A theoretical and numerical study of complex sliding flows of yield-stress fluids is presented. Yield-stress fluids are known to slide over solid surfaces if the tangential stress exceeds the sliding yield stress . The sliding may occur due to various microscopic phenomena such as the formation of an infinitesimal lubrication layer of the solvent and/or elastic deformation of the suspended soft particles in the vicinity of the solid surfaces. This leads to a ‘stick–slip’ law which complicates the modelling and analysis of the hydrodynamic characteristics of the yield-stress fluid flow. In the present study, we formulate the problem of sliding flow beyond one-dimensional rheometric flows. Then, a numerical scheme based on the augmented Lagrangian method is presented to attack these kind of problems. Theoretical tools are developed for analysing the flow/no-flow limit. The whole framework is benchmarked in planar Poiseuille flow and validated against analytical solutions. Then two more complex physical problems are investigated: slippery particle sedimentation and pressure-driven sliding flow in porous media. The yield limit is addressed in detail for both flow cases. In the particle sedimentation problem, method of characteristics – slipline method – in the presence of slip is revisited from the perfectly plastic mechanics and used as a helpful tool in addressing the yield limit. Finally, flows through model and randomized porous media are studied. The randomized configuration is chosen to capture more sophisticated aspects of the yield-stress fluid flows in porous media at the yield limit – channelization.

Incompressible fluid flows are characterised by high correlations between low pressure and high velocity and vorticity. The velocity–pressure correlation is an immediate consequence of fluid acceleration towards low-pressure regions. On the other hand, fluid converging to a low-pressure centre is driven sideways by a resistance due to incompressibility, giving rise to the formation of a strong vortex, hence the vorticity–pressure correlation. Meanwhile, the formation of such a vortex effectively shields the low-pressure centre from incoming energetic fluid. As a result, a local pressure minimum can usually be found at the centre of a vortex where the vorticity is greatest but the velocity is relatively low, hence the misalignment of local pressure minima and velocity maxima. For Navier–Stokes flows, this misalignment has profound implications for extreme momentum growth and maintenance of regularity. This study examines the role of the velocity–pressure correlation in the problem of Navier–Stokes global regularity. On the basis of estimates for flows locally satisfying the critical scaling of the Navier–Stokes system, a qualitative theory of this correlation is considered. The theory appears to be readily quantified, advanced and tested by theoretical, mathematical and numerical methods. Regularity criteria depending on the degree of the velocity–pressure correlation are presented and discussed in light of this theory. The result suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, then regularity is likely to persist. This is the first result that makes use of an explicit measure of the velocity–pressure correlation as a key factor in the maintenance of regularity or development of singularity.

We investigate the two-dimensional (2-D) oscillation of quasi-2-D convection rolls in a liquid metal layer confined by a vessel of aspect ratio five with an imposed horizontal magnetic field. Laboratory experiments were performed in the range of Rayleigh $Ra$ and Chandrasekhar $Q$ numbers of $7.9 \times 10^4 \le Ra \le 1.8 \times 10^5$ and $2.5 \times 10^4 \le Q \le 1.9 \times 10^5$ by decreasing $Q$ at set $Ra$-number intervals to elucidate the features and mechanisms of oscillatory convection. Ultrasonic velocity profile measurements and supplemental numerical simulations show that the 2-D oscillations are caused by oscillations of recirculation vortex pairs between the main rolls, which are intensified by periodic vorticity entrainment from the vortex pair by the main rolls. The investigations also suggest that the oscillations occur at sufficiently large Reynolds $Re$ numbers to induce instabilities on the vortex pair. The $Re$ number is smaller for larger $Q/Ra$ in the 2-D oscillation regime and the variations can be approximated by the effective $Ra$ number; namely, the value reduced by the critical value for the onset of convection depending on $Q$. The variations steepen with further large $Q/Ra$ and approach a scaling law of the velocity reduction as $(Ra/Q)^{1/2}$, which is established assuming that viscous dissipation is dominated by Hartmann braking at the walls perpendicular to the magnetic field. The results suggest that these phenomena are organized by the relationship between buoyancy and magnetic damping due to Hartmann braking.

