Cylindrical Coordinate System Research Articles

Finite-difference scheme and algorithm for solving a two-dimensional system of Maxwell’s equations for inductively coupled plasma

The paper presents a numerical scheme for solving a two-dimensional system of Maxwell equations for a cylindrical coordinate system.

Numerical Simulation of Hydrate Decomposition during the Drilling Process of the Hydrate Reservoir in the Northern South China Sea

The process of drilling in natural gas hydrate reservoirs in sea areas involves problems such as hydrate decomposition and wellbore instability. To study the response behaviors of a reservoir during the drilling process, a two-dimensional numerical model of drilling fluid invading a hydrate reservoir in a cylindrical coordinate system was established to simulate the processes of heat and mass transfer, gas–liquid two-phase flow, and hydrate formation and decomposition in the hydrate reservoir during the drilling process. Based on the hydrate reservoir at station W17, Shenhu area of the South China Sea, the physical property response of the hydrate reservoir under different drilling fluid temperatures and salinity values was studied. The simulation results showed that during the drilling process, the temperature and pressure of the reservoir respond rapidly in a large area, further promoting the hydrate decomposition in the reservoir around the wellbore and leading to secondary hydrate formation. Moreover, a high hydrate saturation zone appears near the decomposed hydrate area in the layer without free gas, which corresponds to the low water saturation and high salinity zone. The hydrate decomposition area in the layer with free gas is larger than that without free gas. The increase in the drilling fluid temperature significantly enhances the hydrate decomposition in both layers of the reservoir. The hydrate decomposition near the wellbore under the high drilling fluid temperature will cause a sharp increase in the pressure in the reservoir, leading to the flow of pore fluid into the wellbore. The increase in drilling fluid salinity has little effect on the range of the hydrate decomposition in the reservoir but significantly increases the salinity of the pore water in the layer with free gas. As the drilling fluid temperature increases, the possibility of the gas invasion from the reservoir into the wellbore will be greatly increased at the early stage.

DETERMINATION OF KINEMATIC CHARACTERISTICS OF METAL PLASTIC FLOW UNDER CONDITIONS OF AXIAL SYMMETRY OF THE DEFORMATION PROCESS

With axisymmetric deformation, the deformable body and the load have a common axis of symmetry. Such deformation occurs in numerous technological operations. About 70% of parts obtained by cold extrusion are deformed under conditions of axial symmetry. The plane problem of the theory of plasticity is reduced to a solution in the framework of a two-dimensional statement, when the motion of points in the section of a workpiece is analyzed. Each point can move only in the section plane, and its velocity can be decomposed into two mutually perpendicular directions along the coordinate axes. For plane deformation, it is assumed that the velocity in the direction of the third coordinate axis is equal to zero. The presence of axial symmetry allows us to confine ourselves to studying the behavior of points located on the plane of the meridional section of the workpiece. In this case, each point can move only in the section plane and its velocity can be decomposed into two orthogonal directions: along the axis and along the radius. The component of the velocity vector in the circumferential direction is equal to zero, so only four of the six independent components of the strain rate tensor remain. In this regard, axisymmetric problems of the theory of plasticity are of considerable interest from the point of view of solving applied problems. The article discusses the possibility of using mixed Euler and Lagrange coordinates to determine the components of the strain rate tensor in plastic deformation processes characterized by the axisymmetric nature of metal plastic flow. The vector field of displacements at each point in space reflects the transformation of the initial (non-deformed) configuration into the current one, and therefore determines the configuration of the deformed body in space in a certain frame of reference. The deformation process is first considered in a single Cartesian coordinate system, and then in a cylindrical coordinate system, the use of which is more appropriate for axisymmetric deformation. It is assumed that the functions of the Euler coordinates from the Lagrange coordinates are obtained by approximating the experimental data.

