Bipolar Coordinate System Research Articles

Burst pressure analysis of pipes with geometric eccentricity and small thickness-to-diameter ratio

Abstract Two analytical models were proposed to predict the burst pressure of pipes characterized by a small thickness-to-diameter ratio and geometric eccentricity. Based on a complex stress function in a bipolar coordinate system, an analytical solution was derived for the stresses of an internally pressurized eccentric pipe, and two burst pressure prediction formulae for restrained and end-capped pipe models were obtained according the von Mises stress, which took the ultimate tensile strength (UTS) of the inner wall as the pipe bursting criterion. Next these formulae were validated by comparing the results obtained with the results of existing elastic–plastic formulae and a nonlinear finite element method. Finally, the effect of eccentricity on the burst pressure was evaluated and simplified as an empirical formula. Our research shows that the burst pressure of restrained and end-capped pipes with geometric eccentricity are nearly equivalent, and the burst pressure of an eccentric pipe is relative to that of a non-eccentric pipe. The theoretical and empirical formulae presented in this paper have broad application in the oil and gas industry to predict the burst pressure of pipes.

The Tangential Force Distribution on Inner Cylinder of Power Law Fluid Flowing in Eccentric Annuli with the Inner Cylinder Reciprocating Axially

Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in bipolar coordinate system, the calculation formulae of tangential force were established, and the relevant numerical calculation method was given. Taking the aqueous solution of partially hydrolyzed polyacrylamides (HPAM) for examples, the tangential forces were calculated by using the formulae and numerical calculation method mentioned above; the curves of the tangential force on the wall of the inner cylinder of HPAM aqueous solution were plotted; and the effects on the tangential force of the flow behavior index of the power law fluid, the stroke and the stroke frequency of the inner cylinder were analyzed.

Numerical modeling for stratified gas–liquid flow and heat transfer in pipeline

A numerical three-dimensional non-isothermal hydrodynamic and thermal modeling based on the unidirectional flow analysis of stratified flow in the circular cross section pipe is developed and tested by experimental data. The model could solve the steady axial momentum equation and heat transfer equation with a low Reynolds number k∼ε model of turbulence for the eddy viscosity. Due to irregular geometry of stratified pipe flow, the equations of the problem are based on a bipolar coordinate system. Grid refinement near the interface and pipe wall is respectively used for an accurate solution near the boundaries. The model is capable of determining pressure drop and liquid height. In addition, wall and interface shear stress, flow field and temperature field for both phases could be predicted successfully. The predicted data of pressure drop, liquid holdup and velocity profile by the model fit well with some available experiment data and one-dimensional model. And the energy equation is solved, and temperature distribution is gained over the flow and compared well with experimental results. In conclusion, the model could have a practical application for estimation of hydrodynamics and thermodynamics in a pipeline.

Electric and magnetic fields of two infinitely long parallel cylindrical conductors carrying a DC current

This paper calculates the electric and magnetic fields and the Poynting vector around two infinitely long parallel cylindrical conductors, carrying a DC current. Also the charges on the surface of the wire are calculated, and their distribution is visualized. The wire is assumed to be perfectly electrically conducting. Furthermore, the Hall effect is ignored. In the literature [S.J. Orfanidis, Electromagnetic waves and antennas, 2008], the problem of determining the electric field is usually tackled using an equivalent model consisting of two line charge densities, producing the same electric field. In this work, the Laplace equation is rigorously solved. The authors found no work explaining the solution of the Laplace equation with boundary conditions for this problem and hence thought it was useful to dedicate a paper to this topic. The method of separation of variables is employed and a bipolar coordinate system is used. After solving the appropriate Sturm-Liouville problems, the scalar potential is obtained. Taking the gradient yields the electric field.

