bluuebunny/crossref_metadata_2025 huggingface dataset:
Resonant interactions between two trains of gravity waves
Authors: Longuet-Higgins M. S.
Published: March 2006

In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).

It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.
Phase velocity effects in tertiary wave interactions
Authors: Longuet-Higgins M. S. , Phillips O. M.
Published: March 2006

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.
The asymptotic form of the laminar boundary-layer mass-transfer rate for large interfacial velocities
Authors: Acrivos Andreas
Published: March 2006

The convective diffusion of matter from a stationary object to a moving fluid stream is distinct from pure heat transfer because of the appearance of a finite interfacial velocity at the solid surface. This velocity is related to the rate of mass transfer by a dimensionless groupBin such a way that for −1 <B< 0 the transfer is from the bulk to the surface while for 0 <B< ∞ the transfer is from the surface to the main stream. In this paper, asymptotic solutions to the two-dimensional laminar boundary-layer equations are developed for the caseB[Gt ] 1, and for rather general systems. It is shown that in most instances the asymptotic expressions for the rate of mass transfer become accurate whenB> 3 and that the transition region between the pure heat-transfer analogy (B∼ 0) and theB[Gt ] 1 asymptote may be described by a simple graphical interpolation. These results may easily be extended to three-dimensional surfaces of revolution by the usual co-ordinate transformations of boundary-layer theory.
The shape and stability of a bubble at the axis of a rotating liquid
Authors: Rosenthal D. K.
Published: March 2006

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.

In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.
The barotropic stability of the mean winds in the atmosphere
Authors: Lipps Frank B.
Published: March 2006

This paper considers the stability of a barotropic current on a beta earth. The motion is assumed to be horizontal, non-divergent and barotropic. The current is taken to be of the formU(y)=Asech2by+B. The perturbations are required to approach zero asyapproaches ± ∞. We introduce the non-dimensional wave-numberland a parameter χ, which is a measure of the rotation effect. χ is inversely proportional to β.

There are only two kinds of perturbations: symmetric disturbances (those with maximum amplitude aty= 0) and antisymmetric disturbances (those with zero amplitude aty= 0). We find the neutral curve in the (χ,l2)-plane for both types of disturbances. The rates of amplification in the immediate vicinity of the neutral curves are also found. It is seen that the beta effect, which is due to the earth's rotation, tends to stabilize the current. For the symmetric disturbances we find a band of unstable wavelengths when χ > 1/2; and for large χ the estimated curve of the maximum value of the imaginary part of the phase velocity is asymptotic to the lower branch of the neutral curve. The antisymmetric disturbances are more stable than the symmetric disturbances.
Transport processes in a combustible turbulent boundary layer
Authors: Kulgein Norman G.
Published: March 2006

Coexistent processes of heat, mass and momentum transfer operative within a combustible turbulent boundary layer have been experimentally investigated. The boundary layer was established on a porous cylinder mounted in a low-speed wind tunnel with its long axis in the flow direction. Methane was transpired into the boundary layer and ignited. Results indicate that the dimensionless transfer numbers corresponding to the three transfer processes can be correlated by the formula 0.038Re−0.2 to within ± 30% of measured values so that a rough numerical analogy exists among all three processes. The effect of mass injection on the skin friction coefficient is reasonably well accounted for by available theory. No effect of mass injection was found on the values of heat and mass transfer parameters. Finally, there was a lack of evidence indicating any sort of reaction-generated turbulence or that the experimentally demonstrated disturbance of the viscous layer by mass injection substantially affected the transport phenomena.
Note on the damping of free-surface oscillations due to drainage
Authors: Miles John W.
Published: March 2006

It is shown that the draining of liquid from a vertical tank produces a small but positive damping of free-surface oscillations. This conclusion is consistent with the experimental observations of Brooks.
Radiative transfer effects in natural convection above fires
Authors: Murgai M. P.
Published: March 2006

This paper describes the results of examining the influence of radiative heat transfer on turbulent natural convection above fires in an atmosphere of constant potential temperature, under both the ‘opaque’ and ‘transparent’ approximations. It turns out that on the basis of the over-all approximations introduced in this investigation, the former case reduces to that of no radiative transfer. For the latter case, the initial fire size, the energy release rate (initial temperature difference) and the absorption coefficient have been regarded as independent parameters. The solution curves presented cover a range over which these parameters are expected to vary in practice.
Effect of thermal radiation on the propagation of plane acoustic waves
Authors: Vincenti Walter G. , Baldwin Barrett S.
Published: March 2006