Алгоритм маршового розрахунку надзвукового обтікання ракети з відхиля-ємою носовою частиною

This paper presents a marching algorithm for the calculation of supersonic flow past a tilting-nose rocket. A feature of the algorithm is that the marching direction of supersonic flow calculation for the nose does not coincide with that for the main part of the rocket surface. Because of this, at first flow past the nose is calculated in a cylindrical coordinate system, the flow field parameters being stored at marching cross-sections. The start and the end of the parameter storage interval in the flow field are determined from the condition of the intersection of the bow shock wave with a plane in which an initial flow field is to be specified for the calculation of flow past the main part of the rocket surface. The flow field is interpolated in two stages, First, in the cylindrical coordinate system bound to the main part of the rocket surface in the initial data plane, the radial coordinates of the bow shock wave are determined at meridional sections. From the radial coordinates of points on the rocket surface and the bow shock wave, new computational grid node coordinates are determined at meridional sections in the cylindrical coordinate system of the main part. Using the new computational grid coordinates specified in the cylindrical coordinate system of the main part, old coordinates specified in the cylindrical coordinate system of the nose are determined with the use of expressions that relate the two coordinate systems to each other. The flow parameters at a point with the calculated coordinates are calculated using linear interpolation of the stored flow field parameters in the cylindrical coordinate system bound to the nose. The calculated flow field is used as initial data for the marching calculation of the main part of the rocket, The paper presents the results of calculation of the aerodynamic characteristics of a tilting-nose rocket in a supersonic flow at different values of the nose angle. The proposed algorithm may be used in a prompt calculation of the aerodynamic characteristics of rockets with tilting elements. In doing so, use may be made of a standard program of rocket flow calculation with an added block for the storage and interpolation of the flow field in a tilted cylindrical coordinate system with a shifted origin, which allows the marching direction to be changed.

Surface 3D Scanner Using Time of Flight Ranging Sensor with Cylindrical Coordinate System

<p class="TTPParagraphothers">3D scanner that uses image sensors requires the role of a computer that includes a data generator, data acquisition, and visual display. In a simply system, it can be designed the sensory system uses non-imagery sensor so the role of the data generator can be handled by the microcontroller. This research aims to make a simple 3D scanner using inexpensive non-imagery Time of Flight VL53L0X sensor and data processing can be processed directly by the microcontroller. The results of sensor distance measurements are processed on the microcontroller and desktop application. The distance and angle values are converted into Cartesian coordinate using cylindrical coordinate system. The scan results of the cubes, prisms and cylinder are similar with the reference object, but the results of the pyramid test at the top cannot be scanned properly due to the narrow surface. The laser beam from the emitter cannot bounce back to the collector properly makes distance reading is inaccurate and causes error in the point cloud conversion. The comparison error between the side of the scan results and the reference object is between 2.54-39.8%. The surface of objects with bright color has a smaller error than those with dark color. The comparison error of the height of the scan results with the reference object is between 5-32%. The angle of the emitter exclusion cone and the collector exclusion cone sensor affects the error at the side and height of the scan results.</p>

Relativistic self-focusing of cosh-Gaussian laser beams in a warm quantum plasma

This paper presents an investigation of the self-focusing of cosh-Gaussian laser beams in warm quantum plasma while taking into account the effects of relativistic nonlinearity. The non-linear differential equation governing the beam width parameter under the weakly relativistic nonlinearity in this study was derived from the cylindrical coordinate system. The equation has been solved numerically by using the fourth-order Runge–Kutta method. Numerical analysis has shown that the present model's parameters, such as the decentered parameter, laser intensity, electron density of plasma, and Fermi temperature of electrons, play a vital role in propagation characteristics. Furthermore, comparisons were made for the cosh-Gaussian laser self-focusing in warm quantum plasma, cold quantum plasma, and classical relativistic plasma. It has also been found that the cosh-Gaussian laser beam has caused better focusing in the warm quantum plasma in comparison with the Gaussian laser beam; a fact that is very important for some applications in high-power laser-dense plasma interactions.

Theory, strict formula derivation and algorithm development for the computation of a geodesic polygon area

The paper presents a coherent, original theory and an effective algorithm to calculate the surface area of a geodesic polygon, i.e. the area on an ellipsoid of revolution limited by geodesics, connecting pairs of subsequent vertices of known geographical coordinates. Using the cylindrical coordinate system and introducing an equatorial geodesic triangle, the novel strict recursive formulas were derived for the longitude on the ellipsoid and the distance along the geodesic and the area of a geodesic polygon. The area is determined by constant coefficients, not the values of finite series. The orthodromic properties of geodesics, which led to the detection of bifurcation lines composed of the double solutions of the inverse geodetic problem were analysed. The novel algorithm presented, integrating the problem with geodesic polygon area calculation, is based on the newly proposed parameterisation with two alternative working unknowns and a new formula to obtain the initial value of the geodesic C parameter. It allows solutions for all cases, including near-antipodal or near-vertices, and offers two solutions if bifurcation occurs. The algorithm presented was tested and verified using numerical examples from known publications. The numerical tests proved the accuracy of 1m2 of the computed geodesic area using standard IEEE 754 Double Floating Point PC arithmetic, for geodesic lengths up to ten thousand km.

