The extensional flow of a polymeric fluid during the fibre spinning process is studied for finite-amplitude stability behaviour. The spinning flow is assumed to be inertialess and isothermal. The nonlinear extensional rheology of the polymer is described with the help of the eXtended Pom-Pom (XXP) model, which is known to exhibit a significant strain hardening effect necessary for fibre spinning applications. The linear stability analysis predicts an instability known as draw resonance when the draw ratio, $\mathit{DR}$, defined as the ratio of the velocities at the two ends of the fibre in the air gap, exceeds a certain critical value, $\mathit{DR}_{c}$. The critical draw ratio $\mathit{DR}_{c}$ depends on the fluid elasticity represented by the Deborah number, $\mathit{De}={\it\lambda}v_{0}/L$, the ratio of the polymer relaxation time to the flow time scale, thus constructing a stability diagram in the $\mathit{DR}_{c}$–$\mathit{De}$ plane. Here, ${\it\lambda}$ is the characteristic relaxation time of the polymer, $v_{0}$ is the extrudate velocity through the die exit and $L$ is the length of the air gap for the spinning flow. In the present study, we carry out a weakly nonlinear stability analysis to examine the dynamics of the disturbance amplitude in the vicinity of the transition point. The analysis reveals the nature of the bifurcation at the transition point and constructs a finite-amplitude manifold providing insight into the draw resonance phenomena. The effect of the fluid elasticity on the nature of the bifurcation and the finite-amplitude branch is examined, and the findings are correlated to the extensional rheological behaviour of the polymer fluid. For flows at small Deborah number, the Landau constant, which captures the role of nonlinearities, is found to be negative, indicating supercritical Hopf bifurcation at the transition point. In the linearly unstable region, the equilibrium amplitude of the disturbance is estimated and shows a limit cycle behaviour. As the fluid elasticity is increased, initially the equilibrium amplitude is found to decrease below its Newtonian value, reaching the lowest value for $\mathit{De}$ when the strain hardening effect is maximum. With further increase in elasticity, the material undergoes strain softening behaviour which leads to an increase in the equilibrium amplitude of the oscillations in the fibre cross-section area, indicating a destabilizing effect of elasticity in this regime. Interestingly, at a certain high Deborah number, the bifurcation crosses over from supercritical to subcritical nature. In the subcritical regime, a threshold amplitude branch is constructed from the amplitude equation.

The release of buoyant harmful gases within enclosed spaces, such as tunnels and corridors, may engender specific health, industrial and transportation risks. For safety, a simple ventilation strategy for these spaces is to impose a flow along the tunnel, whose velocity is defined as ‘critical’, that confines the front of harmful buoyant gases immediately downstream of the source of emission. Determining the critical velocity as a function of the geometrical and dynamical conditions at the source is a fundamental fluid mechanics problem which has yet to be elucidated; this problem concerns the dynamics of non-Boussinesq releases relating to large differences between the densities of the buoyant and the ambient fluids. We have investigated this problem theoretically, by means of a simplified model of a top-hat plume in a cross-flow, and in complementary experiments by means of tests in a reduced-scale ventilated tunnel, examining releases from circular sources. Experimental results reveal: (i) the existence of two flow regimes depending on the plume Richardson number at the source $\unicode[STIX]{x1D6E4}_{i}$, one for momentum-dominated releases, $\unicode[STIX]{x1D6E4}_{i}\ll 1$, and a second for buoyancy-dominated releases, $\unicode[STIX]{x1D6E4}_{i}\gg 1$, with a smooth transition between the two; and (ii) the presence of relevant non-Boussinesq effects only for momentum-dominated releases. All these features can be conveniently predicted by the plume-based model, whose validity is, strictly speaking, limited to releases issuing from ‘small’ sources in ‘weak’ ventilation flows. Analytical solutions of the model are generally in good agreement with the experimental data, even for values of the governing parameters that are beyond the range of validity for the model. The solutions aid to clarify the effect of the source radius, and reveal interesting behaviours in the limits $\unicode[STIX]{x1D6E4}_{i}\rightarrow 0$ and $\unicode[STIX]{x1D6E4}_{i}\rightarrow \infty$. These findings support the adoption of simplified models to simulate light gas releases in confined ventilated spaces.