Numerical Study of Three-Dimensional Mixed Convection in an Eccentric Annulus

Mixed convection of air with a Prandtl number of 0.7 in eccentric horizontal annuli is numerically investigated. The inner cylinder is at a higher temperature, while the inner and outer cylinders can be rotated either clockwise or counterclockwise with a constant angular velocity. The effects of various parameters, such as the eccentricity of the annulus and the Rayleigh number, are studied using a three-dimensional model. The Navier–Stokes equations, along with the energy equation, are solved in an orthogonal structured grid of a bipolar coordinate system employing a finite volume method based on the SIMPLE algorithm. Here, both steady and unsteady results are presented in two separate codes. Three-dimensional considerations lead to a complicated flow pattern, whereas Taylor vortices have a major role on heat and flow characteristics. It is shown that the axial component of velocity in some configurations has a considerable value, therefore, flow pattern and heat transfer characteristics change in comparison with a two-dimensional approach. Eccentricity leads to a different flow pattern in the upper and lower parts of the annulus. Mean and local heat transfer results are obtained. The physics of the flow underlying the observed heat transfer behavior is revealed by the velocity vectors and the isotherm plots of the numerical solutions. Results show that flow and heat transfer are strongly influenced by the eccentricity of the inner cylinder and Raleigh number, and that three-dimensionality consideration is inevitable.

A singular perturbation solution of viscous incompressible fluid flow between two eccentric rotating cylinders

The flow inside two concentric cylinders is one dimensional and an exact solution for quantities is easily found. However, when the cylinders axes are displaced by a small distance, two dimensional effects become obvious. In this research, the equations governing an incompressible viscous flow between two rotating cylinders are considered in polar coordinates that can be simplified by introducing vorticity and stream functions. By taking the curl of the vector form of the momentum equation, the pressure term is omitted. Because of the boundary conditions being in terms of perturbation parameter, a modified bi-polar coordinate system is introduced. This transforms the two eccentric cylinders into two concentric ones. By expanding the quantities in terms of the perturbation parameter up to second-order accuracy and by substitution into the vorticity-stream function, sets of differential equations are obtained to be solved for these functions. At the end, the closed-form velocity components are determined.

Stability Parameter <i>H</i> of the Flow of Power-Law Fluid in Annulus with the Inner Cylinder Executing a Planetary Motion

The governing equations and mathematical model of the stability parameter H are given in bipolar coordinate system when power law fluid flows in annulus with the inner cylinder executing a planetary motion in this paper. The stability parameter H of power law fluid in both wide clearance and thin clearance in annulus with the inner cylinder executing a planetary motion is numerically calculated by finite difference method, the influences of the revolution velocity, rotation velocity, the eccentric distance and the pressure gradient on the stability parameter are analyzed. The results indicate that the eccentricity and the pressure gradient are main influence factors on it. The critical value of stability parameter for the flow of power-law fluid with the inner cylinder executing a planetary motion is 404 by means of experiment of CMC aqueous solution in annulus with the inner cylinder executing a planetary motion.

Adsorption and removal dynamics of polymeric micellar nanocarriers loaded with a therapeutic agent on silica surfaces

Obtaining a better understanding of the adsorption behavior of polymeric nanomedicines is important to properly assess their distribution and fate and to develop successful strategies for minimizing their distribution in the environment. In this study, the adsorption and removal dynamics of a model polymeric nanomedicine of systematically varied sizes – levofloxacin loaded poly(ethyleneglycol-b-e-caprolactone) – on and from silica surfaces are investigated using a quartz crystal microbalance with dissipation (QCM-D). For all the sizes of the nanomedicine investigated (90 nm, 110 nm, 149 nm, 174 nm, 207 nm, and 305 nm), the adsorbate mass on the silica followed a roughly exponential trend with respect to time during the exposure stage, suggesting first-order adsorption kinetics. The adsorption rate constant decreased with the increasing particle size. Furthermore, we have developed a modified Leveque equation to describe the mass transport and adsorption behavior of the nanoparticulate dispersion inside the QCM-D chamber. The derivation involved the application of a bipolar coordinate system to the continuity, Navier–Stokes, and convective–diffusion equations. The adsorption rate constants calculated using the derived equation were found to match the experimental results well. The derived equation is important not only in the field of environmental science but also in many other fields as QCM-D is heavily used to characterize the dynamic adsorption behavior of various types of materials including polymers, proteins, and nanoparticles.

Velocity Distribution of Viscoelastic Fluid Axial Flowing in Eccentric Annulus Based on the Upper-Convected Maxwell Model

After polymer flooding has been put into effect in Daqing oilfield, eccentric wear of sucker rod and tubing has been so serious that it made the rods break. In order to analyze the cause of eccentric wear based on the upper-convected Maxwell model, the flow equation of viscoelastic fluid in eccentric annulus was established in cylinder coordinate system. Then the equation was transformed in bipolar coordinate system and dispersed by control-volume method. The velocity distribution was solved by ADI methods and example was calculated which provides theory basis for solving eccentric wear problems.

