Exact solutions for Stokes flow in and around a sphere and between concentric spheres

2009 ◽  
Vol 631 ◽  
pp. 363-373 ◽  
Author(s):  
P. N. SHANKAR
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Fields ◽  
Concentric Sphere ◽  
Mixing Properties ◽  
Concentric Spheres ◽  
General Method

A general method is suggested for deriving exact solutions to the Stokes equations in spherical geometries. The method is applied to derive exact solutions for a class of flows in and around a sphere or between concentric spheres, which are generated by meridional driving on the spherical boundaries. The resulting flow fields consist of toroidal eddies or pairs of counter-rotating toroidal eddies. For the concentric sphere case the exact solution when the inner sphere is in instantaneous translation is also derived. Although these solutions are axisymmetric, they can be combined with swirl about a different axis to generate fully three-dimensional fields described exactly by simple formulae. Examples of such complex fields are given. The solutions given here should be useful for, among other things, studying the mixing properties of three-dimensional flows.

scholarly journals General Method of Exact Solution of the Concentration?Dependent Diffusion Equation

1960 ◽  
Vol 13 (1) ◽  
pp. 1 ◽  
Author(s):  
JR Philip
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
General Method

Only three forms of D(6) have previously been known to yield exact solutions of the equation

SOME PERIODIC AND BLOW-UP SOLUTIONS FOR LANDAU–LIFSHITZ EQUATION

2011 ◽  
Vol 26 (32) ◽  
pp. 2437-2452 ◽  
Author(s):  
PENGHONG ZHONG ◽  
SHU WANG ◽  
SHENGTAO CHEN
Keyword(s):  
Exact Solution ◽  
Periodic Solution ◽  
Exact Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Three Dimensional ◽  
Ferromagnetic Materials ◽  
Lifshitz Equation ◽  
Dimensional Spacetime ◽  
Vortex Solution

In this paper, we construct the exact solution of two- or three-dimensional spacetime Landau–Lifshitz equation raised in the ferromagnetic materials. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite time blow-up solution, vortex solution and periodic solution. In the end, we sketch some solutions and their spatial curvatures.

Flow Due to Eccentric Rotating a Porous Disk and a Fluid at Infinity

1976 ◽  
Vol 43 (2) ◽  
pp. 203-204 ◽  
Author(s):  
M. Emin Erdogan
Keyword(s):  
Exact Solution ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Uniform Suction ◽  
Asymptotic Profile

An exact solution of the steady three-dimensional Navier-Stokes equations is obtained for the case of flow due to noncoaxially rotations of a porous disk and a fluid at infinity. It is shown that for uniform suction or uniform blowing at the disk an asymptotic profile exists for the velocity distribution.

scholarly journals Three-dimensional exact solutions of the diffusion equation

2021 ◽  
pp. 48-62
Author(s):  
Александр Данилович Чернышов ◽  
Виталий Валерьевич Горяйнов ◽  
Сергей Федорович Кузнецов ◽  
Ольга Юрьевна Никифорова
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
Three Dimensional ◽  
Arithmetic Mean ◽  
Internal Source ◽  
New Exact Solutions ◽  
Fast Expansions ◽  
Free Parameters ◽  
Diffusion Flows

При помощи метода быстрых разложений решается задача диффузии в параллелепипеде с граничными условиями 1-го рода и внутренним источником вещества, зависящим от координат точек параллелепипеда. Получено в общем виде решение, содержащее свободные параметры, с помощью которых можно получить множество новых точных решений с различными свойствами. Показан пример построения точного решения для случая внутреннего источника переменного только по оси OZ . Приведен анализ особенностей диффузионных потоков в параллелепипеде с указанным внутреннем источником. Получено, что концентрация вещества в центре параллелепипеда равна сумме среднеарифметического значения концентраций вещества в его вершинах и амплитуды внутреннего источника умноженного на величину The authors solve the problem of diffusion in a parallelepiped-shaped body with boundary conditions of the 1st kind and an internal source of substance, depending on the parallelepiped points coordinates with the fast expansions method. The proposed exact solution in general form contains free parameters, which can be used to obtain many new exact solutions with different properties. An example of constructing an exact solution with a variable internal source depending on one coordinate z is shown in the work. An analysis of the features of diffusion flows in a parallelepiped with the indicated internal source is given. It was found that the concentration of a substance in the center of a parallelepiped is equal to the sum of the arithmetic mean of the concentration of a substance at its vertices and the amplitude of the internal source multiplied by the value

scholarly journals Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 269 ◽  
Author(s):  
Xiaojing Zhang ◽  
Vladimir Gerdt ◽  
Yury Blinkov
Keyword(s):  
Difference Scheme ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Algebraic Approach ◽  
Second Order ◽  
Algebraic Construction ◽  
Pressure Poisson Equation ◽  
Strongly Consistent

By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet/Gröbner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme’s accuracy.

scholarly journals On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 170
Author(s):  
Panayiotis Vafeas ◽  
Eleftherios Protopapas ◽  
Maria Hadjinicolaou
Keyword(s):  
Stokes Flow ◽  
Cell Model ◽  
Flow Behavior ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Flow Fields ◽  
Particle In Cell ◽  
Embedded Particles ◽  
Type Particle ◽  
Spheroidal Geometry

Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions.

Evolution of wavelike disturbances in shear flows : a class of exact solutions of the Navier-Stokes equations

1986 ◽  
Vol 406 (1830) ◽  
pp. 13-26 ◽  
Keyword(s):  
Exact Solutions ◽  
Shear Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Periodic Disturbances ◽  
Method Of Solution ◽  
Spatially Periodic ◽  
Basic Shear

New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin ( Phil. Mag . 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional ‘ basic ’ shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three- dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero ; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.

scholarly journals Numerical Methods for Simulating the Motion of Porous Balls in Simple 3D Shear Flows Under Creeping Conditions

2017 ◽  
Vol 17 (3) ◽  
pp. 397-412 ◽  
Author(s):  
Aixia Guo ◽  
Tsorng-Whay Pan ◽  
Jiwen He ◽  
Roland Glowinski
Keyword(s):  
Fluid Flow ◽  
Numerical Methods ◽  
Stokes Flow ◽  
Shear Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Results ◽  
Porous Particle ◽  
Particle Interactions ◽  
New Methods

AbstractIn this article, two novel numerical methods have been developed for simulating fluid/porous particle interactions in three-dimensional (3D) Stokes flow. The Brinkman–Debye–Bueche model is adopted for the fluid flow inside the porous particle, being coupled with the Stokes equations for the fluid flow outside the particle. The rotating motion of a porous ball and the interaction of two porous balls in bounded shear flows have been studied by these two new methods. The numerical results show that the porous particle permeability has a strong effect on the interaction of two porous balls.

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