Under certain conditions, experimental values of the tensile strength of water are found to be much lower than theoretical values, even when the water is purified and degassed as much as possible. The discrepancy could be ascribed to stabilized nanobubbles or nanoparticles suspended in the liquid, as such contaminants cannot be eliminated completely from a substantial liquid volume. Thus, the present study aims at elucidating the effects of such nanoscale nuclei on cavitation. A parameter-free mathematical model is derived to predict the cavitation arising from nanoscale nuclei, based on classical nucleation theory. To verify the model, molecular dynamics is used to simulate cavitation at nuclei of different sizes, embedded either in water or in liquid copper at different temperatures. The cavitation pressures calculated from the molecular dynamics results are compared with the predictions of the present mathematical model, with a good agreement between them. The results show that nanoscale nuclei significantly promote cavitation, i.e. the tensile strength is reduced notably by the presence of nanoscale nuclei. The tensile strength decreases when the size of nuclei increases, and the change rule of cavitation pressure is also affected by the liquid properties, such as liquid temperature. The present study may provide an acceptable explanation of the discrepancy between theory and experiment on the cavitation pressure in liquids purified and degassed as much as possible.

Turbulent negatively buoyant jets occur when the buoyancy of a jet directly opposes its momentum, and will decelerate until its mean momentum is reduced to zero. Here, the flow reverses direction and, for an axisymmetric flow originating from a round inlet, returns annularly towards the source, mixing with the opposing fluid and forming a fountain. This investigation focuses on the initial stage of the flow, before the return flow is established. Data are obtained experimentally using two-dimensional particle image velocimetry and planar laser induced fluorescence for saline/freshwater negatively buoyant jets with source Froude number $Fr_o=30$ and Reynolds numbers $5500\lesssim Re_o\lesssim 5900$ at axial locations $18\lesssim z/D\lesssim 30$, and compared to a neutral jet. The development of the mean and turbulence profiles with local $Fr$ are investigated, and it is found that, unlike neutral jets and plumes, the turbulence intensity in negatively buoyant jets does not scale with the mean flow. Additionally, the ratio of widths of the buoyancy and velocity profiles, $\lambda$, increases along the jet. The entrainment coefficient, $\alpha$, was estimated for a negatively buoyant jet, and was found to decrease with local $Fr$, eventually becoming negative, indicating fluid is being ejected from the jet. These observations differ to neutral or buoyant jets and plumes, which approach a constant $\lambda$ and $\alpha$ in the far field. This different behaviour in negatively buoyant jets is a natural consequence of the strongly decelerating mean flow as a result of opposing buoyancy, which is demonstrated in the context of the integral model framework developed by Morton et al. (Proc. R. Soc. Lond. A, vol. 234, no. 1196, 1956, pp. 1–23).

We perform local measurements of both the velocity and the particle volume fraction to study viscous resuspension in non-Brownian suspensions for Shields numbers ranging from $10^{-3}$ to $1$. With this aim, a suspension of polymethacrylate spherical particles dispersed in a lighter Newtonian fluid (Triton X100) is sheared in a vertical Couette cell where both velocity and particle density mappings are performed. We show that the radial profiles of the velocity and of the particle volume fraction are inconsistent in the framework of local rheology of a Newtonian material and that these discrepancies disappear for a neutrally buoyant suspension. The vertical concentration profiles are used to deduce the third particle normal stress, $\varSigma _{33}^p$, by solving the Cauchy equation. The value of $\varSigma _{33}^p$ is shown not to vary linearly with shear rate but rather through a power law with an exponent close to $0.7$, irrespective of the value of the particle volume fraction, in accordance with the recent results of Saint-Michel et al. (Phys. Fluids, vol. 31, 2019, 103301). Finally, we compare our results with the results of previous studies where $\alpha _3=\varSigma _{33}^p/\eta _0\dot {\gamma }$ (with $\eta_0$ the viscosity of the suspending liquid and $\dot{\gamma}$ the shear rate) was deduced from the macroscopic measurement of the height of the resuspended layer. The agreement is satisfactory.

The dynamics of the merging of two distinct holes (hole–hole) and a single hole with a straight rim (hole–rim) are investigated using three-dimensional numerical simulations. Thin liquid sheets with thicknesses ranging from $25$ to $150\ \mathrm {\mu }\textrm {m}$ are considered using air/water conditions. Preliminary simulations of a single hole expansion and the retraction of a bounded liquid sheet edge have shown good agreements with the well-known Taylor–Culick end rim regime. For the hole–hole and the hole–rim cases, our computations reveal that the liquid bridge, formed after the merging, is subjected to three different motions: the extension of the bridge, the mid-plane contraction and vertical/horizontal oscillations of its diameter, with an exponentially decaying amplitude. The mid-plane diameter of the liquid bridge is shown to decrease linearly with time for the hole–rim case while a faster quadratic decrease is observed for the hole–hole case. The small ratio of the extension rate to the capillary contraction rate indicates a slow extension limit of the bridge. However, the liquid bridge does not contract and pinch off on a single point, as predicted in the literature, and its central part forms a cylindrical ligament. Thus, this cylindrical part pinches off at its ends and forms a receding ligament with bulbous ends which can either detach by the end-pinching mechanism or remain attached, recoil and merge into a single large drop. The size of the formed drop, in the case where the ligament does not break, is later expressed as a power-law function of the initial liquid sheet thickness.