A study is made of the propagation of acoustic waves in a semi-infinite expanse of radiating gas on one side of an infinite, plane, radiating wall. A solution is found, in particular, for the case of sinusoidal oscillations in both position and temperature of the wall. The solution is based on a single linear integro-differential equation that plays the same role here as does the classical wave equation in equilibrium acoustic theory. The solution is applicable throughout the range from a completely transparent to a completely opaque gas and from very low to very high temperatures. The solution appears, in general, as the sum of two types of travelling waves: (1) an essentially classical sound-wave, but with a slightly altered speed and a small amount of damping and (2) a radiation-induced wave whose speed and damping may be either large or small, depending on the temperature and absorptivity of the gas. Since the waves are coupled, both types will usually be present together, even in the special cases of pure motion or pure temperature variation of the wall.
Magnetohydrodynamic lubrication flow between parallel rotating disks
Authors: Hughes W. F. , Elco R. A.
Published: March 2006

The motion of an electrically conducting, incompressible, viscous fluid in the presence of a magnetic field is analyzed for flow between two parallel disks, one of which rotates at a constant angular velocity. The specific application to liquid metal lubrication in thrust bearings is considered. The two field configurations discussed are: an axial magnetic field with a radial current and a radial magnetic field with an axial current. It is shown that the load capacity of the bearing is dependent on the MHD interactions in the fluid and that the frictional torque on the rotor can be made zero for both field configurations by supplying electrical energy through the electrodes to the fluid.
Temperature-time histories at the interface between a gas and a solid
Authors: Clarke John F.
Published: March 2006

The effects of compressibility, temperature-jump and gaseous adsorption are considered in an attempt to predict the temperature-time history at the interface between a gas and a solid when both the temperature and the pressure of the gas are suddenly changed. It appears that temperature-jump will be the dominant effect, with adsorption contributing significantly in some circumstances. Compressibility is of minor importance during practically-resolvable time intervals. Some experimental evidence which gives tentative support to the results of the analysis is commented upon.
Some theoretical results concerning the displacement of a viscous oil by a hot fluid in a porous medium
Authors: Fayers F. J.
Published: March 2006

Two simultaneous first-order non-linear equations are derived to give a ’high-flow-rate’ model for the displacement of oil by hot water in a porous medium. The solution of these equations is analysed by the method of characteristics and it is shown that in problems for which thermal capacity dependence on temperature is neglected, the solution will have the properties of a simple wave. The simple wave behaviour gives a rapid method for solving practical systems. When temperature dependence is included in the thermal capacities, it is found that a simultaneous shock in temperature and saturation develops, but the solution will usually approximate quantitatively the simple wave result. Decoupling the equations by using an average saturation in the heat transport equation gives results in reasonable agreement with the coupled case.
Instability in Couette flow of solutions of macromolecules
Authors: Merrill E. W. , Mickley H. S. , Ram A.
Published: March 2006

In Couette flow between coaxial cylinders, ring-shaped vortices, illustrated schematically in figure 1, develop at a Reynolds number which depends strongly on the geometry of the system, as treated analytically for Newtonian fluids by Chandrasekhar (1958). The problem was first considered by Taylor (1923) who, examining simple (micromolecular) liquids, arrived at an equation for the critical Reynolds numberReiabove which instability (vortex formation) would occur when the inner cylinder rotated and the outer is stationary, and when the gap is small compared to the radius.
On the expansion of a gas into vacuum
Authors: Greenspan H. P. , Butler D. S.
Published: March 2006

A study is made of the flow into vacuum of a gas initially at rest in a state of uniform pressure and density; the analysis is based on a continuum model. Among the topics discussed are the motion of the gas-vacuum interface, the reflexion of a plane front off a rigid wall, the propagation of compressive waves within such expansions, the escape from a sphere and the collapse of a spherical cavity.
On the stability of laminar flow of a dusty gas
Authors: Saffman P. G.
Published: March 2006

The equations describing the motion of a gas carrying small dust particles are given and the equations satisfied by small disturbances of a steady laminar flow are derived. The effect of the dust is described by two parameters; the concentration of dust and a relaxation time τ which measures the rate at which the velocity of a dust particle adjusts to changes in the gas velocity and depends upon the size of the individual particles. It is shown that if the dust is fine enough for τ to be small compared with a characteristic time scale associated with the flow, then the addition of dust destabilizes a gas flow; whereas if the dust is coarse so that τ is relatively large, then the dust has a stabilizing action.

