bluuebunny/crossref_metadata_2025 huggingface dataset:
Note on the scattering of long waves in a rotating system
Authors: Williams W. E.
Published: March 2006

It is shown that the solution of the problem of scattering of long waves by a finite barrier in a rotating system may be obtained directly from the solution of the electromagnetic scattering problem for a finite strip. It is shown that the effect of rotation is to produce on both sides of the barrier, but away from the ends, a wave which is propagated without attenuation parallel to the barrier.
The behaviour of clusters of spheres falling in a viscous fluid Part 1. Experiment
Authors: Jayaweera K. O. L. F. , Mason B. J. , Slack G. W.
Published: March 2006

The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re < 0·03. When equal-sized spheres of Re > 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 < Re < 0·16. But, if 0·16 < Re < 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 < Re < 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re > 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.
The behaviour of clusters of spheres falling in a viscous fluid Part 2. Slow motion theory
Authors: Hocking L. M.
Published: March 2006

A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n < 7, which explains why this configuration is not observed for more than six spheres.
A theory for the mode of operation of the Hartmann air jet generator
Authors: Mørch K. A.
Published: March 2006

The Hartmann air jet generator (Hartmann 1939) is sound wave generator in which the sound is generated by oscillations of a shock developed in an over-expanded air jet by means of a blunt body: a resonator. A theory for the instability mechanism is advanced, and it is found that the result calculated from the theory are in good agreement with the experimental observations.
An experimental investigation of the streamlines in viscous fluid expelled from a tube
Authors: Cox B. G.
Published: March 2006

Experimental confirmation of some streamline patterns suggested by Taylor (1961) is described. It is shown, also, that when a viscous fluid is expelled from a tube by an inviscid fluid, the interface has a ‘localized’ effect on the fluid ahead of the interface between the two liquids, that is, the usual Hagen–Poiseuille law is obeyd in the viscous fluid except for the region behind a point about 1½ tube diameters ahead of the bubble.

Use of a recently developed tungsten-iodide lamp is also described.
The generation of symmetrical duct velocity profiles of high uniform shear
Authors: Livesey J. L. , Turner J. T.
Published: March 2006

The logical design method of Owen & Zienkiewicz (1957), for the generation of linear shear flow, is successfully applied to the case of symmetrical velocity profiles in a two-dimensional duct. Shear parameters as high as 0·8 (corresponding to $\hat{V}| \overline {V} = 1\cdot4$) are obtained experimentally. Some slight modification to the theoretical grid spacing is required because of the non-uniformity of the upstream flow.

The decay of the profiles is confined to that due to the developing boundary layer and to the breakdown of the physically impossible condition at the centre, where symmetry implies a change in sign of the uniform shear.
On the propagation of small disturbances in a relaxing gas with heat addition
Authors: Clarke John F.
Published: March 2006

General solutions for the linearized three-dimensional unsteady motion of a relaxing gas, including the effects of direct heat addition, are obtained using a Green's function technique. The possibility of direct heat addition into each separate mode of energy storage must be catered for, and the solutions have some bearing on the question of the heating of a gas by radiation.

Examples are given of initial-value and heat-source solutions in an unbounded domain and the boundary-value problem is exemplified by a study of the motion created by a spherical piston.
Saltation of uniform grains in air
Authors: Owen P. R.
Published: March 2006

The interaction between a turbulent wind and the motion of uniform saltating grains of sand or soil, so massive as to fail to enter into suspension, is examined on the basis of two complementary hypotheses. The first asserts that the effect of the moving grains on the fluid outside the region to which saltation is confined is similar to that of solid roughness of height comparable with the depth of the saltation layer. The second requires the concentration of particles engaging in the saltation to adjust itself so that the shear stress exerted by the wind on the ground—different from that acting on the fluid outside the saltation layer by an amount accountable to the change in horizontal momentum suffered by the particles in their passage through the fluid—is just sufficient to maintain the sand-strewn surface in a mobile state.

Existing experimental data on the wind profiles outside the saltation region and the horizontal flux of particles through it are shown to be consistent with these hypotheses.

