Potential flow past a porous body of arbitrary shape with constant physical permeability k0, as well as the flow in the corresponding porous medium, are analysed by means of a pair of linear Fredholm integral equations of the second kind. As an example for verification of the proposed general method, the case of a two-dimensional porous circular cylinder is worked out in detail.

We investigate the classical problem of the extension of an axisymmetric viscous thread by a fixed applied force with small initial inertia and small initial surface tension forces. We show that inertia is fundamental in controlling the dynamics of the stretching process. Under a long-wavelength approximation, we derive leading-order asymptotic expressions for the solution of the full initial-boundary value problem for arbitrary initial shape. If inertia is completely neglected, the total extension of the thread tends to infinity as the time of pinching is approached. On the other hand, the solution exhibits pinching with finite extension for any non-zero Reynolds number. The solution also has the property that inertia eventually must become important, and pinching must occur at the pulled end. In particular, pinching cannot occur in the interior as can happen when inertia is neglected. Moreover, we derive an asymptotic expression for the extension.

The velocity field and turbulence structure in the near wake of a thick flat plate with a tapered trailing-edge geometry are examined using both hydrogen-bubble flow visualization and hot-film anemometry measurements. Tests were conducted for Re1 = 8.5 × 105 in the region 0 < x+ < 6400 behind the trailing edge. The probe and visualization results indicate a similarity between both (i) velocity and turbulence structure variations wih x+ in the near wake, and (ii) the corresponding changes in similar flow characteristics with y+ within a turbulent boundary layer. In particular, visualization data in the vicinity of the wake centreline reveal the existence of strong streamwise flow structures in the region close (x+ < 270) to the trailing edge. The streamwise orientation of the observed structures diminishes as x+ increases. From hot-film measurements, two separate regions along the wake centreline can be distinguished: (i) a linear growth region which extends over 0 < x+ < 100, wherein the centreline velocity varies linearly with x+; and (ii) a logarithmic growth region for x+ > 270, wherein the centreline velocity varies as log x+. The similarity in behaviour between these regions and the comparable wall region of a turbulent boundary layer suggests the existence of a common functionality. This similarity is demonstrated by a simple linear relationship of the form y+ = Kx+, which is shown to approximately collapse the velocity behaviour both across a turbulent boundary layer and along the wake centreline to a unified set of empirical relationships.

The problem of subsonic flow past micro-electro-mechanical-system-type (MEMS-type) heating elements placed on a flat surface, where the MEMS devices have hump-shaped surfaces, is investigated using triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered in the limit that the Reynolds number is large. The triple-deck problem is formulated, and the linear and nonlinear analysis and results are presented. The current work is a generalisation of the problem discussed by Koroteev & Lipatov (J. Fluid Mech., vol. 707, 2012, pp. 595–605; Z. Angew. Math. Mech., vol. 77, 2013, pp. 486–493), where the MEMS devices have flat-shaped surfaces. The results show that the hump-shaped heating elements enhance large drops in pressure, and peaks and troughs in the skin friction over the centre of the hump compared with the flat-shaped devices, which may be useful for controlling the flow.

The vertical oscillation of a partially submerged body produces a surface wave that carries energy away from the body. The amplitude of this wave, when surface-tension effects are not negligible, depends on the conditions applied at the line of contact between the body and the free surface of the fluid. An edge condition that includes both dynamic variation of the contact angle and contact-angle hysteresis is used in this paper; the condition implies a dissipation of energy at the contact line. The amplitude of the wave and the amount of energy dissipated are calculated for a horizontal circular cylinder and for a simple source-and-plate model. This model is shown to be an adequate representation for the qualitative description of heaving motions and to simplify the calculations considerably. The effects of varying the relative importance of surface tension and gravity, the dynamic behaviour of the contact angle, and the amount of hysteresis, are calculated for the source-and-plate model.

A third-order regular perturbation solution is developed for laminar flow through a straight pipe that is rotating about an axis not aligned with the pipe axis. Coriolis accelerations produce transverse secondary velocities (similar to those in flow through coiled tubes) and modify the axial-velocity profile. The effects of rotation on the velocity fields are shown to depend on two parameters: (i) the product of axial and rotational Reynolds numbers, and (ii) the square of the rotational Reynolds number itself. Even though their strength increases with increases in parameter magnitudes, transverse circulations are qualitatively insensitive to parametric values. The axial profile, on the other hand, can be significantly modified by the rotation; the zeroth-order parabolic axial profile can be skewed toward the outside, dimpled in the centre with maximums on either side of the centreline, or both, depending on the values of the two parameters. The modification of the axial-velocity profile has important ramifications in the design of heat/mass-transfer devices.

