ERROR ANALYSIS AND DIFFERENCE EQUATIONS ON CURVILINEAR COORDINATE SYSTEMS

The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

1. A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions. 2. Differential and Integral Operations on Vector and Scalar Fields. 3. Curvilinear Coordinate Systems. 4. Introduction to Tensors. 5. The Dirac Delta-Function. 6. Introduction to Complex Variables. 7. Fourier Series. 8. Fourier Transforms. 9. Laplace Transforms. 10. Differential Equations. 11. Solutions to Laplace's Equation. 12. Integral Equations. 13. Advanced Topics in Complex Analysis. 14. Tensors in Non-Orthogonal Coordinate Systems. 15. Introduction to Group Theory. Appendix A. The Levi-Civita Identitiy. Appendix B. The Curvilinear Curl. Appendix C. The Double Integral Identity. Appendix D. Green's Function Solutions. Appendix E. Pseudovectors and the Mirror Test. Appendix F. Christoffel Symbols and Covariant Derivatives. Appendix G: Calculus of Variation. Errata List. Bibliography. Index.

Tilted Gaussian beams which are defined by their initial field distribution over a plane generally inclined to the beam axis, serve as the building blocks for the phase-space beam summation method, hence the importance of obtaining asymptotically exact solutions for these wave objects. By applying a novel non-orthogonal ray-centered coordinate system to the inhomogeneous wave equation and using asymptotic (paraxial) considerations, the wave equation is reduced into a new form of a parabolic wave equation. The characteristics of these wave fields are investigated, as well as the novel wave phenomena associated with them.

Despite significant progress in unstructured grid generation and employment, the robust employment of body-conforming coordinate systems promotes the workers to consider it as a major alternative to treat relatively complex flow fields in irregular geometries. Contrary to the laminar flow treatment, there are many ambiguities around treating turbulent flow on body-fitted coordinate system. The ambiguities are mainly originated from the curvature on the boundaries. Considering the past taken efforts in improving the results of treating turbulent flow on curvilinear coordinate system, we have extended a new formulation on body-fitted coordinate system using physical covariant velocities as the dependent variables in our algorithm. The chosen numerical method is finite volume. To suppress nonphysical solution on non-orthogonal grids, the advantages of staggered grid employment are utilized. To evaluate the performance of the newly developed formulation, we solve turbulent flow in complex geometries using different k-omega and k-epsilon turbulence models and compare their results with each other. The comparison indicates that the current formulation performs well using different turbulence models. In some cases, the results of a turbulence model show extra-ordinary advantages comparing with its employment in the other curvilinear coordinate formulations.

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

Rate constants and (12)C/(13)C kinetic isotope effects are calculated by direct dynamics for the OH + CH(4) --> H(2)O + CH(3) reaction. The electronic structure calculations required to generate the implicit potential energy surface were carried out by the high-level multicoefficient Gaussian-3/version-3 (MCG3) method and compared to two other multilevel methods, MC3BB and MC3MPW, and three density functional methods, M06-2X, BB1K, and MPW1K. The rate constants and (12)C/(13)C kinetic isotope effects are shown to depend strongly on the coordinate system used to calculate the frequencies as well as on the method used to account for the torsional anharmonicity of the lowest-frequency vibrational mode of the generalized transition states.

Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface.

In marine seismic exploration, ocean-bottom cable techniques accurately record the multicomponent seismic wavefield; however, the seismic wave propagation in fluid–solid media cannot be simulated by a single wave equation. In addition, when the seabed interface is irregular, traditional finite-difference schemes cannot simulate the seismic wave propagation across the irregular seabed interface. Therefore, an acoustic–elastic forward modeling and vector-based P- and S-wave separation method is proposed. In this method, we divide the fluid–solid elastic media with irregular interface into orthogonal grids and map the irregular interface in the Cartesian coordinates system into a horizontal interface in the curvilinear coordinates system of the computational domain using coordinates transformation. The acoustic and elastic wave equations in the curvilinear coordinates system are applied to the fluid and solid medium, respectively. At the irregular interface, the two equations are combined into an acoustic–elastic equation in the curvilinear coordinates system. We next introduce a full staggered-grid scheme to improve the stability of the numerical simulation. Thus, separate P- and S-wave equations in the curvilinear coordinates system are derived to realize the P- and S-wave separation method.

