Orthogonal Expansions in Curvilinear Coordinates

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

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XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600021982 ◽

1927 ◽

Vol 46 ◽

pp. 194-205 ◽

Cited By ~ 3

Author(s):

C. E. Weatherburn

Keyword(s):

Curvilinear Coordinates ◽

Triple Systems ◽

Orthogonal Curvilinear Coordinates ◽

Orthogonal Systems ◽

Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

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A New Nonlinear Higher-Order Shear Deformation Theory for Nonlinear Vibrations of Laminated Shells

Volume 8: Dynamic Systems and Control, Parts A and B ◽

10.1115/imece2010-39483 ◽

2010 ◽

Author(s):

M. Amabili ◽

J. N. Reddy

Keyword(s):

Shear Deformation ◽

Deformation Theory ◽

Nonlinear Vibrations ◽

Circular Cylindrical Shell ◽

Shear Deformation Theory ◽

Higher Order ◽

Curvilinear Coordinates ◽

Geometrically Nonlinear ◽

Geometric Imperfections ◽

Laminated Shells

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape; taking geometric imperfections into account. The geometrically nonlinear strain-displacement relationships are derived retaining full nonlinear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only nonlinear terms of the von Ka´rma´n type. Results show that inaccurate results are obtained by keeping only nonlinear terms of the von Ka´rma´n type for vibration amplitudes of about two times the shell thickness for the studied case.

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Method for the Construction of Godunov-Type Difference Schemes in Curvilinear Coordinates and its Application to Cylindrical Coordinates

Computational Mathematics and Modeling ◽

10.1007/s10598-014-9228-z ◽

2014 ◽

Vol 25 (3) ◽

pp. 315-333 ◽

Cited By ~ 2

Author(s):

M. V. Abakumov

Keyword(s):

Difference Schemes ◽

Curvilinear Coordinates ◽

Cylindrical Coordinates

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Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

Journal of Applied Mechanics ◽

10.1115/1.3424424 ◽

1978 ◽

Vol 45 (4) ◽

pp. 812-816 ◽

Cited By ~ 1

Author(s):

B. S. Berger ◽

B. Alabi

Keyword(s):

Fourier Transform ◽

Finite Difference ◽

Boundary Conditions ◽

Half Space ◽

Coordinate Transformation ◽

Numerical Results ◽

Three Dimensions ◽

Curvilinear Coordinates ◽

Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

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