The interaction of a hollow droplet impacting a solid surface occurs in several applications, including controllable biomedicine and thermal spray coating. Understanding the physics of the hollow droplet spreading is the key to maintaining the mass transfer process in all relevant applications. In this work, a comprehensive experimental, numerical and theoretical study is performed on water hollow droplets impacting a rigid surface to better understand the flattening process of a hollow droplet. In the numerical part, compressible Navier–Stokes equations are solved using the volume of fluid (VOF) method in a two-dimensional (2-D)-axisymmetric model. The comparison of simulation results with the experimental photographs shows that the numerical solution can correctly predict the hollow droplet shape evolution. The results show that the spreading diameter and height of the counter-jet formed after the hollow droplet impact grow with impact velocity. Investigating the size and location of the entrapped bubble shows an optimum bubble size that facilitates the hollow droplet flattening. It is also shown that the ripples on splats produced by the hollow droplets with a larger bubble size are higher than those of small bubbles. In the end, a theoretical model is developed to analyse the maximum spreading diameter of the hollow droplet impact analytically. Its prediction is in good agreement with the experimental and numerical results.

In a standard Rayleigh–Bénard experiment, a layer of fluid is confined between two horizontal plates and the convection regime is controlled by the temperature difference between the hot lower plate and the cold upper plate. The effect of direct heat injection into the fluid layer itself, for example by light absorption, is studied here theoretically. In this case, the Nusselt number ($Nu$) depends on three non-dimensional parameters: the Rayleigh ($Ra$) and Prandtl ($Pr$) numbers and the ratio between the spatial extension of the heat source ($l$) and the height of the fluid layer ($h$). For both the well-known classical and ultimate convection regimes, the theory developed here gives a formula for the variations of the Nusselt number as a function of these parameters. For the classical convection regime, by increasing $l/h$ from 0 to 1/2, $Nu$ gradually changes from the standard scaling $Nu \sim Ra^{1/3}$ to an asymptotic scaling $Nu \sim Ra^{{\theta }}$, with $\theta =2/3$ or $\theta =1$ by adopting, respectively, the Malkus (Proc. R. Soc. A, vol. 225, 1954, pp. 196–212) theory or the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory. For the ultimate convection regime, $Nu$ gradually changes from $Nu \sim Ra^{1/2}$ scaling to an asymptotic behaviour seen only at very high $Ra$ for which $Nu \sim Ra^{2}$. This theory is validated by the recent experimental results given by Bouillaut et al. (J. Fluid Mech., vol. 861, 2019, R5) and also shows that for these experiments, $Ra$ and $Re$ numbers were too small to observe the ultimate regime. The predictions for the ultimate regime cannot be confirmed at this time due to the absence of experimental or numerical work on convection driven by internal sources and for very large $Ra$ numbers.

We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau–Levich–Derjaguin film. The film thickness $h_f$ is found to decrease with the slip length $\lambda$. By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: $h_fh''_o \sim Ca^{2/3}$ for $\lambda \ll h_f$ and $h_f h''^{3}_o\sim (Ca/\lambda )^2$ for $\lambda \gg h_f$; here, $1/h''_o$ is a characteristic length at the receding contact line and $Ca$ is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.

High-speed vehicles experience a highly challenging environment in which the freestream Mach number and surface temperature greatly influence aerodynamic drag and heat transfer. The interplay of these two parameters strongly affects the near-wall dynamics of high-speed turbulent boundary layers (TBLs) in a non-trivial way, breaking similarity arguments on velocity and temperature fields, typically derived for adiabatic cases. We present direct numerical simulations of flat-plate zero-pressure-gradient TBLs spanning three freestream Mach numbers $[2,4,6]$ and four wall temperature conditions (from adiabatic to very cold walls), emphasising the choice of the wall-cooling parameter to recover a similar flow organisation at different Mach numbers. We link qualitative observations on flow patterns to first- and second-order statistics to explain the decoupling of temperature–velocity fluctuations that occurs at reduced wall temperatures and high Mach numbers. For these cases, we discuss the formation of a secondary peak of thermal production in the viscous sublayer, which is in contrast with the monotonic behaviour of adiabatic profiles. We propose different physical mechanisms induced by wall-cooling and compressibility that result in apparently similar flow features, such as a higher peak in the streamwise velocity turbulence intensity, and distinct features, such as the separation of turbulent scales.

