Publisher's Note: “Separation of variables in an asymmetric cyclidic coordinate system” [J. Math. Phys. 54, 063513 (2013)]

2013 ◽  
Vol 54 (7) ◽  
pp. 079904
Author(s):  
H. S. Cohl ◽  
H. Volkmer
Keyword(s):  
Coordinate System ◽  
Separation Of Variables ◽  
Math Phys

Axisymmetric Potential Flow Over Two Spheres in Contact

1975 ◽  
Vol 42 (4) ◽  
pp. 763-765 ◽  
Author(s):  
R. D. Small ◽  
D. Weihs
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Coordinate System ◽  
Laplace Equation ◽  
Potential Flow ◽  
Separation Of Variables ◽  
Added Mass ◽  
Tangent Sphere

An exact solution for the axisymmetric incompressible potential flow over two touching spheres is presented. A tangent-sphere coordinate system is used to simplify the boundary conditions. The Laplace equation is solved by means of separation of variables and the expression for the added mass obtained.

The Effect of an Imperfect Interface on the Singular Stress Field of an Orthotropic Bimaterial Wedge

2004 ◽  
Vol 261-263 ◽  
pp. 459-464
Author(s):  
Somsak Leungvichcharoen ◽  
Anil C. Wijeyewickrema
Keyword(s):  
Coordinate System ◽  
Separation Of Variables ◽  
Local Coordinate System ◽  
Symmetry Plane ◽  
Stress Singularity ◽  
Imperfect Interface ◽  
Numerical Results ◽  
Displacement Discontinuity ◽  
Singular Stress ◽  
Bimaterial Wedge

In this analysis, the effect of an imperfect interface on the stress singularity of an orthotropic elastic bimaterial wedge subjected to traction free boundary conditions, is investigated, where the planes of symmetry are aligned, and one symmetry plane is along the interface and another symmetry plane coincides with the cross-sectional plane. The Stroh formalism with the method of separation of variables are used to obtain the relevant expressions for displacements and stresses. At the interface, only the interfacial tractions and the displacement normal to the interface are assumed to be continuous. The imperfect bond is modeled using a local coordinate system, where each tangential traction component in this local coordinate system is directly proportional to the corresponding displacement discontinuity and inversely proportional to the distance from the wedge apex. The order of singularity is computed numerically for graphite/epoxy wedges and presented for various imperfect interface conditions. The numerical results agree with the available results for the fully bonded case. It is expected that when the axes of local and global coordinate systems are aligned, that the in-plane and antiplane problems are uncoupled, and this feature also can be seen in the numerical results.

scholarly journals Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

2021 ◽  
Author(s):  
Claudia M. Chanu ◽  
Basel Jayyusi ◽  
Raymond G. Mclenaghan
Keyword(s):  
Helmholtz Equation ◽  
Coordinate System ◽  
Circular Plate ◽  
Riemannian Space ◽  
Euclidean Plane ◽  
Separation Of Variables ◽  
Geometric Theory ◽  
Polar Coordinates ◽  
Uniform Density ◽  
Coordinate Systems

The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic–hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.

An efficient methodology for simulating roll dynamics of a tank vehicle coupled with transient fluid slosh

2016 ◽  
Vol 23 (19) ◽  
pp. 3216-3232 ◽  
Author(s):  
Amir Kolaei ◽  
Subhash Rakheja ◽  
Marc J Richard
Keyword(s):  
Free Surface ◽  
Coordinate System ◽  
Fluid Dynamic ◽  
Separation Of Variables ◽  
Hydrodynamic Pressure ◽  
Computational Effort ◽  
Surface Boundary ◽  
Potential Flows ◽  
Free Surface Boundary Condition ◽  
Cartesian Coordinate

An efficient methodology is proposed for simulation of roll dynamics of a tank vehicle system coupled with transient hydrodynamic forces due to fluid slosh. The transient fluid slosh in a horizontal cylindrical tank is analytically modeled considering simultaneous lateral, vertical and roll excitations assuming potential flows and a linearized free-surface boundary condition. For this purpose, the fluid domain in the Cartesian coordinate system is transformed to the bipolar coordinates, where the Laplace equation could be solved using separation of variables. The resulting hydrodynamic pressure, free-surface elevation and slosh force and roll moment are formulated in the tank-fixed coordinate system. The transient fluid slosh model is subsequently integrated to a dynamic roll plane model of a tank vehicle combination to investigate the effect of transient liquid slosh on the roll stability of the vehicle during steady-turning as well as path-change maneuvers. The analyses are performed for different fluid fill heights considering both variable and constant cargo load conditions. The results suggest that the roll stability of tank vehicles can be efficiently analyzed using the coupled linear slosh and multi-body vehicle models with significantly lower computational effort than the methods employing computational fluid dynamic fluid slosh models.

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