Dense suspensions are non-Newtonian fluids that exhibit strong shear thickening and normal stress differences. Using numerical simulation of extensional and shear flows, we investigate how rheological properties are determined by the microstructure that is built under flows and by the interactions between particles. By imposing extensional and shear flows, we can assess the degree of flow-type dependence in regimes below and above thickening. Even when the flow-type dependence is hindered, non-dissipative responses, such as normal stress differences, are present and characterise the non-Newtonian behaviour of dense suspensions.

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around $47^{\circ }$ , for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below $90^{\circ }$ can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

We present the solution of the idealized steady-state gravity current of height $h$ and density $\unicode[STIX]{x1D70C}_{1}$ that propagates into an ambient motionless fluid of height $H$ and density $\unicode[STIX]{x1D70C}_{2}$ with an upper surface open to the atmosphere (open channel) at high Reynolds number. The current propagates with speed $U$ and causes a depth decrease $\unicode[STIX]{x1D712}$ of the top surface. This is a significant extension of Benjamin’s (J. Fluid Mech., vol. 31, 1968, pp. 209–248) seminal solution for a fixed-top channel ( $\unicode[STIX]{x1D712}=0$ ). In the present case, the determination of $\unicode[STIX]{x1D712}$ is part of the problem. The dimensionless parameters of the problem are $a=h/H$ and $r=\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}$ . We show that a control-volume analysis determines $\tilde{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D712}/H$ and $\mathit{Fr}=U/(g^{\prime }h)^{1/2}$ as functions of $a$ and $r$ , where $g^{\prime }=(r^{-1}-1)g$ is the reduced gravity. The system satisfies balance of volume and momentum (explicitly), and vorticity (implicitly). We develop two branches (models) of solution for the control-volume balances. The first assumes that the outflow velocity $u_{2}$ is $z$ -independent, accompanied by a similarly homogeneous headloss; this model is exactly the counterpart of Benjamin’s solution ( $z$ is the vertical coordinate). The second model assumes that $u_{2}$ increases from $0$ in a layer of finite thickness $\unicode[STIX]{x1D702}$ (say) above the $z=h$ interface, and the headloss occurs in this layer only (a vortex wake). The $\mathit{Fr},\tilde{\unicode[STIX]{x1D712}}$ results of these models are in excellent agreement, and this increases the confidence concerning the physical relevance of the analysis. (The solution is obtained numerically.) The predicted flows are in general dissipative, and thus physically energetically valid only for $a\leqslant a_{max}(r)\approx 0.5r$ , where non-negative dissipation appears. In this range of validity, the open-surface $\mathit{Fr}(a,r)$ is smaller than Benjamin’s $\mathit{Fr}_{b}(a)$ , but the reduction is not dramatic, typically a few per cent. The current with thickness $h=a_{max}H$ is energy conserving. For given $r$ , the energy conserving $a_{max}(r)$ provides the maximum height depression $\tilde{\unicode[STIX]{x1D712}}_{max}(r)\approx 0.45r^{1/4}(1-r)$ . The largest possible depression is $\tilde{\unicode[STIX]{x1D712}}_{max}\approx 0.25$ at $r\approx 0.20$ , and this corresponds to the minimum of $\mathit{Fr}(a,r)/\mathit{Fr}_{b}(a)$ which is approximately 0.75. For $r<0.68$ , the propagation is subcritical; for $r>0.68$ , the subcritical flow condition reduces the applicability slightly to $a\leqslant a_{\mathit{crit}}\approx 0.345$ . For a Boussinesq system with $r\approx 1$ , we obtain $\tilde{\unicode[STIX]{x1D712}}\ll 1$ , and the present $\mathit{Fr}$ , dissipation and $a_{\mathit{crit}}$ results differ only slightly from Benjamin’s classical predictions, as expected.

