Bipolar Cylindrical Coordinates Research Articles

An Elliptical Bipolar Cylindrical Coordinates Based Mixed Domain Near-Field Imaging Algorithm for Scanning 1-D Nonuniform Sparse MIMO Arrays

An Elliptical Bipolar Cylindrical Coordinates Based Mixed Domain Near-Field Imaging Algorithm for Scanning 1-D Nonuniform Sparse MIMO Arrays

Thermocapillary motion of a solid cylinder near a liquid–gas interface

The motion of a solid, infinitely long cylinder perpendicular to a convective liquid–gas interface due to thermocapillarity is investigated via an analytical model. If the cylinder temperature differs from the bulk temperature, a temperature gradient exists along the liquid–gas interface. This results in surface tension gradients at the liquid–gas interface, causing fluid flow around the particle, which induces propulsion. For small particles and, thus, small Péclet and Reynolds numbers, the steady-state equations for temperature and flow fields are solved exactly using two-dimensional bipolar cylindrical coordinates. The velocity of the cylinder as a function of separation distance from the liquid–gas interface is determined for the case of a constant temperature or a constant heat flux on the surface of the cylinder. A larger temperature gradient at the liquid–gas interface in the latter system leads to a larger cylinder velocity and a higher propulsion efficiency. The thermocapillary effect results in larger force on a cylinder than forces arising from other self-propulsion mechanisms.

Stress and pressure fields around two wellbores in a poroelastic medium

The problem of two circular wellbores of different size in a poroelastic medium is considered in the present work. The constitutive behaviour of the poroelastic medium is assumed to comply with the classical Biot model for isotropic porous materials infiltrated by compressible fluid. The wellbores are assumed infinitely long and the fluid flow is taken stationary, thus making it possible to perform a plane strain analysis. Owing to the geometrical layout of the system, bipolar cylindrical coordinates have been adopted. Three different sets of BCs on the pressure field and on the fluid flux have been considered, founding the corresponding forms of the pressure field. Based on Helmholtz representation, a displacement potential has been introduced, and the corresponding stress field in the poroelastic medium has been assessed. However, such a solution does not satisfy the BCs at the edges of the wells. Then, an auxiliary stress function, which allows accomplishing the BCs, is introduced, leading to the complete solution of the problem. The cases of two coaxial wellbores (eccentric annulus), a single hole bored in a poroelastic half plane and two intersecting holes have been considered also. The proposed approach allows evaluating the pore pressure and the stress and strain fields in the system varying the amplitude of the wells and the physical parameters of the porous material. In particular, the evaluation of the peak values of the stress components around the circular boreholes plays a key role in a variety of engineering contexts, with particular reference to the stability analysis of wellbores and tunnels and failure of vascular vessels in biological tissues.

Transport Phenomena in Eccentric Cylindrical Coordinates

Studies in transport phenomena have been limited to a select few coordinate systems. Specifically, Cartesian, cylindrical, spherical, Dijksman toroidal, and bipolar cylindrical coordinates have been the primary focus of transport work. The lack of diverse coordinate systems, for which the equations of change have been worked out, limits the diversity of transport phenomena problem solutions. Here, we introduce eccentric cylindrical coordinates and develop the corresponding equations of change (continuity, motion, and energy). This new coordinate system is unique, distinct from bipolar cylindrical coordinates, and does not contain cylindrical coordinates as a special case. We find eccentric cylindrical coordinates to be more intuitive for solving transport problems than bipolar cylindrical coordinates. Specific applications are given, in the form of novel exact solutions, for problems important to chemical engineers, in momentum, heat and mass transfer. We complete our analysis of eccentric cylindrical coordinates by using the new equations to solve one momentum, one energy, and one mass transport problem exactly. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3563–3581, 2017

Modelling Extraction in Microchannels with Stratified Flow: Channel Geometry, Flow Configuration and Marangoni Stresses

