On the slow motion of two spheres in contact along their line of centres through a viscous fluid

1969 ◽  
Vol 66 (2) ◽  
pp. 407-415 ◽  
Author(s):  
M. B. A. Cooley ◽  
M. E. O'Neill
Keyword(s):  
Exact Solution ◽  
Viscous Fluid ◽  
Stream Function ◽  
Analytic Functions ◽  
Slow Motion ◽  
Range Of Values ◽  
Theoretical Results

An exact solution in terms of analytic functions is given for the Stokes stream function and forces acting on two spheres in contact when the spheres move uniformly with the same velocity along their line of centres. For the case of equal sized spheres, the values of the forces agree with the Faxén limit of the Stimson–Jeffery forces for separated spheres when the minimum clearance between the spheres tends to zero. Numerical values of the forces for arbitrary sized spheres are tabulated and compared with corresponding values when the spheres are separated over a range of values of the minimum clearance. Comparison of the theoretical results with those obtained experimentally for the case of equal sized spheres is also given.

Note on the slow motion of fluid

1939 ◽  
Vol 35 (1) ◽  
pp. 27-43 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Fluid ◽  
Stream Function ◽  
Slow Motion ◽  
Simple Expression ◽  
Two Dimensional ◽  
Fixed Plane ◽  
Uniform Shear

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.

scholarly journals Big Bang Bifurcations in the Tantalus Oscillator Under Biphasic Perturbations

2019 ◽  
Vol 29 (02) ◽  
pp. 1950023
Author(s):  
Humberto Arce ◽  
Araceli Torres ◽  
Augusto Cabrera ◽  
Martín Alarcón ◽  
Carlos Málaga
Keyword(s):  
Bifurcation Diagram ◽  
Limit Cycle ◽  
Excellent Agreement ◽  
Big Bang ◽  
Wide Range ◽  
Coupling Time ◽  
First Time ◽  
Hydrodynamic Oscillator ◽  
Range Of Values ◽  
Theoretical Results

The Tantalus Oscillator is a nonlinear hydrodynamic oscillator with an attractive limit cycle. In this study, we pursue the construction of a biparametric bifurcation diagram for the Tantalus Oscillator under biphasic perturbations. That is the first time that this kind of diagram is built for this kind of oscillator under biphasic perturbations. Results show that biphasic perturbations have no effect when the coupling time is chosen over a wide range of values. This modifies the bifurcation diagram obtained under monophasic perturbations. Now we have the appearance of periodic increment Big Bang Bifurcations. The theoretical results are in excellent agreement with experimental observations.

scholarly journals Bi-harmonic analysis in a perforated strip

1933 ◽  
Vol 232 (707-720) ◽  
pp. 155-222 ◽  
Keyword(s):  
Viscous Fluid ◽  
Harmonic Analysis ◽  
Biharmonic Equation ◽  
Bending Moment ◽  
Slow Motion ◽  
Special Problem ◽  
Pure Bending ◽  
Special Stress ◽  
Straight Lines

A method of solving the biharmonic equation in a region bounded externally by two parallel straight lines and internally by a circle was given by one of the authors in a recent paper. General formulae were developed, but these were restricted to solutions symmetrical about both co-ordinate axes, and were applied to only one special problem of elasticity. In the present paper the analysis is generalized to include unsymmetrical solutions, and the formulae are developed to a point at which it becomes possible to solve any problem of stress within the specified boundaries. Two important special stress-systems—that corresponding to pure bending-moment, and that giving bending-moment with shear—are worked out in detail. A number of other interesting systems may be discussed by the aid of the results given. In addition, only slight modifications are needed to make the equations applicable to the slow motion of a viscous fluid.

scholarly journals The rotation of two circular cylinders in a viscous fluid

1922 ◽  
Vol 101 (709) ◽  
pp. 169-174 ◽  
Keyword(s):  
Viscous Fluid ◽  
Boundary Conditions ◽  
Stream Function ◽  
Elastic Stress ◽  
Steady Motion ◽  
Elastic Equilibrium ◽  
Circular Cylinders ◽  
Two Dimensional ◽  
Concentric Circles ◽  
Hydrodynamical Problem

In a previous communication we employed the solution of the equation ∇ 4 ψ = 0 in bipolar co-ordinates defined by α + iβ = log x + i ( y + a )/ x + i ( y - a ) (1) to discuss the problem of the elastic equilibrium of a plate bounded by any two non-concentric circles. There is a well-known analogy between plain elastic stress and two-dimensional steady motion of a viscous fluid, for which the stream-function satisfies ∇ 4 ψ = 0. The boundary conditions are, however, different in the two cases, and the hydrodynamical problem has its own special difficulties.

Motion of a helical vortex

2017 ◽  
Vol 836 ◽  
Author(s):  
Oscar Velasco Fuentes
Keyword(s):  
Velocity Field ◽  
Cross Section ◽  
Single Point ◽  
Small Circle ◽  
Inviscid Incompressible Fluid ◽  
Helical Vortex ◽  
Angular Velocities ◽  
Infinite Tube ◽  
Range Of Values ◽  
Theoretical Results

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

scholarly journals On exact solution of unsteady MHD flow of a viscous fluid in an orthogonal rheometer

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Muhammad Afzal Rana ◽  
Sadia Siddiqa ◽  
Saima Noor
Keyword(s):  
Exact Solution ◽  
Viscous Fluid ◽  
Mhd Flow ◽  
Orthogonal Rheometer

On asymmetrical slow viscous flows caused by the motion of two equal spheres almost in contact

1969 ◽  
Vol 65 (2) ◽  
pp. 543-556 ◽  
Author(s):  
M. E. O'Neill
Keyword(s):  
Exact Solution ◽  
Viscous Fluid ◽  
Viscous Flow ◽  
Asymptotic Theory ◽  
Viscous Flows ◽  
Slow Viscous Flow ◽  
Angular Velocities ◽  
Well Posed

An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.

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