For plane parallel flow, it is shown that the stability characteristics for a dusty gas are still determined by solutions of the Orr-Sommerfeld equation, but with the basic velocity profile replaced by a modified profile which is in general complex. A simple, although unrealistic, example is used to illustrate some features of the action of dust. It is intended to describe the solution of the modified Orr-Sommerfeld equation for plane Poiseuille flow in a later paper.
Free convection of fluid in a vertical tube filled with porous material
Authors: Wooding R. A.
Published: March 2006

The problem of an unstable fluid overturning in a vertical tube filled with porous material is treated by an approximation of boundary-layer type. It is shown that the fluid can experience a pseudo-inertial effect, in which variations in density across the tube exhibit properties analogous to variations of momentum in an inertial flow. The mean fluid density and mean-square vertical velocity over a horizontal cross-section of the tube are related by a pair of hyperbolic equations, for which there exist two systems of characteristics. It is shown that changes in the mean density of the fluid can be propagated as discontinuities. For discontinuities of finite amplitude, two jump conditions are derived, one of which is found to involve an undetermined parameter λ. The theory is applied to the case of a vertical tube containing porous material saturated with water, which is attached at the top to a reservoir filled with an aqueous solution (an analogue of Taylor's (1954) experiment). The motion of a finite discontinuity which arises at the initial unstable interface is calculated by two approximate methods. These results compare satisfactorily with the data from three experiments, using tubes of circular cross-section, provided that the value of λ is about 0·75. If the theoretical interpretation is correct, it appears that convective flow ceases when the vertical density gradient is slightly less than the neutral value.
Transient gravity wave response to an oscillating pressure
Authors: Miles John W.
Published: March 2006

The gravity-wave response of a semi-infinite liquid to the oscillating pressure Pδ(x) exp (iwt) is given in an asymptotic form that is uniformly valid through the transition zone that separates the dispersion-controlled precursor and the monochromatic steady state. The same problem has been considered previously by Stoker (1957), but his initial conditions were spurious, and he did not seek a uniformly valid asymptotic representation.
Note on the break-up of a charged liquid jet
Authors: Magarvey R. H. , Outhouse L. E.
Published: March 2006

The disintegration of a charged liquid jet is examined, and the break-up mechanism inferred from photographic evidence. Gravitational, molecular and electrical forces all contribute to the segmentation of the jet and determine the drop size distribution. The disintegration process is investigated from the point of view of drop generation. The segmentation of the charged jet differs from the known ways in which an uncharged jet is broken into drops.
Observations on distorted turbulent wakes
Authors: Reynolds A. J.
Published: March 2006

The question considered here is whether turbulent wakes can undergo a simple self-preserving development when convected through a duct of varying crosssection. A flow is described as self-preserving if the distributions of its mean-value properties at successive sections have the same basic shape, differing only in magnitude and lateral extent. In such flows the scales of the mean velocity variation and turbulence have been found to be proportional.

The wakes studied are concentrated near a plane of symmetry through a duct of changing rectangular section but constant area; the portion of the wake near the other plane of symmetry is considered. Measurements in the flows behind three circular cylinders reveal a lack of universality of the scales associated with self-preserving solutions of the momentum equation. The wakes of the smaller cylinders adopt the predicted form, but that of the largest does not. As it moves through the channel, this wake is compressed less rapidly by the imposed lateral strain than is predicted.

Interest is then concentrated on the equation governing the kinetic energy of the turbulence and in particular on the relative size of its two production terms, that representing the action of the mean wake shear and that representing the effect of the distortion applied by the duct. It is found that self-preserving wakes can form only in a few kinds of distorting convecting flow. In the experimental duct, which has exponentially varying walls, one of these is set up. Attention is concentrated on this case. A stability analysis (based on the criterion that a stable wake is one in which the total turbulence production decreases as a result of more rapid wake expansion) suggests that, when production by wake shear is greater than that by distortion, a stable self-preserving development is possible in which the turbulence scale and the velocity defect scale remain proportional. But in the alternative, distortion-dominated, case the velocity defect decreases continually relatively to the turbulence.