Theoretical investigation of a terahertz transmission line in bipolar coordinate system

It has been demonstrated that the dual-wire waveguide (DWW) can be used in terahertz (THz) frequency regime with many advantages. However, the existent research approaches for the DWW are based on the electrostatic theory. In this paper, making use of the bipolar coordinate system (BCS), a rigorous analytical theory of DWW is worked out, and some important physical and optical characters of DWW including the rotating behavior etc. are revealed, the equivalent impendence and the ohmic loss for the gold DWW are calculated. An eigenvalue problem is presented from the point of view of Mathematical-Physics for TE and TM modes. The obtained results will help get a deep-going understanding of DWW and explore its application in high frequency range including THz.

Stokes flow past a compound drop in a circular tube

Microfluidics could generate drops or bubbles with controllable size and frequency at this stage. However, analytical work on such problem is less reported in the literature. In this study, we study the motion of a compound drop, consisting of a fluid drop engulfed in a larger drop, confined in a circular tube. The analysis is based on the low Reynolds number Stokes flow theory. Interfaces are assumed to be spherical due to large surface tension. Stream functions in one bipolar and two cylindrical coordinate systems are developed in series form. Our new contribution is the transformation between cylindrical and bipolar coordinate systems. Flow patterns are mainly dependent on the relative motion and the size of the inner drop. Four types of flow patterns are identified. Drag force on the inner or outer drops is in proportion to the product of the drop radius and viscosity of the phase encapsulating the drop. Drag force on the inner or outer spheres is finally expressed as linear combinations of velocities of the three phases (i.e., the inner drop, the outer drop, and the continuous flow), respectively. Our results show that those coefficients of the linear combinations for the drag forces depend on several parameters: eccentricity of the compound drop, viscosity ratio of two neighboring phases, radius ratio of the inner drop to the outer drop, and the radius ratio of the outer drop to the tube. The two radius ratios have largest effects on the coefficients of the inner or outer drop, respectively. Stability of the compound drop in a circular tube is analyzed. It is found that though the compound drop cannot reach an absolutely steady state, it will enter a quasisteady state where the inner sphere is adjacent to the shell of the outer sphere in practice.

A Closed-Form Solution for the Polarizability of a Dielectric Double Half-Cylinder

Using the bipolar coordinate system, the electrostatic scattering problem of a dielectric double half-cylinder is solved. In particular, an expression for the polarizability of the half-cylinder is obtained in terms of two dilogarithm functions. Also the nonuniqueness of the solution is proven in a certain range of negative values of the permittivity.

Electrostatic Interaction of Spherical Microparticles in Dusty Plasmas

AbstractElectrostatic interaction of two spherical microparticles in a plasma is studied. An investigation of the interaction between a point‐like charge and a conducting spherical body in a plasma shows that plasma screening results in decrease a potential barrier as the point‐like charge approaches the likely charged microparticle, the decrease being more pronounced in case that the microparticle radius is comparable with the Debye screening length. The interaction of two conducting spherical microparticles is considered in the bipolar or bispherical coordinate system for cases of constant charges and constant surface potentials of the microparticles. The latter case firstly considered by us is more suitable to the physics of the electrostatic interaction of microparticles in a plasma where their electrostatic surface potentials are defined by the floating potential of the plasma. The interaction potentials are shown to highly differ in these cases, the electrostatic energy being the interaction potential only in the case of the constant charges independent on the interparticle distance. In the case of the constant surface potentials a work of external sources to sustain surface potentials should be taken into account. By integration of the interaction force calculated using the Maxwell stress tensor, the interaction potential is also defined for the latter case. Approximated analytical expressions for the interaction potential, which are more accurate than the available in the literature, are obtained for both the constant charges and the constant surface potentials (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Numerical Investigation of Flow and Mixed Convection Inside a Vertical Eccentric Annulus

In this article, mixed convection heat transfer is numerically studied in a vertical eccentric annulus. For this purpose, full Navier-Stokes equations along with energy equations are solved in a 3-D orthogonal grid using a bipolar cylindrical coordinate system and finite-volume method based on the SIMPLE algorithm. To investigate the flow and heat transfer in the presence of gravity, the inner wall of the annulus is kept at a constant temperature while the outer wall is assumed to be insulated. The effect of different governing nondimensional parameters consisting of dimensionless eccentricity (E), radius ratio (N), and mixed convection parameter (Gr/Re) on the flow and temperature field are studied in detail. The numerical results obtained are in good agreement with the previous numerical works. Furthermore, the flow and heat transfer field in the presence of internal longitudinal fins on the inner wall of the annulus are studied compared with the finless case. The amount of augmented heat transfer due to internal fins is also discussed.

