Motion Of Fluid Particles Research Articles

Similarity of the mean motion of fluid particles dispersing in a natural channel

The mean longitudinal motion of fluid particles dispersing in a natural river channel for distances up to 800 widths on a small mountain stream and up to approximately 1200 widths on the Missouri River is shown to be different from that predicted by a Taylor type dispersion model. The dispersing cloud is shown to exhibit the characteristics of a self‐similar process, with constant velocity ratios and geometric form, and a dimensionless time‐concentration curve is developed. Based upon the dimensionless coordinates of this form and the linearity of its scaling parameters, a method for prediction of the dispersion pattern is developed.

The slow motion of slender rod-like particles in a second-order fluid

The motion of a slender axisymmetric rod-like particle is investigated theoretically for translation through a quiescent second-order fluid and for rotation in a simple shear flow of the same material. The analysis consists of an asymptotic expansion about the limit of rheologically slow flow, coupled with an application of a generalized form of the reciprocal theorem of Lorentz to calculate the force and torque on the particle. It is shown that an arbitrarily oriented particle with fore-aft symmetry translates, to a first approximation, at the same rate as in an equivalent Newtonian fluid, but that the motion of particles with no fore-aft symmetry may be modified at the same level of approximation. In addition, it is found that freely translating particles with fore-aft symmetry exhibit a single stable orientation with the axis of revolution vertical. In simple shear flow at small and moderate shear rates, the non-Newtonian nature of the suspending fluid causes a drift through Jeffery orbits to the equilibrium orbit C = 0 in which the particle rotates about its axis of revolution. At larger shear rates, the particle aligns itself in the direction of flow and ceases to rotate. Comparison with the available experimental data indicates that the measured rate of orbit drift may be used to determine the second normal stress difference parameter of the second-order fluid model. Finally, in an appendix, some preliminary observations are reported of the motion of slender rod-like particles falling through a quiescent viscoelastic fluid.

Translational and rotational diffusion in liquids. I. Translational single-particle correlation functions

This is the first of two articles which together present a theory for translational and orientational single-particle correlation functions in molecular liquids. The theory is based on the assumption that the single-particle motion in a liquid is determined primarily by the harsh repulsive parts of the intermolecular forces. It is shown that the dynamics due to the repulsive forces can be related to the particle motion in a model rough hard sphere fluid. The dynamics of the model hard sphere system is studied within the context of a binary collision expansion. An approximate formula is derived for the translational self-correlation function of the rough hard sphere fluid. The formula is computationally convenient. Arguments are given which indicate that the formula is accurate. In particular, it is shown that the single-particle distribution function obtained from the approximation is positive; the function is well behaved in the hydrodynamic limit, the approximation becomes exact at low densities; the first three time derivatives of the distribution function at t = 0 are exact for all densities; and the velocity autocorrelation function obtained from the distribution function is the Enskog result when the hard spheres are perfectly smooth. A quantitative criterion is proposed to determine the rough hard sphere model which is most closely related to a real liquid.

SEDIMENT TRANSPORT IN RANDOM WAVES AT CONSTANT WATER DEPTH

Sediment transport in random waves at constant water depth is analyzed by dividing the flow field into two regions-the internal region and the boundary layer region. The suspension and transport of sediment in these two regions are treated separately. The total transport is then obtained as the total of these regions through matching boundary. In the bed layer, the load concentration is assumed to be proportional to the specific weight of the sediment and to the probability that the fluctuating lifting force exceeds the weight of the sediment. The bed load is then transported by the secondary flow (which is unidirectional) in the boundary. In the internal flow region, the sediment suspension is treated as a diffusion problem with the intensity of diffusion to be proportional to the amplitude of fluid particle motion. The transport velocity is assumed to be the same as the mass transport velocity of the wave. The predominant mode of transport is found to be suspended load. The total transport in a wind-generated wave field can be expressed as a power law of wind speed. For the case tested, the power should be of the order of 4.

Effect of Inertia on the Brownian Motion of Rigid Particles in a Viscous Fluid

From Boussinesq's work it is known that the frictional resistance of a particle in a viscous, inert fluid, depends on its history. This plays an important part in Brownian motion. The general theory of fluctuations in fluid dynamics leads to explicit expressions for the autocorrelation function G(t) for the random force acting on a Brownian particle and the autocorrelation function φ(t) for its velocity. It is demonstrated that G(t) contains an inertial term which depends on the velocity distribution in the liquid surrounding the particle. The results obtained lead to the correct value of the diffusivity.

