In this work, we theoretically investigate the swimming velocity of a Taylor swimming sheet immersed in a linearly density-stratified fluid. We use a regular perturbation expansion approach to estimate the swimming velocity up to second order in wave amplitude. We divide our analysis into two regimes of low ($\ll O(1)$) and finite Reynolds numbers. We use our solution to understand the effect of stratification on the swimming behaviour of organisms. We find that stratification significantly influences motility characteristics of the swimmer such as the swimming speed, hydrodynamic power expenditure, swimming efficiency and the induced mixing, quantified by mixing efficiency and diapycnal eddy diffusivity. We explore this dependence in detail for both low and finite Reynolds number and elucidate the fundamental insights obtained. We expect our work to shed some light on the importance of stratification in the locomotion of organisms living in density-stratified aquatic environments.

An experimental study was conducted to measure the growth rates of mechanically generated surface water waves when subjected to a fully developed turbulent channel airflow. The study was designed to test the accuracy of the growth rates predicted by Miles's (1962b) theory. For a series of wave frequencies (from 2·04 to 6·04 Hz at 0·50 Hz increments) and centre-line wind velocities (0·20, 1·12 and 1·84 m/s) wave amplitudes were measured at three stations (2–21, 3–43 and 4·65 m) downwind from a wave generator. In addition, for centre-line velocities of 1–12 and 1·84 m/s, U* (the velocity at the outer edge of the viscous sublayer) and U1, (the shear velocity) were obtained from measured mean velocity and Reynolds stress profiles. The wave amplitude measurements at the wind velocity of 0·20 m/s provided attenuation rate estimates which agreed reasonably well with theoretical attenuation rates based on viscous effects both on the walls and in the bulk of the water. The amplitude measurements at the wind velocities of 1·12 and 1·84m/s provided growth rate estimates which were compared with theoretical growth rates (computed using the wave frequency, U1 and U* predicted by Miles's (1962b) theory. At 1·12m/s Miles's growth rateswere two to five times larger than those measured; at 1·84 m/s Miles's growth rates were about two times larger.

At finite Dukhin numbers, where Smoluchowski’s formula is inapplicable to thin-double-layer electrophoresis, the mobility of non-spherical particles is generally anisotropic. We consider bodies of revolution of otherwise arbitrary shape, where a uniformly applied electric field results in a rectilinear motion in the plane spanned by the field direction and the particle symmetry axis, as well as (for particles lacking fore–aft symmetry) rigid-body rotation about an axis perpendicular to that plane. Focusing upon slender particles, where the ratio $\epsilon $ of cross-sectional and longitudinal scales is asymptotically small, the translational and rotational mobilities are obtained as quadratures which depend upon the lengthwise distribution of the scaled cross-sectional width and the force densities associated with rigid-body motion. These mobility expressions approach finite limits as $\epsilon \rightarrow 0$, yielding closed-form expressions for specific particle geometries.

A formulation based on direct and adjoint parabolized equations is developed to account for the spatial evolution of perturbations in swept attachment-line boundary layers. For sweep Reynolds numbers larger than Re = 100 the dynamics is dominated by a lift-up mechanism which is responsible for large energy amplification by transforming spanwise vortices into spanwise streaks. This mechanism favours steady perturbations with a chordwise scale that quantitatively matches its counterpart for classical Blasius boundary layers.

The flow in a plane liquid jet from an aperture is obtained by direct simulation of the Navier–Stokes equations. The gas–liquid interface is tracked using the level set method. Flows are calculated for different Reynolds and Weber numbers. When We = ∞, the maximum value of the discharge coefficient appears around Re = O(100). The regions that are vulnerable to cavitation owing to the total stress are identified from calculations based on Navier–Stokes equations and viscous potential flow; the two calculations yield similar results for high Reynolds numbers. We prove that the classical potential flow solution does not give rise to a normal component of the rate of strain at the free streamline. Therefore, the normal component of the irrotational viscous stresses also vanishes and cannot change the shape of the free surface. The results of calculations of flows governed by the Navier–Stokes equations are close to those for viscous potential flow outside the vorticity layers at solid boundaries. The Navier–Stokes solutions for the axisymmetric aperture are also given for two values of Reynolds numbers. The results for axisymmetric and planar apertures are qualitatively similar, but the axisymmetric apertures have a lower discharge coefficient and less contraction.