Group scattering of an initially compact two-dimensional configuration of N point vortices with strengths of different signs in an unbounded ideal fluid is investigated. This phenomenon manifests itself in a permanent increase of the vortex cloud, accompanied by mixing, clustering and emission of vortices. Possible mechanisms for such an evolution are considered, and the necessary condition is formulated. An invariant function of relative motion, F, is introduced, which is a combination of the first integrals of the system and can be expressed through 2(N−2) independent phase variables. We use this function to construct vortex equilibria and to investigate their nonlinear stability. Cases of N = 3 and N = 5 (for vortex configurations which possess central symmetry), when F depends on two variables only, are examined in detail. In addition to the familiar self-similar scattering (or collapse), we found a regime of asymptotically self-similar scattering (but not collapse) which occurs from any trajectory inside a certain finite domain in the phase space. A similar domain exists also for the case of dipole emission. A periodic absolute motion consisting of alternating stages of expansion and convergence of vortices is discussed. These mechanisms also work during group scattering. Simple examples of this phenomenon are given for the case of five vortices. Finally, group scattering is demonstrated for an initially compact cloud of 1009 vortices with strengths randomly distributed in the range (−1, 1). The area of the main part of the vortex cloud grows with time according to an almost linear law characteristic of self-similar motion.

We employ reconnection-capable, vortex filament methods and finite-volume, Navier–Stokes flow solvers to investigate the topological and helicity dynamics of vortex links for medium and high Reynolds numbers. Our vortex-dynamical model is based on discretization of vortex tubes into bundles of numerical analogues of vortex lines. Due to their nearly singular nature, the numerical vortex lines have topological writhe but not twist. By means of our reconnecting vortex tube model, it is shown that the helicity of a vortex link is conserved during the unknotting process. The dynamics of linking and writhe topological measures indicate that most of the initial linking becomes writhe during the post-reconnection evolution. The helicity spectra of the vortex link present alternating-sign helicity fluctuations at all (potential flow) scales up to the vortex core. At pre-reconnection times, these fluctuations are damped by Biot–Savart vortex stretching and helicity becomes single signed. The post-reconnection spectra indicate an inverse helicity cascade corresponding to the creation of a homogenized vortex blob, a process reminiscent of coherent structure formation in turbulence. An accompanying Navier–Stokes calculation of vortex link dynamics at Reynolds numbers $Re=1500$ confirms the post-reconnection transformation of linking into different topological measures, the pre-reconnection damping of helicity spectra fluctuations and the spectral shift to low wavenumbers at post-reconnection times. Due to viscous dissipation action, however, this shift is accompanied by progressive reduction of helicity peak values.

This study examines the flow characteristics of a turbulent boundary layer over different sand-grain roughness created by P$24$ and P$36$ and P$60$ sandpapers. The experimental dataset is acquired with high-resolution planar particle image velocimetry in the streamwise–wall-normal plane for a range of Reynolds number between $\delta ^+=1200\text {--}6300$, which consists of a number of transitionally and fully rough flow conditions where $30\leq \delta /k_s\leq 111$. The conditions formed over different rough surfaces (having identical surface morphology) enable us to compare rough flows at matched $k_s^+$ or $\delta ^+$ (roughness Reynolds number and Kármán number, respectively), including matched conditions from other studies in the literature. For all the cases, the friction velocity is determined from the direct wall shear-stress measurements using a floating-element drag balance. Mean streamwise velocity profiles exhibit a logarithmic behaviour in the inertial region, and their defect forms are observed to collapse in the outer layer even for the transitionally rough cases at relatively low Reynolds numbers. However, the diagnostic plot of the streamwise velocity intensity suggests that the wall similarity only holds for $k_s^+\geq 75(\Delta U^+\geq 7)$. Analyses at several matched $\delta ^+$ cases show that the mean streamwise velocity defect and turbulence profiles (streamwise and wall-normal velocity variances and the Reynolds shear stress) are self-similar in the outer layer independent of the surface roughness. This similarity extends closer to the wall for the wall-normal velocity variances and Reynolds shear-stress profiles for the weaker roughness (lower $k_s$), which could be a result of higher $\delta /k_s$ for these cases compared with the P$24$ grit sandpaper. For the matched $k_s^+$ conditions, all the profiles were observed to collapse better for fully rough conditions. However, in the transitionally rough regime, the current turbulence statistics are observed to deviate in the outer layer from those reported in other studies (Squire et al., J. Fluid Mech., vol. 795, 2016, pp. 210–240; Morrill-Winter et al., Phys. Rev. Fluids, vol. 2, 2017, 054608). Higher values of roughness function, turbulence intensity and Reynolds shear stress in the current study could be due to overstimulation of the boundary layer. Despite the differences in the turbulence profiles observed, the average large-scale structures across all wall-normal locations are found to be independent of $k_s^+$ and $\delta ^+$.