The instability of the interface between a dielectric and a conducting liquid, excited by a spatially homogeneous interface-normal time-periodic electric field, is studied based on experiments and theory. Special attention is paid to the spatial structure of the excited Faraday waves. The dominant modes of the instability are extracted using high-speed imaging in combination with an algorithm evaluating light refraction at the liquid–liquid interface. The influence of the liquid viscosities on the critical voltage corresponding to the onset of instability and on the dominant wavelength is studied. Overall, good agreement with theoretical predictions that are based on viscous fluids in an infinite domain is demonstrated. Depending on the relative influence of the domain boundary, the patterns exhibit either discrete modes corresponding to surface harmonics or boundary-independent patterns. The agreement between experiments and theory confirms that the electrostatically forced Faraday instability is sufficiently well understood, which may pave the way to control electrostatically driven instabilities. Last but not least, the analogies to classical Faraday instabilities may enable new approaches to study effects that have so far only been observed for mechanical forcing.

A stably stratified fluid-filled two-dimensional square cavity is subjected to harmonic horizontal oscillations with frequencies less than the buoyancy frequency. The static linearly stratified state, which is an equilibrium of the unforced system, is not an equilibrium for any non-zero forcing amplitude. As viscous effects are reduced, the horizontally forced flows computed from the Navier–Stokes–Boussinesq equations tend to have piecewise constant or piecewise linear vorticity within the pattern of characteristic lines originating from the corners of the cavity. These flows are well described in the inviscid limit by a perturbation analysis of the unforced equilibrium using the forcing amplitude as the small perturbation parameter. At first order, this perturbation analysis leads to a forced linear inviscid hyperbolic system subject to boundary conditions and spatio-temporal symmetries associated with the horizontal forcing. A Fredholm alternative determines the type of solutions of this system: either the response is uniquely determined by the forcing, or it is resonant and corresponds to an intrinsic mode of the cavity. Both types of responses are investigated in terms of a waveform function satisfying a set of functional equations and are related to the behaviour of the characteristics of the hyperbolic system. In particular, non-retracing (ergodic) characteristics may lead to fractal responses. Models of viscous dissipation are also formulated to adjust the linear inviscid model for viscous effects obtained in the viscous nonlinear simulations.

Internal shear layers generated by the longitudinal libration of the inner core in a spherical shell rotating at a rate $\varOmega ^*$ are analysed asymptotically and numerically. The forcing frequency is chosen as $\sqrt {2}\varOmega ^*$ such that the layers issued from the inner core at the critical latitude in the form of concentrated conical beams draw a simple rectangular pattern in meridional cross-sections. The asymptotic structure of the internal shear layers is described by extending the self-similar solution known for open domains to closed domains where reflections on the boundaries occur. The periodic ray path ensures that the beams remain localised around it. Asymptotic solutions for both the main beam along the critical line and the weaker secondary beam perpendicular to it are obtained. The asymptotic predictions are compared with direct numerical results obtained for Ekman numbers as low as $E=10^{-10}$. The agreement between the asymptotic predictions and numerical results improves as the Ekman number decreases. The asymptotic scalings in $E^{1/12}$ and $E^{1/4}$ for the amplitudes of the main and secondary beams, respectively, are recovered numerically. Since the self-similar solution is singular on the axis, a new local asymptotic solution is derived close to the axis and is also validated numerically. This study demonstrates that, in the limit of vanishing Ekman numbers and for particular frequencies, the main features of the flow generated by a librating inner core are obtained by propagating through the spherical shell the self-similar solution generated by the singularity at the critical latitude on the inner core.

