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.

scholarly journals ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

2021 ◽  
pp. 68-80
Author(s):  
Т. В. Денисова ◽  
А. П. Рыбалко
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Classical Method ◽  
Separation Of Variables ◽  
Fourier Method ◽  
Coordinate Systems ◽  
Linear Combinations ◽  
Conjugation Conditions ◽  
Boundary Functions ◽  
Classical Boundary

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Confinement Effects on Added Mass of Cylindrical Structures in a Potential Flow

2017 ◽  
Author(s):  
R. Capanna ◽  
G. Ricciardi ◽  
C. Eloy ◽  
E. Sarrouy
Keyword(s):  
Mechanical Behaviour ◽  
Potential Flow ◽  
Nuclear Power ◽  
Stokes Equations ◽  
Reactor Core ◽  
Axial Flow ◽  
Added Mass ◽  
Seismic Load ◽  
Slender Body Theory ◽  
Computational Costs

Efficient modelling and accurate knowledge of the mechanical behaviour of the reactor core are needed to estimate the effects of seismic excitation on a nuclear power plant. The fuel assemblies (in the reactor core) are subjected to an axial water flow which modifies their dynamical behaviour. Several fluid-structure models simulating the response of the core to a seismic load has been developed in recent years; most of them require high computational costs. The work which is presented here is a first step in order to simplify the fluid forces modelling, and thus to be able to catch the main features of the mechanical behaviour of reactor core with low computational costs. The main assumption made in this work is to consider the fluid flow as an inviscid potential flow. Thus, the flow can be described only using one scalar function (velocity potential) instead of a vector field and strongly simplifies the fluid mechanics equations, avoiding the necessity to solve Navier-Stokes equations. The pressure distribution around a cylinder is first solved in Fourier space for different values of the parameters (wavenumber, confinement size). The method is applied to a simple geometry (cylinder in an axial flow with a variable confinement) in order to test its effectiveness. The empirical model is then compared to simulations and reference works in literature. The configuration with large confinement has been solved, and results were in agreement with Slender Body Theory. The dependency on the confinement size strongly depends on the wavenumber, but in any case added mass increases as the confinement size decreases. Finally, future perspectives to extend the model to a group of cylinders and to improve the model are discussed (i.e. add viscosity to the model).

scholarly journals New Application for the Generalized Incomplete Gamma Function in the Heat Transfer of Nanofluids via Two Transformations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Hibah S. Alhawiti
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Gamma Function ◽  
Transfer Equation ◽  
Nonlinear Differential Equations ◽  
Incomplete Gamma Function ◽  
Heat Transfer Equation ◽  
Layer Flow ◽  
The One

The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with infinity boundary conditions. These boundary conditions at infinity are transformed into classical boundary conditions via two different transformations. Accordingly, the original heat transfer equation is changed into a new one which is expressed in terms of the new variable. The exact solutions have been obtained in terms of the exponential function for the stream function and in terms of the incomplete Gamma function for the temperature distribution. Furthermore, it is found in this project that a certain transformation reduces the computational work required to obtain the exact solution of the heat transfer equation. Hence, such transformation is recommended for future analysis of similar physical problems. Besides, the other published exact solution was expressed in terms of the WhittakerM function which is more complicated than the generalized incomplete Gamma function of the current analysis. It is important to refer to the fact that the analytical procedure followed in our project is easier and more direct than the one considered in a previous published work.

Analysis of Deep Beams

1951 ◽  
Vol 18 (2) ◽  
pp. 163-172
Author(s):  
H. D. Conway ◽  
L. Chow ◽  
G. W. Morgan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Stress Distribution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Normal Stress ◽  
Stress Function ◽  
Beam Theory ◽  
Deep Beam ◽  
Stress Functions

Abstract This paper presents a method of analyzing the stress distribution in a deep beam of finite length by superimposing two stress functions. The first stress function is chosen in the form of a trigonometric series which satisfies all but one of the boundary conditions—that of zero normal stress on the ends of the beam. The principle of least work is then used to obtain a second stress function giving the distribution of normal stress on the ends which is left by the first stress function. By superimposing the two solutions, all the boundary conditions are satisfied. Two particular cases of a given type of loading are solved in this way to investigate the stresses in a deep beam and their deviation from the ordinary beam theory. In addition, an approximate solution by the numerical method of finite difference is worked out for one of the two cases. Results from the two methods are compared and discussed. A method of obtaining an exact solution to the problem is given in an Appendix.

Hydrodynamic Coupling of Viscous and Non-Viscous Numerical Wave Solutions Within the Open-Source Hydrodynamics Framework REEF3D

2021 ◽  
Author(s):  
Weizhi Wang ◽  
Csaba Pákozdi ◽  
Arun Kamath ◽  
Tobias Martin ◽  
Hans Bihs
Keyword(s):  
High Resolution ◽  
Boundary Conditions ◽  
Open Source ◽  
Laplace Equation ◽  
Scale Model ◽  
Surface Boundary ◽  
Cfd Model ◽  
Hydrodynamic Coupling ◽  
Coupling Strategy ◽  
Numerical Wave

Abstract A comprehensive understanding of the marine environment in the offshore area requires phase-resolved wave information. For the far-field wave propagation, computational efficiency is crucial, as large spatial and temporal scales are involved. For the near-field extreme wave events and wave impacts, high resolution is required to resolve the flow details and turbulence. The combined use of a computationally efficient large-scale model and a high-resolution local-scale solver provides a solution the combines accuracy and efficiency. This article introduces a coupling strategy between the efficient fully nonlinear potential flow (FNPF) solver REEF3D::FNPF and the high-fidelity computational fluid dynamics (CFD) model REEF3D::CFD within in the open-source hydrodynamics framework REEF3D. REEF3D::FNPF solves the Laplace equation together with the boundary conditions on a sigma-coordinate. The free surface boundary conditions are discretised using high-order finite difference methods. The Laplace equation for the velocity potential is solved with a conjugated gradient solver preconditioned with geometric multi-grid provided by the open-source library hypre. The model is fully parallelised following the domain decomposition strategy and the MPI protocol. The waves calculated with the FNPF solver are used as wave generation boundary condition for the CFD based numerical wave tank REEF3D::CFD. The CFD model employs an interface capturing two-phase flow approach that can resolve complex wave structure interaction, including breaking wave kinematics and turbulent effects. The presented hydrodynamic coupling strategy is tested for various wave conditions and the accuracy is fully assessed.

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