The experimental results are in accord with these predictions. The transition from shear-dominated to distortion-dominated wakes appears to take place when the production terms are about equal. The failure of the one experimental wake to adopt a simple self-preserving form can be attributed to the relatively slight organization of its turbulence when distortion is begun.
The ‘starting plume’ in neutral surroundings
Authors: Turner J. S.
Published: March 2006

The advancing front of a buoyant plume which is being established in uniform surroundings has some properties in common with the plume, while in other respects it behaves more like a ‘thermal’ released from rest. The solutions for these two cases cannot be matched directly, since the dependence of velocity on height is different. It is shown here that a similarity solution, which is consistent with the equations describing both parts of the flow, can be obtained once it is recognized that the velocity of the front may be less than that of the steady plume. The cap moves with a constant fraction of the plume velocity at the same level, and the total buoyancy in the cap is increasing, so a modification of the simple relations for thermals is required.

This prediction is verified experimentally, and numerical values for the ratio of the velocities and the rate of increase of the cap radius with height determined. The extreme front of plume cap advances at about 0·6 times the mean velocity on the axis of a steady plume, and it spreads at just over half the angle of a thermal. This implies a smaller rate of entrainment and therefore a smaller rate of dilution per unit height compared with a thermal, especially since about half of the fluid mixed into the cap comes from the plume below. The model allows one to estimate the time necessary for convection from a known steady source on the ground to lead to the formation of a cloud.
Supersonic flow past cones of general cross-section
Authors: Stocker P. M. , Mauger F. E.
Published: March 2006

The differential equations representing the supersonic flow of a gas past a cone of any cross-section are integrated numerically, using a method similar to those used for bluff-body problems. A stream function is used as one of the independent variables and this is particularly suitable for determining the singular ‘vortical layer’. The method is here applied to the cases of elliptic cones at zero yaw and circular cones at incidence. The results are compared with experiment and with other numerical solutions.
The creeping motion of a non-Newtonian fluid past a sphere
Authors: Caswell B. , Schwarz W. H.
Published: March 2006

The equations of motion and continuity are solved together with the slow-flow stress equations for an incompressible Rivlin-Ericksen fluid. The boundary conditions for slow flow past a sphere are satisfied by matching inner (Stokes) and outer (Oseen) Reynolds-number expansions of the stream function. The terms in the inner expansion are the solutions of non-linear partial differential equations which are solved approximately by expanding in terms of a non-Newtonian parameter λ. The drag force on the sphere is obtained from the solution.
A note on the inviscid Orr-Sommerfeld equation
Authors: Miles John W.
Published: March 2006

The inviscid Orr-Sommerfeld equation for ϕ(y) in y > 0 subject to a null condition as y → ∞ is attacked by considering separately the intervals (0, y1) and (y1, ∞), such that the solution in (0, y1) can be expanded in powers of the wave-number (following Heisenberg) and the solution of (y1, ∞) regarded as real and non-singular. Complementary variational principles for the latter solution are determined to bound an appropriate parameter from above and below. It also is shown how the original differential equation may be transformed to a Riccati equation in such a way as to facilitate both the Heisenberg expansion of the solution in (0, y1) and numerical integration in (y1, ∞). These methods are applied to a velocity profile that is linear in (0, y1) and asymptotically logarithmic as y → ∞, and it is found that the mean of the two variational approximations is in excellent agreement with the results of numerical integration of the Riccati equation.
On the generation of surface waves by shear flows. Part 4
Authors: Miles John W.
Published: March 2006

The transfer of energy from wind to short surface waves through the viscous Reynolds stress in the immediate neighbourhood of the surface is explored. Resonance between the Tollmien-Schlichting waves for a given wind profile and the free-surface waves is shown to be an important possibility. Numerical results are given for water wave generation by a wind having a velocity profile that is linear in the neighbourhood of the surface and asymptotically logarithmic.
Electrode conduction processes in air plasmas
Authors: George D. W. , Messerle H. K.
Published: March 2006

Using an electrically driven shock tube with initial pressures of 0.1 to 1.0 mm Hg and shock speeds of about Mach 12 to 15, the resistance of an air plasma between two parallel probes has been measured by two different techniques and the results compared. In one, external voltages of from 0 to 100 V were applied to the probes and in the other, electromagnetically induced voltages of from 0 to 25 V were produced by the plasma's motion in a magnetic field of up to 3500 G. In either case the resistance was found to decrease as the current flow increased and was consistent with the equilibrium electronic conductivity of the air plasma at high current densities.
Magnetohydrodynamic pipe flow Part2. High Hartmann number
Authors: Shercliff J. A.
Published: March 2006