Destabilizing resonances of precessing inertia-gravity waves.

Instabilities in stratified precessing fluid are investigated. We extend the study by Mahalov [Phys. Fluids A 5, 891 (1993)0899-821310.1063/1.858635] in the stably stratified Boussinesq framework, with an external Coriolis force (with rate Ω_{p}) altering the base flow through the distortion of the circular streamlines of the unperturbed axially stratified rotating columns (with constant vorticity 2Ω.) It is shown that the inviscid part of the modified velocity flow (0,Ωr,-2ɛΩrsinφ) and buoyancy with gradient N^{2}(-2ɛcosφ,2ɛsinφ,1) are an exact solution of Boussinesq-Euler equations. Here (r,φ,z) is a cylindrical coordinate system, with ɛ=Ω_{p}/Ω being the Poincaré number and N the Brunt-Väisälä frequency. The base flow is transformed into a Cartesian coordinate system, and the stability of a superimposed perturbation is studied in terms of Fourier (or Kelvin) modes. The resulting Floquet system for the Fourier modes has three parameters: ɛ, N=N/Ω, and μ, which is the angle between the wave vector k and the solid-body rotation axis in the limit ɛ=0. In this limit, there are inertia-gravity waves propagating with frequency ±ω and the resonant cases are those for which 2ω=nΩ, n being an integer. We perform an asymptotic analysis to leading order in ɛ and characterize the destabilizing resonant case of order n=1 (i.e., the subharmonic instability) which exists and for 0≤N<Ω/2. In this range, the subharmonic instability remains the strongest with a maximal growth rate σ_{m}=[ɛ(5sqrt[15]/8)sqrt[1-4N^{2}]/(4-N^{2})]. Stable stratification acts in such a way as to make the subharmonic instability less efficient, so as it disappears for N≥0.5Ω. The destabilizing resonant cases of order n=2,3,4,5 are investigated in detail by numerical computations. The effect of viscosity on these instabilities is briefly addressed assuming the diffusive coefficients (kinematic and thermal) are equal. Likewise, we briefly investigate the case where N^{2}<0 and show that the instability associated to the mode with k_{3}=0 is the strongest.

Inverse boundary problem in estimating heat transfer coefficient of a round pulsating bubbly jet: design of experiment

ABSTRACT An inverse algorithm is developed to estimate the transient convective coefficient distribution of a pulsating bubbly jet impinging on a cylindrical thermal mass. Cooling of the thermal mass is simulated by solving the two-dimensional transient heat equation in a cylindrical coordinate system using the finite difference method. The sum of squared differences between calculated and measured temperature data is the objective functional. As this is a nonlinear IHCP, the adjoint method is employed to derive the gradient components of the objective function, needed for the implementation of CGM. The procedure is carried out for six gas Reynolds numbers changing continuously through ten seconds of the jet impingement. The inverse scheme is validated using noiseless temperature data. The accuracy and convergence rate of the algorithm is then assessed by simulating the cooling process of thermal mass with four different materials. Moreover, the effects of sensors’ location, as well as initial guess on the estimation, are evaluated. Time-varying heat transfer coefficients are then estimated in the presence of Gaussian noise, having four standard deviations. Results demonstrate that, despite the estimation is affected by the noise level, the adjoint method is most efficient and rapidly convergent for the thermal mass with moderate thermal conductivity having a noisy temperature field of 1 .