The second hypothesis implies a self-balancing mechanism for controlling the concentration of saltating particles. For if the concentration is too low the shear stress at the surface rises above the value required merely to secure mobility and more particles are encouraged to leave the surface; conversely, too large a concentration depresses the surface stress, and the consequent loss of surface mobility inhibits saltation and reduces th concentration of particles until equilibrium is restored.
Asymptotic solutions in non-equilibrium nozzle flow
Authors: Blythe P. A.
Published: March 2006

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.
On the non-steady motion of visco-plastic liquids in porous media
Authors: Gheorghitza St. I.
Published: March 2006

The previous works on the motions of visco-plastic fluids in porous media are generalized in order to include the phenomenon of hysteresis. Only slowly varying motions are studied to any extent, when all inertial terms can be neglected without appreciable error. A general equation is deduced and some features of the motion are discussed. Simple particular cases without free or seepage surfaces are considered, when the fluid moves in the whole porous medium. A plane example points out the existence of different regions of motion during the variation of the heads in upper and lower water reservoirs. The necessity of further research on the flow of visco-plastic fluids through porous media is emphasized.
Modes of instability in spiral flow between rotating cylinders
Authors: Schwarz K. W. , Springett B. E. , Donnelly R. J.
Published: March 2006

A study of the stability of flow between concentric cylinders, with the inner one rotating, distinguished three kinds of instabilities: the familiar axisymmetric mode, an azimuthal mode with the predicted exp i(θ—ωt) angular dependence, and a completely non-symmetric instability which apparently arises from the interaction of the other two. The effect of small axial flow upon all of these modes was to give an approximately parabolic dependence of the critical Taylor number on the axial flow rate. In the case of the axisymmetric mode, agreement with the theory of Krueger & Di Prima (1964) was found to be excellent.
Turbulent flow characteristics of viscoelastic fluids
Authors: Metzner A. B. , Park M. Graham
Published: March 2006

In this paper the turbulent flow characteristics of viscoelastic fluids are investigated quantitatively. The outstanding property of these flow fields is seen to be a very pronounced suppression of turbulence, accompanied by major reductions in the turbulent drag coefficients. Careful measurements of the rheological properties of the several fluids used suggest that the observed turbulence suppression is a function of the ratio of the elastic to the viscous forces developed in the fluid. An empirical correlation of the results, based upon this observation, is proposed; the present data, while indicative, are not sufficiently extensive to verify conclusively the existence of a quantitative correlation.

In a number of respects, the observed reduction in drag is similar to that which may be obtained through promotion of ‘slip’ at the tube wall or by addition of particulate matter. It is shown that slip phenomena are clearly distinct from those studied in the present work but that particulate effects (albeit of much greater magnitude than observed heretofore) cannot be ruled out as contributory mechanisms. Further studies are thus required to determine the relative importance of continuum (viscoelastic) and particulate effects.
On the non-existence of subcritical instabilities in fluid layers heated from below
Authors: Sani R.
Published: March 2006

Using some recent results it is established that, for very general boundary conditions, time-independent subcritical instabilities do not exist for the non-linear thermoconvective stability problem.
Note on a heterogeneous shear flow
Authors: Miles John W. , Howard Louis N.
Published: March 2006

Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleigh's band to zero as the Richardson number increases from 0 to ¼.
An experimental investigation of the instability of a two-dimensional jet at low Reynolds numbers
Authors: Sato Hiroshi , Sakao Fujihiko
Published: March 2006

An experimental investigation was made of the stability of a two-dimensional jet at low Reynolds numbers with extremely small residual disturbances both in and around the jet. The velocity distribution of a laminar jet is in agreement with Bickley's theoretical result. The stability and transition of a laminar jet are characterized by the Reynolds number based on the slit width and the maximum velocity of the jet. When the Reynolds number is less than 10, the whole jet is laminar. When the Reynolds number is between 10 and around 50, periodic velocity fluctuations are found in the jet. They die out as they travel downstream without developing into irregular fluctuations. When the Reynolds number exceeds about 50, periodic fluctuations develop into irregular, turbulent fluctuations. The frequency of the periodic fluctuation is roughly proportional to the square of the jet velocity.

The stability of the jet against an artificially imposed disturbance was also investigated. Sound was used as an artificial disturbance. The disturbance is either amplified or damped in the jet depending on its frequency. The conventional stability theory was modified by considering the streamwise increase of Reynolds number. The experimental results are in agreement with the theoretical results.
On the viscous hypersonic blunt body problem
Authors: Bush William B.
Published: March 2006

