with respect to the body's major axis. The Green's function based solution, employs Havelock's formula for the ultimate image singularity system which avoids redundant surface integrations and solving integral equations. Numerical simulations of the linearized Kelvin-Neumann problem for the hydrodynamic forces and moments exerted on the moving spheroid for different depths of submergence and flow parameters are presented and compared against the existing data, given mainly for spherical shapes. The present method is claimed to be simpler to implement and more versatile, yet more accurate with respect to existing codes. Aside from free surface hydrodynamics, it can be also extended to other harmonic practical problems involving interacting ellipsoidal geometries in confined domain.

Unsteady three-dimensional sources in deep water with an elastic cover and their applications

AbstractThe velocity potential is derived for a transient source of arbitrary strength undergoing arbitrary three-dimensional motion. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogeneous. The upper surface of the fluid is covered by a thin layer of elastic material of uniform density with lateral stress. The linearized initial boundary-value problem is formulated within the framework of the potential-flow theory, and the Laplace transform technique is employed to obtain the solution. The potential of a time-harmonic source with forward speed is obtained as a particular case. The far-field wave motion at long time is determined via the method of stationary phase. The problems of radiation (surge, sway and heave) of the flexural–gravity waves by a submerged sphere advancing at constant forward speed are investigated. The method of multipole expansions is used. Numerical results are obtained for the wave-making resistance and lift, added-mass and damping coefficients. The effects of an ice sheet and broken ice on the hydrodynamic loads are discussed in detail.

The analytic solution for hydrodynamic diffraction by submerged prolate spheroids in infinite water depth

This study investigates the linear hydrodynamic scattering problem by stationary prolate spheroidal bodies and aims at providing an analytic solution for the associated boundary value problem. It extends the work of the present author on the hydrodynamics of oblate spheroidal bodies following the same procedure. The structural model under consideration is a spheroid with its polar axis greater than its equatorial diameter, subjected to the action of monochromatic incident waves. The polar axis is assumed to be perpendicular to the free surface that leads to the axisymmetric case concept. The analytic solution is sought using the method of multipole expansions constructed by employing Thorne’s formulas (Multipole expansions in the theory of surface waves. Proc Cam Philos Soc 49:707–716, 1953) that describe the velocity potential at singular points within a fluid domain with free upper surface and infinite water depth. The final stage of the solution process is the application of the zero velocity condition on the wetted surface of the spheroid. Inevitably this task requires the transformation of the involved velocity potentials, originally expressed with respect to spherical and polar coordinates, into prolate spheroidal coordinates. To this end, the appropriate addition theorems are derived, which recast Thorne’s expressions into infinite series of associated Legendre functions.

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.

Exciting forces due to interaction of water waves with a submerged sphere in an ice-covered two-layer fluid of finite depth

Abstract Scattering of water waves by a sphere in a two-layer fluid, where the upper layer has an ice-cover modelled as an elastic plate of very small thickness, while the lower one has a rigid horizontal bottom surface, is investigated within the framework of linearized water wave theory. The effects of surface tension at the surface of separation is neglected. There exist two modes of time-harmonic waves – the one with lower wave number propagating along the ice-cover and the one with higher wave number along the interface. Method of multipole expansions is used to find the particular solution for the problem of wave scattering by a submerged sphere placed in either of the layers. The exciting forces for vertical and horizontal directions are derived and plotted against different values of the wave number for different submersion depths of the sphere and flexural rigidity of the ice-cover. When the flexural rigidity and the density of the ice-cover are taken to be zero, the numerical results for the exciting forces for the problem with free surface are recovered as particular cases.

Diffuse X-Ray scattering from crystalline systems with ellipsoidal quantum dots

A theory of diffuse X-ray scattering from a semiconductor system with ellipsoidal quantum dots (QDs) has been developed. The elastic strains outside a QD are calculated using the method of multipole expansions. An expression for the lattice displacement field is presented accurate to within the quadrupole term of expansion. Using the proposed approach, an analytical solution for the diffuse scattering from a crystalline medium with ellipsoidal inclusions is obtained. Reciprocal-space maps of the scattering intensity distribution are obtained by numerical calculations for QDs with various ratios of the height to lateral radius.

