Bipolar Coordinate System Research Articles

Analytical solution to some boundary value problems of elasticity in bipolar coordinates

Analytical solutions of two-dimensional statics problems of elasticity are presented in bipolar coordinates for homogeneous isotropic bodies bounded by the coordinate lines of a bipolar coordinate system. In particular, various boundary problems for an eccentric circular ring, a half-plane with circular holes, and others are considered. The equilibrium equations and Hooke’s law are expressed using bipolar coordinates. This paper does not address the static equilibrium requirement of the external load at each circular boundary of the study area. This requirement, which significantly limits the range of problems that can be solved, typically appears in papers dealing with the aforementioned problems. In addition, the proposed method for obtaining an exact (analytical) solution is much simpler compared to the traditional approach. The exact solutions are derived using the method of separation of variables. By utilizing MATLAB software, the numerical results of some boundary value problems for an eccentric semi-ring are obtained, and the corresponding diagrams are presented.

Generalized Reynolds equation for microscale lubrication between eccentric circular cylinders based on kinetic theory

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi,Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Parabolic Cylindrical Description of Velocity and Acceleration Tensor in Bipolar Coordinate Using Tensor Analysis

Instantaneous velocity and acceleration are often studied and expressed in Cartesian, Circular Cylindrical, and Spherical Coordinate systems but it is a well-known fact that some bodies cannot be perfectly described in these coordinate systems, so they required some other curvilinear systems such as oblate spheroidal, parabolic cylindrical, elliptic cylindrical bipolar, and others. It is of important interest in theoretical physics to establish equations of motion in Bipolar Coordinate system, which is essentially suitable to accurately describe the motion of bipolar bodies such as hyperbolas, ellipses and other curves in the universe. In this work, we derived a new expression for the instantaneous velocity and acceleration vector of bodies and test particles in the bipolar coordinate system using tensor analysis. The Laplacian of a scalar field in this bipolar coordinate system was also derived which has applications in mechanics.

The Stokes thermocapillary motion of a spherical droplet in the presence of an interface

Using a spherical bipolar coordinate system, an exact analytical solution is obtained for the thermocapillary motion of a spherical droplet settling in the presence of a plane interface formed by the contact of two immiscible viscous fluids. A uniform temperature gradient is applied to the system in a direction perpendicular to the plane interface. The appropriate field equations of energy and momentum are solved in the quasi-steady limit under the conditions of small Péclet and Reynolds numbers. In this study, we also assumed that the capillary numbers at the interface of the droplet or at the separating plane surface are respectively small to maintain the spherical shape of the droplet and that the flat shape of the separating surface is permanent during the motion. The novelty of these three fluid phases problem is to find the thermocapillary velocity of the droplet and study the effect of the separating surface and the thermal conductivities of the system on the thermocapillary velocity. It is found that the interaction between the droplet and the separating surface can be strong in the case when the droplet is almost in contact with the separating surface. Several limiting cases are discussed and compared with the relevant cases available in the literature. The motivation of the study is its potential applications in many branches of chemical, biomedical, and environmental engineering, such as the locomotion of microswimmers near an interface.

On Derivations of Basic Governing Equations for Solid Bodies in Bipolar Cylindrical Coordinate System

On Derivations of Basic Governing Equations for Solid Bodies in Bipolar Cylindrical Coordinate System

Particle-surface interactions in a uniform electric field.

The electrostatic force on a spherical particle near a planar surface is calculated for the cases of a uniform electric field applied in either normal or tangential direction to the surface. The particle and suspending media are assumed to be weakly conducting, so that that the leaky dielectric model applies. The Laplace equationfor the electric potential is solved in bipolar coordinate system and the potential is obtained in terms of a series expansion of Legendre polynomials. The force on the particle is calculated using the Maxwell tensor. We find that in the case of normal electric field, which corresponds to a particle near an electrode, the force is always attractive but at a given separation it varies nontrivially with particle-suspending medium conductivity ratio; the force on a particle that is more conducting than the suspending medium is much larger compared to the force on a particle less conducing than the suspending medium. In the case of tangential electric field, which corresponds to a particle near an insulating boundary, the force is always repulsive.