The viscous hypersonic flow past an axisymmetric blunt body is analysed based upon the Navier-Stokes equations. It is assumed that the fluid is a perfect gas having constant specific heats, a constant Prandtl number, P, whose numerical value is of order one, and a viscosity coefficient varying as a power, ω, of the absolute temperature. Limiting forms of solutions are studied as the free-stream Mach number, M, and the free-stream Reynolds number based on the body nose radius, R, go to infinity, and ε = (γ − 1)/(γ + 1), where γ is the ratio of the specific heats, and δ = 1/(γ − 1) M2 go to zero.
The unsteady flow within a spinning cylinder
Authors: Wedemeyer E. H.
Published: March 2006

A theoretical analysis is given of the unsteady flow of a liquid within a cylinder of finite length started suddenly so as to spin about its axis. It is found that a secondary flow, caused by the end walls of the cylindrical container, has a strong effect on the generation of spin in the liquid. In the vicinity of the end walls the fluid motion is characterized by a boundary-layer flow, which can be either laminar or turbulent. The fluid within the boundary layers rotates faster than that at a large distance from the end walls, and therefore is thrown, by centrifugal forces, radially outwards. The radial outflow in the boundary layer creates a slow secondary motion within the spinning liquid. Due to the secondary flow, the transport of angular momentum from the walls to the interior is accomplished by convection rather than diffusion. A treatment is given for both laminar and turbulent end-wall boundary layers. The theoretical results are compared with experimental observations and good agreement is found.
On convection velocities in turbulent shear flows
Authors: Wills J. A. B.
Published: March 2006

The problem of defining an effective velocity of convection for turbulent fluctuations in a shear flow is considered, and the definitions adopted by various workers are discussed. An experiment in the shear layer of a circular jet has shown that the usual definitions, based on peaks of the space-time correlations of the fluctuations, yield convection velocities whose magnitudes depend on the value of space or time separation chosen. An alternative approach shows that, by considering the turbulent field as a superposition of harmonic travelling waves, a wave-number/velocity spectrum can be defined that lends itself to the definition of a wave-number-dependent convection velocity and an overall convection velocity, both of which have real physical significance. An experimental technique is described for obtaining the spectrum, and results are presented for one position in the shear layer of the jet.
Aligned-fields magnetogasdynamic wakes
Authors: Fan D. N.
Published: March 2006

The effect of compressibility is included here in a study of wakes created in two-dimensional, steady, aligned-fields, magnetogasdynamic flow past obstacles. The gas is assumed to be viscous, resistive, and thermally conducting. With the Oseen type of approximation as well as the magnetogasdynamic boundary-layer approximation, a great simplification in the formulation of wakes results. The boundary-layer equations, although linearized, still retain the coupling between the velocity, the magnetic, and the temperature fields. The solution of the magnetogasdynamic wake, in general, is a superposition of three individual-wake components, each satisfying a diffusion type of equation. Only one of them is capable of extending upstream. Hence the wake picture is generally characterized by a conventional downstream wake with also the possibility of the existence of an upstream one.

To be sure, the general features of the magnetogasdynamic wakes in aligned-fields flows are similar to those of an incompressible fluid; however, the flow now, instead of just being subalfvénic, must be subcritical for the upstream wake to occur. That is, the flow condition corresponding to the occurrence of the upstream wake is such that the sum of the square of the Mach number M∞ and the square of the Alfvén number A∞ is less than unity. Evidently it is the mechanism of magneto-sonic-wave propagation that modifies the transition of wakes from the Alfvénic point A∞ = 1 (for an incompressible fluid) to the subcritical are $A^2_\infty + M^2_\infty = 1$ in the Taniuti-Resler diagram.
The ‘piston problem’ with thermal radiation
Authors: Wang Kuo Chang
Published: March 2006

The classical problem of the motion of a one-dimensional unsteady shock generated by a piston moving with velocity vp = ctn is extended to take into account thermal radiation effects by the similarity method of Taylor and Sedov. Gray gas and local thermodynamic equilibrium are assumed and a modification of the Schuster-Schwarzschild differential equation for the heat flux is adopted. The optical thickness is not restricted to be thin or thick, and the absorption coefficient is assumed to vary with the density and temperature. Numerical results indicate that the pressure and velocity are not affected much by the radiation, but the density, temperature and radiant heat flux are changed considerably.
The radial spread of a liquid jet over a horizontal plane
Authors: Watson E. J.
Published: March 2006

When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated.
Experiments on the lift and drag of spheres suspended in a Poiseuille flow
Authors: Eichhorn R. , Small S.
Published: March 2006