Long-term marginal stability of a new family of isolated oceanic vortices is analysed. Sign reversal of the radial gradient of the potential vorticity anomaly, as implied by the isolation requirement, leads to vortex unsteadiness but does not break the coherence of the vortex, which remains marginally stable even for high absolute Rossby numbers $Ro\simeq 0.8$ . The marginally stable vortices are characterized by a zero amount of potential vorticity anomaly on every isopycnal. The marginally stable final state is an unsteady vortex whose inner one-signed potential vorticity anomaly experiences revolution, rotation, precession and nutation.

In this paper, drop impact onto a sphere is numerically investigated at moderate Reynolds and Weber numbers. It is naturally expected that the aspect ratio of the sphere to the drop, $\unicode[STIX]{x1D706}_{r}$ , would make a big difference to drop spreading and retraction on the sphere, compared with drop impact onto a flat substrate. To quantitatively assess the effect of $\unicode[STIX]{x1D706}_{r}$ , a diffuse-interface immersed-boundary method is adopted after being validated against experiments. With the help of numerical simulations, we identify the key regimes in the spreading and retraction, analyse the results by scaling laws, and quantitatively evaluate the effect of $\unicode[STIX]{x1D706}_{r}$ on the impact dynamics. We find that the thickness of the liquid film spreading on the sphere can be well approximated by $h_{L,\infty }(1+3/4\unicode[STIX]{x1D706}_{r}^{-3/2})$ , where $h_{L,\infty }$ represents the film thickness of drop impact on a flat substrate. At the early stage of spreading, the temporal variation of the wetted area is independent of $\unicode[STIX]{x1D706}_{r}$ when the time is rescaled by the thickness of the liquid film. Drops are observed to retract on the sphere at a roughly constant speed, and the predictions of theoretical analysis are in good agreement with numerical results.

The cascading process of turbulent kinetic energy from large-scale fluid motions to small-scale and lesser-scale fluid motions in isotropic turbulence may be modelled as a hierarchical random multiplicative process according to the multifractal formalism. In this work, we show that the same formalism might also be used to model the cascading process of momentum in wall-bounded turbulent flows. However, instead of being a multiplicative process, the momentum cascade process is additive. The proposed multifractal model is used for describing the flow kinematics of the low-pass filtered streamwise wall-shear stress fluctuation $\unicode[STIX]{x1D70F}_{l}^{\prime }$ , where $l$ is the filtering length scale. According to the multifractal formalism, $\langle {\unicode[STIX]{x1D70F}^{\prime }}^{2}\rangle \sim \log (Re_{\unicode[STIX]{x1D70F}})$ and $\langle \exp (p\unicode[STIX]{x1D70F}_{l}^{\prime })\rangle \sim (L/l)^{\unicode[STIX]{x1D701}_{p}}$ in the log-region, where $Re_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number, $p$ is a real number, $L$ is an outer length scale and $\unicode[STIX]{x1D701}_{p}$ is the anomalous exponent of the momentum cascade. These scalings are supported by the data from a direct numerical simulation of channel flow at $Re_{\unicode[STIX]{x1D70F}}=4200$ .

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

Here, we examine the conditions for layer formation in a stratified shear flow. We reformulate the conditions for amplification of small perturbations to a uniform stratification first proposed by Phillips (Deep Sea Research and Oceanographic Abstracts, vol. 19, 1972, pp. 79–81, Elsevier) and Posmentier (J. Phys. Oceanogr., vol. 7 (2), 1977, pp. 298–300) using the sorted buoyancy coordinates introduced by Nakamura (J. Atmos. Sci., vol. 53 (11), 1996, pp. 1524–1537) and Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193). We consider the possible dependence of the effective diffusivity on three non-dimensional parameters, the gradient Richardson number, the buoyancy Reynolds number and the Prandtl number, and obtain a simple expression for conditions favourable for layer formation. The new framework is applied to direct numerical simulations of stratified shear flow. We then apply a recent multi-parameter parameterization developed by Salehipour et al. (Geophys. Res. Lett., vol. 43 (7), 2016, pp. 3370–3379), which suggests that layer formation is favoured for large Prandtl numbers and moderate to large values of the gradient Richardson and buoyancy Reynolds numbers.