Two-phase binding of drug in a three-layered arterial wall following stent implantation

The present study explores the distribution and the binding of drug eluted from an embedded drug-eluting stent (DES), having struts of the circular cross-section. The arterial tissue is composed of three distinct layers, namely, the intima, the media and the adventitia, with varying diffusivity, and at the interfaces between different layers, a jump condition is imposed. The transport of free drug is modelled by the convection–diffusion-reaction process while the binding of drug is modelled by a nonlinear saturable reversible chemical reaction only. This model incorporates two types of binding mechanisms, namely, the specific binding caused by the interaction of drug and receptors, and the non-specific binding due to the gripping of drug in the extracellular medium (ECM). Furthermore, a constant as well as time-dependent release kinetics have been considered. All the governing equations along with suitable initial, boundary and jump interface conditions in cylindrical polar coordinate system are solved successfully by using the finite difference method. In this study, several clinical key factors like the degree of strut embedment, inter-strut distance (ISD), release mechanism are examined thoroughly. The study reveals that the inclusion of different layers along with jump interface conditions has a remarkable impact on stent-based delivery. Simulated results predict that a higher concentration profile for free drug and rapid saturation of binding sites take place for a half-embedded stent as compared to a quarterly and well-apposed stent. An intriguing feature is to be noted that the inter-strut distance through which the transmural plasma filtration takes place, plays a pivotal role in the distribution and retention of drug in the therapeutic domain considered. Furthermore, the steady-state simulation predicts the influence of strut dimension on the distributions of all drug forms.

A powerful matrix formalism for stress singularities in anisotropic multi-material corners. Homogeneous (orthogonal) boundary and interface conditions

A computational code based on a semianalytic procedure for the determination of the characteristic exponents and the singular stress and displacement fields in multi-material corners is developed. Linear elastic anisotropic materials under generalized plane strain state are considered. This code is a universal computational tool able to analyze both open and closed (periodic) corners, composed of one or multiple materials with isotropic, transversely isotropic or orthotropic constitutive laws, covering both mathematically non-degenerate and degenerate materials in the framework of Stroh formalism. In multi-material corners, material junctions with perfectly bonded or frictionless sliding interfaces can be studied. The considered homogeneous boundary conditions cover stress free and fixed faces, or faces with some restricted or allowed direction of displacements, defined either in the reference frame aligned with the cylindrical coordinate system or in an inclined reference frame. The code is developed in MATLAB and it is based on the Stroh matrix formalism for anisotropic elasticity, the concept of transfer matrix for single material wedges, and on the matrix formalism for homogeneous (orthogonal) boundary conditions. The comparison of the characteristic exponents obtained by the present code and by the solution of closed-form eigenequations available in the literature, has a two-fold objective, first to exhaustively check the general computational implementation of the matrix formalism presented, and second to check the closed-form expressions of eigenequations for relevant specific cases published in the literature.

Cylindrical wave spectrum decomposition method for evaluating the expansion coefficients of the shaped beams in spherical coordinates

In the analysis of light scattering by spherical particles, the incident beam is expanded into an infinite series of spherical wave functions weighted by the beam shape coefficients (BSCs). Different techniques have been developed for evaluating the BSCs in the context of generalized Lorenz-Mie theory (GLMT). In this work, the spectrum decomposition method which we call cylindrical wave spectrum decomposition (CWSD) is recast and is extended to beams having simple dependence on the azimuth angle. The method first expands the beam field in the cylindrical coordinate system and then expands the cylindrical waves into the spherical waves based on the relations between the cylindrical and spherical waves. The CWSD method offers a deep insight into the effects of the beam distribution and the beam's location to the BSCs. Numerical calculations are performed to verify the method. The CWSD method may serve as a powerful tool for studying the light scattering, in the similar way to the angular spectrum decomposition (ASD) method.

Simplified Mutual Inductance Calculation of Planar Spiral Coil for Wireless Power Applications.

In this paper, a simplified method for the calculation of a mutual inductance of the planar spiral coil, motivated from the Archimedean spiral, is presented. This method is derived by solving Neumann’s integral formula in a cylindrical coordinate system, and a numerical tool is used to determine the value of mutual inductance. This approach can calculate the mutual inductances accurately at various coaxial and non-coaxial distances for different coil geometries. The calculation result is compared with the 3D finite element analyses to verify its accuracy, which shows good consistency. Furthermore, to confirm it experimentally, Litz wire is used to fabricate the sample spiral coils. Finally, the comparison of a simplified method is also studied relative to the coupling coefficient. The accuracy of the calculation results with the simulation and the measurement results makes it a good candidate to apply it in wireless power applications.