An experimental investigation of the fluid dynamic forces on spheres suspended in a Poiseuille flow was performed. Small spheres of polystyrene, nylon, and Lucite, having diameters ranging from 0.061 in. to 0.126 in. were suspended in Poiseuille flows in a 0.419 in. diameter tube. Variations in particle size and density, the fluid properties, and the angle of inclination of the tube, resulted in a sphere Reynolds number (based on particle diameter and approach velocity) ranging from 80 to 250. The results are presented as curves which include the coefficients of lift and drag, and the dimensionless rotation speed plotted versus Reynolds number and a dimensionless shear parameter.
Long waves on the continental shelf: an experiment to separate trapped and leaky modes
Authors: Munk Walter , Snodgrass Frank , Gilbert Freeman
Published: March 2006

Random fluctuations in sea level, ζ, in the frequency range 0·1-60 cycles per hour were measured along the coast near Oceanside, California, where the coastline and bottom contours are fairly straight and parallel for 30 km. The two-dimensional covariance
$R(\eta, \tau) = \langle \zeta (y,t) \zeta (y + \eta, t+ \tau) \rangle$
was computed for points separated by various distances η along the coast. The Fourier transform
$S(f,n) = \int \int R(\eta, \tau)exp [2\pi i (n \eta + f \tau)]d \eta d \tau$
gives the contribution towards the ‘energy’ $\langle \zeta ^2 \rangle$ per unit temporal frequency f per unit spacial frequency (long-shore component) n. It is found that most of the energy is confined to a few narrow bands in (f, n) space, and these observed bands correspond very closely to the gravest trapped modes (or edge waves) computed for the actual depth profile. The bands are 0·02 cycles per km wide, which equals the theoretical resolution of the 30 km array. Very roughly S(f,n) ≈ S(f, -n), corresponding to equal partition of energy between waves travelling up and down the coast. Theory predicts ‘Coriolis splitting’ between the lines f± (n) corresponding to these oppositely travelling waves, but this effect is below the limit of detection. The principal conclusion is that most of the low-frequency wave energy is trapped.
The wall-jet in a moving stream
Authors: Kruka V. , Eskinazi S.
Published: March 2006

The wall-jet is the flow of fluid emanating from a narrow slot and blowing over a rigid wall. The configuration of the turbulent wall-jet is that of a very narrow, plane, turbulent half-jet investigated by Liepmann & Laufer (1947). The width of the slot is of the order of the boundary layer on the infinite wall and the jet, in this case, mixes with a stream moving with constant velocity. This flow has drawn considerable basic and applied interest in the past few years for it has the characteristics of both a boundary-layer and a free-mixing flow.
The Taylor column problem
Authors: Jacobs S.J.
Published: March 2006

We consider here flow past an obstacle on the lower of two rotating horizontal planes which bound a viscous fluid. It is found that when the Taylor number is large viscous effects are confined to Ekman boundary layers on the solid surfaces and to a free shear layer coincident with the vertical cylinder circumscribing the bottom obstacle. The flow in the main body of the fluid outside the cylinder proves to be two-dimensional with zero relative vorticity, while inside the cylinder the fluid is stagnant in the rotating frame. The shear layer provides a continuous transition between the exterior and interior of the cylinder, and in addition provides a means by which fluid from the Ekman layers on the horizontal planes is exchanged with fluid outside the Ekman layers and exterior to the circumscribing cylinder. The predicted flow proves to be in agreement with many of the experimental results.
Axial flow in trailing line vortices
Authors: Batchelor G. K.
Published: March 2006

A characteristic feature of a steady trailing line vortex from one side of a wing, and of other types of line vortex, is the existence of strong axial currents near the axis of symmetry. The purpose of this paper is to account in general terms for this axial flow in trailing line vortices. the link between the azimuthal and axial components of motion in a steady line vortex is provided by the pressure; the radial pressure gradient balances the centrifugal force, and any change in the azimuthal motion with distance x downstream produces an axial pressure gradient and consequently axial acceleration.

It is suggested, in a discussion of the evolution of an axisymmetric line vortex out of the vortex sheet shed from one side of a wing, that the two processes of rolling-up of the sheet and of concentration of the vorticity into a smaller cross-section should be distinguished; the former always occurs, whereas the latter seems not to be inevitable.

In § 4 there is given a similarity solution for the flow in a trailing vortex far downstream where the departure of the axial velocity from the free stream speed is small. The continual slowing-down of the azimuthal motion by viscosity leads to a positive axial pressure gradient and consequently to continual loss of axial momentum, the asymptotic variation of the axial velocity defect at the centre being as x−1 log x.