The paper presents an improved, second approximation for the laminar motion of a conducting liquid at high Hartmann number in non-conducting pipes of arbitrary cross-section under uniform transverse magnetic fields. A satisfactory comparison with the author's previously experimental pressure gradient/flow results is made for the case of a circular pipe.
A periodic boundary-layer flow in magnetohydrodynamics
Authors: Turcotte D. L. , Lyons J. M.
Published: March 2006

It is the purpose of this paper to solve a boundary-value problem posed by induction electromagnetic pumps and generators. Solutions are obtained by an expansion technique and a momentum method for the laminar, incompressible flow problem. For large values of the interaction parameter (μ2σH20λ/ρμe viscous effects are shown to be restricted to periodic boundary layers. In regions of high-field strength a local Hartmann solution is valid. Where the applied field is weak an inertial boundary layer is present which thickens in the upstream direction.

A logical explanation of this phenomenon is given. The condition that a boundary-layer type flow exist is obtained and is shown to be in general satisfied. The results show that inviscid theory may be used to calculate the overall performance of electromagnetic pumps and generators while the boundary-layer theory developed here may be used to obtain the wall shear stress.
Small-scale advection and the neutral wind profile
Authors: Taylor R. J.
Published: March 2006

The flow of air in the lowest few metres of the atmosphere is examined in the case when a neutrally stratified logarithmic profile encounters a sudden change in surface roughness. Examination of the available evidence suggests that the ratio of new to old friction velocity is given by the ratio of the roughness lengths raised to a power about equal to 0.09. Some field observations and wind-tunnel measurements over a change in roughness indicate that vertical displacement of the streamlines at all heights within the range considered commences very near the surface transition. Calculations are made of the fetch necessary for the new wind profile to be established up to a given height and indicate that its ratio to the height is not constant but varies with the height and amount of change in roughness. Over a fairly wide range this ratio is about 100-150.
On Oseen's approximation
Authors: Chester W.
Published: March 2006

An investigation is made into the validity of the Oseen equations, for incom-pressible, viscous flow past a body, as an approximation to the Navier-Stokes equations. It is shown that, when the body is such that a reversal of the uniform flow at infinity merely reverses any component of the force on the body without changing its absolute magnitude, that component can be determined correctly to the first order in the Reynolds number, though the detailed velocity field is not correct to this order. Moreover, this force can be deduced simply from a knowledge of the force on the body according to Stokes's approximation.

The analysis is also generalized to include the magneto-hydrodynamic effects when the fluid is conducting and the flow takes place in the presence of a magnetic field.
Hydromagnetic waves in a cylindrical plasma: an experiment
Authors: Jephcott D. F. , Stocker P. M.
Published: March 2006

In a previous paper (Woods 1962) a theory is presented which leads to a general dispersion relation for hydromagnetic waves in a dissipative plasma contained in a cylindrical tube. In the present paper experimental observations are compared with the predictions of this theory.

An experiment is described in which torsional hydromagnetic waves are excited in a gas discharge. Measurements of the wave velocity and damping are compared with solutions of the dispersion equation which are computed from measured values of the plasma parameters. The results are consistent with the theory, and good numerical agreement is obtained by assuming a loss of particles to the tube walls. There is evidence of a cut-off in wave propagation in the region of ion cyclotron resonance.
A round jet in an ambient co-axial stream
Authors: Maczyński J. F. J.
Published: March 2006

Measurements were made of the mean velocity profiles in a jet immersed in a stream whose velocity was in the same direction as that of the jet. They showed a mean velocity on the jet axis falling inversely as the distance downstream of the jet origin, a behaviour demonstrably inconsistent with any theory which assumes that the turbulence at each section of the jet is in local equilibrium. A mixing length, LM, and a length, Ld, characterizing the rate of dissipation are defined and it is shown that LM increases with distance downstream more rapidly than the jet width, and Ld less rapidly.
The Preston tube as a means of measuring skin friction
Authors: Head M. R. , Rechenberg I.
Published: March 2006

Preston's method of measuring skin friction, which makes use of a Pitot tube resting on the surface, depends upon the assumption of a region of flow similarity, adjacent to the wall, common to fully developed turbulent pipe flow and the turbulent boundary layer. Experiments performed elsewhere have cast considerable doubt on the validity of this assumption, and the present investigation was undertaken to establish whether or not it is justified.

Experiments were carried out in a short length of large-diameter pipe which could either form part of a very much longer pipe, giving fully developed turbulent pipe flow, or could be preceded by a conventional contraction and screens, giving a developing turbulent boundary layer.

