scholarly journals III. On the motion of a sphere in a viscous liquid

1888 ◽  
Vol 179 ◽  
pp. 43-63 ◽  
Keyword(s):  
Viscous Liquid ◽  
Steady Motion ◽  
Constant Force ◽  
Two Dimensional ◽  
Sufficient Time ◽  
Small Oscillations ◽  
Uniform Velocity ◽  
Straight Line ◽  
Principal Axes ◽  
The Subject

1. The first problem relating to the motion of a solid body in a viscous liquid which was successfully attacked was that of a sphere, the solution of which was given by Professor Stokes in 1850, in his memoir “On the Effect of the Internal Friction of Fluids on Pendulums,” ‘Cambridge Phil. Soc. Trans.,’ vol. 9, in the following cases: (i.) when the sphere is performing small oscillations along a straight line; (ii.) when the sphere is constrained to move with uniform velocity in a straight line; (iii.) when the sphere is surrounded by an infinite liquid and constrained to rotate with uniform angular velocity about a fixed diameter: it being supposed, in the last two cases, that sufficient time has elapsed for the motion to have become steady. In the same memoir he also discusses the motion of a cylinder and a disc. The same class of problems has also been considered by Meyer and Oberbeck, the latter of whom has obtained the solution in the case of the steady motion of an ellipsoid, which moves parallel to any one of its principal axes with uniform velocity. The torsional oscillations about a fixed diameter, of a sphere which is either filled with liquid or is surrounded by an infinite liquid when slipping takes place at the surface of the sphere, forms the subject of a joint memoir by Helmholtz and Piotrowski. Very little appears to have been effected with regard to the solution of problems in which a viscous liquid is set in motion in any given manner and then left to itself. The solution, when the liquid is bounded by a plane which moves parallel to itself, is given by Professor Stokes at the end of his memoir referred to above; and the solutions of certain problems of two-dimensional motion have been given by Stearn. In the present paper I propose to obtain the solution for a sphere moving in a viscous liquid in the following cases:—(i.) when the sphere is moving in a straight line under the action of a constant force, such as gravity; (ii.) when the sphere is surrounded by viscous liquid and is set in rotation about a fixed diameter and then left to itself.

scholarly journals IV. On the motion of a sphere in a viscous liquid

1888 ◽  
Vol 43 (258-265) ◽  
pp. 174-175
Keyword(s):  
Internal Friction ◽  
Viscous Liquid ◽  
Solid Body ◽  
Steady Motion ◽  
Constant Force ◽  
Small Oscillations ◽  
Straight Line ◽  
Initial Motion

The determination of the small oscillations and steady motion of a sphere which is immersed in a viscous liquid, and which is moving in a straight line, was first effected by Professor Stokes in his well-known memoir “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums;” and in the appendix he also determines the steady motion of a sphere which is rotating about a fixed diameter. The same subject has also been subsequently considered by Helmholtz and other German writers; but, so far as I have been able to discover, very little appears to have been effected with respect to the solution of problems in which a solid body is set in motion in a viscous liquid in any given manner, and then left to itself. In the present paper I have endeavoured to determine the motion of a sphere which is projected vertically upwards or downwards with given velocity, and allowed to ascend or descend under the action of gravity (or any constant force), and which is surrounded by a viscous liquid of unlimited extent, which is initially at rest excepting so far as it is disturbed by the initial motion of the sphere.

On the steady motion of viscous liquid in a corner

1949 ◽  
Vol 45 (3) ◽  
pp. 389-394 ◽  
Author(s):  
W. R. Dean ◽  
P. E. Montagnon
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Sharp Corner

1. In a steady two-dimensional motion of viscous liquid in the sharp corner formed by the rigid straight boundaries θ = 0, α, where r, θ are plane polar coordinates, it is found that, near enough to the corner, the most important term in the stream-function is of the form rmf(θ). The index m is evaluated in §§ 2–4 for values of α between 360 and 90°, and is found to be complex if α is less than about 146°; the limiting form of the stream-function when α is small is considered in § 5.

Slow motion of viscous liquid in a semi-infinite channel

1951 ◽  
Vol 47 (1) ◽  
pp. 127-141 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Pressure Gradient ◽  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Slow Motion ◽  
Biharmonic Function ◽  
Two Dimensional ◽  
Calculated Velocity ◽  
Straight Lines ◽  
Stream Lines

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.

Helicopters

1922 ◽  
Vol 26 (142) ◽  
pp. 390-407
Author(s):  
John Case
Keyword(s):  
Experimental Work ◽  
Steady Motion ◽  
Forward Motion ◽  
Stability Equation ◽  
Aerodynamic Efficiency ◽  
Helicopter Flight ◽  
Stability Derivatives ◽  
Straight Line ◽  
The Subject ◽  
The Stability

SummaryThe present article is an account of some calculations on helicopters.Airscrews have been calculated for different conditions according to two theories: (1) the multiplane interference theory; (2) Glauert's vortex theor). According to both there should be no difficulty in designing a screw to give a good lift at a reasonable rate of climb, and the ceiling should also be quite good. When we consider the speed of falling, with the screws free-wheeling, the two theories give widely different results, and the practicability of the helicopter depends largely upon this question being settled. Simple airscrew theory shows that at least moderate speeds should be obtainable by inclining the airscrew axis. For many reasons it seems desirable that the screws should have at least four blades; gyroscopic couples on the whole machine are eliminated; the forces are widely fluctuating during forward motion with only two blades; the stability derivatives and equations are simplified; but the aerodynamic efficiency will be impaired. In general the stability equation is of the tenth degree, and the lateral and longitudinal stabilities are not separable when the machine, in a state of steady motion in a straight line, receives an asymmetrical disturbance.The following notes are the results of an attempt to investigate the theoretical possibilities of the helicopter, and generally to develop some branches of the theory of helicopters. In this country extremely little work on the subject seems to have been published, and the only experiments I have been able to find are those given in the Report of the Advisory Committee for 1917-18 (Vol. II.). Several articles have appeared in “ L'Aérophile ” from time to time, notably by Lamé, Touissaint and Margoulis, and some experimental work has been done by Eiffel. But it is extremely difficult to find adequate experimental results with which to compare any theory ; for instance it is very rare to find results of tests on airscrews working under helicopteral conditions, and also the aerodynamic data of the aerofoil sections used. In the matter of stability I do not know of any experimental work at all.I shall give first the results of my investigations into airscrews for helicopters, and then proceed to the consideration of the dynamics of helicopter flight and the development of the stability equations.