The concept of the drag associated with the core of a trailing vortex is introduced, and the drag is expressed as an integral over a transverse plane which is independent of x. This drag is related to the arbitrary constant appearing in the above similarity solution.
The conservation equations for a non-equilibrium plasma
Authors: Appleton J. P. , Bray K. N. C.
Published: March 2006

The paper is concerned with formulation of the gas dynamic conservation equations for the individual species in a non-equilibrium partially ionized gas mixture. As an example, the conservation equations for the electrons and the overall conservation equations are developed for a three component plasma consisting of electrons, singly-ionized positive ions and neutral atoms. Non-elastic collisions are represented by the collisional-radiative decay mechanism of Bates, Kingston & McWhirter (1962a, b). Maxwellian velocity distributions are assumed, but the electrons are allowed to have a temperature different from the heavier particles and to drift relative to them. Particular attention is given to the electron energy balance equation which differs from that used by other investigators.
On the transient motion of a contained rotating fluid
Authors: Greenspan H. P.
Published: March 2006

This paper considers the transient motion of a viscous fluid in a container rotating with constant angular velocity. The principal objective is to study the manner in which an arbitrary initial state of motion becomes a rigid rotation. In order to concentrate on the effects of viscosity, only the spherical container is studied here in great detail. A general theory will be presented in a subsequent publication.

Several sources of non-uniform behaviour make the analysis difficult and complex. In particular, there are three important time scales, viscous boundary layers, boundary-layer resonances at critical latitudes and intricate side-wall effects. The basic aproach consists of an expansion procedure by means of which the general inviscid solution is corrected for viscous effects and is made uniformly valid in time through the critical spin-up phase. Uniform validity is effected through the elimination of secular terms with unacceptable growth rates arising from the asymptotic perturbation series.

The interior (inviscid) motion leads to a non-self-adjoint partial differential equation eigenvalue problem with many intriguing properties. The general expansion theorem, orthogonality relationships, and viscous decay factors are deduced and used to solve the arbitrary intial-value problem. It is shown that the depth averaged circulation about circular contours, x2 + y2 = r2, is extracted from the fluid in the spin-up time scale T = L(Ωv)½. This is accomplished by a secondary non-oscillatory convective motion produced by suction into the Ekman layer. The excess circulation not eliminated in this way excites inviscid inertial oscillations which are also caused to decay by the boundary layers in the same time scale. Some very small residual effects decay in the ordinary viscous diffusion time, but all the essential processes are concluded in the much shorter interval. All modal oscillations in a sphere are determined and several specific calculations of frequency and decay rate are made and compared with experimental data. Perhaps the most important of these concerns the mode corresponding to rigid internal motion about another axis which can be produced by impulsively changing the rotation axis of the container. Agreement between theory and experiment is very good in all cases compared thus far.
The viscous incompressible flow inside a cone
Authors: Ackerberg Robert C.
Published: March 2006

The steady, axisymmetric, converging motion of a viscous incompressible fluid inside an infinite right circular cone is considered. It is shown that the exact solution of the Navier-Stokes equation for the stream function Ψ is of the form Ψ(r,θ) = AF(rv/A,θ), where (r, θ) are spherical polar co-ordinates chosen so r = 0 is the apex and θ = 0 is the axis of the cone, 2πA is the volumetric flow rate, and v the kinematic viscosity of the fluid. Asymptotic expansions of the stream function are found for large and small rv/A.

For large rv/A, Stokes's method for slow motions is generalized to obtain a complete asymptotic expansion. Except for cones of special angles, all terms in this expansion may theoretically be found.

For small rv/A a solution is constructed in two parts, namely, an inner expansion which starts from boundary-layer type equations as well as the no-slip condition at the wall, and an outer expansion in unstretched variables rv/A and cosθ which satisfies the boundary conditions at the axis of the cone. The condition that the inner solution merge with the outer solution with an exponentially small error requires an outer solution near the apex which is not potential sink flow, as might perhaps have been expected from the solution for two-dimensional flow in a wedge. The simplest outer flow satisfying the requirement is a vortex motion. Complete inner and outer expansions are developed and it is shown that they contain only six undetermined constants which must be determined by joining this solution numerically to the Stokes solution upstream. The inclusion of logarithmic terms in these expansions has not been found necessary.
Transition in circular Couette flow
Authors: Coles Donald
Published: March 2006

Two distinct kinds of transition have been identified in Couette flow between concentric rotating cylinders. The first, which will be called transition by spectral evolution, is characteristic of the motion when the inner cylinder has a larger angular velocity than the outer one. As the speed increases, a succession of secondary modes is excited; the first is the Taylor motion (periodic in the axial direction), and the second is a pattern of travelling waves (periodic in the circumferential direction). Higher modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This kind of transition may be viewed as a cascade process in which energy is transferred by non-linear interactions through a discrete spectrum to progressively higher frequencies in a two-dimensional wave-number space. At sufficiently large Reynolds numbers the discrete spectrum changes gradually and reversibly to a continuous one by broadening of the initially sharp spectral lines.