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative–dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

A long-wavelength weakly nonlinear analysis is used to investigate the possibility for resonant energy exchange between low-mode internal waves and counter-rotating roll vortices known as Langmuir circulation. The analysis is based on a two-layer ocean model in which the Langmuir circulation is confined to the upper layer and counter-propagating internal waves travel along the sharp thermocline normal to the axes of the vortices. An asymptotically consistent description of the slow-time behaviour is obtained by making a WKBJ approximation to treat the comparatively high-frequency internal-wave reflections identified in Part 1. When the vortices and waves are modelled as linearly neutral modes, the resulting dynamics take the form of nonlinear oscillations. The theory suggests that Langmuir cells may transiently lose stability to standing internal-wave disturbances whose nodes are aligned with the cell downwelling zones. An exact solution of the Langmuir-circulation–standing-wave interaction is used to gain insight into the nonlinear instability mechanism. As in Part 1, the modification of the linear internal-wave dynamics by the Craik–Leibovich ‘vortex force’ is found to be crucial to the interaction.

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.

An examination of the Newtonian inviscid theory of hypersonic flow past a sphere is made in the neighbourhood of the singular point which occurs at θ = 60°. A uniformly valid theory is developed in this neighbourhood which exhibits the limiting detached ‘free-layer’ behaviour postulated by Hayes and Lighthill in the limit as γ → 1. A complete solution is obtained which gives details of streamline and shock wave shape, pressure distribution, etc. for γ − 1 small.

Reasons are given why the empirical ‘modified’ Newtonian theory of Lees proves to be a good approximation to experimentally determined pressure distributions.

A novel check to the theory is provided by a simple power-law relation between the pressure on the sphere surface and the distance of the shock wave away from sphere at the same point.

We study analytically the asymptotics of the upper bound on energy dissipation for the two-dimensional plane Couette flow considered numerically in Part 1 of this work, in order to identify the mechanisms underlying the variational approach. With the help of shape functions that specify the variational profiles either in the interior or in the boundary layers, it becomes possible to quantitatively explain all numerically observed features, from the occurrence of two branches of minimizing wavenumbers to the asymptotic parameter scaling with the Reynolds number. In addition, we derive a new variational principle for the asymptotic bound on the dissipation rate. The analysis of this principle reveals that the best possible bound can only be attained if the variational profiles allow the shape of the boundary layers to change with increasing Reynolds number.

Three-dimensional laminar flow profiles were measured at each of three axial stations downstream from the inlet of a slowly tapering tube of constant flow cross-section. It was found that strong inlet wall shear induces significant secondary swirl, whose downstream growth is initially consistent with an inviscid model of vorticity transfer. This secondary flow may act to retard the onset of local flow separation by convecting axial momentum into the boundary layer adjacent to the diverging wall of the tube. These observations could account for the similar geometric shape of major bronchial airways: the suppression of flow separation in pulmonary bifurcations would promote moire effective airway ventilation in the human lung.

Liquid-metal flows in rectangular ducts having electrically insulating tops and bottoms and perfectly conducting sides and in the presence of strong, polar, transverse magnetic fields are examined. Solutions are presented for the boundary layers adjacent to the sides that are parallel to the magnetic field. Overshoots in the radial velocity profiles show that the side layers have zero displacement thickness and do not perturb the inviscid core. Very weak secondary flows involve four significant vortices, as reflected in the polar velocity profiles.

The method of contour dynamics (CD) is applied to several inviscid prototype flows typical of the motions found in the transition region of the free shear layer. Examples of the interaction between the fundamental streamwise-layer perturbation and its first subharmonic are presented that illustrate the events of pairing and tearing of two rolled-up cores and also the coalescence of three rolled-up cores. The present simulations of the temporally unstable two-dimensional layer, at effectively infinite Reynolds number, support the hypothesis that the dynamics of the large-scale roll-up is only weakly dependent on Reynolds number. However, we find fine-scale structure that is not apparent in previous simulations at moderate Reynolds number. Spiral filaments of rotational fluid wrap around the rolled-up vortex cores producing ‘spiky’ vorticity distributions together with the entanglement of large quantities of irrotational fluid into the layer. Simulations proceeded only until the first such event because we were unable to resolve the fine detail generated subsequently. The inclusion of prescribed vortex stretching parallel to the vortex lines is found to accelerate the initial roll-up and to enhance the production of spiral vortex filaments. In the fundamental-subharmonic interaction, vortex stretching slows but does not prevent pairing.

Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1H(σy/h1), 0 [les ] y < ∞, are calculated through an asymptotic expansion in σ/kh1 on the assumptions that σ [Lt ] 1 and kh1 = O(1). Approximations to the free-surface displacement in an inner domain that includes the singular point at h = 0 and the turning point near gh ≈ ω2/K2 and to the eigenvalue λ ≡ ω2/σgh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to Λ are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from σ to 0 as h increases from 0 to h1 and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = h1 and with the geometrical-optics approximation of Shen, Meyer & Keller (1968).