By analysing a database (Lozano-Durán & Jiménez, Phys. Fluids, vol. 26, 2014, 011702) of fully developed turbulent channel flow at the friction Reynolds number $Re_\tau =4179$, we investigate the sustaining mechanism of a hierarchy of coherent structures in the turbulence. For this purpose, we decompose the turbulent fields into different scales by band-pass filters and quantify the real-space energy transfer. Visualizations of the hierarchy of vortices and velocity in the filtered fields show that the largest-scale structures at each distance from the wall are composed of quasi-streamwise vortices and low-speed streaks. These are similar to well known coherent structures in the buffer layer and they are maintained by a hierarchical self-sustaining process. Quantitatively, however, the energy production rate of the largest-scale structures is different in the log and buffer layers. This difference explains the change of the scaling of the Reynolds stress as a function of the Reynolds number. In contrast to the largest-scale structures, vortices smaller than the distance from the wall distribute isotropically, and they are generated by an energy cascading process. The energy of these small-scale structures is transferred predominantly from twice-larger-scale structures and reduced by half-scale ones through the vortex stretching and contraction, respectively. Turbulent advection from the wall hardly contributes to the maintenance of small-scale structures in the log layer.

We derive an analytical solution for the acoustic streaming inside a rigid tube resulting from a pseudo-standing wave field, generated by two counterpropagating travelling waves. We solve the second-order axisymmetric problem that follows from the perturbation expansion of the governing equations. In the process, we impose no restriction on the diameter of the tube with respect to the thickness of the viscous boundary layer and acoustic wavelength. The derived solution is then used to study the evolution of streaming patterns inside the tube when geometrical and material parameters are varied. We show how the Schlichting streaming torus at the wall bounds the Rayleigh streaming near the axis of the tube. Decreasing the ratio ($\varXi$) of the tube radius to the viscous boundary layer thickness gradually expands the Schlichting streaming, suppressing the Rayleigh streaming. Considering the average mass transport velocity, the Rayleigh streaming vanishes at the critical ratio $\varXi _{{S}}^{{M}}=6.2$. The critical ratio is independent of fluid properties in the limit of large acoustic wavelength relative to the radius of the tube ($\varLambda$). When $\varLambda$ decreases towards unity, large-scale Eckart-like streaming develops near the axis, superseding the Rayleigh streaming, while the Schlichting streaming remains at the wall. In addition, we demonstrate the relevance of the compressibility of the streaming flow and of the full inclusion of the spatial variation of the Reynolds stress that acts as the streaming source. The study is especially relevant for microfluidic systems, wherein the viscous boundary layer can reach significant thicknesses.

We present a novel theoretical framework for the emission and absorption of two-dimensional internal waves in a density stratified medium. Our approach uses a weakly nonlinear perturbation expansion of a streamfunction field that exposes the harmonic structure emitted from a flexible boundary of infinite extent. We report the discovery of a special symmetry in polychromatic waves that share a common horizontal component of phase velocity. Under these conditions, there can be no wave–wave interactions in the domain interior, and therefore all harmonic generation is from the boundary. By activating polychromatic waves on this same flexible surface, we then consider the equivalent inverse problems of emission of a prescribed harmonic signature and absorption of wave energy from a given flow field. Specialising to monochromatic waves, to calculate the amplitudes and phases of the harmonics generated by a monochromatic boundary displacement and to find the explicit form of the absorbing boundary condition for a monochromatic internal wave, we present algorithms that refine lengthy algebraic processes down to a set of executable instructions valid for arbitrary order in the small parameter of the expansion. Finally, we compare our theoretical predictions up to third order with a sophisticated, digitally controlled experimental realisation that we call a ‘magic carpet’, and we find that harmonic analysis of the flow field convincingly supports our theory.