These periodic flows are not uniquely determined by the Reynolds number. For the case of the inner cylinder rotating and the outer cylinder at rest, as many as 20 or 25 different states (each state being defined by the number of Taylor cells and the number of tangential waves) have been observed at a given speed. As the speed changes, theso states replace each other in a repeatable but irreversible pattern of transitions; vortices appear or disappear in pairs, and waves are added or subtracted. More than 70 such transitions have been found in the speed range up to about 10 times the first critical speed. Regardless of the state, however, the angular velocity of the tangential waves is nearly constant at 0.34 times the angular velocity of the inner cylinder.

The second kind of transition, which will be called catastrophic transition, is characteristic of the motion when the outer cylinder has a larger angular velocity than the inner one. At a fixed Reynolds number, the fluid is divided into distinct regions of laminar and turbulent flow, and these regions are separated by interfacial surfaces which may be propagating in either direction. Under some conditions the turbulent regions may appear and disappear in a random way; under other conditions they may form quite regular patterns. One common pattern of particular interest is a spiral band of turbulence which rotates at very nearly the mean angular velocity of the two walls without any change in shape except possibly an occasional shift from a right-hand to a left-hand pattern. One example of this spiral turbulence is being studied in some detail in an attempt to clarify the role played in transition by interfaces and intermittency.
Laminar convection in a heated vertical tube rotating about a parallel axis
Authors: Morris W. D.
Published: March 2006

This paper presents an investigation into the influence of rotation on the laminar asymptotic velocity and temperature profiles obtained when fluid flows through a vertical tube which rotates about a parallel axis with uniform angular velocity, and which is subjected to a uniform temperature gradient. Rotation induces secondary free convection flow in the plane perpendicular to the axis resulting in non-symmetrical axial-velocity and temperature profiles which modify the solved using a series expansion in ascending powers of the rotational Rayleigh number, the resulting solution being approximate and valid only for low rates of heating.
On the mathematics of fluidization Part 1. Fundamental equations and wave propagation
Authors: Murray J. D.
Published: March 2006

When the upward flow of a fluid through a bed of particles of appropriate and almost uniform size is rapid enough so that the drag on each particle is as great as the particles buoyant weight, the particles do not remain close packed and the bed is said to be fluidized. Industrial uses of fluidized beds in the chemical and petroleum industries in particular are already extensive. Uses in the atomicenergy industry are being developed.

In this paper a mathematical model which describes the phenomena on a continuum basis is deduced. With this model we find that the system is unstable to small internal disturbances. Alternatively, we find that surface waves can be propagated (with attenuation) in the composite fluid and these waves for fluidized beds with a high ratio of solids density to fluid density are stable. These results are in agreement with experiment. Hot beds, where strongly exothermic reactions may be taking place, centrifugal beds (beds fluidized within a rotating system), and electromagnetic beds (those in which the particulate phase is electrically conducting) are all shown to be unstable to small internal disturbances.

The equations derived here may also be used as approximate equations for dispersed particle two-phase flow.
Determination of the sonic line in hypersonic flow past a blunt body
Authors: Bloor M. I. G.
Published: March 2006

The Newtonian theory of inviscid hypersonic flow is extended to obtain a solution uniformly valid in the subsonic region, and that is used to determine the position and shape of the sonic line. The main modification to the theory has to be made near the body surface and an expansion, essentially in terms of the stream function, is employed.