In this work, we present a hierarchy of mathematical models for extraction in microchannels with two-phase stratified flow. A flat stable inter-fluid interface is considered. We present three models for rectangular channels, of varying degrees of complexity: 2D, 1D Laminar and 1D Plug models. The predictions of these models are compared with each other and with experiments reported in the literature. Conditions under which a simpler 1D model can replace the complex 2D model are identified. Next, a detailed model for channels with a circular cross-section is developed using bipolar cylindrical coordinates. We also show how a much simpler description can be obtained by applying the 1D rectangular channel models to a circular channel. The relative performance of co-current and counter-current flow configurations is analysed next, using the 1D Plug flow model. Finally, the model is extended to account for Marangoni stresses that result from the variation of interfacial tension with solute concentration. The effect of Marangoni stresses on extraction in stable stratified flow is found to be insignificant. This is because the flow induced by Marangoni stresses is too weak to control the pressure-driven primary flow.

Full-wave finite-difference time-domain analysis of the invisibility cloak mapped to a line segment with isotropic complementary media

A dispersive full-wave finite-difference time-domain model is used to study the performance of point mapped and line-segment mapped complementary invisibility cloaking devices. We have used the permittivity and the permeability tensors for conventional elliptic and bipolar cylindrical invisibility cloaks obtained from an effective medium approach in general relativity. In the case of a line-segment mapped cloak we also employ the mapping of the σ-axis in bipolar cylindrical coordinates. In these cloaks, we employ the complementary media both horizontally and vertically. Cloaks with horizontally or vertically arranged complementary media mapped to a point show good performance of cloaking in any case. On the other hand, cloaks with horizontally arranged complementary media mapped to a line-segment, do not show cloaking performance. However, for cloaks with vertically arranged complementary media mapped to a line-segment, cloaking works very well in any cases. These results show improved cloaking performance over the conventional cloaks with perfect electrical conductor mapped to a line-segment. On the other hand, realistic cloaking materials with loss still show cloaking but attenuated backscattering waves exist.

Viscous dissipation in plastic pipe extrusion

We study the temperature distribution of a power-law fluid in a pressure-driven axial flow between eccentric cylinders in bipolar cylindrical coordinates. We begin our analysis by writing the equation of energy in bipolar cylindrical coordinates. We then obtain a dimensionless algebraic analytic solution for temperature profiles under a steady, laminar, incompressible, and fully developed flow for an adiabatic core and an isothermal outer cylinder (Eq. (59)). We find that the dimensionless temperature profile depends upon the radius ratio of the inner to outer cylinders, the eccentricity, the angular position, and the power-law exponent n. The temperature is a strong function of the gap between the cylinders. Finally, dimensionless maximum temperatures are plotted to help pipe manufacturing engineers prevent excessive heating during production. POLYM. ENG. SCI., 53:2205–2218, 2013. © 2013 Society of Plastics Engineers

Transient Conduction From Parallel Isothermal Cylinders

The transient heat conduction from two parallel isothermal cylinders is studied using the naturally fit bipolar cylindrical coordinates system. The energy equation is expanded in a Fourier series using appropriate basis functions to eliminate one of the physical coordinates. The resulting modes of the expansion are solved using a finite difference scheme. It is shown that, as is the case with a single isothermal cylinder in an infinite medium, steady states for two isothermal cylinders are not possible and heat transfer changes indefinitely with time.

Viscous dissipation of a power law fluid in axial flow between isothermal eccentric cylinders

We study the temperature distribution of a power law fluid in a pressure-driven axial flow between isothermal eccentric cylinders in bipolar cylindrical coordinates. We begin our analysis by writing the equation of energy in bipolar cylindrical coordinates. We then obtain a dimensionless algebraic analytic solution for temperature profiles under a steady, laminar, incompressible and fully developed flow [Eq. (64)]. We find that the dimensionless temperature profile depends upon the radius ratio of the inner to outer cylinders, the eccentricity, the angular position, and the power law exponent n. The temperature is a strong function of the gap between the cylinders. The temperature profiles are flat in the middle of the gap and then, near the wall, suddenly drop to the wall temperature.