On the Mechanism of Finite-Amplitude Entropic Waves

Initial results of an investigation of finite-amplitude entropic waves are presented. The study is based upon the thermodynamic J-function methods as applied to adiabatic pressure waves with friction. It is shown that a strong coupling exists between the energy of the propagated acoustic wave and that of fluid particle motion. This coupling mechanism causes the energy of the frictional finite-amplitude pressure wave to be dissipated in the convected motion of entropic perturbations. The convection characteristics of these entropic perturbations depend directly upon the frictional flow effects and indirectly upon the thermodynamic-state variables. In particular, when the adiabatic exponent has the value 1.4, the convection velocities of these perturbations are shown to be 0.750 U for small Mach numbers, decreasing to 0.599 U at Mach 2.5, where U represents the fluid velocity. It is also shown that the governing characteristic equations for the thermodynamic states of fluid particles have the following properties: along the finite-amplitude pressure wave, the entropy function depends on pressure only, whereas along the convected entropic “wave” the entropy function depends upon two thermodynamic state variables—and this expression reduces to the isentropic acoustic-wave formula in the last stages of fluid motion as the flow Mach number tends to zero, i.e., as the frictional effects become negligible. [This study has been supported by the Unsteady Aerodynamic Branch of NASA-MSFC, Huntsville, Alabama.]

Forces and Motions of a Flexible Floating Barrier

The problem of determining the motions, the structural and the hydrodynamic forces on a flexible, floating barrier are considered. Such barriers are frequently used as containment devices for floating liquid pollutants such as oil. Local hydrodynamic forces on a barrier are approximated by those for a straight barrier at angles to the waves and current equal to those of the local section. This affords a simplification in the theory that allows a solution in closed form to the nonlinear problem of a barrier in a current and a numerical solutions for the linearized problem of a barrier in waves. These problems are treated additively inasmuch as the interaction effects between waves and currents on a barrier are not yet known. The degree to which the barrier follows the fluid particle motion in waves is known to be particularly important. Poor following not only leads to degradation of the barrier as a containment device, but also results in enormous forces on the barrier. Application of the method of solutions for barriers to the case of a towed cable in waves is discussed. Previously unpublished force coefficients for a two-dimensional flat plate are given.

The effect of finite boundaries on the motion of particles in non-Newtonian fluids

A general formula is derived for the effect of container boundaries on the motion of a particle immersed in a non-Newtonian fluid. The result is similar to the corresponding formula of Brenner[1] for Newtonian fluids except that it is valid only for the correction of the velocity and not of the force. For a sphere falling axially in a cylinder, a formula is given which is the non-Newtonian equivalent of the Faxen wall correction formula. The experimental results of Turian[14] are analyzed with this formula and the value of the zero-shear viscosity of his polymer solution is found by an extrapolation based on the theory of third order Rivlin-Ericksen fluids.

Motion of a tracer particle surface in strata of finite length

We consider some characteristic features that occur in the solution of problems of unsteady motion of tracer fluid particles in strata of finite dimensions and in strata consisting of portions of different permeability. The behavior of a water-copper sulfate solution interface at the boundary of two media of different permeability was investigated experimentally in [1]. In that study, depending on the conditions, either breaks or discontinuities of the exudation interval type were observed in the fluid interface line at this boundary of the media. It appears that certain of the observed effects may be described by techniques similar to those presented in the following.

A Proposed Mechanics for the Investigation of Surface Runoff From Small Watersheds: 1, Development

The difficulties of predicting runoff rates and volumes owing to the inadequacy of existing functional forms of the prediction equation and a lack of analytic definitions for the pertinent parameters related to the runoff process are discussed. An analogy is established between the path of water on the watershed surface and the phenomenon of Brownian motion. Probabilistic approaches are used to model a watershed surface and to obtain a stochastic surface impressed force, which acts upon a fluid particle and is found to be a statistically homogeneous function of position. Analogously to Brownian motion theory, a Langevin‐type equation containing the stochastic surface impressed force is postulated as a model for fluid particle motion on an impermeable watershed surface. The influence of surface roughness on the amount of energy available to induce and perpetuate fluid motion on a surface is considered by examining the rate of available energy degradation for a particle collection undergoing random motion induced by surface‐impressed forces.

