The effect of viscous dissipation in natural convection is appreciable when the induced kinetic energy becomes appreciable compared to the amount of heat transferred. This occurs when either the equivalent body force is large or when the convection region is extensive. Viscous dissipation is considered here for vertical surfaces subject to both isothermal and uniform-flux surface conditions. A perturbation method is used and the first temperature perturbation function is calculated for Prandtl numbers from 10−2 to 104. The magnitude of the effect depends upon a dissipation number, which is not expressible in terms of the Grashof or the Prandtl number.

In experiments on the stability of a submerged axisymmetric jet at Reynolds numbers large compared with unity, Reynolds (1962) observed axisymmetric ‘condensations’ which appear to grow spontaneously, whereas Batchelor & Gill (1962) have shown that infinitesimal disturbances of this type do not grow in an inviscid fluid. Here it is shown that axisymmetric disturbances do not grow in a slightly viscous fluid either, and the solution for which the rate of damping is smallest is found. It is suggested that the growth of small but finite disturbances is responsible for the condensations observed. The order of magnitude of the disturbance velocity at which non-linear effects could produce growth of disturbance is found to depend on the wave-number of the disturbance. The smallest velocity which a ‘finite’ disturbance may have is found to be of order $R^{-\frac{2}{3}$, and corresponds to a disturbance whose wave-number is of order $R^{\frac{1}{3}$, R being the Reynolds number based on the local radius and maximum velocity of the jet. On the assumption that some disturbances whose velocity is of this order will grow, deductions are made as to the size, position, wave-number, and point of appearance of condensations. The deductions appear to agree with the experimental results.

The classical theory of the instability of laminar flow predicts the growth (or decay) rate and phase velocity of two-dimensional small disturbance waves. In order to study the growth and dispersion of an originally localized spot-like disturbance, the initial disturbance is built up from all possible simple-harmonic waves. The propagation velocity and amplification rate for each vector wave-number then follows from the two-dimensional theory by Squire's generalization. The initial development can be solved explicitly by a power series in time, and the asymptotic behaviour is also predicted. For times between initial and final periods, exact numerical calculations have been made using an IBM 709 electronic computer. The role which localized disturbances can play in ultimate transition to turbulent motion is also indicated.

Some general properties of one-dimensional deflagration waves in a non-conducting inviscid gas at rest are discussed when ionization of the gas takes place across a shock wave which precedes the flame front, and electromagnetic fields are present. The direction of wave propagation, the electric field and magnetic field are taken as a mutually orthogonal triad of vectors. The jump relationships across the gas-ionizing shock wave and magnetogasdynamic combustion wave are investigated and the two Hugoniot curves analysed in detail in the pressure-specific volume plane. The possible types of wave are indicated for arbitrary magnitudes of the upstream electromagnetic field. It is shown that weak gasionizing shock waves cannot exist. For suitably chosen electromagnetic field strenghts the density ratio across the shock wave may be greater than the ordinary gasdynamic limit and, in such cases, the pressure and density ratios are related in an inverse manner, in contrast to the behaviour for ordinary gasdynamic or magnetogasdynamic shock waves. The magnetogasdynamic combustion wave has similar properties to that in ordinary gasdynamics.

The Navier-Stokes equations for a viscous, incompressible fluid are considered for a steady, axisymmetric flow composed of a strong rotation combined with radial sink flow which exhausts axially inside a finite radius. The equations are reduced to two coupled partial differential equations in terms of the stream function and circulation. The equations contain three dimensionless parameters: the radial Reynolds number, a characteristic ratio of mass flow per unit lenght to circulation, and a characteristic ratio of an axial dimension to a radial dimension. The product of these last two dimensionless parameters is used as a new expansion parameter for generating an asymptotic series solution. To zeroth order in this parameter, the solution for the stream function is a linear distribution between two axial boundary values. First-order correction terms are calculated for a specific example.