For simplicity the solution is limited to the cases of axially- and plane-symmetric flows. As an illustration of the theory the flows past a sphere and a circular cylinder are treated in some detail. Comparison with the numerical results of Garabedian and Lieberstein gives favourable agreement.
A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe
Authors: Gill A. E.
Published: March 2006

It is found that only a small change in either of the undisturbed velocity profiles concerned is required to change them from stable profiles to unstable profiles. The change must be such as to produce a local maximum in the magnitude of the vorticity, or in the case of the pipe, in the magnitude of the vorticity divided by the radius. The actual change in the vorticity (or vorticity/radius) need only be small, but the gradient of the vorticity (or vorticity/radius) must be finite. Viscosity will tend to damp out the distortion in the mean flow that is responsible for the instability, so that if the flow is to become turbulent, non-linear effects must become important before the distortion of the mean flow is reduced to an ineffective level. This requirement leads to the determination of critical Reynolds numbers which depend on the initial (small) distortion of the mean flow and the initial (smaller) amplitude of periodic disturbances. These critical Reynolds numbers are large.
Vorticity amplification in stagnation-point flow and its effect on heat transfer
Authors: Sutera S. P.
Published: March 2006

Recently a mathematical model was proposed (Sutera, Maeder & Kestin 1963) to demonstrate that vorticity amplification by stretching was an important mechanism underlying the sensitivity of stagnation-point heat transfer on cylinders to free-stream turbulence. According to the model, vorticity of a scale larger than a certain neutral scale and appropriately oriented can undergo amplification as it is convected towards the boundary layer. Such vorticity, present in the oncoming flow with small intensity, can reach the boundary layer with a greatly magnified intensity and induce substantial three-dimensional effects therein. The mean temperature profile was shown to be much more responsive to these effects than the mean velocity profile and very large increases in the wall-heat-transfer rate were calculated for Prandtl numbers 0·74 and 7·0.

In this work a second, more general, case is treated in which the approaching flow carries vorticity of scale 1·5 times the neutral. By means of iterative procedures applied on an electronic analogue computer, an approximate solution to the full Navier–Stokes equation is generated. The heat-transfer problem is solved simultaneously for Pr = 0·70, 7·0 and 100. It is found that a vorticity input which increases the wall-shear rate by less than 3% is capable of increasing the wall-heat-transfer rate by as much as 40%. The sensitivity of the thermal boundary layer depends on Prandtl number. In the three cases investigated it is greatest for Pr = 7·0 and least for Pr = 100.
The dispersion and attenuation of helicon waves in a uniform cylindrical plasma
Authors: Klozenberg J. P. , McNamara B. , Thonemann P. C.
Published: March 2006

A systematic account is given of the derivation of the dispersion relation for helicon waves in a uniform cylindrical plasma bounded by a vacuum. By retaining finite resistivity in the equations, boundary conditions present no difficulties, since the wave magnetic field is continuous through the plasma-vacuum interface. Two unexpected results are found. First, the wave attenuation remains finite in the limit of vanishing resistivity. This is due to the energy dissipated at the interface by the surface currents required to match the plasma wave field to the vacuum wave field. Zero wave attenuation for zero resistivity is recovered if electron inertia is included. Secondly, it is found that waves with azimuthal numbers m of opposite sign propagate differently, but the sense of polarization at the axis of the cylinder is independent of the sign of m.

The argument of the dispersion function is complex and numerical results were obtained using a computer. The method of programming is described, and results are given applicable to propagation in metals at low temperatures, or in a typical gas discharge plasma for the m = 0 and m = ± 1 modes.

An example of the amplitude of the wave fields as a function of radius is given for the axisymmetric mode, and of amplitude and phase for the m = ± 1 modes.
The instability of a fluid layer with time-dependent heating
Authors: Lick Wilbert
Published: March 2006

The problem of the stability and growth of disturbances in a fluid with time-dependent heating is investigated. The analysis is restricted to the case when the temperature gradient is large in a layer which is narrow by comparison with the overall depth of the fluid. An approximate method of solution is presented. Results of computations are also presented which illustrate the method of solution and the essential features of the problem.
Shock waves in a gas with several relaxing internal energy modes
Authors: Clarke John F. , Rodgers J. B.
Published: March 2006

The structure of plane steady shock waves in a gas with several internal energy modes which relax in parallel is investigated. Transport effects are neglected. Conditions for continuity and monotonicity of the velocity profile are discussed; when all modes have constant specific heats and relaxation times it is established that velocity must decrease monotonically. Internal mode energy contents may overshoot their local equilibrium values.