Relative Dispersion of Particles in a Stratified Rotating Atmosphere

Abstract An analysis of the kinetics and dynamics of the relative dispersion of particles in a stratified rotating fluid is made. The expressions for the relative displacement tensor, and the power- and cross-spectra of the relative velocity are derived. Their characteristics for large and small diffusion times are examined. The governing equations for the motion of marked fluid particles are separated into two sets of equations, one governing the motion of the center of mass and the other governing the motion of individual particles relative to the center of mass. Discussions of the concentration distribution in clusters of marked fluid particles are made. A turbulent diffusion model is constructed for the estimate of the effects of thermal stratification and rotation on the dispersion of particles in the atmosphere.

Particle dynamics in centrifugal fields

The trajectories in centrifugal force fields of particles with diameters from one to 50 microns, were examined both theoretically and experimentally. Equations were developed and solved by analog computer for the non-steady, two-dimensional motion of particles in viscous fluids. An experimental centrifugal classifier was developed that permitted initial conditions and types and strengths of centrifugal fields to be varied widely. Experimental results are shown to be in excellent agreement with theoretical predictions for a wide range of particle diameters when a function generator fit of experimental drag coefficient data is employed.

The Development of Clast Fabric in Mudflows

ABSTRACT Computer simulation of the motion of particles in a viscous fluid in laminar flow shows that strong long-axis fabrics may develop in mudflows in a short period of time. The rate at which the fabric develops in any one situation depends on the velocity gradient. The viscosity of the mudflow one of the main factors controlling the velocity gradient, is increased considerably by the clasts contained in the matrix Fabric development is cyclical and begins as a vague girdle dipping upstream. At the same time a single mode develops, centered on the girdle, and gradually increases in intensity until a maximum is reached when the plunge of the mode is horizontal. The fabric from this point degenerates in reverse order, first to a weak girdle dipping downstream, and then to a fabric similar to the initial fabric when the particles have completed a half cycle of motion. The process then begins again. The strength of the fabric depends on the instant at which the mudflow is arrested. Even though most mudflows travel in a turbulent manner, the fabric is developed in the final stages as the velocity of the flow decreases, and it passes to a laminar phase before coming to rest. Limited experiments and simulations show that a C-axis fabric should also develop with a single vertical mode and girdle transverse to the flow direction. Settling of the clasts under gravity after the flow has come to rest probably leads to a strengthening of the C-axis fabric, as clasts would tend to reorient themselves with their plane of maximum projection (the A, B plane) normal to the force of gravity. The Permian Pagoda Formation in Antarctica consists largely of tillite but contains a small number of beds deposited from mudflows. Four A-axis fabrics were measured from the mudflow units and were found to be similar to long-axis fabrics produced by the simulations. All four have dipping girdles, and two display a central mode. When compared with till fabrics from the same section the mudflow fabrics can be distinguished because of the girdles. Had the mudflows been arrested at a stage of maximum fabric development, it would be impossible to distinguish them from most till fabrics. In this situation the problem may be clarified by measuring several fabrics at different levels in the flow. Because the velocity gradient decreases upward the strength of the mudflow fabrics should be dif erent at different levels. An attempt to exploit this method in the field met with very limited success, probably because the change in velocity gradient upward through the mudflow was too small. C-axis fabrics developed by mudflows have vertical modes while their counterparts in till fabrics plunge steeply in a downstream direction. It is concluded that clast fabric could be a valuable tool in determining the origin of diamictites, but more field data are needed.

Transmission Patterns of Cardiac Murmurs

Aknowledge of blood flow pathways within the heart and great vessels is extremely useful in understanding the transmission patterns of cardiac murmurs. This is true because murmurs are maximally transmitted along the direction of flow of the jet or column of blood associated with their production. This observation is in keeping with the newer vortex-shedding theory of murmur genesis and less consistent with the older turbulence theory of murmur production (Fig 1). 1-3 Turbulence represents a breakdown of laminar flow and is associated with random movement of fluid particles. Although turbulent flow regularly occurs in the cardiovascular system, the energy levels involved are rarely, if ever, capable of producing audible sound. Also, the turbulence theory fails to account for other characteristics of murmurs, such as their variation in pitch and intensity. Vortex shedding involves the formation of vortices or eddies which arise as a column of fluid passes an obstruction.