Experimental measurements of the displacements of shock accelerated microparticles from shortly after shock interaction to the particle relaxation time show time-dependent drag coefficients ( $C_{D}$ ) that are much higher than those predicted by quasi-steady and unsteady drag models. Nylon particles with mean diameter of $4~\unicode[STIX]{x03BC}\text{m}$ , accelerated by one-dimensional normal shocks (Mach number $M_{s}=1.2$ , 1.3 and 1.4), have measured $C_{D}$ values that follow a power-law behaviour. The drag is a function of the time-dependent Knudsen number, $Kn^{\ast }=M_{s}/Re_{p}$ , where the particle Reynolds number ( $Re_{p}$ ) is calculated using the time-dependent slip velocity. Some portion of the drag can be attributed to quasi-steady forces, but the total drag cannot be predicted by current unsteady force models that are based on the Basset–Boussinesq–Oseen equation and pressure drag. The largest contribution to the total drag is the unsteady component ( $C_{D,us}$ ) until the particle attains $Kn^{\ast }\approx 0.5{-}1.0$ , then the unsteady contribution decays. The quasi-steady component ( $C_{D,qs}$ ) increases almost linearly with $Kn^{\ast }$ , intersects the $C_{D,us}$ at $Kn^{\ast }\approx 2$ and becomes the primary contributor to the drag towards the end of the relaxation zone as $Re_{p}\rightarrow 0$ . There are currently no analytical models that are able to predict the nonlinear behaviour of the shock accelerated particles during the relaxation phase of the flow.

Cavitation bubbles are nucleated at a high repetition rate in water by delivering a pulsed laser through a fibre optic. Continuous high-frequency cavitation drives a stream away from the fibre tip. Using high-speed photography and particle image velocimetry, the stream is characterised as a synthetic jet, generated by trains of vortices induced by non-spherical bubble collapse. At low laser power, the bubbles collapse before the arrival of a subsequent laser pulse. Yet, by increasing the laser power, a system of bubbles is formed which leads to complex bubble–bubble interactions. The synthetic jet is observed regardless of the bubble formation regime, demonstrating the stability of the phenomenon. Synthetic jet generation by repetitive bubble collapse extends the well-studied acoustic streaming from small-amplitude bubble oscillations.

We derive an alternative formulation of the turbulent kinetic energy equation for flows with strong near-wall density and viscosity gradients. The derivation is based on a scaling transformation of the Navier–Stokes equations using semi-local quantities. A budget analysis of the semi-locally scaled turbulent kinetic energy equation shows that, for several variable property low-Mach-number channel flows, the ‘leading-order effect’ of variable density and viscosity on turbulence in wall bounded flows can effectively be characterized by the semi-local Reynolds number. Moreover, if a turbulence model is solved in its semi-locally scaled form, we show that an excellent agreement with direct numerical simulations is obtained for both low- and high-Mach-number flows, where conventional modelling approaches fail.

An assessment of the turbulent boundary layer flow structure, which is coherent with the near-wall region, is carried out through a spectral coherence analysis. This spectral method is applied to datasets comprising synchronized two-point streamwise velocity signals at a near-wall reference position and a range of wall-normal positions spanning a Reynolds-number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{6})$ . Within each dataset, a self-similar structure is identified from the coherence between the turbulence in the logarithmic region and at the near-wall reference position. This self-similarity is described by a streamwise/wall-normal aspect ratio of $\unicode[STIX]{x1D706}_{x}/z\approx 14$ , where $\unicode[STIX]{x1D706}_{x}$ and $z$ are the streamwise wavelength and wall-normal distance respectively.

A comparison between swimming by flapping and by periodic contractions is conducted. Swimming by flapping is approximated as a pitching plate while swimming by periodic contractions is approximated as clapping plates. A direct comparison is made between the two propulsion mechanisms by utilizing a machine that can operate in either a flapping or a clapping mode between Reynolds numbers of 1880 and 11 260 based on the average plate tip velocity and span. The average thrust generated and the average input power required per cycle are compared between cases where the total sweep angle and the total sweep time are identical. Variation of the kinematics results in a similar thrust between the two mechanisms, but a greater power is required for clapping. Variation of the flexibility results in a consistent decrease in the required power for clapping and a decrease in thrust at high flexibility. Variation of the duty cycle for clapping rigid plates results in a significant increase in thrust and a significant decrease in the required power. Overall, the results suggest that flapping propulsion is the more effective propulsion mechanism within the range of Reynolds numbers tested.