Analytic solution for wave diffraction of a submerged hollow sphere with an opening hole

An investigation is carried out on the interaction of surface waves with a submerged sphere having an opening hole in finite-depth water in this article. Based on the linear wave theory, 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 an infinite set. In terms of the body surface boundary condition and the matching condition between the inner and outer flows at the hole, the complex matrix equations for the coefficients of the series are established. The infinite sets of matrix equations are solved by truncating the series at a finite number. The hydrodynamic pressure on the structure surface and the exciting forces acting on the structure are graphically presented. The dynamic pressure on the wave front surface of the sphere varies slightly with angle of opening hole increasing, while that on the wave back surface does obviously. When the angles of opening hole are increasing, the absolute values of the complex exciting forces tend to fall as a whole.

Simulation of stress-strain state in SiGe island heterostructures

The problem of simulation of the stress-strain state in SiGe island heterostructures is considered. The analytic-numerical method of multipole expansions is used to obtain an approximate solution. The problem of the stressed state influence on the diffusion mobility of atoms adsorbed on the heterostructure free surface is also discussed.

Application of the method of multipole expansions in the 3D‐elasticity problem for a medium with ordered system of spherical pores

AbstractA problem of determination of the stress‐strain state in an elastic isotropic 3D‐space containing an ordered system of spherical voids in a bounded region subjected to hydrostatic pressure is considered. A method based on expansion of unknown solution in terms of multipoles located at pores centers is proposed for solving the problem. Expansion coefficients (multipoles intensities) are found by using an alternation method which implies successive satisfying the boundary conditions for each void of the system. Considered examples of application of the proposed method for structures with various number of pores show that the method gives a good approximation to the solution of the problem.

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.

Nonlinear interactions in electrophoresis of ideally polarizable particles

In the classical analysis of electrophoresis, particle motion is a consequence of the interfacial fluid slip that arises inside the ionic charge cloud (or Debye screening layer) surrounding the particle surface when an external field is applied. Under the assumptions of thin Debye layers, weak applied fields, and zero polarizability, it can be shown that the electrophoretic velocity of a collection of particles with identical zeta potential is the same as that of an isolated particle, unchanged by interactions [L. D. Reed and F. A. Morrison, “Hydrodynamic interaction in electrophoresis,” J. Colloid Interface Sci. 54, 117 (1976)]. When some of these assumptions are relaxed, nonlinear effects may also arise and result in relative motions. First, the perturbation of the external field around the particles creates field gradients, which may result in nonzero dielectrophoretic forces due to Maxwell stresses in the fluid. In addition, if the particles are able to polarize, they can acquire a nonuniform surface charge, and the action of the field on the dipolar charge clouds surrounding them drives disturbance flows in the fluid, causing relative motions by induced-charge electrophoresis. These two nonlinear effects are analyzed in detail in the prototypical case of two equal-sized ideally polarizable spheres carrying no net charge, using accurate boundary-element simulations, along with asymptotic calculations by the method of twin multipole expansions and the method of reflections. It is found that both types of interactions result in significant relative motions and can be either attractive or repulsive depending on the configuration of the spheres.

Wave scattering by a horizontal circular cylinder in a two-layer fluid with an ice-cover

In a two-layer fluid wherein the upper layer is of finite depth and bounded above by a thin but uniform layer of ice-cover modelled as a thin elastic sheet and the lower layer is infinitely deep below the interface, time-harmonic waves with a given frequency can propagate with two different wavenumbers. The wave of smaller wavenumber propagates along the ice-cover while wave of higher wavenumber propagates along the interface. In this paper problems of wave scattering by a horizontal circular cylinder submerged in either the lower or in the upper layer due to obliquely as well as normally incident wave trains of both the wave numbers are investigated by using the method of multipole expansions. The effect of the presence of ice-cover on the various reflection and transmission coefficients due to incident waves at the ice-cover and the interface is depicted graphically in a number of figures.

Transmission and Polarization of Elastic Waves in Irregular Structures

Propagation of elastic waves in an infinite elastic medium containing a finite array of circular cylindrical cavities is considered. It is assumed that the cavities are equally spaced within each layer, but the cylinder radii, as well as the distance between the layers, may vary from layer to layer. Our objective is to analyze the effect of such an irregularity on scattering properties of the system. Using the method of multipole expansions we evaluate for the case of normal incidence the reflection and transmission matrices for a stack of layers, and find a range of frequencies for which polarization or complete reflection of an incident wave takes place. Presented results highlight the effect of variation in radii and distance between the layers on propagation of elastic waves in such structures.