We numerically investigate the effects of non-Newtonian fluid properties, including shear thinning and elasticity, on the locomotion of Taylor’s swimming sheet with arbitrary amplitude. Our results show that elasticity hinders the swimming speed, but a shear-thinning viscosity in the absence of elasticity enhances the speed. The combination of the two effects, modelled using a Giesekus constitutive equation, hinders the swimming speed. We find that the swimming speed of an infinitely long waving sheet in an inelastic shear-thinning fluid has a maximum, whose value depends on the sheet undulation amplitude and the fluid rheological properties. The power consumption, on the other hand, follows a universal scaling law.

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading-order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers that adjusts the normal derivative of the temperature fluctuation to zero. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading-order reduced system of governing equations is therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading-order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

We present experimental evidence that a minute amount of polymer additives can significantly enhance heat transport in the bulk region of turbulent thermal convection. The effects of polymer additives are found to be the enhancement of coherent heat fluxes and suppression of incoherent heat fluxes. The enhanced heat transport is associated with the increased coherency of thermal plumes, as a result of the suppression of small-scale turbulent fluctuations by polymers. The incoherent heat flux, arising from turbulent background fluctuations, makes no net contribution to heat transport. The fact that polymer additives can increase the coherency of thermal plumes is supported by the measurements of a number of local quantities, such as the extracted plume amplitude and width, the velocity autocorrelation functions and the velocity–temperature cross-correlation coefficient. The results from local measurements also suggest the existence of a threshold value for the polymer concentration, only above which significant modification of the plume coherent properties and enhancement of the local heat flux can be observed. Estimation of the plume emission rate suggests a stabilization of the thermal boundary layer by polymer additives.

Many micro-organisms find themselves immersed in fluids displaying non-Newtonian rheological properties such as viscoelasticity and shear-thinning viscosity. The effects of viscoelasticity on swimming at low Reynolds numbers have already received considerable attention, but much less is known about swimming in shear-thinning fluids. A general understanding of the fundamental question of how shear-thinning rheology influences swimming still remains elusive. To probe this question further, we study a spherical squirmer in a shear-thinning fluid using a combination of asymptotic analysis and numerical simulations. Shear-thinning rheology is found to affect a squirming swimmer in non-trivial and surprising ways; we predict and show instances of both faster and slower swimming depending on the surface actuation of the squirmer. We also illustrate that while a drag and thrust decomposition can provide insights into swimming in Newtonian fluids, extending this intuition to problems in complex media can prove problematic.

Backwater curves denote the depth profiles of steady flows in a shallow open channel. The classification of these curves for turbulent regimes is commonly used in hydraulics. When the bottom slope $I$ is increased, they can describe the transition from fluvial to torrential regimes. In the case of an infinitely wide channel, we show that laminar flows have the same critical height $h_{c}$ as that in the turbulent case. This feature is due to the existence of surface slope singularities associated to plug-like velocity profiles with vanishing boundary-layer thickness. We also provide the expression of the critical surface slope as a function of the bottom curvature at the critical location. These results validate a similarity model to approximate the asymptotic Navier–Stokes equations for small slopes $I$ with Reynolds number $Re$ such that $Re\,I$ is of order 1.

We present new experiments to measure the rate of entrainment of ambient fluid into a high Reynolds number, axisymmetric, turbulent gravity current. The current is produced by the rapid release of a finite volume of aqueous salt solution from a lock of length $r_{o}$ into a diverging channel, $r>0$ , of angle $9.5^{\circ }$ , filled with a finite depth, $H$ , of fresh water. Using light attenuation we measure the evolving density of the flow, and using dye studies we illustrate the process of mixing between the current and ambient fluid. After an initial adjustment, a circulation develops in the head of the flow: current fluid reaches the nose of the flow, rises up and moves backwards relative to the nose. We find that, owing to the mixing, the volume of the current increases as $V\sim 0.2r_{n}^{7/4}r_{o}^{1/4}H$ while the maximum depth of the head decreases as $h_{n}\sim 0.5H(r_{o}/r_{n})^{1/4}$ , where $r_{n}$ is the location of the front of the current. Combining these results, we estimate that the recirculating current fluid mixes with a fraction $E=0.33\pm 0.09$ of the ambient fluid that is directly ahead of the current and displaced upwards by it. Some of the mixed fluid supplies the tail of the flow, while the remainder recirculates into the head, which becomes progressively more dilute. In accord with Huppert & Simpson (J. Fluid Mech., vol. 99, 1980, pp. 785–799), we find that the position of the front increases with time as $r_{n}\approx (1.28\pm 0.05)B^{1/4}t^{1/2}$ , where $B$ is the total buoyancy of the flow. We also find that the maximum value of the vertical integral of the buoyancy $(\overline{g^{\prime }}h)_{n}$ decreases with the position of the nose according to the relation $(\overline{g^{\prime }}h)_{n}\approx (0.89\pm 0.12)Br_{n}^{-2}$ , consistent with a Froude number $0.86\pm 0.07$ . We compare our measurements with a new idealised self-similar solution of the depth-averaged equations that accounts for the mixing at the nose, the vertical shear in the velocity and the lateral stratification of the buoyancy within the current.