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

We develop a generalized Rotne–Prager–Yamakawa approximation for the dipolar components of the inverse friction matrix and use it for calculating the intrinsic viscosity of rigidly connected bead conglomerates. Such bead models are commonly used in the calculation of hydrodynamic properties of macromolecules. We consider both the case of non-overlapping constituent beads as well as overlapping beads of different sizes. We demonstrate the accuracy of the approximation in two test cases and show that it performs well even if the distances between the beads are small or if the beads overlap. Robust performance of this approximation in the case of overlapping beads stems from its correct limiting behaviour at a complete overlap, with one sphere fully immersed in the other. The generalized Rotne–Prager–Yamakawa approximation is thus well suited for evaluation of intrinsic viscosity, which is a key quantity in characterizing molecular conformations of biological macromolecules.

We demonstrate that the critical magnetic Reynolds number $Rm_{c}$ for a turbulent non-helical dynamo in the limit of low magnetic Prandtl number $Pm$ (i.e. $Pm=Rm/Re\ll 1$ ) can be significantly reduced if the flow is subjected to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor of more than $10^{3}$ . This strong decrease in the onset is attributed to the transfer of energy to the large scales, forming a large-scale condensate, and the reduction in the turbulent fluctuations that cause the flow to have a much larger cutoff length scale than in a non-rotating flow of the same Reynolds number. The dynamo thus behaves as if it is driven just by the large scales that act as a laminar flow (i.e. it behaves as a high $Pm$ dynamo) even though the actual Reynolds number is much higher than the magnetic Reynolds number (i.e. low $Pm$ ). Our finding thus points to a new paradigm for the design of new experiments on liquid metal dynamos.

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

The main objective of this work is to study experimentally the primary instability of non-Newtonian film flows down an inclined plane. We focus on low-concentration shear-thinning aqueous solutions obeying the Carreau law. The experimental study essentially consists of measuring wavelengths in marginal conditions, which yields the primary stability threshold for a given slope. The experimental results for neutral curves presented in the $(Re,f_{c})$ and $(Re,k)$ planes (where $f_{c}$ is the driving frequency, $k$ is the wavenumber and $Re$ is the Reynolds number) are in good agreement with the numerical results obtained by a resolution of the generalized Orr–Sommerfeld equation. The long-wave asymptotic extension of our results is consistent with former theoretical predictions of the critical Reynolds number. This is the first experimental evidence of the destabilizing effect of the shear-thinning behaviour in comparison with the Newtonian case: the critical Reynolds number is smaller, and the ratio between the critical wave celerity and the flow velocity at the free surface is larger.

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.

It was previously observed by Krug et al. (J. Fluid Mech., vol. 765, 2015, pp. 303–324) that the surface area $A_{\unicode[STIX]{x1D702}}$ of the turbulent/non-turbulent interface (TNTI) in gravity currents decreases with increasing stratification, significantly reducing the entrainment rate. Here, we consider the multiscale properties of this effect using direct numerical simulations of temporal gravity currents with different gradient Richardson numbers $Ri_{g}$ . Our results indicate that the reduction of $A_{\unicode[STIX]{x1D702}}$ is caused by a decrease of the fractal scaling exponent $\unicode[STIX]{x1D6FD}$ , while the scaling range remains largely unaffected. We further find that convolutions of the TNTI are characterized by different length scales in the streamwise and wall-normal directions, namely the integral scale $h$ and the shear scale $l_{Sk}=k^{1/2}/S$ (formed using the mean shear $S$ and the turbulent kinetic energy  $k$ ) respectively. By recognizing that the anisotropy implied by the different scaling relations increases with increasing $Ri_{g}$ , we are able to model the $Ri_{g}$ dependence of $\unicode[STIX]{x1D6FD}$ in good agreement with the data.