The forces on two spherical particles moving in a fluid are investigated by the method of matched asymptotic expansions in the small Reynolds number, for the case when the particles are within each other's inner region of expansion. The particular case in which the distance l between the sphere centres is very much larger than the sphere radii a and b is studied in detail. The asymptotic expansion of the force on one of the spheres for small a/l and b/l is obtained. Some properties of the force, not to be expected from the Stokes equation, are revealed.

The envelope equation of Dysthe (1979), which provides an extension of the nonlinear Schrödinger equation (NLS) to fourth order in wave steepness, is used to discuss higher-order modulation effects on the long-time evolution of solitary wave envelopes in deep water. The Dysthe equation admits solitary-wave solutions, similar to those of the NLS. Using perturbation methods, it is shown that an initial disturbance in the form of a solitary wave group of the NLS evolves to a solitary wave of the Dysthe equation having lower peak amplitude and moving with higher speed than the original wave; the increase in wave speed is caused by a downshift in wavefrequency. Asymptotic expressions are derived for this amplitude decrease and frequency downshift, which are consistent with numerical and experimental results.

The linearized boundary-value problem for surface waves of frequency ω in a closed basin of variable depth is reduced to a non-self-adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue κ = ω2/g is constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to κ), also is constructed; it offers significant advantages vis-à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small. These general results are illustrated by comparison with Rayleigh's (1899) results for a semicircular channel, Sen's (1927) inverse results for a family of circular basins, and Lamb's (1932) results for a shallow circular paraboloid. The eigenvalue for the dominant mode in the paraboloid is determined through O(δ5), where δ = depth/radius.

We consider the resonant excitation of surface waves inside a rectangular wave tank of arbitrary water depth with a flap-type wavemaker on one side. Depending on the length and width of the tank relative to the sinusoidal forcing frequency of the wave paddle, three classes of resonant mechanisms can be identified. The first two are the well-known synchronous, resonantly forced longitudinal standing waves, and the subharmonic, parametrically excited transverse (cross) waves. These have been studied by a number of investigators, notably in deep water. We rederive the governing equations and show good comparisons with the experimental data of Lin & Howard (1960). The third class is new and involves the simultaneous resonance of the synchronous longitudinal and subharmonic cross-waves and their internal interactions. In this case, temporal chaotic motions are found for a broad range of parameter values and initial conditions. These are studied by local bifurcation and stability analyses, direct numerical simulations, estimations of the Lyapunov exponents and power spectra, and examination of Poincaré surfaces. To obtain a global criterion for widespread chaos, the method of resonance overlap (Chirikov 1979) is adopted and found to be remarkably effective.

An experimental investigation of high Mach number free shear layers has been undertaken. The experiments were performed using a Mach 7 gun tunnel facility and a planar duct with injection from the base of a central strut producing a Mach 3 flow parallel to the gun tunnel stream. This configuration is relevant to the development of efficient scramjet propulsion, and the gun tunnel Mach number is significantly higher than the majority of previous supersonic turbulent mixing layer investigations reported in the open literature. Schlieren images and Pitot pressure measurements were obtained at four different convective Mach numbers ranging from 0 to 1.8. Only small differences between the four cases were detected, and the relatively large high-speed boundary layers at the trailing edge of the struct injector appear to strongly influence the shear layer development in each case. The Pitot pressure measurements indicated that, on average, the free shear layers all spread into the Mach 3 stream at an angle of approximately 1.4°, while virtually no spreading into the Mach 7 stream was detected until all of the low-speed stream was entrained. The free shear layers were simulated using a PNS code; however, the experimentally observed degree of spreading rate asymmetry could not be fully predicted with the k−ε turbulence model, even when a recently proposed compressibility correction was applied.