The geometry and methods of forming right conoids are considered in the article. Right conoids are formed by the movement of a rectilinear generatrix perpendicular to a fixed straight line – the axis of the conoid, along which the rectilinear generatrix moves. When moving, the rectilinear generatrix rotates according to a given law around the directrix curve. Usually, the generatrix line is connected to some reference curve that the generatrix line touches. The coordinate grid along the generatrix lines is also connected with the reference curve. In this case, a non-orthogonal coordinate system of a right conoid is formed. Coefficients of quadratic forms of surfaces with a non-orthogonal coordinate system, usually are complex. Methods for analyzing shells with a non-orthogonal coordinate system of the middle surface of the shell are becoming more complicated. The article considers the possibility of forming right conoids with an orthogonal coordinate system, including with directrix reference curves. A vector equation is given for specifying right conoids with an orthogonal nonconjugate coordinate system. Formulas of coefficients of quadratic forms characterizing the internal geometry and curvature of right conoidal surfaces in space with the help of this vector equation are obtained. Based on the vector equations of the surface assignment, drawings of right conoids with various geometric parameters are constructed in the MathCAD program. The difference in the construction of right conoids with orthogonal and non-orthogonal curved coordinate systems is clearly demonstrated, so with different methods of specifying the angles of inclination of rectilinear generatrix of surfaces.

Numerical simulation was performed to study the hydrodynamics and heat transfer characteristics of a curved pipe. The section of a uniformly heated pipe with a ratio of the bend radius to the pipe’s inner diameter being equal to 25 is considered. The differential conservation equations of a moving continuous medium are formulated in a curvilinear coordinate system. The system of equations for a cylindrical coordinate system, which includes additional terms that arise in the ring coordinate system, is taken as a basis. The problem’s geometry remains unchanged; it is still a straight pipe in cylindrical coordinates. In the simulation, a structured computational grid is used, and the influence of thermogravitational convection is considered. Experimental studies were carried out to investigate heat transfer in a liquid metal flow in a curved pipe with an inner diameter of 19 mm and a bend radius of 0.5 m. The experiments were conducted on a mercury loop at JIHT RAS in the range of the Reynolds number between 11,000 and 80,000 and at different heat loads. A probe with a correlation sensor that consists of two micro-thermopairs was used. The developed mechanism permits continuous movement of the probe in the pipe’s section, which is at a distance of 76 calibers to the heating zone. In the section placed in the stabilized region of the non-isothermal turbulent flow, the averaged velocity and temperature fields were measured in detail. The parameters, presented in dimensionless form, are compared with similar numerical simulation results. It was found that inertial and gravitational forces have a strong influence on the velocity and temperature profiles of the flow, which results in significant heterogeneity in the distribution of the wall temperature around the perimeter of the considered section of the pipe. The comparison shows good agreement between experimental and calculated data.