Given a passive tracer distribution $f(x,y)$, what is the simplest unstirred pattern that may be reached under incompressible advection? This question is partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which the patterns of a topological invariant called the field line helicity greatly simplify until reaching a relaxed state. We test two approaches: a variational method with minimal constraints, and a magnetic relaxation scheme where the velocity is determined explicitly by the pattern of $f$. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of $f$, and is numerically more robust. We therefore apply the latter method to two complex mixed patterns modelled on the field line helicity of 3-D magnetic braids. The unstirring separates $f$ into a small number of large-scale regions determined by the initial topology, which is well preserved during the computation. Interestingly, the velocity field is found to have the same large-scale topology as $f$. Similarity to the simplification found empirically in 3-D magnetic reconnection simulations supports the idea that advection is an important principle for field line helicity evolution.

This study investigates the transitional wake dynamics of a simultaneously pitching– heaving airfoil in the low Reynolds number regime. The transition from Kármán to reverse Kármán wakes and the subsequent wake deflection are known to take place entirely in the periodic regime as the dynamic heave velocity $\kappa h$ (proportionally, the amplitude-based Strouhal number $St_A$) is gradually increased. However, further increase in $\kappa h$ may result in the loss of dynamic stability of the deflected vortex street and an eventual transition to chaos. This phenomenon has largely remained unexplored in the existing literature. The present study attempts to fill this gap and aims to establish a dynamic link between the near- and far-field wake transitions. It is shown that the deflected jet undergoes a switching of deflection direction in the far-field as the near-field encounters a quasi-periodic transition with the increase in $\kappa h$. The quasi-periodic behaviour of the near-field is seen to get interspersed with intermittent aperiodic windows, resulting in a complete reversal of the deflection direction through flipping of the immediate vortex couple. Eventually, the wake topology becomes fully chaotic through a series of rapid aperiodic jet-switching. To the best of the authors’ knowledge, this is the first study that investigates the role of this aperiodic jet-switching in ushering in chaos in the wake of a pitch–heave flapping system. The mechanism of jet-switching and the role of leading-edge vortex are also revealed. Flow field dynamics during these changes and the underlying vortex interactions are analysed using nonlinear dynamical tools.

Spontaneous capillary flow of liquids in narrow spaces plays a key role in a plethora of applications including lab-on-a-chip devices, heat pipes, propellant management devices in spacecrafts and flexible printed electronics manufacturing. In this work we use a combination of theory and experiment to examine capillary-flow dynamics in open rectangular microchannels, which are often found in these applications. Scanning electron microscopy and profilometry are used to highlight the complexity of the free-surface morphology. We develop a self-similar lubrication-theory-based model accounting for this complexity and compare model predictions to those from the widely used modified Lucas–Washburn model, as well as experimental observations over a wide range of channel aspect ratios $\lambda$ and equilibrium contact angles $\theta _0$. We demonstrate that for large $\lambda$ the two model predictions are indistinguishable, whereas for smaller $\lambda$ the lubrication-theory-based model agrees better with experiments. The lubrication-theory-based model is also shown to have better agreement with experiments at smaller $\theta _0$, although as $\theta _0\rightarrow {\rm \pi}/4$ it fails to account for important axial curvature contributions to the free surface and the agreement worsens. Finally, we show that the lubrication-theory-based model also quantitatively predicts the dynamics of fingers that extend ahead of the meniscus. These findings elucidate the limitations of the modified Lucas–Washburn model and demonstrate the importance of accounting for the effects of complex free-surface morphology on capillary-flow dynamics in open rectangular microchannels.

This is an ocean motivated study which investigates the impacts of sinusoidal bottom topography on baroclinic instability of zonal vertically sheared flows in the two-layer quasigeostrophic model. The corresponding linear stability problem is solved by assuming Fourier-mode solutions in both the zonal and meridional directions. In the presence of variable topographic features, the Fourier modes become coupled due to phase shifts in the wavevectors. The spectral discretisation method used in this work retains the primary relationship between different Fourier modes; thus, the linear stability eigenproblem can be solved for any periodic topography. Moreover, this method does not need any additional assumptions, such as considering small-amplitude or large-scale bottom irregularities, as in some previous studies. In this work, the eigenproblem is solved for a range of topographic amplitudes and wavenumbers, and their effects on the growth rates and shapes of the most unstable eigenmodes are discussed. In general, both the zonal and meridional variations in topography tend to suppress the baroclinic instability. However, it is found that only meridionally varying topography affects the magnitudes of the fastest growth rates. In this instance, unstable modes appear to form two clusters well separated in the zonal wavenumber axis and growth rate maxima occur at two distinct zonal wavenumbers. Dependencies of the characteristics of these clusters on the values of topography amplitude and ridge width are reviewed. Finally, doubly periodic numerical simulations are used to verify the results from the linear stability analysis.

