A quantitative analogy between fully developed turbulent flows in curved pipes and orthogonally rotating pipes will be described through similarity arguments, the use of experimental data and computational results. A pair of similarity parameters will be derived for each turbulent flow, so that they have the same dynamical meaning as those of laminar flows. When the second parameter for each flow is large enough, it will be shown that friction factors, as well as heat transfer rates, of the two flows coincide for equal values of the fundamental parameters. Computed contours of velocity and temperature will also reveal strong similarities between the two flows.

The evolution of a tripolar vortex under the influence of a parabolic topography – like the free surface of a rotating fluid – is studied experimentally and with a point-vortex model. Laboratory experiments reveal that tripoles generated off-axis become asymmetric and the whole structure travels towards the centre of the tank along an anticyclonic spiral. During this translation the structure rotates quasi-periodically with the core pairing alternately with one of the satellites. An asymmetric point-vortex tripole (with the central vortex located at a distance ε from the middle point of the configuration) displays a periodic motion which is qualitatively similar to the motion of the laboratory tripoles. The exchange of fluid between the three vortices as a function of the perturbation parameter ε is studied using the lobe-dynamics technique. A point-vortex tripole modulated on the basis of conservation of potential vorticity reproduces quantitatively the trajectories of the individual vortices measured in the laboratory. As in the experiments, the model shows that fluid is strongly stirred in the region surrounding the vortex cores and that the tripole carries a finite amount of fluid.

Stability and transition to turbulence are studied in a simple incompressible two-dimensional bounded swirling flow with a rectangular planform – a vortex in a box. This flow is unstable to three-dimensional disturbances. The instability takes the form of counter-rotating swirls perpendicular to the axis which bend the vortex into a periodic wave. As these swirls grow in amplitude the primary vorticity is compressed into thin vortex layers. These develop secondary instabilities which roll up into vortex tubes. In this way the flow attains a turbulent state which is populated by intense elongated vortex tubes and weaker vortex layers which spiral around them. The flow was computed at two Reynolds numbers by spectral methods with up to 2563 resolution. At the higher Reynolds number broad three-dimensional shell-averaged energy spectra are found with nearly a decade of Kolmogorov k−5/3 law and small-scale isotropy.

This paper is concerned with the impact of a jet of arbitrary cross-section onto a rigid plane. At the initial stage of the impact the liquid motion is described within the framework of the acoustic approximation. Behind the shock front which is generated under the impact, the pressure distribution is calculated analytically for an arbitrary cross-section of the jet. It is shown that a quarter of the total impact energy transfers into the internal energy of the compressed liquid. The focusing of the compression and relief waves in the axisymmetrical case is discussed.

The onset of the folding effect characteristic of highly viscous liquid films (plane jets) slowly impinging on a wall is studied. Nonlinear quasi-one-dimensional equations are derived to describe the flow. In the linear approximation they reduce to the eigenvalue problem, whose solution predicts that instability (the onset of folding) sets in when the length of the film exceeds a critical value. The critical folding heights and the oscillation frequencies at the onset of instability are predicted as a function of flow parameters. Theoretical results are compared with Cruickshank's (1988) experimental data. Agreement is quite good only in the range of parameters where the quasi-one-dimensional approximation is applicable (thin films at the onset of folding).

The paper studies certain effects of non-symmetry on vortex/wave interactions, for inviscid inflexional waves interacting nonlinearly with the vortex component of the mean flow in boundary-layer transition at large Reynolds number. Two types of non-symmetry are investigated, namely for unequal input wave amplitudes and for small cross-flows. These lead to coupled integro-differential equations for spatial development of the wave amplitudes, which are examined in an essentially equivalent differential form for various degrees of the non-symmetry present. Each type of non-symmetry can have a significant influence on the nonlinear interaction properties. Special emphasis is given to bounded solutions, and numerous interesting new flow responses are found analytically and computationally. The theory provides a basis for tackling enhanced non-symmetry in the input or stronger cross-flows.

