We derive equations of motion that describe the dynamics of a fluid confined within an elastic nanotube subject to periodic bending deflections. We use the principle of least action applied to a continuous open system at constant temperature. We solve the equations analytically in two limiting situations: when the tube oscillations are so small that they do not affect the fluid motion, but this one affects the tube dynamics; and when the flow magnitude is so small that it has no influence on the tube dynamics, but this one affects fluid motion. In the first case, we find out that the characteristic bending frequency spectrum of the tube depends not only on the magnitude of flow velocity, as previously stated in the literature, but also on the fluid velocity profile. This could constitute the basis of a strategy for indirect determination of the slip length in carbon nanotubes conveying flow via measurement of the buckling speed. In the second case, we find that tube vibrations can modify the dynamics of the fluid. Particularly, for a fluid subject to a constant pressure gradient, the tube motion induces an oscillatory motion in the fluid with twice the frequency of the tube. Moreover, the amplitude of the oscillatory fluid motion persists at high frequencies. This could constitute a strategy to generate high-frequency flows at nanoscales. Our results open up a panorama to control flow across nanotubes via tube vibrations, which could be complementary to chemical functionalization of nanostructures.

The geometric phase techniques for swimming in viscous flows express the net displacement of a swimmer as a path integral of a field in configuration space. This representation can be transformed into an area integral for simple swimmers using the Stokes theorem. Since this transformation applies for any loop, the integrand of this area integral can be used to help design these swimmers. However, the extension of this Stokes theorem technique to more complicated swimmers is hampered by problems with variables that do not commute and by how to visualise and understand the higher-dimensional spaces. In this paper, we develop a treatment for each of these problems, thereby allowing the displacement of general swimmers in any environment to be designed and understood similarly to simple swimmers. The net displacement arising from non-commuting variables is tackled by embedding the integral into a higher-dimensional space, which can then be visualised through a suitability constructed surface. These methods are developed for general swimmers and demonstrated on three benchmark examples: Purcell's two-hinged swimmer, an axisymmetric squirmer in free space and an axisymmetric squirmer approaching a free interface. We show in particular that, for swimmers with more than two modes of deformation, there exists an infinite set of strokes that generate each net displacement. Hence, in the absence of additional restrictions, general microscopic swimmers do not have a single stroke that maximises their displacement.

As a simplified model to study continental drift over the convective mantle, a thermally insulating plate is introduced to float freely over a bottom-heated thermally convecting fluid. Results of numerical simulations reveal that plate size plays an essential role in plate motion and the flow--structure interaction. The insulating plate exerts a thermal blanket effect on the flow underneath by locally accumulating heat and modifying the flow structure. As the thermal blanket effect enhances with plate size, the plate becomes increasingly capable of overcoming the impeding flow and continue its movement. As plate size increases, plate motion transitions from long durations of stagnancy over cold downwelling, to short-term stagnancy over newly formed hot upwelling and eventually to a unidirectionally moving mode (UMM) with no stagnancy. For the UMM, a hot ‘propelling plume’ is always observed near the trailing edge of the plate, moving with the plate and generating a convection cell that propels the plate. However, it moves toward the plate centre as the plate extends to cover almost the entire flow domain. The average plate speed is revealed to be proportional to the multiplication of the average plume distance and the open surface area. The critical plate sizes for the UMM predicted by a simplified model agree well with results of the full numerical simulations. Moreover, spectra of the streamfunction field show that an increase in plate size induces longer-wavelength structure, and an increase in plate speed is accompanied by an enhanced dominance of longer-wavelength structure.

This paper presents a study on the dynamics of a thin liquid film flowing down a vertical cylindrical fibre under a streamwise thermal gradient. Previous works on isothermal flows have shown that the inlet flow and fibre geometry are the main factors that determine a transition from the absolute to the convective instability flow regimes. Our experiments demonstrate that an irregular wavy pattern and bead coalescence, which are commonly seen in the convective regime, can also be triggered by applying a thermal gradient along the fibre. We develop a lubrication model that accounts for gravity, temperature-dependent viscosity and surface tension to describe the thermal effects on downstream bead dynamics. Numerical simulations of the model show good agreement between the predicted droplet coalescence dynamics and the experimental data.

In the present study, we quantify the vorticity interactions in a bluff body stabilized turbulent combustor during the transition from combustion noise to thermoacoustic instability via intermittency using complex networks. To that end, we perform simultaneous acoustic pressure, high-speed particle image velocimetry (PIV) and high-speed chemiluminescence measurements during the occurrence of combustion noise, intermittency and thermoacoustic instability. Based on the Biot–Savart law, we construct time-varying weighted spatial networks from the flow fields during these different regimes of combustor operation. We uncover that the turbulent networks display weighted scale-free behaviour intermittently during the different regimes of combustor operation, with the strong vortical structures acting as the hubs. Further, we discover two optimal locations for injecting steady air jets to successfully suppress the thermoacoustic oscillations. The amplitude of the acoustic pressure fluctuations of the suppressed state is comparable to that during the occurrence of combustion noise. However, the weighted scale-free network topology during the suppressed state is not as dominant as compared with the state of combustion noise.