Motion of discrete particles in a turbulent fluid

Various approximations to Basset's equation for the motion of a particle in a viscous fluid have been applied to the complex phenomenon of dispersion in a turbulent fluid. The deviations of the particle motion from the fluid motion, as predicted by the various approximations, is explored, and the frequencies for which this deviation is large are described. The approximations are found to be invalid for such cases as sediment transport and motion of gas bubbles in liquids. For small, 7 micron, liquid or solid particles in air, however, all approximations are shown to be valid for turbulent frequencies below 812 cps.

Fluid-Particle Motion During Rotary Sloshing

This paper presents the results of a theoretical and experimental investigation of fluid-particle motion in a partially filled cylindrical tank. The body of fluid is assumed to have a steady-state motion consisting of first nonsymmetric sloshing mode only (lowest J1 mode), which gives rise to a free-surface wave that rotates around the tank at the sloshing natural frequency. It was found that theory predicts a net transport motion of the fluid particles in the direction of the free-surface waves when nonlinear terms are retained in differential equations that describe the fluid-particle displacements. Experimental measurements of the particle motion are compared with theoretical predictions. Fluid angular momentum was computed using the theoretical fluid motion and compared with the angular momentum that the fluid would possess if the fluid moved as a rigid body at the same rate as the free-surface waves. It was found that an upper bound to the ratio of the transport angular momentum to the rigid-body angular momentum was equal to 0.8 (η/a)2, where η is the peak wave height of the free surface waves and a is the tank radius.

Harmonic Distortion in Cochlear Models

The envelope over the train of waves traveling along the cochlear partition is asymmetrical: the distal slope being steeper than the proximal one [G. von Békésy, J. Acoust. Soc. Am. 19, 452 (1947)]. In experiments on models, this asymmetry was seen to increase with intensity. Since the mean revolving velocity of Békésy's “eddies” also varies with intensity, artificial “eddies” were introduced into the model while the intensity remained constant. Thereby the asymmetry of the envelope was also increased. Propulsion of the “perilymphatic” fluids by the “eddies” converts particle motion (ordinarily in closed orbits) [Juergen Tonndorf, J. Acoust. Soc. Am. 5, 558 (1957)] into cycloids. As the eddies accelerate along the partition (see the paper by Tonndorf), the cycloids expand with distance, its velocity attaining a sawtooth-like pattern. The asymmetry of the envelope can be explained satisfactorily by the combined effect upon the partition of the distorted cycloids in both “perilymphatic scalae.” However, the same theoretical considerations lead to asymmetry of the displacement pattern of the partition and to an effect similar to peak clipping, both of which increase with distance. This distorted displacement pattern in the distal region of the partition in turn affects particle motion in the adjacent fluids, whence the distortion is propagated proximally by the eddies to be resolved in the usual way (see the paper by Tonndorf) along the cochlear partition. [This research was supported in whole or in part by the U. S. Air Force, under Contract No. AF-41 (657)148, monitored by the School of Aviation Medicine, UASF, Randolph Air Force Base, Texas.]

The Theory of Potentiometric Models

Abstract The detailed analogy between flow systems in porous media and thecorresponding potentiometric model systems is developed under conditions whereit may be desirable to take into account variable pay thickness, variableporosity, and permeability, and also the dependence of the fluid density onpressure. It is shown that in such models it is only necessary that theelectrolyte thickness be made everywhere proportional to the millidarcy-feet ofthe formation. In contrast to the iso-vol type model, previously suggested as abasis for the analogy, the porosity does not enter directly in the constructionof the model. It is introduced only in translating the electrical voltagegradient measurements into the equivalent fluid travel times. A discussion ofthis procedure is given. Introduction It is now some 50 years since it was pointed out that on the basis ofDarcy's law Laplace's equation must govern the steady state flow of homogeneousfluids through porous media. It is 16 years since the obvious implication ofthis fact, namely, that such steady state homogeneous flow systems in porousmedia could be simulated by electrical analogies, was first applied to problemsof practical interest with respect to oil production. In these initial studiesmajor emphasis was placed on the use of electrolytic models, made of blottingpaper, to give a direct and graphic history of the fluid particle motion inregular and infinite well networks. However, it was also noted there that thebasic requirement of the model was that it give a potential distributionsimilar to the pressure distribution in the flow system, and that from theelectrical measurement of the potential distribution the fluid particle motioncould be graphically or numerically determined. This was demonstrated byapplication to the fivespot infinite network, for which a conducting metallicsheet was used to establish the equipotential contours. As anticipated, thefluid particle motion computed from these contours agreed well with that givendirectly by the blotting paper model. For irregular well distributions it was found more convenient to useelectrolytic bath analogs rather than metallic sheets. Several investigations, as applied specifically to cycling well patterns, have been reported with thesemodels, which have become known as "potentiometric models." However, the extremely laborious nature of both the electrical measurements by directprobing and the associated interpretive computations retarded the widespreaduse of these procedures. The electrical measurements can be accelerated byusing a four-probe electrode, which provides a means for simultaneouslydetermining the streamline paths along which the fluid particles must move andthe voltage gradients along these paths which are proportional to the fluidvelocities. With the increasing frequency of discovery of condensate pools, asdrilling depths are becoming greater, the applicability of the potentiometricmodel to production problems, and especially to the study of well patterns forcycling, has been given a fresh impetus. T.P. 2490