The evaporation and combustion of a single-component fuel droplet which is moving slowly in a hot oxidant atmosphere have been analysed using perturbation methods. Results for the flow field, temperature and species distributions in each phase, inter-facial heat and mass transfer, and the enhancement of the mass burning rate due to the presence of convection have all been developed correct to second order in the translational Reynolds number. This represents an advance over a previous study which analysed the problem to first order in the perturbation parameter. The primary motivation for the development of detailed analytical/numerical solutions correct to second order arises from the need for such a higher-order theory in order to investigate fuel droplet ignition and extinction characteristics in the presence of convective flow. Explanations for such a need, based on order of magnitude arguments, are included in this article. With a moving droplet, the shear at the interface causes circulatory motion inside the droplet. Owing to the large evaporation velocities at the droplet surface that usually accompany drop vaporization and burning, the entire flow field is not in the Stokes regime even for low translational Reynolds numbers. In view of this, the formulation for the continuous phase is developed by imposing slow translatory motion of the droplet as a perturbation to uniform radial flow associated with vigorous evaporation at the surface. Combustion is modelled by the inclusion of a fast chemical reaction in a thin reaction zone represented by the Burke–Schumann flame front. The complete solution for the problem correct to second order is obtained by simultaneously solving a coupled formulation for the dispersed and continuous phases. A noteworthy feature of the higher-order formulation is that both the flow field and transport equations require analysis by coupled singular perturbation procedures. The higher-order theory shows that, for identical conditions, compared with the first-order theory both the flame and the front stagnation point are closer to the surface of the drop, the evaporation is more vigorous, the droplet lifetime is shorter, and the internal vortical motion is asymmetric about the drop equatorial plane. These features are significant for ignition/extinction analyses since the prediction of the location of the point of ignition/extinction will depend upon such details. This article is the first of a two-part study; in the second part, analytical expressions and results obtained here will be incorporated into a detailed investigation of fuel droplet ignition and extinction. In view of the general nature of the formulation considered here, results presented have wider applicability in the general areas of interfacial fluid mechanics and heat/material transport. They are particularly useful in microgravity studies, in atmospheric sciences, in aerosol sciences, and in the prediction of material depletion from spherical particles.

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. The effects of gravity are considered to be negligible, and the contact line is assumed to have a stationary circular or elliptical shape. In the absence of shear flow, the drop assumes a hydrostatic shape with constant mean curvature. Families of hydrostatic shapes, parameterized by the drop volume and aspect ratio of the contact line, are computed using an iterative finite-difference method. The results illustrate the effect of the shape of the contact line on the distribution of the contact angle around the base, and are discussed with reference to contact-angle hysteresis and stability of stationary shapes. The transient deformation of a drop whose viscosity is equal to that of the ambient fluid, subject to a suddenly applied simple shear flow, is computed for a range of capillary numbers using a boundary-integral method that incorporates global parameterization of the interface and interfacial regriding at large deformations. Critical capillary numbers above which the drop exhibits continued deformation, or the contact angle increases beyond or decreases below the limits tolerated by contact angle hysteresis are established. It is shown that the geometry of the contact line plays an important role in the transient and asymptotic behaviour at long times, quantified in terms of the critical capillary numbers for continued elongation. Drops with elliptical contact lines are likely to dislodge or break off before drops with circular contact lines. The numerical results validate the assumptions of lubrication theory for flat drops, even in cases where the height of the drop is equal to one fifth the radius of the contact line.

A theory is described for the nonlinear waves on the surface of a thin film flowing down an inclined plane. Attention is focused on stationary waves of finite amplitude and long wavelength at high Reynolds numbers and moderate Weber numbers. Based on asymptotic equations accurate to the second order in the depth-to-wavelength ratio, a third-order dynamical system is obtained after changing to the frame of reference moving at the wave propagation speed. By examining the fixed-point stability of the dynamical system, parametric regimes of heteroclinc orbits and Hopf bifurcations are delineated. Extensive numerical experiments guided by the linear analyses reveal a variety of bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. These include homoclinic bifurcations which lead to homoclinic orbits corresponding to well separated solitary waves with one or several humps, some of which occur after passing through chaotic zones generated by period-doublings. There are also cases where chaos is the ultimate state following cascades of period-doublings, as well as cases where only limit cycles prevail. The dependence of bifurcation scenarios on the inclination angle, and Weber and Reynolds numbers is summarized.

A nonlinear analysis is performed numerically for the motion of an electrically conducting fluid between parallel plates in relative motion when a transverse magnetic field is applied. It is found that steady three-dimensional finite-amplitude solutions exist even when the linear analysis predicts an infinite critical Reynolds number.

We consider the solidification of a binary alloy in a mushy layer and analyse the linear stability of a quiescent state with specific interest in identifying an oscillatory convective instability. We employ a near-eutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. We consider also the limit of large Stefan number, which incorporates a key balance necessary for the existence of the oscillatory instability. The model we consider here contains no double-diffusive effects and no region in which a statically stable density gradient exists. The mechanism underlying the oscillatory instability we discover is instead associated with a complex interaction between heat transfer, convection and solidification.

Dilute mixtures of 3He in superfluid 4He have Prandtl numbers easily tunable between those of liquid metals and water: 0.04 < Pr < 2. Moreover, owing to the tight coupling of the temperature and concentration fields, superfluid mixture convection is closely analogous to classical Rayleigh–Bénard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids, well described by the Boussinesq equations. This work has two goals. The first is to put the theory of superfluid mixture convection on a firmer basis. We accomplish this by combining experiment and analysis to measure superfluid effects on the onset of convection. In the process, we demonstrate quantitative control over superfluid effects and, in particular, that deviations from classical convective behaviour can be made small or at worst no larger than finite aspect ratio effects. The size of superfluid effects at convective onset can be less than a few percent for temperatures 1 < T < 2 K. Comparison of the measured properties of superfluid mixture roll instabilities above the onset of convection (e.g. skewed varicose, oscillatory, and particularly near the codimension-2 point) to the properties predicted by Boussinesq calculations further verifies that superfluid mixtures convect as classical fluids.