Acoustic Wave Propagation in a Borehole with a Gas Hydrate-Bearing Sediment

A knowledge of wave propagation in boreholes with gas hydrate-bearing sediments, a typical three-phase porous medium, is of great significance for better applications of acoustic logging information on the exploitation of gas hydrate. To study the wave propagation in such waveguides based on the Carcione–Leclaire three-phase theory, according to the equations of motion and constitutive relations, a staggered-grid finite-difference time-domain (FDTD) scheme and a real axis integration (RAI) algorithm in a two-dimensional (2D) cylindrical coordinate system are proposed. In the FDTD scheme, the partition method is used to solve the stiff problem, and the nonsplitting perfect matched layer (NPML) scheme is extended to solve the problem of the false reflection waves from the artificial boundaries of the computational region. In the RAI algorithm, combined with six boundary conditions, the displacement potentials of waves are studied to calculate the borehole acoustic wavefields. The effectiveness is verified by comparing the results of the two algorithms. On this basis, the acoustic logs within a gas hydrate-bearing sediment are investigated. In particular, the wave field in a borehole is analyzed and the amplitude of a Stoneley wave under different hydrate saturations is studied. The results indicate that the attenuation coefficient of the Stoneley wave increases with the increase of gas hydrate saturation. The acoustic responses in a borehole embedded in a horizontally stratified hydrate formation are also simulated by using the proposed FDTD scheme. The result shows that the amplitude of the Stoneley wave from the upper interface is smaller than that from the bottom interface.

Simulating Interaction of Liquid Steel with Gate Wall at Harmonic Motion

The problem of determining the forces of interaction of a viscous fluid with the cylindrical pipe wall is considered. It is assumed that near the pipe wall, the fluid motion is completely determined by viscous forces. The pipe moves along the streamline. The annular fluid element motion law is a special case of the Navier–Stokes equation in a cylindrical coordinate system. The equation is solved by the Fourier method in Bessel functions. Considering the orthogonality of the eigenfunctions, an equation for the squared norm is found. As an example, the case is considered when the pipe is subjected to vibration. Equations have been obtained for the velocities and viscous friction forces in the laminar sublayer. It has been found that when the pipe moves harmonically, the velocities and shear stresses at the pipe wall do not reach their maximum synchronously. The distribution of velocities and stresses in the section of the steel-pouring ladle gate channel has been considered for three vibration modes. The solution provided can be, in particular, used to determine the fluid–pipe wall interaction forces when the pipe is technologically affected by vibration, impulse, etc., as well as study moving joints such as piston, plunger, etc.

Transition of natural convection of liquid metal in an annular enclosure

Natural convection of a low-Prandtl-number fluid in an annular enclosure with a square cross section and radius ratio of the annulus of 0.5 is numerically studied using a finite-difference method in the cylindrical coordinate system. The fluid in the annular enclosure is heated from an inner vertical wall and cooled from an opposing vertical outer wall, both isothermally, whereas the two horizontal walls are adiabatic. Characteristics of the present natural convection are investigated in detail, with one of the annular computational domains divided into several equal parts. Three-dimensional (3D) regular oscillatory flow appears for a range of Rayleigh numbers when the computational domain angle is set to 90° or less. It is revealed that both the average Nusselt number and kinetic energy obtained from 3D flow are smaller than those obtained from a two-dimensional (2D) computation for the same Rayleigh number. When considered for an annular domain divided into 20 equal parts, flow transition of natural convection occurs gradually on the order of 2D steady, 3D regular steady, 3D regular oscillatory, and 3D irregular oscillatory flows, as the Rayleigh number increases. Most of the 3D regular oscillatory flows exhibit half-period symmetry in time and space. Both steady and oscillatory disturbance components coexist during the growing process from an initial state. In a fully developed stage of the 3D steady flow, only the steady disturbance component survives, whereas in that of the 3D regular half-symmetric oscillatory flow, only the oscillatory disturbance component survives.

An Original 2-D Analytical Model for Investigating Coupled Vibrations of Finite Piezoelectric Resonators.