When the Galileo number is below the first bifurcation, the instability and transition of a vertical ascension or the fall of a free sphere affected by a vertical magnetic field are investigated numerically. A compact model is used to explain that the magnetic field can destabilize the fluid–solid system. When the interaction parameter exceeds a critical value, the sphere trajectory is transitioned from a steady vertical trajectory to a steady oblique one. Furthermore, the trajectory will remain vertical at a sufficiently large magnetic field because of a double effect of the magnetic field on the fluid–solid system. Under the influence of an external vertical magnetic field, four wake patterns at the rear of the sphere are found and the physical behaviour of the free sphere is independent of the density ratio. The wake or trajectory of the free sphere is only determined by the Galileo number $G$ and the interaction parameter $N$. A close relationship between the streamwise vorticity and the sphere motion is found. An interesting ‘agglomeration phenomenon’ is also found, which shows that the vertical velocities are agglomerated into a point for a certain magnetic field regardless of the Galileo number and satisfy a scaling law $V_{z}\sim N^{-1/4}$, when $N>1$. The principal results of the present work are summarized in a map of regimes in the $\{G,N\}$ plane.

In a stationary weak sound wave the four gas dynamical quantities, entropy, stagnation enthaply, mass flow per unit area and impulse per unit area are constant. The six processes in which pairs of these four variables are kept constant are studied for the case of any single phase fluid or mixture of fluids in equilibrium, and it is shown that the remaining variables are stationary at sonic points. Such points are shown to occur once only in each process and are identified as maxima or minima, on the assumption that the fluid is a normal one which expands on heating at constant pressure and for which (∂2p/∂v2)s is positive.

Newton's theory of sound assumed that the fluid temperature was invariant. Across a stationary Newtonian sound wave, the four quantities, temperature, mass flow and impulse per unit area, and a dynamical variant of the Gibbs function are constant. The six processes in which pairs of these four variables are kept constant are studied, and it is shown that the remaining variables are stationary at points where the fluid speed equals Newton's sound velocity. These points are shown to occur once only in each process and are identified as maxima or minima with the further proviso that (∂2p/∂v2)T is positive.

The two sets of processes have one member in common, that usually referred to as the Rayleigh line. The Fanno line and the usual isentropic ‘nozzle’ process also belong to the first set.

Finally, the variation of stagnation temperature and pressure in some of the processes and in stationary shocks is investigated.

The dynamics of the subgrid-scale (SGS) stress and scalar flux in the convective atmospheric surface layer are studied through the budgets of the SGS turbulence kinetic energy (TKE), the SGS stress and the SGS temperature flux using field measurements from the Advection Horizontal Array Turbulence Study (AHATS). The array technique, which employs sensor arrays to perform filter operations to obtain the SGS velocity and temperature, is extended to include pressure sensors to measure the fluctuating pressure, enabling separation of the resolvable- and subgrid-scale pressure, and therefore for the first time allowing for measurement of the pressure covariance terms and the full SGS budgets. The non-dimensional forms of the SGS budget terms are obtained as functions of the stability parameter $z/ L$ and the ratio of the wavelength of the spectral peak of the vertical velocity to the filter width, ${\Lambda }_{w} / {\Delta }_{f} $. The results show that the SGS TKE budget is a balance among the production, transport and dissipation. The SGS shear stress budget and the SGS temperature flux budgets are dominated by the production and pressure destruction, with the latter causing return to isotropy. The budgets of the SGS normal stress components are more complex. Most notably the pressure–strain-rate correlation includes two competing processes, return to isotropy and generation of anisotropy, the latter due to ground blockage of the large convective eddies. For neutral surface layers, return to isotropy dominates. For unstable surface layers return to isotropy dominates for small filter widths, whereas for large filter widths the ground blockage effect dominates, resulting in strong anisotropy. The results in the present study, particularly for the pressure–strain-rate correlation, have strong implications for modelling the SGS stress and flux using their transport equations in the convective atmospheric boundary layer.