The deformation of an initially circular miscible blob in a rectilinear displacement is investigated numerically for porous media when the blob is more viscous than the displacing fluid. We find in the parameter space spanned by the Péclet number and log-mobility ratio the existence of a new lump-shaped instability zone between two distinct regimes of comet and viscous fingering (VF) deformations. The more viscous circular blob is destabilized by VF only over a finite window of log-mobility ratio, contrary to the displacement of a more viscous finite slice with planar interfaces. This difference is attributed to the initial curvature of the miscible blob.

It is shown theoretically how the scaling of coherent structures in shear flows changes their asymptotic development at large Reynolds number. Based on the theory a family of nonlinear self-sustained states at Kolmogorov microscale is numerically identified on the laminar–turbulent boundary of shear flows. Theoretically and numerically the states connect to known asymptotic states existing at larger scale. The asymptotically very small amplitude of the new states may explain why strongly sheared, linearly stable laminar flows can cause a turbulent transition by small disturbances. The numerically obtained Kolmogorov-scale solutions can be used to describe the theoretically minimal self-sustained structures appearing in various shear flows.

We perform direct numerical simulations of an unstably stratified turbulent channel flow to address the effects of buoyancy on the boundary layer dynamics and mean field quantities. We systematically span a range of parameters in the space of friction Reynolds number ( $\mathit{Re}_{{\it\tau}}$ ) and Rayleigh number ( $\mathit{Ra}$ ). Our focus is on deviations from the logarithmic law of the wall due to buoyant motion. The effects of convection in the relevant ranges are discussed, providing measurements of mean profiles of velocity, temperature and Reynolds stresses as well as of the friction coefficient. A phenomenological model is proposed and shown to capture the observed deviations of the velocity profile in the log-law region from the non-convective case.

The spatial and orientational distribution in a dilute active suspension of non-Brownian run-and-tumble spherical swimmers confined between two planar hard walls is calculated theoretically. Using a kinetic model based on coupled bulk/surface probability density functions, we demonstrate the existence of a concentration wall boundary layer with thickness scaling with the run length, the absence of polarization throughout the bulk of the channel, and the presence of sharp discontinuities in the bulk orientation distribution in the neighbourhood of orientations parallel to the wall in the near-wall region. Our model is also applied to calculate the swim pressure in the system, which approaches the previously proposed ideal-gas behaviour in wide channels but is found to decrease in narrow channels as a result of confinement. Monte Carlo simulations are also performed for validation and show excellent quantitative agreement with our theoretical predictions.

Evaporating suspensions of colloidal particles lead to the formation of a variety of patterns, ranging from a left-over ring of a dried coffee drop to uniformly distributed solid pigments left behind wet paint. To characterize the transition between rings and uniform deposits, we investigate the dynamics of a drying droplet via a multiphase model of colloidal particles in a solvent. Our theory couples the inhomogeneous evaporation at the evolving droplet interface to the dynamics inside the drop. This includes the liquid flow, local variations of the particle concentration leading to a cross-over between dilute and dense suspensions, and the resulting propagation of the deposition front. A dimensionless parameter combining the capillary number and the droplet aspect ratio captures the formation conditions of different pattern types while correctly accounting for the transition from Stokes flow to Darcy flow at high solute concentrations.

This study revisits the problem of a steadily propagating semi-infinite hydraulic fracture in which the processes of toughness-related energy release, viscous dissipation and leak-off compete on multiple length scales. This problem typically requires the solution of a system of integro-differential equations with a singular kernel, which is complicated by the need to capture extremely disparate length scales. In this study the governing equations are rewritten in the form of one non-singular integral equation. This reformulation enables the use of standard numerical techniques to capture the complete multiscale behaviour accurately and efficiently. This formulation also makes it possible to approximate the problem by a separable ordinary differential equation, whose closed-form solution captures the multiscale behaviour sufficiently accurately to be used in practical applications. We also consider a similar reformulation of the equations governing the propagation of a buoyancy-driven semi-infinite hydraulic fracture. The resulting numerical solution is able to capture the near-tip multiscale behaviour efficiently and agrees well with published solutions calculated in the large-toughness limit.