Numerical quasi-three dimensional modeling of stratified oil-water flow in horizontal circular pipe

The stratified oil-water two-phase flow is commonly encountered in offshore oil production systems, and is considered to be among the fundamental flow configuration in two-phase systems. A numerical quasi-three dimensional model for the calculations of the axial pressure gradient, interface height, water holdup, and the flow field in a fully developed oil-water two-phase stratified pipe flow has been developed. The oil-water curved interface configuration is predicted by using minimum energy model. The model solves the governing equations of the steady axial momentum and mass conservation equations together with a low Reynolds number k～ε model of turbulence for the eddy viscosity, performed in the bipolar coordinate system, for convenient describing the curved interface and mapping of the physical domain. The predicted dynamic parameters, such as interface height of close to the pipe wall and in the middle of the pipe cross-section, water holdup, pressure gradient and flow filed are compared with the experiments. At the same time, this model is utilized to study the variation of these dynamic parameters with increasing superficial oil and water velocities. In addition, the numerical model with concave oil-water interface gives more accurate results than a flat interface. It indicates that the interfacial configuration effects are significant.

Simplified model for static stress analysis of straight eccentric lug subjected to cosine distribution pressure

A simplified analytical method is presented for predicting static stress distribution of the straight eccentric lug subjected to cosine distribution pressure. The method of analysis is only suitable for the situation where the predicted peak stress in the straight eccentric lug remains at or below the yield stress of the steel material used for it. In this method, the Castigliano’s theorem is used to determine the internal force distribution of the straight eccentric lug, and the bipolar coordinate system is introduced to analyze the stress state of the straight eccentric lug under bending moment. By comparing with the FEM’s results, it is proved that the present method can give a good precision in predicting the stress distribution of the straight eccentric lug subjected to the cosine distribution pressure. And the peak circumferential tension stress of the straight eccentric lug is observed at the edge of lug hole above the pin. Moreover, the effects of inner radius/outer radius ratio, distance between the two circle centers, and lug thickness to the peak tension stress are investigated. Based on contrastive analysis, it is concluded that the peak tension stress can be reduced either by increasing the inner radius/outer radius ratio and lug thickness, or by decreasing the distance between the two circle centers.

Flow Past a Pair of Separated Porous Spheres using Brinkman model

In this work, we obtained an analytical solution to the problem of viscous fluid flow past a pair of porous separated spheres. Stokesian approximation of the Navier-Stokes equation for the viscous fluid governs the flow in the region outside the two spheres, whereas Brinkman's model describes the flow in the porous region (within the spheres). Since the bipolar coordinate system is the most convenient system to represent separated spheres' geometry, we formulated this problem in the bipolar coordinate system. We then eliminated the pressure term from the equations governing the flow in the region outside the spheres, and they got reduced to a separable equation in terms of the stream function. Further, the flow governing equations inside the porous spheres gave rise to the Helmholtz equation. As the Helmholtz equation is not separable in the bipolar system, we used the spherical coordinate system to describe the fluid flow within each separated spheres and solved the resulting problem in the spherical coordinate system. Because of this, the flow variables on either side of the interface (spheres) are in different coordinate systems. To match the values of the field variables at the boundary, we used the transformation equations between the bipolar and spherical systems and transformed all the variables into the bipolar coordinate system. We then solved the governing equations with appropriate boundary conditions for the arbitrary constants and derived expressions for the stream and pressure functions. We plotted the respective functions for various values of the mathematical model's parameters to understand the flow pattern and pressure distribution in the flow domain and noted our observations. This study revealed an intuiting insight that the pressure distribution inside the porous spheres is independent of the non-dimensional parameter related to the medium's permeability and the fluid's viscosity.