We examine the free surface flow over a fixed bed covered by rigid sinusoidal dunes. The mean current velocity is near the critical value at which the linearized theory predicts unbounded response. By allowing transients we examine the instability of the steady and nonlinear solution of Mei (1969) and the possibility of chaos when the current has a small oscillatory component.

By using and extending earlier results (Liu & Ricca, J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

This paper describes particle transport in Stokes flow in a two-dimensional corner whose walls oscillate, which is a simple model for particle transport in the pulmonary alveoli. Formally speaking, the wall motion produces a perturbation to the well-known Moffatt corner eddies. However, this ‘perturbation’ is dominant as the corner is approached. The motion of particles is regular near to the corner. Far from the corner, chaotic motion within the main part of the flow is restricted to very small regions. We deduce that there is competition between the far-field motion that generates eddies and the wall motion. The relative strengths of these two motions determines whether a given particle moves regularly or chaotically. Consequently, there is an intermediate region in which chaotic transport is maximized.

It is shown that in an incompressible fluid in which the magnetic diffusivity λ is much less than the kinematic viscosity ν, certain magnetic field distributions of limited spatial extent (conveniently described as magnetic eddies) can exist on a length scale such that the associated Reynolds number and magnetic Reynolds number are respectively small and large compared with unity. The Lorentz forces are in equilibrium with the dynamic forces associated with the fluid motion. The boundary condition imposed on this motion is that at a large distance from a magnetic eddy the velocity field should be a uniform axisymmetric irrotational straining motion. The eddies are steady in the limit λ → 0, but decay slowly in a fluid of finite conductivity. Two particular eddies are considered in detail: a disk-shaped eddy in a compressive straining motion, and a spherical eddy in an extensive straining motion. Possible applications to turbulence in interstellar gas clouds are qualitatively considered.

Global modes of the thermal convection type in the Rayleigh–Bénard–Poiseuille (RBP) system are analysed for the case of non-uniform heating of the lower wall. Specifically, a single two-dimensional ‘hot spot’ or ‘temperature bump’ giving rise to a finite region of local instability is considered. For the case of the lower wall temperature varying slowly on the scale of the RBP cell height, i.e. for a gentle temperature bump, WKBJ asymptotics are used to construct an analytical approximation of the linear global mode. At the same time, an analytical selection criterion for the critical global mode is derived from the breakdown of the WKBJ expansion at a double turning point located at the top of the temperature bump. The analytical construction and the underlying assumptions are supported by comparison with direct numerical simulations for Gaussian temperature bumps of elliptical planform not necessarily aligned with the mean flow. From these comparisons, it is concluded that the proposed analytical construction indeed yields the most amplified global mode, which is characterized by an essentially transverse orientation (normal to the mean flow direction) of the convection rolls, independent of the planform of the temperature bump. The paper concludes with preliminary DNS results on the saturated global mode shape and a discussion of possible connections to the ‘steep’ fully nonlinear global modes found for one-dimensional inhomogeneities of the basic state.

Streamline patterns and their bifurcations in two-dimensional Navier–Stokes flow of an incompressible fluid near a non-simple degenerate critical point close to a stationary wall are investigated from the topological point of view by considering a Taylor expansion of the velocity field. Using a five-order normal form approach we obtain a much simplified system of differential equations for the streamlines. Careful analysis of the simplified system gives possible bifurcations for non-simple degeneracies of codimension three. Three heteroclinic connections from three on-wall separation points merge at an in-flow saddle point to produce two separation bubbles with opposite rotations which occur only near a non-simple degenerate critical point. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity.