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude ${\it\gamma}_{0}$ , we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope ${\it\alpha}$ of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux $P$ associated with a certain power-law spectrum. We propose an effective and novel way to evaluate $P$ in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux $P$ and the total energy dissipation rate ${\it\Gamma}$ , which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of $P$ which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of ${\it\gamma}_{0}$ only. The value of the spectral slope ${\it\alpha}$ at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of ${\it\gamma}_{0}$ , in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of ${\it\gamma}_{0}$ .

We analyse the low-Reynolds-number flow generated by a cylinder (of radius $a$ ) in a stream (of velocity $U_{\infty }$ ) which has a uniform through-surface blowing component (of velocity $U_{b}$ ). The flow is characterized in terms of the Reynolds number $Re$ ( $=2aU_{\infty }/{\it\nu}$ , where ${\it\nu}$ is the kinematic viscosity of the fluid) and the dimensionless blow velocity ${\it\Lambda}$ ( $=U_{b}/U_{\infty }$ ). We seek the leading-order symmetric solution of the vorticity field which satisfies the near- and far-field boundary conditions. The drag coefficient is then determined from the vorticity field. For the no-blow case Lamb’s (Phil. Mag., vol. 21, 1911, pp. 112–121) expression is retrieved for $Re\rightarrow 0$ . For the strong-sucking case, the asymptotic limit, $C_{D}\approx -2{\rm\pi}{\it\Lambda}$ , is confirmed. The blowing solution is valid for ${\it\Lambda}<4/Re$ , after which the flow is unsymmetrical about ${\it\theta}={\rm\pi}/2$ . The analytical results are compared with full numerical solutions for the drag coefficient $C_{D}$ and the fraction of drag due to viscous stresses. The predictions show good agreement for $Re=0.1$ and ${\it\Lambda}=-5,0,5$ .

We experimentally determine the phase diagram for impacting ethanol droplets on a smooth, sapphire surface in the parameter space of Weber number $\mathit{We}$ versus surface temperature $T$ . We observe two transitions, namely the one towards splashing (disintegration of the droplet) with increasing $\mathit{We}$ , and the one towards the Leidenfrost state (no contact between the droplet and the plate due to a lasting vapour film) with increasing $T$ . Consequently, there are four regimes: contact and no splashing (deposition regime), contact and splashing (contact–splash regime), neither contact nor splashing (bounce regime), and finally no contact, but splashing (film–splash regime). While the transition temperature $T_{L}$ to the Leidenfrost state depends weakly, at most, on $\mathit{We}$ in the parameter regime of the present study, the transition Weber number $\mathit{We}_{C}$ towards splashing shows a strong dependence on $T$ and a discontinuity at $T_{L}$ . We quantitatively explain the splashing transition for $T<T_{L}$ by incorporating the temperature dependence of the physical properties in the theory by Riboux & Gordillo (Phys. Rev. Lett., vol. 113(2), 2014, 024507; J. Fluid Mech., vol. 772, 2015, pp. 630–648).

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law ( ${\it\theta}(t)\sim t^{-2}$ ), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely two-dimensional plane Poiseuille flow. Exact solutions of the Navier–Stokes equations are computed numerically and tracked in the streamwise wavenumber–Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane Couette flow.

We study the behaviour of elongated puddles deposited on non-wetting substrates. Such liquid strips retract and adopt circular shapes after a few oscillations. Their thickness and horizontal surface area remain constant during this reorganization, so that the energy of the system is only lowered by minimizing the length of the contour (and the corresponding surface energy); despite the large scale of the experiments (several centimetres), motion is driven by surface tension. We focus on the retraction stage, and show that its velocity results from a balance between the capillary driving force and inertia, due to the frictionless motion on non-wetting substrates. As a consequence, the retraction velocity has a special Taylor–Culick structure, where the puddle width replaces the usual thickness.

We present measurements of the azimuthal rotation velocity $\dot{{\it\theta}}(t)$ and thermal amplitude ${\it\delta}(t)$ of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. Both $\dot{{\it\theta}}(t)$ and ${\it\delta}(t)$ exhibit clear oscillations at the modulation frequency ${\it\omega}$ . Fluid acceleration driven by oscillating Coriolis force causes an increasing phase lag in $\dot{{\it\theta}}(t)$ when ${\it\omega}$ increases. The applied modulation produces oscillatory boundary layers and the resulting time-varying viscous drag modifies ${\it\delta}(t)$ periodically. Oscillation of $\dot{{\it\theta}}(t)$ with maximum amplitude occurs at a finite modulation frequency ${\it\omega}^{\ast }$ . Such a resonance-like phenomenon is interpreted as a result of optimal coupling of ${\it\delta}(t)$ to the modulated rotation velocity. We show that an extended large-scale circulation model with a relaxation time for ${\it\delta}(t)$ in response to the modulated rotation provides predictions in close agreement with the experimental results.