On a Study of Flow Past Non-Newtonian Fluid Bubbles

In the current work, we aim at finding an analytical solution to the problem of fluid flow past a pair of separated non-Newtonian fluid bubbles. These bubbles are assumed to be spherical and non-permeable with the non-Newtonian fluid, viz. the couple stress fluid filling their interior. Further, the bubbles are presumed to be static in the flow domain, where a Newtonian fluid streams past these bubbles with a constant velocity (U) along the negative x-direction. We developed a mathematical model in the bipolar coordinate system for the fluid flow outside the bubbles and the spherical coordinate system inside the bubbles to derive a separable solution for their respective governing equations. Furthermore, to evaluate the model's applicabilities on the industrial front, the data on some widely used industrial fluids are given as inputs to the model, such as density, the viscosity of air or water for the fluid flow model developed for the region outside the fluid bubbles and the data on Cyclopentane or DIDP (non-Newtonian) for that within the bubbles. Some interesting findings are: the pressure in the outer region of the bubbles is higher when filled with low viscous industrial fluid, Cyclopentane, than a high viscous fluid, DIDP. Furthermore, an increase in the viscosity of Cyclopentane did not alter the pressure distribution in the region outside the bubbles. However, there is a considerable effect on this pressure in the case of DIDP bubbles.

Exact solution for the slow motion of a spherical particle in the presence of an interface with slip regime

An analytical and numerical study for the creeping flow caused by a solid spherical particle with a slip-flow surface is considered in the presence of a fluid–fluid plane interface. The particle rotating about or translating along an axis perpendicular to the interface. The motion is investigated in the limit of low capillary number where in this situation the interface is of negligible deformation. Using a bipolar coordinate system, the stream functions are constructed for both fluid phases as Reynolds number tends to zero. The novelty of this work is allowing the slip on the surface of the particle. The matching boundary conditions at the plane interface and the slip boundary condition on the particle’s surface are applied to the truncated solutions to specify the unknown coefficients. A comparison is made between the results of the analytical solution and the results obtained from a boundary collocation method. The torque and drag force exerted on the particle are calculated using both techniques, which are found in perfect agreement. In addition to compression with collocation techniques, we also studied the predicted changes in the drag force and torque due to the presence of the plane interface and the slippage at the surface of the particle. Our results of the drag force and torque are compared with the available data in the literature for the special cases. The work is motivated by its possible application as an analytical tool in the study of locomotion of microswimmers near an interface such as synthetic swimmers and microorganisms.

ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

The influence of surface effects in the problem of theory of elasticity for a circular hole in a half-plane

Surface effects are important for modeling structures, such as nanofilms, nanoporous materials, and other nanoscale constructions. In the current study, we consider the problem of the theory of elasticity - the problem of a half-plane containing a circular hole, stretched by constant stresses applied at infinity, and take into account surface effects such as surface elasticity and surface stresses. The problem solution has been obtained by expanding the Fourier series with the variables written in the bipolar coordinate system (which simplifies the problem solution because one of the coordinates becomes a constant on the hole contour), where the stress components are expressed through a bi-harmonic stress function. The parametric coefficients involved in the solution, namely in the Fourier series, are determined in order to satisfy the boundary conditions on the hole contour. To solve the problem, in addition to the equations of the theory of elasticity, the equations of surface elasticity were used, in particular, by applying the generalized Young-Laplace’s law and the Shuttleworth’s law; the surface stress on the hole contour has been calculated directly. Using recurrence relations for the stress components at the boundary, stress concentration values have been obtained. The resulting expressions can be considered as a generalized solution of the problem in case of the classical elasticity. The stress concentrations are compared for the cases with and without taking into account surface effects at various points on the hole contour. The contribution caused by the surface effects depending on the relative distance between the hole and the half-plane boundary is studied. It is shown that despite a quite simple geometry, owing to the fairly small distance between the hole and the half-plane boundary, the stress concentration with and without taking into account the surface stress are significantly different from each other, due to the significant contribution of surface effects.

Prediction of casing wear depth and residual strength in highly-deviated wells based on modeling and simulation.