The stability of the flow of interstitial liquid in a mushy layer driven by expansion or contraction upon solidification is analysed. The full perturbation equations are reduced in a particular aymptotic limit that allows the principal mechanisms controlling instability to be identified. Comparisons are made with the acid-etching instabilities in porous rocks. The full equations are then solved to determine the parametric dependences of the instability. It is found that, though the potential for instability exists, it is unlikely to occur in practice.

This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios [Gcy ]. The horizontal cross-section of each layer is circular, and [Gcy ] is defined by [Gcy ] = D/2d where D is the diameter of the layer. We report results for 2.4 [les ] [Gcy ] [les ] 16 and for Prandtl numbers Pr spanning 0.5 [lsim ] Pr [lsim ] 0.9 These results are presented in terms of the slope N1 = RcdN/dR evaluated just above the onset of convection at Rc. We find that N1 is only a slowly increasing function of [Gcy ] in the range 6 [lsim ] [Gcy ] [lsim ] 16, and that it has a value there which is quite close to 0.72. This value of N1 is in good agreement with variational calcuations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for [Gcy ] [lsim ] 6, we find additional small-scale structure in N1 associated with changes in the number of convection rolls with changing [Gcy ]. An additional test of the linearzied hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.

An investigation is made of the scattering effect of a random ocean bottom of constant average depth upon the propagation of shallow water waves. Of particular concern is the case of long-distance propagation, in which the conventional perturbation schemes fail to apply. The approximation scheme employed is basically one of selective summation of the type used in other areas of physics such as the theory of many-body interactions. Results are obtained for the average wave and the two-point correlation function of the wave field for the case when the ocean statistics are homogeneous and isotropic. The application of the results to the case of a tsunami is discussed.

Standing radial cross-waves in an annular wave tank are investigated using Whitham's average-Lagrangian method. For the simplest case, in which a single radial cross-wave is excited, energy is transferred from the wavemaker to the cross-wave through the spatial mean motion of the free surface, as described by Garrett (1970) for a purely transverse cross-wave in a rectangular tank. In addition, energy is transferred through spatial coupling since, in contrast to the purely transverse cross-wave in the rectangular tank, the (non-axisymmetric) radial cross-wave is three-dimensional. It is shown in an Appendix that this spatial coupling does occur for a three-dimensional cross-wave in a rectangular tank. The equations that govern this single-mode resonance are isomorphic to those that govern the Faraday resonance of surface waves in a basin of fluid subjected to vertical excitation (Miles 1984a).

It is found that the second-order Stokes-wave expansion for deep-water, standing gravity waves, which is regular for rectangular containers, may become singular for circular containers (Mack (1962) noted these resonances for finite-depth, standing gravity waves in circular containers). The evolution equations that govern two distinct types of resonant behaviour are derived: (i) 2:1 resonance between a radial cross-wave and a resonantly forces axisymmetric wave, corresponding to approximate equality among the driving frequency, a natural frequency of the directly forced wave, and twice the natural frequency of a cross-wave; (ii) 2:1 internal resonance between a radial cross-wave and a non-axisymmetric second harmonic, corresponding to approximate equality among the driving frequency, the natural frequency of a non-axisymmetric wave of even azimuthal wavenumber, and twice the natural frc.quency of the cross-wave. The axisymmetric, directly forced wave in (i) is resonantly excited and exchanges energy with the subharmonic cross-wave through spatial coupling, whereas the cross-wave in (ii) is parametrically excited and exchanges energy with the non-axisymmetric second harmonic through spatial coupling. The equations governing case (i) are shown to exhibit chaotic motions; those governing (ii) are shown to be isomorphic to the equations governing 2:1 internal resonance in the Faraday problem (Miles 1984a, §6), which have been shown to exhibit chaotic motions (Gu & Sethna 1987).