scholarly journals On the steady motion of a cylinder through infinite viscous fluid

1923 ◽  
Vol 102 (719) ◽  
pp. 766-778 ◽  
Keyword(s):  
Circular Cylinder ◽  
Viscous Fluid ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Steady Motion ◽  
Three Dimensions ◽  
Essential Difference ◽  
Two Dimensional ◽  
Uniform Velocity ◽  
General Physical

There are a good many known solutions of problems of the three-dimensional motion of an infinite viscous fluid disturbed by a moving solid. The simplest of the corresponding two-dimensional problems, that of a circular cylinder moving with uniform velocity, was shown by Stokes to be impossible, when the equations of motion are simplified by the omission of the so-called “inertia terms”; and a general physical argument, given by him to explain the essential difference between the cases of two- and three-dimensions, suggests that the problem is insoluble for a cylinder of any ordinary form, if the “inertia terms” are neglected.

Molecular Dynamics Simulations of Shock-Defect Interactions in Two-Dimensional Nonreactive Crystals

1992 ◽  
Vol 296 ◽  
Author(s):  
Robert S. Sinkovits ◽  
Lee Phillips ◽  
Elaine S. Oran ◽  
Jay P. Boris
Keyword(s):  
Molecular Dynamics ◽  
Hexagonal Lattice ◽  
Square Lattice ◽  
Two Dimensional ◽  
Connection Machine ◽  
Defect Sites ◽  
Principal Axes ◽  
Shock Motion ◽  
Defect Interactions ◽  
Dynamics Simulations

AbstractThe interactions of shocks with defects in two-dimensional square and hexagonal lattices of particles interacting through Lennard-Jones potentials are studied using molecular dynamics. In perfect lattices at zero temperature, shocks directed along one of the principal axes propagate through the crystal causing no permanent disruption. Vacancies, interstitials, and to a lesser degree, massive defects are all effective at converting directed shock motion into thermalized two-dimensional motion. Measures of lattice disruption quantitatively describe the effects of the different defects. The square lattice is unstable at nonzero temperatures, as shown by its tendency upon impact to reorganize into the lower-energy hexagonal state. This transition also occurs in the disordered region associated with the shock-defect interaction. The hexagonal lattice can be made arbitrarily stable even for shock-vacancy interactions through appropriate choice of potential parameters. In reactive crystals, these defect sites may be responsible for the onset of detonation. All calculations are performed using a program optimized for the massively parallel Connection Machine.

A modified tangent-gas approximation for two-dimensional steady flow

1958 ◽  
Vol 4 (6) ◽  
pp. 600-606 ◽  
Author(s):  
G. Power ◽  
P. Smith
Keyword(s):  
Least Squares ◽  
Steady Flow ◽  
Variational Approach ◽  
Least Squares Method ◽  
Two Dimensional ◽  
Subsonic Flows ◽  
Von Karman ◽  
Straight Line ◽  
Volume Relationship ◽  
Pressure Volume Relationship

A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956).

scholarly journals The formation of vortices from a surface of discontinuity

1931 ◽  
Vol 134 (823) ◽  
pp. 170-192 ◽  
Keyword(s):  
Steady State ◽  
Plane Surface ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Small Oscillations ◽  
Approximate Form ◽  
History Of ◽  
State Increase ◽  
Liquid Surfaces

Helmholtz was the first to remark on the instability of those “liquid surfaces” which separate portions of fluid moving with different velocities, and Kelvin, in investigating the influence of wind on waves in water, supposed frictionless, has discussed the conditions under which a plane surface of water becomes unstable. Adopting Kelvin’s method, Rayleigh investigated the instability of a surface of discontinuity. A clear and easily accessible rendering of the discussion is given by Lamb. The above investigations are conducted upon the well-known principle of “small oscillations”—there is a basic steady motion, upon which is superposed a flow, the squares of whose components of velocity can be neglected. This method has the advantage of making the equations of motion linear. If by this method the flow is found to be stable, the equations of motion give the subsequent history of the system, for the small oscillations about the steady state always remain “small.” If, however, the method indicates that the system is unstable, that is, if the deviations from the steady state increase exponentially with the time, the assumption of small motions cannot, after an appropriate interval of time, be applied to the case under consideration, and the equations of motion, in their approximate form, no longer give a picture of the flow. For this reason, which is well known, the investigations of Rayleigh only prove the existence of instability during the initial stages of the motion. It is the object of this note to investigate the form assumed by the surface of discontinuity when the displacements and velocities are no longer small.

Two-dimensional buoyant plumes in a uniform co-flow

2021 ◽  
Vol 932 ◽  
Author(s):  
Gary R. Hunt ◽  
Jamie P. Webb
Keyword(s):  
Theoretical Investigation ◽  
Analytical Solutions ◽  
Richardson Number ◽  
Relative Magnitude ◽  
Dynamical Behaviour ◽  
Finite Width ◽  
Two Dimensional ◽  
Buoyant Plumes ◽  
The Subject ◽  
Flow Case

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

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