Casing wear is a serious problem in highly-deviated wells because serious wear will lead to casing deformation, drilling tool sticking and failure of subsequent operations. The purpose of this paper is to predict casing wear depth and evaluate its effect on casing strength degradation in highly-deviated well drilling operation. Special attention has been given to the algorithm to achieve the prediction and evaluation. The effect of tool joint on contact force distribution is considered in contact force model. Then a wear depth prediction model and its solution method are proposed based on crescent-shaped wear morphology and wear-efficiency model. Besides, strength degradation of worn casing is analyzed in bipolar coordinate system and the model is verified by finite element method. Therefore, the technology of casing wear prediction and residual strength evaluation is completed systematically. Then, to apply casing wear prediction and residual strength evaluation technologies to an actual highly-deviated well, casing wear experiment and friction coefficient experiment are carried out to obtain wear coefficient and friction coefficient. Finally, based on the established models as well as experimental results, the distribution of casing wear is predicted and residual strength is evaluated. The method presented in this paper will contribute greatly to casing wear prediction and evaluation in highly-deviated wells.

Influence of Surface Effects in the Problems of the Theory of Elasticity for Domains Bounded by Non-Concentric Circles

The article deals with some problems of the theory of elasticity for domains bounded by non-concentric circles by taking into account surface effects such as surface elasticity and surface stresses. The solutions are obtained by expanding variables written in a bipolar coordinate system into Fourier series. The values of surface stresses and stress concentrations that are of our interest are obtained using recurrence relations. The contribution made by surface effects is considered.

Analytical solution of higher order modes of a dielectric-lined eccentric coaxial cable

AbstractThis study provides an analytic method for the calculation of the cutoff frequencies and waveguide modes of a partially filled eccentric coaxial cable. The method is based on the expressions of the involved electromagnetic fields in bipolar coordinate systems and the validity range of the solution is discussed. It is shown how the waveguide geometry and dielectric parameters may be selected to engineer the lined waveguide's spectral response. Numerical results are included which show good agreement with the corresponding results from full-wave simulations by commercial software.

Semi analytical cum numerical analysis of two-dimensional elasticity problem of plate with eccentric hole in bipolar coordinate system

A semi analytical cum numerical method for analyzing 2D elasticity problem of plate with eccentric hole in bipolar coordinate systems is given. Complete formulation of the plate problem is derived in bipolar coordinate system. A novel numerical integration technique is adopted to obtain the numerical results. Few parameters are identified that adversely affects the stresses and displacements of isotropic and FG plate with eccentric hole. Verification of the mathematical models with numerical results such as finite element models (COMSOL Multiphysics) and wide variety of applicability of the developed models for hypothetical/real field problems is seen from the present work.

The singular hydrodynamic interactions between two spheres in Stokes flow

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

A New Analytical Method for Calculating the Cutoff Frequencies of an Eccentrically Dielectric-Loaded Circular Waveguide

This letter provides an analytical method to calculate the cutoff frequencies and waveguide modes of conducting circular waveguide eccentrically loaded with a dielectric material based on bipolar coordinate system (BCS). A circular waveguide partially filled with an eccentric dielectric cylinder supports hybrid electric (HE) and hybrid magnetic (EH) modes with nonzero longitudinal electric and magnetic fields, respectively. The cutoff frequencies of the HE and EH modes have been calculated and compared to benchmark results. An analytical expression for the electromagnetic fields has been also proposed and the validity range of the solution has been discussed. An excellent agreement between the calculated cutoff frequencies with those obtained by more rigorous numerical methods has been observed. Our analytical method provides better physical intuition and is fast, accurate, and easy to implement.

Comments on “Elasto-plastic solution for shallow tunnel in semi-infinite space”

This paper considers the approach proposed by Zou et al. for determining the plastic zone around a shallow circular tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. The paper aims to analyze the correctness of the analytic expressions for elastic and plastic stress, as well as the reasonableness of the procedure building for solving the elasto-plastic interface. Finally, a promising way of solving this problem more exactly is presented.