Numerical results for waves in a hypothetical gas with two relaxing modes are presented for purposes of illustration.
A computational method for viscous flow problems
Authors: Pearson Carl E.
Published: March 2006

A digital computer method for solving certain problems involving two-dimensional incompressible viscous flow is described. The time-dependent case is treated; the mathematical problem is thus that of solving a non-linear fourth-order partial differential equation in three variables. The choice of difference equations, of relaxation procedure, the kind of approximation to boundary conditions, and the resulting computational stability, speed, and accuracy are considered. Most experience so far has been for a rectangular region for which boundary velocities are prescribed as certain functions of time; an example of one such problem showing vortex formation and break-up is given.
Stability of spiral flow between concentric circular cylinders at low axial Reynolds number
Authors: Datta Subhendu K.
Published: March 2006

The stability of a viscous liquid between two concentric rotating cylinders with an axial flow has been investigated. Attention has been confined to the case when the cylinders are rotating in the same direction, the gap between the cylinders is small and the axial flow is small. A perturbation theory valid in the limit when the axial Reynolds number R → 0 has been developed and corrections have been obtained for Chandrasekhar's earlier results.
Thermal conductivity of argon at high temperatures
Authors: Camac Morton , Feinberg Robert M.
Published: March 2006

An infra-red heat-transfer gauge was used in a shock tube for end-wall measurements of the convective heat transfer from argon behind the reflected shock. The thermal conductivity of neutral (un-ionized) argon was measured before the ionization-relaxation time, and was fitted with the power-law temperature dependence 4·2 × 10−5(T/300)0·76±0·03cal/sec cm°K, whereTis measured in °K, and ±0·03 refers to the probable error The free-stream temperature ranged from 20,000 to 75,000°K, corresponding to incident-shock velocities from 3 to 6mm/μsec. At later times, after the free stream established equilibrium ionization, the convective-heat-transfer rate remained the same as the initial rate with neutral argon. Theoretical predictions of Fay & Kemp (1965), assuming equilibrium-boundary-layer conditions, are 20–30% below the experimental values. Also reported in this paper are measurements of the ionization times behind the reflected shock, and these are in agreement with an extrapolation of the Petschek & Byron (1957) measurements behind the incident shock. There is a discussion of the large changes in the gas conditions behind the reflected shock due to the ionization process. The final equilibrium conditions are reached abruptly, as indicated by the continuum-radiation emission which becomes constant immediately after ionization relaxation.
Space-charge waves carried by a plasma trapped in a potential well
Authors: Kahn F. D.
Published: March 2006

A mono-energetic electron plasma is trapped in a one-dimensional parabolic potential well. In the undisturbed state a suitable background charge provides space-charge neutrality. It is shown that stationary disturbances of infinitesimal amplitude can exist in the plasma for certain critical values of the parameter $\Lambda = 4\overline{n}e^2|m \omega^2$, where $\overline{n}$ is the mean electron density and ω/2π is the frequency of oscillation of the electrons in the well. The first few critical values are Λ = 0, 4·12, 8·2.

The boundary conditions at the end of the plasma are non-linear. As a result stationary disturbances of finite amplitude in a given mode, say the rth, require that Λ shall exceed Λr. Further it can be shown that disturbances of small amplitude in the rth mode are unstable when Λ exceeds Λr. This applies even for Λr = 0; in this case there exist nearby an unstable even and an unstable odd mode.

It seems likely that these results can be extended to all cases in which the potential well is symmetrical.
The steady separated flow past a circular cylinder at large Reynolds numbers
Authors: Acrivos Andreas , Snowden D. D. , Grove A. S. , Petersen E. E.
Published: March 2006

This paper is concerned with deducing the most important features of the steady separated flow past a circular cylinder in the limit of vanishing viscosity. First of all, it is shown that the experimental results reported in an earlier article cannot be reconciled with the notion that, as the Reynolds number Re is increased, the flow becomes inviscid everywhere and that viscous effects remain confined within infinitesimally thin shear layers. In contrast, the limiting solution is visualized as exhibiting three essential features: a viscous, closed ‘wake bubble’ of finite width but with a length increasing linearly with Re in which inertial and viscous effects are everywhere of equal order of magnitude; an outer inviscid flow; and, separating the two regions, a diffuse viscous layer covering large sections of the external field. Further properties of this asymptotic solution include: a finite form drag, a negative rear pressure coefficient at the rear stagnation point of the cylinder, and a Nusselt number for heat transfer which becomes independent of Re along the non-wetted portion of the cylinder surface. This model is shown to be consistent with all the experimental data presently available, including some new heat transfer results that are presented in this paper.

An approximate technique is also proposed which takes into account the asymptotic character of the flow in the vicinity of the cylinder and which predicts the pressure distribution around the cylinder in good agreement with the experiments.