Assumed periodicity and dynamic shear stress transduction in rheometry

In viscoelastic property measurements, material is subjected to time unsteady deformations using a rheometer. In step shear strain experiments, for example, the shear strain suddenly jumps to a steady value. In this paper, we develop a method to study the dynamic response of a shear stress transducer in a sliding plate rheometer for any time unsteady rheological test. This general method is developed by first considering a special case of step shear strain for a fluid sample with material ingress in the annular transducer gap. Both the fluid sample and its ingress obey the generalized Maxwell model. Our main mathematical trick is assumed periodicity where a single step in shear strain is treated as the first step in a reciprocating square wave. After solving the problem in the frequency domain, we recover our single step by taking the limit as the period goes to infinity. The transducer eccentricity following step shear is then determined analytically by solving the force balance on the transducer active face using bipolar cylindrical coordinates. Dimensionless graphs are provided for estimating transducer dynamic response in step shear strain. Finally, a worked example illustrates the application of these results.

A numerical study of momentum and forced convection heat transfer around two tandem circular cylinders at low Reynolds numbers. Part I: Momentum transfer

This work presents a computational study of the steady, axisymmetric, viscous flow around two circular cylinders in tandem. The vorticity–stream function formulation of the Navier–Stokes equations was chosen. Numerical solutions have been obtained in bipolar cylindrical coordinates. A compact, high-order, finite difference method was used to discretize the model equations. Iterative algorithms were tested to solve the discrete equations. Different cylinders spacing and sizes were considered for the upstream cylinder Reynolds number up to 30. Vorticity and pressure distributions on the cylinders surfaces and drag coefficients are presented and compared with those calculated for an isolated cylinder.

A numerical study of momentum and forced convection heat transfer around two tandem circular cylinders at low Reynolds numbers. Part II: Forced convection heat transfer

This work presents a computational study of the forced convection heat transfer around two circular cylinders in tandem. Axisymmetric, steady, laminar flow around the cylinders was assumed. The temperature inside the cylinders is considered spatially uniform but not constant in time. Numerical solutions have been obtained in bipolar cylindrical coordinates. The finite difference method was used to discretize the equations of the mathematical model. The influence of the model parameters on the heat transfer rate was analysed for the upstream cylinder Reynolds number, Re, varying from 1 to 30 and fluid phase Prandtl number equal to 0.1, 1, 10 and 100.

AXIAL FLOW BETWEEN ECCENTRIC CYLINDERS

Plastic pipe production must use annular dies with eccentricity to compensate for gravity flow in the cooling chamber. Here an algebraic analytic solution is obtained in bipolar cylindrical coordinates for Newtonian and power-law liquids. Dimensionless average velocity profiles around eccentric dies are provided to help plastic pipe engineers determine the needed eccentricity. In curved hose manufacture, the die centerpiece is made eccentric to cause curvature. Dimensionless radii of curvature are plotted to help curved hose manufacturers select proper die eccentricity.

Hybrid Approach to Uniform Flow Past Two Cylinders

An approximate approach that transforms an analytical solution for uniform flow past two circular cylinders into a solution for uniform flow past two cylinders of arbitrary shape is proposed in this study. This approximate approach is a combination of the analytical solution, which is obtained in bipolar cylindrical coordinates, and the numerical pseudoconformal transformation. The method applied in this study is not only simpler than conventional numerical methods, but also considers the Kutta condition. The numerical transformation is examined by the Joukowski transformation. The streamlines of uniform flow past a symmetric Joukowski airfoil with an ellipse in front of it shows good agreement with the Hele-Shaw cell experiment. The lift force on the airfoil is also compared with the lift force of a single airfoil.

Unipolar wire-to-plane corona-a definitive computation

It is shown that the unipolar wire-to-plane space charge problem can be calculated for wires of small radius, using bipolar cylindrical coordinates and an iterative technique. Proper attention must be paid to the dilute charge density at large distances and to careful handling of the transport equation. It is found that the voltage falls off asymptotically, as predicted by Felici, that the experimental results are reproduced, and that the charge density varies strongly around the wire, both when heavy injection occurs and for thicker wires. This calculation should serve as a benchmark for the more approximate methods needed to handle more complicated transmission line geometries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>