The Electrolytic Model and Its Application to the Study of Recovery Problems

Abstract It is possible by means of the electrolytic model to simulate water- floodingor gas recycling systems involving input and output wells, and also theencroactment of a natural water drive. The results are obtained pictorially, and by simple measurements and calculations the percentage of recovery isobtained qualitatively as a function of the total input. Introduction The increased demand for the petroleum products, together with the decline indiscovery rate and the higher cost of recovery as deeper formations are broughtinto production, has stimulated interest in methods for studying and predictingefficiencies of various production programs. Among methods of recovery thathave become of increased significance recently are gas recycling in condensatefields and secondary recovery by water-flooding. The electrolytic model is auseful device for studying such methods in the laboratory, at least in aqualitative way. The original experiments with electrolytic models and the development of theirapplication to the study of secondary recovery problems were published in 1933.Subsequently other investigators continued the development and modification ofthe basic method, with resultant improvements and extension of itsapplications. The method and equipment to be described here are thought to represent stillfurther improvement in the results obtainable, with increased validity in theirapplication to field problems. Summary of Principles of Operation The underlying theory of the operation of the model is given in references 1,2, and 3. However a brief statement of principles involved is given herewith.The applicability of the model is based on the fact that ohm's law and d'Arcy? slaw are exactly analogous; whence electrical flow through a permeable medium.If two electrodes are placed in a conducting (ionized) solution and connectedto the positive and negative terminals of a source of direct current, thepositive ions of the solution will move toward the negative terminal. Since thevelocity of an ion in an electrolytic medium is proportional to the electricalgradient, just as the velocity of a fluid particle in a porous medium isproportional to the pressure gradient, the motion of the ions in the potentialfield will be the same as the motion of fluid particles in a porous medium ifthe positive and negative terminals (input and output wells) and boundaryconditions are made equivalent in shape and distribution. In order for the analogy to hold exactly, the sand must be assumed to haveuniform porosity and permeability and the input and output fluids to haveidentical viscosities and densities. In gas recycling in condensate fields andin water-flooding in oil fields, the variations in density and viscosityusually are not great enough to introduce serious discrepancies intoresults. T.P. 1945

流體の場の變動方程式とその應用

§1. IntroductionThe so-called equations of fluid motion show that the fluid particle is accelerated by pressure gradient, which is assumed to be a force exterior to the fluid system. But, in reality, the pressure gradient is not an “exterior” force, but is influenced by the fluid motion itself.V. Bjerknes, famous theorem states that circulation is accelerated by baroclinic field. But, as mentioned just now, this statement must be supplemented by the idea that the baroclinic field may also grow as a result of circulation, because this theorem is introduced from the equations of motion.The deformation or change of pressure (temperature, etc.) field is described analytically in the following section.§2. Analysis of changing fields (Press., Temp., Pot. Temp., Density).The motion of fluid particles is analysed, and equations of the changing field of f are introduced, f being any physical quantity. That is:-and two other equations of y and z directions.This is a well-known differential formula.§3. -Equations of various changing fields.When we take density (ρ) as f, this becomes as follows by using the equation of continuity:-where_??_is the velocity vector.When the fluid undergoes polytropic change, we get for pressure (P), temperature (T) and potential temperature (θ) fields:§4. Equations of changing baroelinic fields.The baroclinic field develops as a result o fluid motion.§5. Vorticity and the baroclinic field -their growth.This problem has been discussed by many authors. But many of them treat only equations of motion. In this paragraph it is shown that these problems must be discussed by using two systems of equations, one of which is of motion, the other of changing fields.And as a result of this analysis, the following scheme of the genesis of vorticity and the baroclinic field has been obtaind for only a short time at initial growth.