In discussing these equations the limitations of the exact solutions due to Donaldson & Sullivan (1960) and Long (1961) are noted. These exact solutions are contrasted with the approximate treatment of this type of vortex originated by Einstein & Li (1951) and generalized by Deissler & Perlmutter (1958).

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.

Standing surface waves in an inviscid incompressible fluid of finite depth are considered, taking into account the effect of capillary forces. Perturbation solutions for the surface profile, velocity potential, frequency of oscillation, and pressure are found to third order in the amplitude of the waves. A graph is given showing the regions in which the frequency of oscillation increases with amplitude and those in which it decreases with amplitude. These regions are defined as a function of the depth of the fluid and a parameter called the relative capillarity. A graph is also given showing the surface profile of a wave.

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.

In earlier papers Phillips (1960) and Longuet-Higgins (1962) have investigated phase velocity effects and possible resonances associated with the interactions of gravity waves. In this note the problem is discussed from a different viewpoint which demonstrates more clearly the energy-sharing mechanism involved. Equations governing the time dependence of the resonant modes are obtained, rather than the initial growth rate as has been found previously.

The effect of radiative transfer on the thermal stability of an ionized fluid, in the presence of uniform vertical magnetic field has been examined from hydromagnetic approximations. Two asymptotic cases (i) when the fluid is optically thin and (ii) when it is optically thick, have been examined in detail. The principle of exchange of stabilities and the concept of over-stability have been discussed and the physical conditions under which the former will hold, obtianed. Also the conditions as to which type of instability will arise first are derived and it has been shown that in the presence of large magnetic field over-stability arises earlier than convection under the first approximation.

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.

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 $\sim Z^{-\frac{3}{4}$.

This paper describes an experiment in which a cylindrical vortex, formed in a long tube, was used to study the ‘vortex breakdown’ that has been previously reported in investigations of the flow over slender delta wings. By varying the amount of swirl that was imparted to the fluid before it entered the tube, it was found that the breakdown was the intermediate stage between the two basic types of rotating flows, that is, those that do and those that do not exhibit axial velocity reversal. In addition, it was shown that an unusual flow pattern was established after the breakdown and that certain features of this pointed to it being a ‘critical’ phenomenon. The tests were concluded by measuring the swirl angle distribution a short distance ahead of the breakdown and comparing these results with the prediction of Squire's theory (1960).

The boundary layer on a semi-infinite flat plate placed in a two-dimensional, unbounded, steady, constant shear flow of a viscous incompressible fluid is examined on the basis of the constant-pressure assumption. An asymptotic solution is obtained first for large vorticity numbers. Then an approximate solution is found for arbitrary vorticity numbers that gives good agreement with exact calculations for the extreme cases of small and large vorticity numbers. The present calculations are limited to the boundary layer on the top side of the plate only.

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.

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 phenomenon examined is the abrupt structural change which can occur at some station along the axis of a swirling flow, notably the leading-edge vortex formed above a delta wing at incidence. Contrary to previous attempts at an explanation, the belief demonstrated herein is that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone. It is instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in openchannel flow. A set of properties essential to such a transition, corresponding to a set shown to provide a complete explanation for the hydraulic jump, is demonstrated with wide generality for axisymmetric vortex flows; and the interpretation covers both the case of mild transitions, where an undular structure is developed without the need arising for significant energy dissipation, and the case of strong ones where a region of vigorous turbulence is generated. An important part of the theory depends on the calculus of variations; and the comprehensiveness with which certain properties of conjugate flow pairs are demonstrable by this analytical means suggests that present ideas may be useful in various other problems.

Some general stability criteria for non-dissipative swirling flows are derived, and extended to the case of an electrically conducting fluid in the presence of axial magnetic field and current. In particular it is shown that the analogy between a rotating and a stratified fluid holds in this case, and that an important determinant of stability is a ‘Richardson number’ based on the analogue of the density gradient and the shear in the axial flow.

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.

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.