Final results showed that for a given skin friction the Pitot tube reading was the same for both boundary layer and pipe flows, thus vindicating Preston's method and confirming the existence of a universal region of wall similarity. Initial experimental difficulties were found to be due to unexpectedly large circumferential variations in skin friction in the growing boundary layer.
A note on the bathtub vortex
Authors: Sibulkin Merwin
Published: March 2006

Observations were made of the vortex motion which occurs when a liquid is draining from a vessel through a hole in its bottom. It was found that, for relatively quiescent initial conditions, the direction of rotation reversed as the liquid surface approached the bottom of the vessel. An explanation of this phenomenon is proposed which is based upon the vorticity associated with the circumferential component of the flow in the boundary layer on the bottom of the vessel.
The effect of viscosity near the cylindrical boundaries of a rotating fluid with a horizontal temperature gradient
Authors: Rogers Ruth H.
Published: March 2006

Under certain conditions, the motion caused in an annulus of fluid by rotating it about its (vertical) axis of symmetry and at the same time subjecting it to a radial temperature gradient has been shown by Hide (1958) to be mostly concentrated in a narrow jet stream which meanders between the inner and outer cylindrical boundaries of the fluid in a regular wave pattern: this wave pattern has a small angular velocity relative to the cylindrical walls containing the fluid. A theoretical solution has been found by Davies (1959) which is valid in the main body of the fluid: this solution neglects viscosity (which is permissible except near the boundaries of the fluid), and is related to the absolute angular velocity of the wave pattern. The present paper introduces viscous boundary layers between the main body of the fluid and the cylindrical walls, in an attempt to find a relation between the angular velocity of the wave pattern and that of the walls. That this is only partially successful is due to the presence of the boundary layer at the rigid surface at the bottom of the fluid (which is rotating with the same angular velocity as the cylindrical walls): this layer is ignored in the present theory. In addition to this contribution towards a complete explanation of the steady motion, the theory describes qualitatively certain periodic oscillations (vacillation) which were observed by Hide in his experiments.
The movement of single large bubbles in closed vertical tubes
Authors: Goldsmith H. L. , Mason S. G.
Published: March 2006

The flow and deformation of single large air and liquid bubbles suspended in wetting liquids when moving under gravity in closed vertical tubes have been studied. The flow patterns in both phases and thickness of the liquid film surrounding the bubbles and the shape of the bubble ends were determined by direct observation through a microscope.

Quantitative agreement with a theory relating bubble velocity to film thickness in viscous flow, and to the tube radius in inertial flow was obtained. In viscous flow the deformation at a given interfacial tension was independent of bubble length and of the viscosity of the suspending liquid.

In viscous flow the end of the sedimenting bubbles were spheroidal, with the leading ends prolate and the trailing ends generally oblate. The axis ratio of the leading ends increased with increasing interfacial tension. Wave disturbances occurred at the trailing ends of the bubbles. In inertial flow the leading ends of rising air bubbles were hemispherical.
On driving a viscous fluid out of a tube
Authors: Cox B. G.
Published: March 2006

Two problems are considered. First, it is shown experimentally that the amount of viscous fluid left on the walls of a horizontal tube, when it is expelled by an inviscid fluid, reaches an asymptotic value of 0.60 of the amount required to fill the tube, when the parameter μU/T is increased, μ and T being the coefficients of viscosity and interfacial surface tension respectively, and U the velocity of the interface between the two fluids. Secondly, by neglecting the inertia terms in the equations of motion and the effect of gravity, a theory for the passage of this type of bubble is presented, together with experimental results in support of the theory. It is shown that such a solution is only valid under certain other restrictions, and then only to within half a tube diameter of the nose of the bubble.
The non-linear interaction of two disturbances in the thermal convection problem
Authors: Segel Lee A.
Published: March 2006

In the thermal convection problem with free boundaries, the interaction of two ‘roll’ disturbances is considered. The problem is reduced to a pair of non-linear ordinary differential equations, which should also provide a model for the interaction of two disturbances in more general situations than that for which these equations have been derived. The equations contain several parameters which necessitates a discussion of various possible types of solution. Some representative results are: (1) under certain circumstances, an equilibrium state may be composed of a mixture of a linearly stable disturbance and a linearly unstable disturbance; (2) for the thermal convection problem, when the Rayleigh number is slightly above the minimum critical value, the equilibrium state will contain only one of two linearly unstable disturbances. These and other results are compared with experimental observations.
Behaviour of macroscopic rigid spheres in Poiseuille flow Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams
Authors: Segré G. , Silberberg A.
Published: March 2006

An apparatus is described for determining particle passages through any selected point on a tube cross-section. The method depends on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region. The coincidence is registered and counted electronically. At higher particle concentrations coincidences are also registered arising from a pair-wise occupation of the light beams by two particles. An analysis is presented showing that these pair coincidences can be allowed for exactly in terms of experimentally measurable quantities.

