Effect of an infinite plane wall on the motion of a spherical Brownian particle

1982 ◽  
Vol 76 (6) ◽  
pp. 3193-3197 ◽  
Author(s):  
Toshiyuki Gotoh ◽  
Yukio Kaneda
Keyword(s):  
Brownian Particle ◽  
Plane Wall ◽  
Infinite Plane

Three-dimensional flow of Jeffrey fluid along an infinite plane wall with periodic suction

Meccanica ◽  
2017 ◽  
Vol 52 (11-12) ◽  
pp. 2705-2714 ◽  
Author(s):  
A. M. Siddiqui ◽  
M. Shoaib ◽  
M. A. Rana
Keyword(s):  
Three Dimensional ◽  
Plane Wall ◽  
Jeffrey Fluid ◽  
Infinite Plane ◽  
Three Dimensional Flow ◽  
Dimensional Flow

The boundary layer due to rectilinear vortex

1978 ◽  
Vol 359 (1697) ◽  
pp. 167-188 ◽  
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Inviscid Flow ◽  
Plane Wall ◽  
Infinite Plane ◽  
Unsteady Boundary Layer ◽  
Short Period ◽  
Time Dependent Problem ◽  
Layer Flow ◽  
Layer Solution

The boundary layer created by the motion of a single rectilinear vortex filament above an infinite plane wall is considered. In a frame of reference which moves uniformly with the vortex the inviscid motion is steady; however, the possibility of a corresponding steady boundary-layer solution can be ruled out and it is concluded that the boundary-layer flow is inherently unsteady for all time. To investigate the nature of the unsteady boundary-layer flow, a time-dependent problem, corresponding to the sudden insertion of the plane wall at time t = 0, is considered; separation in the boundary layer is found to take place in a short period of time and the solution shows possibly explosive features as t increases. It is conjectured that an eventual eruption of the boundary-layer flow is to be expected along with a major modification of the inviscid flow. The theory compares favourably with experiments on the flow induced near the ground by trailing aircraft vortices.

Stokes flow through periodic orifices in a channel

1994 ◽  
Vol 263 ◽  
pp. 207-226 ◽  
Author(s):  
Y. Zeng ◽  
S. Weinbaum
Keyword(s):  
Aspect Ratio ◽  
Stokes Flow ◽  
Three Dimensional ◽  
Plane Wall ◽  
Channel Height ◽  
Infinite Plane ◽  
Electron Microscopic ◽  
Dimensional Solution ◽  
Aspect Ratios ◽  
Flow Through

This paper develops a three-dimensional infinite series solution for the Stokes flow through a parallel walled channel which is obstructed by a thin planar barrier with periodically spaced rectangular orifices of arbitrary aspect ratio B’/d’ and spacing D’. Here B’ is the half-height of the channel and d’ is the half-width of the orifice. The problem is motivated by recent electron microscopic studies of the intercellular channel between vascular endothelial cells which show a thin junction strand barrier with discontinuities or breaks whose spacing and width vary with the tissue. The solution for this flow is constructed as a superposition of Hasimoto's (1958) general solution for the two-dimensional flow through a periodic slit array in an infinite plane wall and a new three-dimensional solution which corrects for the top and bottom boundaries. In contrast to the well-known solutions of Sampson (1891) and Hasimoto (1958) for the flow through zero-thickness orifices of circular or elliptic cross-section or periodic slits in an infinite plane wall, which exhibit characteristic viscous velocity profiles, the present bounded solutions undergo a fascinating change in behaviour as the aspect ratio B’/d’ of the orifice opening is increased. For B’/d’ [Lt ] 1 and (D’ –- d’)/B’ of O(1) or greater, which represents a narrow channel, the velocity has a minimum at the orifice centreline, rises sharply near the orifice edges and then experiences a boundary-layer-like correction over a thickness of O(B’) to satisfy no-slip conditions. For B’/d’ of O(1) the profiles are similar to those in a rectangular duct with a maximum on the centreline, whereas for B’/d’ [Gt ] 1, which describes widely separated channel walls, the solution approaches Hasimoto's solution for the periodic infinite-slit array. In the limit (D’ –- d’)/B’ [Lt ] 1, where the width of the intervening barriers is small compared with the channel height, the solutions exhibit the same behaviour as Lee & Fung's (1969) solution for the flow past a single cylinder. The drag on the zero-thickness barriers in this case is nearly the same as for the cylinder for all aspect ratios.

On the slow motion of a sphere parallel to a nearby plane wall

1967 ◽  
Vol 27 (4) ◽  
pp. 705-724 ◽  
Author(s):  
M. E. O'Neill ◽  
K. Stewartson
Keyword(s):  
Stokes Flow ◽  
Slow Motion ◽  
Plane Wall ◽  
Rigid Sphere ◽  
Infinite Plane ◽  
Direction Parallel ◽  
Outer Solution ◽  
Velocity Gradients ◽  
Leading Term ◽  
Nearest Points

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.

Electrophoresis of a colloidal sphere along the axis of a circular orifice or a circular disk

1991 ◽  
Vol 224 ◽  
pp. 305-333 ◽  
Author(s):  
Huan J. Keh ◽  
Liang C. Lien
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Plane Wall ◽  
Boundary Effects ◽  
Double Layers ◽  
Infinite Plane ◽  
Dual Integral Equations ◽  
Conducting Plane ◽  
Colloidal Sphere ◽  
Electrophoretic Velocity

The axisymmetric electrophoretic motion of a dielectric sphere along the axis of an orifice in a large conducting plane or of a conducting disk is considered. The radius of the orifice or the disk may be either larger or smaller than that of the sphere. The assumption of thin electrical double layers at the solid surfaces is employed. To solve the electrostatic and hydrodynamic governing equations both the electric and the flow fields are partitioned at the plane of the orifice or the disk. For each field, separate solutions are developed on both sides of the plane that satisfy the boundary conditions in each region and the unknown functions for the field at the fluid interface. The continuities of the electric current flux and the fluid stress tensor at the matching interface lead to sets of dual integral equations which are solved analytically to determine the unknown functions for the fields at the matching interface. Then, a boundary–collocation technique is used to satisfy the boundary conditions on the surface of the sphere.The numerical solutions for the electrophoretic velocity of the colloidal sphere are presented for various values of a/b and a/d, where a is the particle radius, b is the radius of the orifice or the disk, and d is the distance of the particle centre from the plane of the wall. For the limiting case of electrophoresis of a sphere perpendicular to an infinite plane wall, our results for the boundary effects agree very well with the exact calculations using spherical bipolar coordinates. Interestingly, the electrophoretic velocity of a sphere approaching an orifice of a larger radius increases when the sphere is close to the orifice, and this velocity can be even larger than that for an identical sphere undergoing electrophoresis in an unbounded fluid. If the sphere has a radius larger than that of the orifice, or if the sphere has a smaller radius and is located sufficiently far from the orifice, its electrophoretic mobility will decrease with the increase of the spacing parameter a/d. For the electrophoretic motion of a sphere along the axis of and close to a disk of finite radius, the resistance to the particle movement can be stronger than that for an equal sphere undergoing electrophoresis normal to an infinite plane wall at the same value of a/d. As the particle approaches the disk wall, its mobility decreases steadily and vanishes at the limit a/d → 1. The boundary effects on the particle mobility and the fluid flow pattern in electrophoresis differ significantly from those of the corresponding sedimentation problem with which a comparison is made.

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