We study bubble motion in a vertical capillary tube under an external flow. Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188) has shown that, without external flow, a bubble can spontaneously rise when the Bond number (${Bo} \equiv \rho g R^2 / \gamma$) is above the critical value ${Bo}_{cr}=0.842$, where $\rho$ is the liquid density, $g$ the gravitational acceleration, $R$ the tube radius and $\gamma$ the surface tension. It was then shown by Magnini et al. (Phys. Rev. Fluids, vol. 4, issue 2, 2019, 023601) that the presence of an imposed liquid flow, in the same (upward) direction as buoyancy, accelerates the bubble and thickens the liquid film around it. In this work we carry out a systematic study of the bubble motion under a wide range of upward and downward external flows, focusing on the inertialess regime with Bond numbers above the critical value. We show that a rich variety of bubble dynamics occurs when an external downward flow is applied, opposing the buoyancy-driven rise of the bubble. We reveal the existence of a critical capillary number of the external downward flow (${Ca}_l \equiv \mu U_l/\gamma$, where $\mu$ is the fluid viscosity and $U_l$ is the mean liquid speed) at which the bubble arrests and changes its translational direction. Depending on the relative direction of gravity and the external flow, the thickness of the film separating the bubble surface and the tube inner wall follows two distinct solution branches. The results from theory, experiments and numerical simulations confirm the existence of the two solution branches and reveal that the two branches overlap over a finite range of ${Ca}_l$, thus suggesting non-unique, history-dependent solutions for the steady-state film thickness under the same external flow conditions. Furthermore, inertialess symmetry-breaking shape profiles at steady state are found as the bubble transits near the tipping points of the solution branches, which are shown in both experiments and three-dimensional numerical simulations.

Biological swimmers frequently navigate in geometrically restricted media. We study the prescribed-stroke problem of swimmers confined to a planar viscous membrane embedded in a bulk fluid of different viscosity. In their motion, microscopic swimmers disturb the fluid in both the membrane and the bulk. The flows that emerge have a combination of two-dimensional (2-D) and three-dimensional (3-D) hydrodynamic features, and such flows are referred to as quasi-two-dimensional. The cross-over from 2-D to 3-D hydrodynamics in a quasi-2-D fluid is controlled by the Saffman length, a length scale given by the ratio of the 2-D membrane viscosity to the 3-D viscosity of the embedding bulk fluid. We have developed a computational and theoretical approach based on the boundary element method and the Lorentz reciprocal theorem to study the swimming of microorganisms for a range of values of the Saffman length. We found that a flagellum propagating transverse sinusoidal waves in a quasi-2-D membrane can develop a swimming speed exceeding that in pure 2-D or 3-D fluids, while the propulsion of a 2-D squirmer is slowed down by the presence of the bulk fluid.

The evolution of an instability in a transitional hypersonic boundary layer and its effects on aerodynamic cooling are investigated over 260-mm-long flared cone models with smooth and porous surfaces. Experiments are conducted in a Mach 6 wind tunnel using Rayleigh-scattering flow visualization, fast-response pressure sensors and an infrared camera. Calculations are performed based on both direct numerical simulations (DNS) and linear stability theory (LST). The unit Reynolds number is $9.7 \times 10^6\ \textrm {m}^{-1}$. It is confirmed that a cooled region appears downstream of the local heat peak as the second-mode instability evolves over the smooth-surface model, as found in other studies. Comparisons between the DNS and LST results show that the nonlinear interaction of the second mode causes the phase difference $\phi _{p\theta }$ to change between the periodic pressure and dilatation waves. This forms a negative cycle-averaged pressure dilatation near the wall and creates the cooled region. Further, by using porous steel to modify the sound admittance of the model surface, it is possible to artificially obtain negative cycle-averaged pressure dilatation near the wall, and thus reduce the surface heat flux by approximately 28 %. These results indicate the possibility of precisely controlling the pressure-dilatation-induced aerodynamic heating through the modification of surface sound admittance.