Preferential flow in a porous medium is commonly encountered in many practical applications. Our previous studies have discovered the preferential flow-induced non-monotonic wettability effect on displacement (J. Fluid Mech, vol. 942, 2022a, R5), but whether this non-monotonic rule is consistent for different disordered media and the impact of the interplay between the disorder and wettability under preferential flow conditions is still not well understood. Here, we combine microfluidic experiments, pore-scale simulations and theoretical analysis to study the impact of the disorder on the invading process where wettability varies from strongly water wet to strongly oil wet. Even though the strongly disordered matrix varies to a uniform state, the generality of the preferential flow-induced non-monotonic wettability effect is still proved. However, the previous pore-scale dynamics based on the strongly disordered matrix cannot explain the invading behaviour in the uniform matrix under preferential flow conditions. New mechanisms for the uniform matrix are further investigated by pore-scale modelling, which indicates that the balance of microscopic imbibition ability and the macroscopic interfacial stability dominate the invading process. We derive a theoretical model to describe the wettability effect and predict the optimal contact angle, which fits well with experimental and simulation results. Our work extends the understanding of the impact of preferential flow conditions on the wettability effect and is also of practical significance for engineering applications, such as geological CO2 sequestration, enhanced hydrocarbon recovery, soil wetting, liquid-infused material fabrication and microfluidic device design.

The motion of a slightly rarefied gas in a long straight two-dimensional channel caused by a discontinuous surface temperature is investigated on the basis of kinetic theory with a special interest in the fluid-dynamic description. More precisely, the channel is longitudinally divided into two parts and each part is kept at a uniform temperature different from each other, so that the surface temperature of the whole channel has a jump discontinuity at the junction. Under the assumption that the amount of jump in the surface temperature is small, the steady behaviour of the gas induced in the channel is studied on the basis of the linearized Boltzmann equation and the diffuse reflection boundary condition in the case where the Knudsen number, defined by the ratio of the molecular mean free path and the width of the channel, is small. Using a matched asymptotic expansion method combined with Sone’s asymptotics, a Stokes system describing the overall macroscopic behaviour of the gas inside the channel is derived, with a new feature of the ‘slip boundary condition’ for the flow velocity due to the jump discontinuity in the surface temperature of the channel. This condition takes the form of a diverging singularity with source and sink located at the point of discontinuity, with a multiplicative factor determined through the analysis of a spatially two-dimensional Knudsen-layer (or a Knudsen-zone) problem. Some numerical demonstrations based on the Bhatnagar–Gross–Krook equation are also presented.

Motivated by design challenges related to aerospace propulsive jets, an experimental investigation has been conducted of the high Mach number jet plume flow field from a round convergent nozzle at under-expanded shock-containing conditions. Hot jets up to a total temperature ratio of 3 were considered. Laser doppler anemometry (LDA) measurements in the jet near field (first 15 nozzle exit diameters) captured the turbulent mixing process in detail, enabling the separate effects of compressibility and static temperature ratio (t) on the development of the velocity and turbulence profiles to be identified. Compressibility dominated in the initial shear layer region, whereas temperature effects controlled the downstream jet merging zone. Analysis of shear layer development demonstrated that, at all temperature ratios, a similar, but significantly stronger, damping effect was observed as in planar shear layers (correlated well by convective Mach number Mc). Consideration of the interaction of compressibility and temperature ratio – which reduce/enhance turbulent mixing respectively – provided for the first time a rational explanation of the observation that increasing jet temperature influenced flow development only up to a static temperature ratio t ~ 1.5, after which further increase has little effect. Measurements of the potential core length (Lp) were analysed to produce an empirical correlation that also illustrated the diminishing effects of heat addition at all jet Mach numbers. The data provide the improved understanding and empirical design techniques essential for developing technologies for jet noise and infra red (IR) signature reduction and represent an important validation test case for computational fluid dynamics (CFD) modelling.