The reliability and reproducibility of the method is discussed and illustrated by examples from sphere suspensions in Poiseuille flow.
Behaviour of macroscopic rigid spheres in Poiseuille flow Part 2. Experimental results and interpretation
Authors: Segré G. , Silberberg A.
Published: March 2006

It is shown that a rigid sphere transported along in Poiseuille flow through a tube is subject to radial forces which tend to carry it to a certain equilibrium position at about 0.6 tube radii from the axis, irrespective of the radial position at which the sphere first entered the tube. It is further shown that the trajectories of the particles are portions of one master trajectory and that the origin of the forces causing the radial displacements is in the inertia of the moving fluid. An analysis of the parameters determining the behaviour is presented and a phenomenological description valid at low Reynolds numbers is arrived at in terms of appropriate reduced variables. These phenomena have already been described in a preliminary note (Segré & Silberberg 1961). The present more complete analysis confirms the conclusions, but it appears that the dependence of the effects on the particle radius go with the third and not the fourth power as was then reported.

It is also shown that the description of the phenomena becomes more complicated at tube Reynolds numbers above about 30.
Higher approximations in boundary-layer theory Part 1. General analysis
Authors: Van Dyke Milton
Published: March 2006

Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.
Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer
Authors: Willmarth W. W. , Wooldridge C. E.
Published: March 2006

Measurements of the turbulent pressure field at the wall beneath a thick (5-inch) turbulent boundary layer produced by natural transition on a smooth surface are reported. The data include the mean-square pressure, parallel to the stream, and spatial correlation of the pressure transverse to the stream.

The root-mean-square wall pressure was 2.19 times the wall shear stress. The power spectra of the pressure were found to scale with the free-stream speed and the boundary-layer displacement thickness. A few tests with a rough surface showed that the increase in root-mean-square wall pressure was greater than the increase in wall shear stress.

The space-time correlation measurements parallel to the stream direction exhibit maxima at certain time delays corresponding to the convection of pressure-producing eddies at speeds varying from 0.56 to 0.83 times the stream speed. The lower convection speeds are measured when the spatial separation of the pressure transducers is small, or when only the pressure fluctuations at high frequencies are correlated. Higher convection speeds are observed when the spatial separation of the pressure transducers is large, or when only low frequencies are correlated. The result that low-frequency pressure fluctuations have the highest convection speed is in agreement with the measurements of Corcos (1959, 1962) in a fully turbulent tube flow. Analysis of these measurements also shows that both large- and small-scale pressure-producing eddies decay after travelling a distance proportional to their scale. More precisely, a pressure-producing eddy of large or small wavelength λ decays and vanishes after travelling a distance of approximately 6λ.

The transverse spatial correlation of the wall-pressure fluctuations was measured and compared with the longitudinal scale. Both the transverse and the longitudinal scale of the pressure fluctuations were of the order of the boundary-layer displacement thickness. The transverse and longitudinal scales of both large- and small-scale wall-pressure fluctuations were also measured and were also found to be approximately the same.
On spatially-growing finite disturbances in plane Poiseuille flow
Authors: Watson J.
Published: March 2006

Possible solutions of the Navier-Stokes equations are given representing certain finite disturbances in plane Poiseuille flow which vary withdistanceparallel to the bounding walls. These solutions are based on infinitesimal disturbances which vary exponentially withdistance(upstream or downstream) instead of with time, and they are more closely related to the disturbances investigated experimentally than the corresponding ‘time-dependent’ solutions.
A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability
Authors: Gaster M.
Published: March 2006

The frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially-growing wave having the same wave-number. For small rates of amplification it is shown that the frequencies are equal to a high order of approximation, and that the spatial growth is related to the time growth by the group velocity.
Three-dimensional viscous wakes
Authors: Steiger Martin H. , Bloom Martin H.
Published: March 2006