This article deals with 2-D modeling of coupled vibrations of finite piezoelectric resonators. A general solution for all the physical quantities in the Cartesian and cylindrical coordinate systems is deduced from the governing equations by expansion in series summation of trigonometric functions of thickness coordinate and trigonometric or Bessel functions of the lateral one. The essential difference between this model and the earlier ones is that instead of expressing mainly in the thickness coordinate and integration through the thickness, the solutions are expressed in the form of double Fourier series augmented by single Fourier or Fourier-Bessel series, which contributes to better satisfy the mechanical and electrical boundary conditions. The dynamic stiffness matrix of the system is developed. Electrical impedances of a typical piezoelectric parallelepiped under stress-free and symmetrical loading conditions and its frequency spectrum for different width-to-thickness ratios are calculated using our model as well as by the finite element method. A comparison shows an excellent agreement. Finally, theoretical and measured electrical impedances of a piezoelectric parallelepiped and a piezoelectric disk are compared and discussed. The 2-D theoretical model proposed here is shown to be accurate and efficient for coupled vibration analysis of piezoelectric resonators and is applicable for any set of finite dimensions and crystal symmetry.

Derivative of Spatial Variables in Orthogonal Curvilinear System with Respect to the Ones in Cartesian Coordinate System

Based on the transformation of the spatial variables between the orthogonal curvilinear coordinate system and the Cartesian coordinate system, the derivation of spatial variables in orthogonal curvilinear system to the ones in Cartesian coordinate system is derived. The derivatives of the spatial variables in cylindrical, spherical, and elliptic cylindrical coordinate system are derived respectively.

Vibration analysis of permanent magnet synchronous motor using coupled finite element analysis and optimized meshless method

Conventional finite element analysis (FEA) performed in electromagnetic-vibration coupling calculation of motor suffers from several significant drawbacks, such as large memory space, high computational cost and heavy reliance on mesh quality for accurate solution. With the traditional meshless method, special attention needs to be paid to correctly impose the boundary condition like FEA. Besides, the matrix is easily prone to be ill-conditioned due to introducing large amount of higher-order basis functions. We propose a novel multi-physical coupling method combining FEA and optimized meshless method. The proposed methodology is further evaluated on the vibration analysis of a 12-slot 10-pole permanent magnet synchronous motor (PMSM). Firstly, 2D stator electromagnetic force is simulated and derived based on the local Jacobian derivative method through FEA. The electromagnetic force spectrum is calculated using FFT analysis and further imported into commercial meshless structural simulation software SimSolid for stator harmonic response analysis. Correct force boundary condition and data mapping between meshless and FEA simulation interface are key to the accuracy of the proposed combined multi-physical modeling methodology. This is achieved by introducing new high-dimensional ramp function in the transition region between FEA and meshless domains, which are defined with the shape functions composed of the FEA and meshless method. This function satisfies the continuity and consistency of the displacement function and ensures the convergence of coupled FEA-meshless method. Subsequently, construction of basis function is key to the establishment of convention meshless discrete equation for the elastic problem of rotating machinery. This is designed by using moving least square theory in cylindrical coordinate system. A harmonic response with meshless method is analyzed by using the mode superposition method to obtain detailed mode shape data, acceleration and displacement distribution of stator. Finally, the tangential continuity and robustness are not well considered in the traditional simulation with FEA coupled meshless method. To mitigate this problem, we propose an optimized meshless method based on modified local basis functions to recalculate the harmonic response motion. Then, the coupling electromagnetic-vibration simulation results of traditional coupled FEA-meshless method, optimized coupled FEA-meshless method and complete FEA coupled method are compared. It is worth noting that the optimized method significantly improves accuracy, robustness and computational speed at the same time. In short, the proposed electromagnetic-structure coupling calculation method provides a novel alternative for the multi-physical coupling calculation of rotating machinery combining FEA and meshless simulation methods.

Thin liquid films on a rotary bell atomizer in surface-following coordinates

This article analyzes a published formulation of the Navier–Stokes equations cast into surface-following coordinates and provides some additional mathematical background to follow the article. Ubiquitous in the paint shops of automotive plants around the world, a high-speed rotary bell is succinctly described as a rapidly spinning concave axisymmetric surface with liquid paint supplied from a port coinciding with the center of rotation. The spinning surface transfers momentum to the paint film causing it to flow outward. Upon reaching the bell periphery, it is flung off, subsequently forming an atomized spray transferred to an automotive body through advection and electrostatics. Common analytical frameworks of rotating films were spherical or cylindrical coordinate systems where the wetted surface profile of the bell was constrained to follow a coordinate axis. This led to solutions for films modeled with conical, disk-like, or partial hemispherical profiles. An alternative was a more general case using a surface-following orthogonal curvilinear coordinate system along with its derived vector operators. In the unique case of a thin film, these results validated a simpler pattern found in common coordinate systems.