Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

2003 ◽  
Vol 54 (2) ◽  
pp. 304-327 ◽  
Author(s):  
L. Elasmi ◽  
M. Berzig ◽  
F. Feuillebois
Keyword(s):  
Stokes Flow ◽  
Plane Wall ◽  
Spherical Particles ◽  
Axisymmetric Motion

The motion of particles in the Hele-Shaw cell

1994 ◽  
Vol 261 ◽  
pp. 199-222 ◽  
Author(s):  
C. Pozrikidis
Keyword(s):  
Integral Equations ◽  
Stokes Flow ◽  
Single Layer ◽  
Surface Force ◽  
Spherical Particles ◽  
Fredholm Integral Equations ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Single Plane ◽  
Boundary Velocity

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

Interception of two spheres with slip surfaces in linear Stokes flow

2007 ◽  
Vol 581 ◽  
pp. 129-156 ◽  
Author(s):  
H. LUO ◽  
C. POZRIKIDIS
Keyword(s):  
Integral Equations ◽  
Stokes Flow ◽  
Fourier Coefficients ◽  
Tangential Velocity ◽  
Critical Value ◽  
Element Distribution ◽  
Spherical Particles ◽  
Polar Coordinate System ◽  
Simple Shear Flow ◽  
Time Step

The interception of two spherical particles with arbitrary size in an infinite linear ambient Stokes flow is considered. The particle surfaces allow for slip according to the Navier–Maxwell–Basset law relating the shear stress to the tangential velocity. At any instant, the flow is computed in a frame of reference with origin at the centre of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the centre of the second particle. Taking advantage of the axial symmetry of the boundaries of the flow in the particle coordinates, the problem is formulated as a system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The force and torque exerted on each particle are determined by the zeroth and first Fourier coefficients, while the stresslet is determined by the zeroth, first, and second Fourier coefficients. The derived integral equations are solved with high accuracy using a boundary element method featuring adaptive element distribution and automatic time step adjustment according to the inter-particle gap. The results strongly suggest the existence of a critical value for the slip coefficient below which the surfaces of two particle collide after a finite interception time. The critical value depends on the relative initial particle positions. The particle stress tensor and coefficients of the linear and quadratic terms in the expansion of the effective viscosity of a dilute suspension in terms of the concentration in simple shear flow are discussed and evaluated. Surface slip significantly reduces the values of both coefficients and the longitudinal particle self-diffusivity.

scholarly journals The 3D Happel model for complete isotropic Stokes flow

2004 ◽  
Vol 2004 (46) ◽  
pp. 2429-2441 ◽  
Author(s):  
George Dassios ◽  
Panayiotis Vafeas
Keyword(s):  
Angular Velocity ◽  
Stokes Flow ◽  
Total Pressure ◽  
Mechanical Energy ◽  
Creeping Flow ◽  
Spherical Particles ◽  
Mathematical Treatment ◽  
Normal Velocity ◽  
Tensor Fields ◽  
Concentric Spheres

The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere-in-cell model. The mathematical treatment is based on the two-concentric-spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non-axisymmetric formulation are similar to those of the 2D sphere-in-cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich-Neuber differential representation of the solutions for Stokes flow, which is valid in non-axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self-sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided.

scholarly journals Evaluation of the acoustic impedance of a screen

1977 ◽  
Vol 20 (1) ◽  
pp. 46-61 ◽  
Author(s):  
C. Macaskill ◽  
E. O. Tuck
Keyword(s):  
Numerical Computation ◽  
Stokes Flow ◽  
Acoustic Impedance ◽  
Plane Wall ◽  
Regular Array

AbstractA direct numerical computation is provided for the impedance of a screen consisting of a regular array of slits in a plane wall. The problem is solved within the framework of oscillatory Stokes flow, and results presented as a function of porosity, frequency and viscosity.

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.

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