The velocity fields of three-dimensional viscous wakes are examined with the use of the boundary-layer approximations, Oseen's linearization of the convective terms, and the assumption of constant fluid properties. Transform methods yield solutions for general types of initial conditions. As an illustration, the axial velocity distribution of a wake whose initial isovels (lines of constant velocity) are of elliptic shape and their decay to axial symmetry are demonstrated. Both laminar and turbulent flows are considered.
Similarity in swirling wakes and jets
Authors: Reynolds A. J.
Published: March 2006

Both the net linear momentum and the net angular momentum of a developing swirling flow can play important parts in determining its ultimate form. To illustrate this the turbulent wake with both axial and swirl components of mean velocity is discussed, in particular for the two limiting cases of domination by linear momentum and domination by angular momentum. The dual conservation of axial and angular momentum implies that in general the mean swirl component decreases more rapidly downstream than does the defect in the mean axial velocity. Hence wakes with non-zero momentum flux ultimately have the familiar length scale $\sim Z^{\frac{1}{3}$ and velocity defect scale $\sim Z^{-\frac{2}{3}$. But in the wake of a self-propelled body the net drag is negligible and a swirl-dominated development can persist with length scale $\sim Z^{\frac{1}{4}$ and swirl velocity scale $\sim Z^{-\frac{3}{4}$.
Effect of natural convection on stability of flow in a vertical pipe
Authors: Scheele George F. , Hanratty Thomas J.
Published: March 2006

If water is heated or cooled while flowing through a vertical pipe with a laminar motion, the velocity profile will differ from the parabolic shape for isothermal flow due to density variations in the fluid. If a constant heat flux is used at the wall and if the changes in temperature affect only the density appearing in the gravity term of the equations of motion, a condition is attained far downstream in the heat-transfer section such that there is no further change in the velocity profile. The shape of this fully developed velocity profile depends on the ratio of the heat flux to the flow rate. The stability of flow in an electrically heated pipe 762 diameters long was studied by detecting temperature fluctuations in the effluent. By use of a carefully designed entry and a long isothermal section prior to the heat exchange section, inlet disturbances were eliminated and transition to an unsteady flow resulted from a natural instability of the distorted profiles. It was found that the stability depends primarily on the shape of the velocity profile and only secondarily on the value of the Reynolds number, if at all. For upflow heating the flow first becomes unstable when the velocity profiles develop points of inflexion. Transition to an unsteady flow involves the gradual growth of small disturbances and therefore it is quite possible to have unstable flows without observing transition because the pipe is not long enough for the disturbances to attain a measurable amplitude. For downflow heating the flow instability is associated with separation at the wall. Transition to an unsteady flow is sudden and therefore transition occurs shortly after an unstable flow occurs. It is suggested that a change from a steady symmetrical to a steady unsymmetrical flow occurs in downflow when the profile develops points of inflexion.
The instability to long waves of unbounded parallel inviscid flow
Authors: Drazin P. G. , Howard L. N.
Published: March 2006

Formulas for the determination of the instability characteristics of unbounded parallel flow are obtained for the case of long waves, and applied, together with some general results, to give a qualitative description of the different modes of instability of such flows. It is found that there is a finite number of different modes unstable to long waves, essentially one for each relative maximum and minimum of the velocity profile. These modes appear to become stable when the wavelength is sufficiently small, reducing to neutral solutions associated with inflexion points as stability is approached. The formulas are also useful for quantitative calculation of instability characteristics.
The motion of rigid particles in a shear flow at low Reynolds number
Authors: Bretherton F. P.
Published: March 2006

According to Jeffery (1923) the axis of an isolated rigid neutrally buoyant ellipsoid of revolution in a uniform simple shear at low Reynolds number moves in one of a family of closed periodic orbits, the centre of the particle moving with the velocity of the undisturbed fluid at that point. The present work is a theoretical investigation of how far the orbit of a particle of more general shape in a non-uniform shear in the presence of rigid boundaries may be expected to be qualitatively similar. Inertial and non-Newtonian effects are entirely neglected.

The orientation of the axis of almost any body of revolution is a periodic function of time in any unidirectional flow, and also in a Couette viscometer. This is also true if there is a gravitational force on the particle in the direction of the streamlines. There is no lateral drift. On the other hand, certain extreme shapes, including some bodies of revolution, will assume one of two orientations and migrate to the bounding surfaces or to the centre of the flow. In any constant slightly three-dimensional uniform shear any body of revolution will ultimately assume a preferred orientation.