Biorthogonal Series Solution of Stokes Flow Problems in Sectorial Regions

1996 ◽  
Vol 56 (1) ◽  
pp. 19-39 ◽  
Author(s):  
S. A. Khuri
Keyword(s):  
Stokes Flow ◽  
Series Solution ◽  
Flow Problems ◽  
Biorthogonal Series

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Stokes Flow in a Driven Sector by Two Different Methods

1980 ◽  
Vol 47 (3) ◽  
pp. 482-484 ◽  
Author(s):  
J. Sanders ◽  
V. O’Brien ◽  
D. D. Joseph
Keyword(s):  
Finite Difference ◽  
Series Expansion ◽  
Stokes Flow ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Mutual Support ◽  
Biorthogonal Series ◽  
Total Angle

A biorthogonal series expansion and a numerical finite-difference approximation are applied to the problem of steady Stokes flow in a driven sector of 10° total angle, providing mutual support of the theoretical techniques. For this problem the method of biorthogonal series is faster, cheaper, and more accurate.

Application of the Cosserat Spectrum Theory to Stokes Flow

1999 ◽  
Vol 66 (3) ◽  
pp. 811-814
Author(s):  
W. Liu ◽  
A. Plotkin
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Laplace Equation ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Low Reynolds Number ◽  
Flow Problems ◽  
Cosserat Spectrum ◽  
Linear Shear ◽  
Flow Past A Sphere

This paper presents an application of the Cosserat spectrum theory in elasticity to the solution of low Reynolds number (Stokes flow) problems. The velocity field is divided into two components: a solution to the vector Laplace equation and a solution associated with the discrete Cosserat eigenvectors. Analytical solutions are presented for the Stokes flow past a sphere with uniform, extensional, and linear shear freestream profiles.

scholarly journals Symplectic Exact Solution for Stokes Flow in the Thin Film Coating Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yan Wang ◽  
Zi-Chen Deng ◽  
Wei-Peng Hu
Keyword(s):  
Thin Film ◽  
Hamiltonian System ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Separation Of Variables ◽  
Hamiltonian Function ◽  
Film Coating ◽  
Thin Film Coating ◽  
Flow Problems ◽  
Dual Variables

The symplectic analytical method is introduced to solve the problem of the stokes flow in the thin film coating applications. Based on the variational principle, the Lagrangian function of the stokes flow is established. By using the Legendre transformation, the dual variables of velocities and the Hamiltonian function are derived. Considering velocities and stresses as the basic variables, the equations of stokes flow problems are transformed into Hamiltonian system. The method of separation of variables and expansion of eigenfunctions are developed to solve the governing equations in Hamiltonian system, and the analytical solutions of the stokes flow are obtained. Several numerical simulations are carried out to verify the analytical solutions in the present study and discuss the effects of the driven lids of the square cavity on the dynamic behavior of the flow structure.

scholarly journals Comparison of differential representations for radially symmetric Stokes flow

2004 ◽  
Vol 2004 (4) ◽  
pp. 347-360 ◽  
Author(s):  
George Dassios ◽  
Panayiotis Vafeas
Keyword(s):  
Stokes Flow ◽  
Cell Model ◽  
Spherical Geometry ◽  
Analytical Models ◽  
Spherical Particles ◽  
Particle In Cell ◽  
Flow Problems ◽  
Spherical Surfaces ◽  
Type Boundary ◽  
Connection Formulae

Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and Usha (PNAU) proposed two different representations of the velocity and the pressure fields in Stokes flow, in terms of harmonic and biharmonic functions, which form a practical tool for many important physical applications. One is the particle-in-cell model for Stokes flow through a swarm of particles. Most of the analytical models in this realm consider spherical particles since for many interior and exterior flow problems involving small particles, spherical geometry provides a very good approximation. In the interest of producing ready-to-use basic functions for Stokes flow, we calculate the PNAU and the PN eigensolutions generated by the appropriate eigenfunctions, and the full series expansion is provided. We obtain connection formulae by which we can transform any solution of the Stokes system from the PN to the PNAU eigenform. This procedure shows that any PNAU eigenform corresponds to a combination of PN eigenfunctions, a fact that reflects the flexibility of the second representation. Hence, the advantage of the PN representation as it compares to the PNAU solution is obvious. An application is included, which solves the problem of the flow in a fluid cell filling the space between two concentric spherical surfaces with Kuwabara-type boundary conditions.

Topology optimization of large scale stokes flow problems

2007 ◽  
Vol 35 (2) ◽  
pp. 175-180 ◽  
Author(s):  
Niels Aage ◽  
Thomas H. Poulsen ◽  
Allan Gersborg-Hansen ◽  
Ole Sigmund
Keyword(s):  
Topology Optimization ◽  
Stokes Flow ◽  
Large Scale ◽  
Flow Problems

Method of regularized sources for axisymmetric Stokes flow problems

2016 ◽  
Vol 26 (3/4) ◽  
pp. 1226-1239 ◽  
Author(s):  
Kai Wang ◽  
Shiting Wen ◽  
Rizwan Zahoor ◽  
Ming Li ◽  
Božidar Šarler
Keyword(s):  
Analytical Solution ◽  
Fundamental Solution ◽  
Boundary Conditions ◽  
Stokes Flow ◽  
Regularization Parameter ◽  
Fine Grid ◽  
Content Type ◽  
Flow Problems ◽  
Singular Source ◽  
Analytical Expressions

Purpose – The purpose of this paper is to find solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient non-singular method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space. Design/methodology/approach – Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions. Findings – An axisymmetric driven cavity numerical example and the flow in a hollow tube and flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary. Originality/value – Analytical expressions for the axisymmetric blobs are derived. The method of regularized sources is for the first time applied to axisymmetric Stokes flow problems.

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