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Heat Conduction
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
In the previous section, standard coordinate systems including Cartesian and radial coordinates were presented. For more complex geometrical configurations, curvilinear coordinates may be used. Curvilinear coordinates refer to a coordinate system in which the coordinate lines may be curved. The coordinates can be obtained by a mapping from Cartesian coordinates by a coordinate transformation that is locally invertible (a one-to-one mapping from any point in the Cartesian coordinates to curvilinear coordinates). Common examples of curvilinear coordinate systems include cylindrical and spherical coordinates. These coordinates are illustrated in Figure 2.10. Other examples include ellipsoidal, toroidal, bispherical, and oblate spheroidal coordinates. Ellipsoidal coordinates are the most generalized coordinate system from which other curvilinear coordinate systems are special cases.
Orthogonal Coordinate Systems
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
ABSTRACT The purpose of this chapter is to present coordinate systems in such a way that it improves the ability of readers to use them to describe the electric field in a better way. The goal is to put forward the coordinate system as a useful language for describing, visualizing and understanding the concepts that are central to electric field analysis. Many mathematical operators related to electric field analysis have simple forms in Cartesian coordinates and are easy to remember and evaluate. In fact, all the problems could be solved using the Cartesian coordinate system. However, the resulting expressions might be unnecessarily complex. While dealing with problems that have high degree of symmetry, it is helpful to use a coordinate system that exploits the symmetry. As a result, it is useful to have a general method for expressing mathematical operators in non-Cartesian forms. Orthogonal curvilinear coordinate system is a useful approach by which the matching of the symmetry of a given physical configuration could be done and simplified mathematical models could be built to clarify the field concepts clearly.
Analysis of Stress
Published in Abdel-Rahman Ragab, Salah Eldin Bayoumi, Engineering Solid Mechanics, 2018
Abdel-Rahman Ragab, Salah Eldin Bayoumi
In curvilinear coordinates, the space axes and coordinates are all curved and the curvature from one point to another is not usually constant. They are the most general form of coordinates, and it will be observed that Cartesian, cylindrical, and spherical coordinates are special cases of curvilinear coordinates. The space coordinates consist of three families of curves, α, β, and γ, which correspond to certain characteristics in the body. In orthogonal curvilinear coordinates these three families of curves meet mutually at right angles. Orthogonal curvilinear coordinates are utilized in some solid mechanics problems such as thin-walled shells and slip-line field analysis.
Study of Bohr–Mottelson with minimal length effect for Q-deformed modified Eckart potential and Bohr–Mottelson with Q-deformed quantum for three-dimensional harmonic oscillator potential
Published in Molecular Physics, 2020
A. Suparmi, C. Cari, M. Ma’arif
The quadratic momentum operator, which is influenced by minimal length, is shown by Equation (8). The collective geometrical model of nucleus is expressed by [2,5] Equation (9) is in curvilinear coordinates, where x is a curved space and is a metric tensor. In the axial symmetry case, the nucleus has three degrees of freedom: , so the metric tensor is given by where is a variable that corresponds to nucleus deformation in the radial direction, and and are part of Euler angles. The Laplacian operator is obtained from Equations (7) to (8) [2,5] with g and being determinant and inverse of the matrix , respectively. From Equation (10), we get the determinant of the matrix and the inverse of the matrix is By using Equations (11)–(13) we get the Laplacian operator as
Thermophysical Parameter Estimation of a Neutron Source Based on the Action of Broadband Radiation on a Cylindrical Target
Published in Fusion Science and Technology, 2023
Victor V. Kuzenov, Sergei V. Ryzhkov
In the numerical solution of such problems,12 movable grids are often used, which adapt to the peculiarities of the numerical solution. This approach allows one to obtain results of increased accuracy on relatively coarse computational grids. The use of dynamically adaptive grids leads to the need to transfer from an axisymmetric coordinate system to arbitrary moving (in 2D and three-dimensional cases to curvilinear) coordinates (). Therefore, in what follows, the mathematical formulation of the first fractional step and the hyperbolic Eq. (1) use the coordinate system ().
Effects of electro-osmotic and double diffusion on nano-blood flow through stenosis and aneurysm of the subclavian artery: numerical simulation
Published in Waves in Random and Complex Media, 2022
A. M. A. Moawad, A. M. Abdel-Wahab, Kh. S. Mekheimer, Khalid K. Ali, N. S. Sweed
Constituent equation of Bingham–Papanastasiou fluid which deformed as [1] is defined as The second invariant of rate-of-strain-tensor is given as Assuming that the velocity vector is expressed in curvilinear coordinates and the components of strain-tensor rate in an axisymmetric flow are determined in curvilinear coordinates Now, the second invariant of rate-of-strain-tensor in curvilinear coordinates can be deformed as The charge density in a unit volume of the fluid is defined as are the densities number of negative and positive ions, is the valence of ions, is the charge of electron, is the constant of Boltzmann and is the local absolute electronic solution temperature and positive or negative ion concentration in the bulk volume. In addition, the Debaye–Huckel linearization principle is used in Equation (14) and we get where is a parameter of Debaye–Huckel which determines the characteristic thickness of EDL.