Investigation on the Flow Stability of Newtonian Fluid in Eccentric Annulus Via the Axial Reciprocation of the Inner Tube

The governing equation, together with the formulas of the averaged flux and the stability parameter for Newtonian fluid flowing in eccentric annulus via the axial reciprocation of the inner tube was established in the bipolar coordinate system. Numerical calculation was conducted for the measurement of water flow using the formulas above, of which the results agree well with the experimental data. It is shown that the flow instability can be induced by the increase of the stroke, the frequency, and the eccentricity of annulus, making the sucker rod more susceptible to partial abrasion. Some protecting methods, including adding centralizers, adjusting the stroke and frequency, are proposed to protect the sucker rod from the partial abrasion, and satisfactory results have been achieved in the oilfields.

Radiation characteristics of a coaxial waveguide with eccentric inner conductor for application in hyperthermia and microwave reflex therapy

Abstract. This paper examines the radiation characteristics of a contact emitter conceived for application in hyperthermia and microwave reflex therapy. It is important to analyse the distribution of power density in the near field area, as the radiator's therapeutic sphere of activity is localized here. The contact emitter is a coaxial radiator with an eccentric course of the inner conductor. According to Huygens principle, a theoretical view of the near field radiation characteristics is made by determining the equivalent current densities in the emitter aperture. It is shown that by an eccentric shift of the inner conductor, an almost isotropic near field radiation pattern and power density can be achieved. For this, the electromagnetic field in the emitter aperture is determined by using a Bipolar coordinate system. This calculation considers only the fundamental TEM mode of the contact emitter. Besides the theoretical results near and far fields are simulated using the programme system Ansoft HFSS.

Pressure distribution on the wall of the inner cylinder reciprocating axially to the unsteady flow of viscoelastic fluid in eccentric annulus

In this article, the governing equations for the unsteady flow of viscoelastic fluid in the eccentric annulus with the inner cylinder reciprocating axially and the expression of the pressure distribution on the wall of the inner cylinder of the annulus are established and derived, respectively, under the bipolar coordinate system. The equations and the expression are solved and calculated numerically using the finite difference method, respectively. The curves of the pressure distribution on the wall of the inner cylinder of the aqueous solution of Hydrolyzed Polyacrylamide (HPAM) are plotted and the influences of annular eccentricity, stroke, and stroke frequency on the pressure distribution are analyzed.

The exact solution of the time-harmonic wave equation for a linear velocity profile

SUMMARY We present the exact solution of the acoustic wave equation in the frequency domain in two space dimensions for a velocity that increases linearly with depth and a density that is a power law in depth. The geometric properties of the solution are described by the bipolar coordinate system. The asymptotic behaviour for large frequencies matches the result obtained by ray theory. The 3-D solution was already known.

Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

A finite-difference method for solving three-dimensional time-dependent incompressible Navier–Stokes equations in arbitrary curvilinear orthogonal coordinates is presented. The method is oriented on turbulent flow simulations and consists of a second-order central difference approximation in space and a third-order semi-implicit Runge–Kutta scheme for time advancement. Spatial discretization retains some important properties of the Navier–Stokes equations, including energy conservation by the nonlinear and pressure-gradient terms. Numerical tests cover Cartesian, cylindrical-polar, spherical, cylindrical elliptic and cylindrical bipolar coordinate systems. Both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method.

Exact asymptotic behaviour of fermion correlation functions in the massive Thirring model

We obtain an exact asymptotic expression for the two-point fermion correlation functions in the massive Thirring model (MTM) and show that, for $\beta^2=8\pi$, they reproduce the exactly known corresponding functions of the massless theory, explicitly confirming the irrelevance of the mass term at this point. This result is obtained by using the Coulomb gas representation of the fermionic MTM correlators in the bipolar coordinate system.