Preliminary experiments on standing radial cross-waves are reported in an Appendix, and theoretical predictions of mode stability are in qualitative agreement with these experiments. For the single-mode theory, the interaction coefficient that is a measure of the energy exchange between the wavemaker and the cross-wave is evaluated numerically for a particular wavemaker. The maximum interaction coefficient for a fixed azimuthal wavenumber of the cross-wave typically occurs for that radial mode number for which the turning point of the cross-wave radial profile is nearest the wavemaker. The present experiments for standing radial cross-waves are compared with those of Tatsuno, Inoue & Okabe (1969) for progressive radial cross-waves.

The two-dimensional problem of wave diffraction by a strip of arbitrary width is investigated here in the context of linearized theory of water waves by reducing it to a pair of Carleman-type singular integral equations. These integral equations have been solved earlier by an iterative process which is valid only for a sufficiently wide strip. A new method is described here by which solutions of these integral equations are determined by solving a set of four Fredholm integral equations of the second kind, and the process is valid for a strip of arbitrary width. Numerical solutions of these Fredholm integral equations are utilized to obtain fairly accurate numerical estimates for the reflection and transmission coefficients. Previous numerical results for a wide strip are recovered from the present analysis. Additional results for the reflection coefficient are presented graphically for moderate values of the strip width which exhibit a less oscillatory nature of the curve than the case of a wide strip.

In strongly stratified geophysical fluids such as the stratosphere and the ocean, the vertical mixing of tracers is largely due to patches of turbulence that are intermittent in time and space. Heuristic models for this type of mixing are studied which extend that of Dewan (1981a). The recognition that, in these models, fluid particles undergo continuous-time random walks allows the derivation of closed-form results for the particle-position statistics. The particle dispersion is shown generally to be diffusive in the long-time limit. However, the early-time, non-diffusive regime is also analysed, since a time-scale estimate indicates its practical importance, in particular for stratospheric mixing.

Because the restratification of fluid patches previously homogenized by turbulence takes a finite time, the probability for a fluid region to become turbulent may depend on the time elapsed since it has last been turbulent. This introduces a ‘memory effect’ whose consequences for the tracer mixing are analysed in detail using a simple non-Markovian model.

The heuristic models studied allow the large-scale dispersive effects of the turbulent patches to be inferred from the properties of individual patches. This highlights those properties that might most usefully be determined from investigations of the dynamics of the turbulent patches themselves.

We have carried out high-precision measurements of the heat transport in intermediate-size rectangular layers of convecting normal liquid 4He with Prandtl numbers of 0.52 and 0.70. The containers used for these experiments had horizontal dimensions, in units of the height d, of 13.4 × 5.95 (cell I) and 18.2 × 8.12 (cell II). The slopes N1 of the Nusselt curves were 0.56 and 0.70 respectively for cell I and cell II. These values are significantly lower than predictions for N1 for horizontally unbound layers, but comparable with results obtained in cylindrical containers of liquid helium with roughly the same number of convection rolls. For the two containers, the onset of the first instability after the onset of convection occurred at Rayleigh numbers R1 that were in reasonable quantitative agreement with the predictions of Busse and Clever for the skewed-varicose instability. For both containers, the transition at R1 was characterized by long transients ranging from ∼ 102 to ∼ 103 vertical-thermal-diffusion times. A decrease in the Nusselt number was also observed. As the Rayleigh number was increased above R1, a new steady state evolved and then additional transitions were observed. These transitions occurred at Rayleigh numbers labelled R2, R3,…, with a total of five transitions seen in cell I and a total of three transitions seen for cell II. The transition for each cell at R2 can be related quantitatively to the skewed-varicose instability, and the transition at R3 is associated with an oscillatory instability. For cell II, the time-dependence beginning at R3 persisted to the highest Rayleigh number studied, R = 11.7Rc. However, for container I, two more regimes of time-independent flow were observed; the last of these was at an unexpectedly high Rayleigh number of 6.7Rc. This work extends to lower Prandtl number recent studies made on moderate-size rectangular layers of convecting water and alcohol.