With superfluid effects understood and under control, the second goal, presented in Part 2, is to exploit the unique Pr range of superfluid mixtures and the variable aspect ratio (Γ) capabilities of our experiment to survey convective instabilities in the broad, and heretofore largely unexplored, parameter space 0.12 < Pr < 1.4 and 2 < Γ < 95. The aim is to identify and characterize time-dependence and chaos, and to discover new dynamical behaviour in strongly nonlinear convective flows.

Dilute mixtures of 3He in superfluid 4He have Prandtl numbers easily tunable between those of liquid metals and water: 0.04 < Pr < 2. Moreover, superfluid mixture convection is closely analogous to classical Rayleigh–Bénard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids. This work has two goals. The first, accomplished in Part 1, is to experimentally validate the superfluid mixture convection analogue to Rayleigh–Bénard convection.

With superfluid effects understood and under control, the second goal is to identify and characterize time-dependence and chaos and to discover new dynamical behaviour in strongly nonlinear convective flows. In this paper, Part 2, we exploit the unique Pr range of superfluid mixtures and the variable aspect ratio (Γ) capabilities of our experiment to survey convective instabilities in the broad, and heretofore largely unexplored, parameter space 0.12 < Pr < 1.4 and 2 < Γ < 95. Within this large parameter space, we have focused on small to moderate Γ and Pr and on large Γ with Pr ≈ 1. The novel behaviour uncovered in the survey includes the following. Changing attractors: at Γ = 6.0 and Pr = 0.3, we observe intermittent bursting destabilizing a fully developed chaotic state. Above the onset of bursting the average length of a burst-free interval and the average length of a burst vary as power laws. At Γ = 4.25 and Pr = 0.12 we observe a particularly novel reversible switching transition involving two chaotic attractors. Instability competition: near the codimension-2 point at the crossing of the skewed-varicose and oscillatory instabilities we find that the effects of instability competition greatly increase the complexity and multiplicity of states. A heat-pulse method allows selection of the active state. Decreasing Γ suppresses the available complexity. Superfluid turbulence: we find that the large-amplitude noisy states, previously believed due to superfluid turbulence, are confined to small values of Γ and Pr and are not consistent with superfluid turbulence. Changing instabilities: at Pr = 0.19 a wavevector detuning changes the type of secondary instability from oscillatory to saddle-node, with an unusual 3/4 exponent time scaling. Very large Γ: at Pr = 1.3 for Γ increasing from 44 to 90, we observe the onset of convection changing from ordered and stationary to disordered and time-dependent. At the beginning of the crossover there are hysteretic transitions to coherent oscillations close to the onset of convection. By the end of the crossover convection is time-dependent and irregular at onset with the fluctuation amplitude correlated with the mean Nusselt number.

A matched asymptotic analysis is presented that describes the mechanical response of the vestibular semicircular canals to rotation of the head and includes the fluid–structure interaction which takes place within the enlarged ampullary region of the duct. New theoretical results detail the velocity field in a fluid boundary layer surrounding the cupula. The governing equations were linearized for small perturbations in fluid displacement from the prescribed motion of the head and reduced asymptotically by exploiting the slender geometry of the duct. The results include the pressure drop around the three-dimensional endolymphatic duct and through the transitional boundary layers within the ampulla. Results implicitly include the deflected shape of the cupular partition and provide an expression for the dynamic boundary condition acting on the two surfaces of the cupula. In this sense, the analysis reduces the three-dimensional fluid dynamics of the endolymph to a relatively simple boundary condition acting on the surfaces of the cupula. For illustrative purposes we present specific results modelling the cupula as a simple viscoelastic membrane. New results show that the multi-dimensional fluid dynamics within the enlarged ampulla has a significant influence on the pointwise deflection of the cupula near the crista. The spatially averaged displacement of the cupula is shown to agree with previous macromechanical descriptions of endolymph flow and pressure that ignore the fluid–structure interaction at the cupula. As an example, the model is applied to the geometry of the horizontal semicircular canal of the toadfish, Opsanus tau, and results for the deflection of the cupula are compared to individual semicircular canal afferent responses previously reported by Boyle & Highstein (1990). The cupular-shear-angle gain, defined by the angular slope of the cupula at the crista divided by the angular velocity of the head, is relatively constant at frequencies from 0.01 Hz up to 1 Hz. Over this same range, the phase of the cupular shear angle aligns with the angular velocity of the head. Near 10 Hz, the shear-angle gain increases slightly and the phase shows a lead of as much a 30°. Results are sensitive to the cupular stiffness and viscosity. Comparing results to the afferent responses represented within the VIIIth nerve provides additional theoretical evidence that the macromechanical displacement of the cupula accounts for the behaviour of only a subset of afferent fibres.