Geophysical granular flows, such as snow avalanches and debris flows, represent a great hazard to man and infrastructures. These natural phenomena are characterized by the rapid flow of a granular solid phase, embedded in an ambient fluid. Since such events occur with little warning, in addition to investigating the triggering conditions, it is of crucial interest to predict their propagation and run-out distances. The main purpose of the present research is to better understand the flow dynamics of dry granular flows, in the presence of no-slip bottom boundary conditions, and predict their propagation by using a depth-averaged approach. In this case, a stratification of different flow regimes, such as quasi-static and dense collisional, occurs and makes the utilization of the classical single layer depth-averaged models insufficient, because of important uncertainties about the velocity and shear stress distributions along the flow depth. In the first part of the present work, an experimental-numerical study on dam-break flows of dry granular material is reported. It has been found that a modification to the formula, proposed in the Savage-Hutter model for calculating the earth-pressure coefficient, leads to an improved agreement with experimental data, in presence of no-slip bottom conditions. In the second part, an experimental study on steady state velocity profiles of dry granular flows is reported. The granular material, used in this experimental research, was Ottawa sand (ASTM C-778 20/30). The velocity profiles at the side walls and free surfaces have been obtained, through granular PIV techniques. The measurements are in accordance with other experimental works on different granular materials and suggest the occurrence of a rheological stratification along the flow depth, in case of no-slip bottom condition. In order to better describe such a complex flow dynamics, a two-layer depth-averaged model has been proposed. The dynamics of the two layers, ideally corresponding to dense-collisional and quasi-static regimes, have been considered independently. As well, mass exchanges between the layers have calculated through a physically based closure equation. The well-known mathematical issue of the hyperbolicity loss in two-layer models has been carefully addressed. In order to overcome it, a local modification of source terms of the original mathematical model has been proposed. Such a treatment consists of introducing an extra resistance at the interface, if necessary, to avoid the hyperbolicity loss. This approach has been found to be robust and yields reasonable results, as long as the asymptotic steady state solution and boundary conditions fulfil the hyperbolicity requirements of the original model. Through comparisons with experimental data, the two-layer approach turns out to be very promising to describe the complex flow dynamics in case of no-slip bottom boundary conditions, although a more detailed description of the basal shear stress seems to be required. In the third part of the present dissertation, the same two-layer approach has been used in order to rewrite the mathematical model in a curvilinear coordinates system, attached to the interface between the two layers. The model equations have been obtained, by using the same approach proposed in Tai and Kuo (2008). In the model derivation, the physically negligible quantities are identified in a more rational way than the previous model, by means of a scaling approximation.

24 - Systems of Orthogonal Coordinates

The position and attitude information obtained by the position and orientation system (POS) are defined by using the navigation coordinate system, but the actual production application is generally carried out under the map projection frame. Therefore, it is necessary to transform POS data into the map projection frame before it is applied in practice. Considering the complexity of the existing algorithms, and the geocentric coordinate systems that are currently widely used, such as WGS84 and CGCS2000, this letter proposes a simplified transformation method (ST-method) for the POS attitude transformation. It is obviously different from the classical Legat's method which is widely used at present. The algorithm in this letter can be called one-step transformation method, in which the geocentric coordinate system and the local tangent plane coordinate system are not involved. The conversion process has been deeply analyzed and the conversion formula is deduced, directly from navigation coordinate system to projection coordinate system. Compared with the classical methods, the most important contribution of this letter is to simplify the conversion process, and the experimental results show that the proposed method can achieve the same effect as the existing methods with no loss of accuracy.

In order to simulate the bed scouring around piers, many researchers have developed sophisticated numerical algorithms. In this work, the numerical modeling of souring around cylindrical piers are performed by solution of the shallow water equations flow and depth averaged turbulent model equations coupled with the empirical relations for cohesive and granular beds. The shallow water flow equations include to the depth average continuity and motion and depth average turbulent model equations in horizontal Cartesian coordinate systems. For computation of bed erosion at each time step, the numerical results of turbulent flow model are combined with SRICOS empirical bed erosion empirical relations, which are derived from laboratory experiments. The governing partial differential equations of depth average turbulent flow are converted to discrete form using a vertex base finite volume method suitable for triangular unstructured meshes. Therefore, scouring due to flow around piers with arbitrary shape can be modeled. Several test cases of scouring on the cohesive and sandy beds around circular piers are modeled using the developed numerical model and the computed results are compared with the laboratory measurements reported for scouring in cohesive and granular bed around circular piers by the previous experimental workers. Despite of the simplicity and light computational work load of the introduced algorithm, the computed results show promising agreements with the anticipated trends.

A concept of generating and adapting grids according to flow streamlines is employed in the prediction of complex fluid flows. In general, this method retains the merits of body-fitted coordinates in the near-wall regions. Moreover, this approach generates new coordinate lines according to the streamline patterns of the preceding solution based on either Cartesian or curvilinear grid systems and adapts sufficient grids in regions where high resolutions are needed so that the difficulty of numerical diffusion is significantly reduced, To give systematic coordinate transformations, conservation equations applicable in the general nonorthogonal curvilinear coordinate system are derived by the method of generalized tensors. A treatment of near-wall turbulence suitable in the general curvilinear coordinate system is also presented. The superiorities of the present approach are demonstrated by studying the laminar/turbulent flow over a backward facing step and the flow induced by a two-dimensional turbulent jet...