In the present work the effect of spanwise oscillations on the most unstable Görtler vortex is studied. The wavenumber of the most unstable disturbance is very large, so that the parabolic character of the problem is eliminated. After the disturbances are expanded in a Fourier series in time, the eigenvalue problem is solved for various amplitudes and frequencies of the oscillation. Due to numerical difficulties the treatment of very high amplitudes and/or very low frequencies becomes prohibitively expensive. Therefore an approximate formulation was developed for high magnitudes of the spanwise speed of the oscillation $E$. The results, which were obtained in all cases, indicate that the disturbances move away from the wall in a logarithmic way, reducing the effects of the oscillation. Consequently, it is impossible to stabilize the flow completely. However, it is shown that a large reduction of the growth rate can be achieved even for moderate values of $E$.

The velocity field of a binary mixture of incompressible miscible liquids is non-solenoidal when the densities of the two liquids are different. If the mixture density is linear in the volume fraction, as in the case of simple (ideal) mixtures or very nearly for glycerin and water, then the velocity can be decomposed into a solenoidal and an expansion part. In the context of this theory, we derive a new solution which describes the smoothing of an initial plane discontinuity in concentration across a channel bounded by sidewalls. The requirement that the velocity vanishes on the sidewall introduces a different initial discontinuity not present in the solenoidal theory. The problem may be reduced to a partial differential equation in two similarity variables, one for the smoothing of a concentration discontinuity without sidewalls and the other for the smoothing the velocity discontinuity at the sidewall. The similarity equations are solved explicitly in a special case.

By adapting the general flame-acoustic interaction theory developed in Wu et al. (J. Fluid Mech., vol. 497, 2003, pp. 23–53), a systematic analysis is carried out for the interaction of a stable premixed flame in a duct with vortical disturbances superimposed on the oncoming mixture. A small-amplitude vortical perturbation, assumed to be a convecting gust with a frequency ω, induces a hydrodynamic field in the vicinity of the flame, causing an initially planar flame to wrinkle. The unsteady heat release resulting from the increased surface area of the wrinkling flame then generates a sound wave with frequency 2ω. When 2ω coincides with the natural frequency of an acoustic mode of the duct, a flame-acoustic resonance takes place, through which the flame-induced sound may attain an amplitude sufficiently large to modulate the flame through the unsteady Rayleigh–Taylor effect. A novel evolution system is derived to describe this two-way coupling for two cases: (a) a flame with a fixed mean position and (b) a moving flame. Numerical solutions show that for (a), the mutual flame-acoustic interaction initiates a violent subharmonic parametric instability, and the flame-acoustic system quickly evolves into a fully nonlinear regime, which probably corresponds to a state of self-sustained oscillation. This finding presents a peculiar instability scenario: a small-amplitude vortical perturbation may, by initiating acoustic-flame resonance, completely destabilize an otherwise stable planar flame. For a moving flame, the flame-acoustic resonance is of transient nature. The acoustic pressure gains substantially, but the parametric flame instability is induced only when the vortical disturbance exceeds a finite threshold.

Magnetic reconnection at an X-type neutral point of a two-dimensional magnetic field is studied in an incompressible viscous resistive fluid whose flow is assumed to be slow enough that its inertia is negligible. In both the ideal and resistive magnetohydrodynamic approximations current singularities appear at the X-point and along the separatrices. It is shown here analytically that the combined effect of viscosity and resistivity can resolve these singularities with the flow crossing the separatrices. A wide class of exact solutions describing the structure of the flow and current density distribution is found. The results suggest that reconnection may occur with localized distributions of strong current density restricted to finite regions.

Mixing of a passive scalar in the peripheral region close to a wall is investigated by means of accurate direct numerical simulations of both a three-dimensional Couette channel flow at low Reynolds numbers and a two-dimensional synthetic flow. In both cases, the resulting phenomenology can be understood in terms of the theory recently developed by Lebedev & Turitsyn (Phys. Rev. E, vol. 69, 2004, 036301). Our results prove the robustness of the identified mechanisms responsible for the persistency of scalar concentration close to the wall with important consequences in completely different fields ranging from microfluidic applications to environmental dispersion modelling.