We report direct numerical simulations of the flow around a spanwise-flexible wing in forward flight. The simulations were performed at $Re=1000$ for wings of aspect ratio 2 and 4 undergoing a heaving and pitching motion at Strouhal number $St_c\approx 0.5$. We have varied the effective stiffness of the wing $\varPi _1$ while keeping the effective inertia constant, $\varPi _0=0.1$. It has been found that there is an optimal aerodynamic performance of the wing linked to a damped resonance phenomenon, that occurs when the imposed frequency of oscillation approaches the first natural frequency of the structure in the fluid, $\omega _{n,f}/\omega \approx 1$. In that situation, the time-averaged thrust is maximum, increasing by factor 2 with respect to the rigid case with an increase in propulsive efficiency of approximately 15 %. This enhanced aerodynamic performance results from the combination of larger effective angles of attack of the outboard wing sections and a delayed development of the leading edge vortex. With increasing flexibility beyond the resonant frequency, the aerodynamic performance drops significantly, in terms of both thrust production and propulsive efficiency. The cause of this drop lies in the increasing phase lag between the deflection of the wing and the heaving/pitching motion, which results in weaker leading edge vortices, negative effective angles of attack in the outboard sections of the wing, and drag generation in the first half of the stroke. Our results also show that flexible wings with the same $\omega _{n,f}/\omega$ but different aspect ratio have the same aerodynamic performance, emphasizing the importance of the structural properties of the wing for its aerodynamic performance.

The objective of this experimental investigation is to characterize the gas–surface interaction under different flow conditions. Therefore, the mass flow rates driven by a pressure gradient under isothermal conditions and by only a temperature gradient under constant pressure conditions are measured in the same microchannel for five different gases: helium, neon, nitrogen, argon and krypton. The pressure driven experiments are carried out in the hydrodynamic and slip flow regimes, $0.0016< \text{Knudsen number } (Kn) <0.12$, while the temperature driven experiments in the slip and transitional flow regimes have $0.05< Kn<0.45$. Using a previously developed methodology, the velocity and thermal slip coefficients are derived from the measured mass flow rates. By adopting the classical Maxwell boundary condition, the accommodation coefficients are found to be very different for both types of flows, with a significantly lower value for polyatomic nitrogen in the case of temperature gradient driven flows. An attempt to calculate the tangential momentum and normal energy accommodation coefficients in the frame of the Cercignani–Lampis model was successful only for the tangential momentum accommodation coefficient, which was found to be very close to that derived with the Maxwell model. However, it was not possible to obtain the values of the normal energy accommodation coefficient due to a lack of numerical results which connect the thermal slip and normal energy accommodation coefficients for very low values of the latter.

Three-dimensional, steady laminar flow structures with mixing, chemical reaction, normal strain and shear strain representative of turbulent combustion are analysed. A mixing layer is subjected to counterflow in the transverse $y$- and $z$-directions providing the important practical interaction of shear-strain rate with normal-strain rate. Larger consequences for mixing rates and burning rates occur than would appear with shear strain or normal strain alone. The three characteristic times for chemical reaction, normal strain and shear strain are cast through two ratios: a Damköhler number based on rate of shear strain and a ratio of rate of normal strain to rate of shear strain. Reduction to a one-dimensional similar form is obtained with density and property variations. A generalization is found extending the Crocco integral for non-unitary Prandtl number and for imposed normal strain. A diffusion flamelet model with combined shear and normal strains is developed. Another similar solution is obtained for a configuration with a dominant diffusion flame and a weaker fuel-rich premixed flame. A conserved scalar is cast as the independent variable giving an alternative description. The imposed normal strain decreases mixing-layer thickness and increases scalar gradients and transport rates. Diffusion control is possible for partially premixed flames in the multi-branched flame situation. The imposition of shear strain and thereby vorticity on the counterflow can have a substantial consequence, indicating the need for flamelet models with both shear strain and normal strain.