The linear stability of finite-amplitude capillary waves on inviscid sheets of fluid is investigated. A method similar to that recently used by Tiron & Choi (J. Fluid Mech., vol. 696, 2012, pp. 402–422) to determine the stability of Crapper waves on fluid of infinite depth is developed by extending the conformal mapping technique of Dyachenko et al. (Phys. Lett. A, vol. 221 (1), 1996a, pp. 73–79) to a form capable of capturing general periodic waves on both the upper and the lower surface of the sheet, including the symmetric and antisymmetric waves studied by Kinnersley (J. Fluid Mech., vol. 77 (02), 1976, pp. 229–241). The primary, surprising result is that both symmetric and antisymmetric Kinnersley waves are unstable to small superharmonic disturbances. The waves are also unstable to subharmonic perturbations. Growth rates are computed for a range of steady waves in the Kinnersley family, and also waves found along the bifurcation branches identified by Blyth & Vanden-Broeck (J. Fluid Mech., vol. 507, 2004, pp. 255–264). The instability results are corroborated by time integration of the fully nonlinear unsteady equations. Evidence is presented for superharmonic instability of nonlinear waves via a collision of eigenvalues on the imaginary axis which appear to have the same Krein signature.

Direct numerical simulations of stably stratified homogeneous turbulence, with and without mean shear, are used to investigate the three-dimensional structure, evolution and energetic significance of density overturns. Although the flow conditions are idealized, examination of the full-field simulation data provides insight into flow energetics and mixing which may assist in the interpretation of physical measurements, typically limited to one-dimensional vertical profiles. Overturns, defined here through the density field as contiguous regions of non-zero Thorpe displacement, are initially generated by the stirring action of coherent vortex structures present in the flow and further develop through merging with adjacent overturns. During this growth phase, overturns exhibit irregular spatial structure in unsheared flow and elongated structure with distinct orientation in shear flow. Although most of the available potential energy (APE) and buoyancy flux are associated with stable (non-overturning) regions in the flow, young overturns actively contribute to the flow energetics. In particular, overturn peripheries are sites of high levels of APE, buoyancy flux and diapycnal mixing. A collapse phase may follow the growth phase in the absence of adequately strong mean shear. During this phase, buoyancy gradually assumes control of the overturns and their vertical scale steadily decreases. The energetic significance of the overturns diminishes, although high APE and diapycnal mixing continue to occur near their boundaries. In the final phase of their evolution, overturns contribute negligibly to the energetics. The remaining overturns are characterized by a viscous–buoyant balance which maintains their vertical scale. The overturns eventually vanish due to homogenization of their internal density distribution by diffusion. Activity diagrams, sampled at different points of flow evolution, show significant variation in overturn Reynolds and Froude numbers which may have implications for vertical sampling of a turbulent event.

An analysis is presented of the three-dimensional creeping flow in and around a porous sphere, modelled as a generalized Brinkman medium, near a smooth plane where the sphere (i) translates uniformly without rotating in an otherwise quiescent Newtonian fluid, (ii) rotates uniformly without translating in an otherwise quiescent Newtonian fluid, and (iii) is fixed in a shear field, which is uniform in the far field and has a linearly increasing velocity profile with increasing distance from the plane. The linear superposition of these three flow regimes is also considered for the special case of the free translational and rotational motion of a neutrally buoyant porous sphere in a shear field that is uniform in the far field. Exact series solutions to the momentum equations are derived for the velocity and pressure fields in the Brinkman and Stokes-flow regions. Coefficients in the series solutions for each flow regime are determined using recursion relations derived from the continuity equations in the Brinkman and Stokes-flow regions, from the interfacial boundary conditions on the porous spherical surface, and from the no-slip condition on the plane. Results are presented in terms of the drag force on the porous sphere and torque about the sphere centre as a function of the dimensionless clearance distance between the sphere and the rigid plane for several values of the dimensionless hydraulic permeability of the Brinkman medium. The free motion of the neutrally buoyant sphere is calculated by requiring that the net hydrodynamic drag force and torque acting on the sphere vanish. Results for this case are presented in terms of the dimensionless translational and rotational speeds of the porous sphere relative to the far-field shear rate as a function of the dimensionless clearance distance for several values of the dimensionless hydraulic permeability. The work is motivated by insights it offers into the behaviour of porous agglomerates, and by its potential utility in industrial, biological, biophysical, medicinal and environmental applications wherever gas or liquid suspensions of porous agglomerates might arise.