This paper presents a class of turbulence models written in terms of fractional partial differential equations (FPDEs) with stochastic loads. Every solution of these FPDE models is an incompressible velocity field and the distribution of solutions is Gaussian. Interaction of the turbulence with solid walls is incorporated through the enforcement of various boundary conditions. The various boundary conditions deliver extensive flexibility in the near-wall statistics that can be modelled. Reproduction of both fully developed shear-free and uniform shear boundary layer turbulence are highlighted as two simple physical applications; the first of which is also directly validated with experimental data. The rendering of inhomogeneous synthetic turbulence inlet boundary conditions is an additional application, motivated by contemporary numerical wind tunnel simulations. Calibration of model parameters and efficient numerical methods are also conferred upon.

The buoyancy-driven instabilities triggered by neutralization reaction were studied experimentally in a miscible two-layer system placed in a vertically oriented Hele-Shaw cell. The initial density stratification was always set to exclude the development of the Rayleigh–Taylor instability. The problem was examined for a few reactant pairs formed by a strong acid and a strong base. To classify the numerous experimental observations we introduced a dimensionless parameter, namely, a reaction-induced buoyancy number ${K}_{\rho }$, which defines the density of the reaction zone relative to that of the upper layer. We show that, depending on the value of this parameter, one of two global scenarios develops in the system right after the layers came into contact. If ${K}_{\rho }>1$, the process is governed mainly by diffusion, which results later on in the development of relatively weak convective motion caused by a differential-diffusion effect. Besides the irregular finger-type flow structures, reported earlier in numerous studies, we found a new type of instability, called the concentration-dependent diffusion instability, which is characterized by the formation of a regular cell-type convective pattern. In the case ${K}_{\rho }\le 1$, the unstable density stratification above the reaction front leads to the development of vigorous convection in the upper layer, forcing the reaction front to move downwards fast, so that it takes just a few minutes for reagents to burn out. We show that a new parameter can be used to plot the stability maps, which allows us to predict reaction–diffusion–convection processes in similar systems prior to experiment.

The buoyancy-driven chemoconvection induced by a neutralization reaction is theoretically studied for a system consisting of nitric acid and sodium hydroxide aqueous solutions placed in a vertically oriented Hele-Shaw cell. This pair of reactants is a representative case of reacting miscible acid–base systems investigated experimentally in Part 1 of this work (Mizev et al., J. Fluid Mech., vol. 916, 2021, A22.). We showed that the list of the possible instabilities in this system is much richer than previously thought. A new scenario for pattern formation depends on a single parameter denoted by $K_{\rho }$, the reaction-induced buoyancy number defined in Part 1. In this paper, the theoretical analysis complementing the experimental observations provides the conceptual insights required for a full understanding of the mechanisms of the observed phenomena. The mathematical model we develop consists of a system of reaction–diffusion–advection equations governing the evolution of concentrations coupled to the Navier–Stokes equation. The system dynamics is examined through transient linear stability analysis and numerical simulation. If $K_{\rho }>1$, then a statically stable potential well appears adjacent to the reaction front. As a result, a Rayleigh–Bénard-like cellular pattern can arise in this depleted density region. If $K_{\rho }\leqslant 1$, then a potential well collapses, and a shock-wave-like structure with an almost planar front occurs. This wave propagates fast compared with the diffusion time and acts as a turbulent bore separating immobile fluid and an area of intense convective mixing. Finally, we determine the place of the above instabilities in an extended classification of known instability types.

We study the steady force acting on a rigid spherical particle immersed in an ideal gas and impinged by a standing acoustic wave. The acoustic wavelength and particle radius are assumed much larger and smaller, respectively, than the molecular mean-free path. To analyse the system, an asymptotic scheme is constructed, combining an inviscid continuum description in the far field with a free-molecular formulation for the near gas–surface interaction. The computation yields a closed-form expression for the steady acoustic force. The free-molecular solution is compared with a formerly derived result in the continuum regime (Doinkov, Proc. R. Soc. Lond. A, vol. 447, 1994, pp. 447–466), and the latter is found characteristically larger by an order of magnitude at a given ratio between the particle radius and the acoustic wavelength. Markedly, the size of the acoustic force at ballistic flow conditions may become up to four orders of magnitude larger than the typical gravitational force, suggesting the feasibility of nano-particle acoustic levitation.

The combined effect of Reynolds number ($Re$) and localised wall suction applied through a porous strip on a fully rough wall turbulent boundary layer (TBL) is investigated using hot-wire anemometry. The measurements show that the response of the TBL to suction is modulated by the ratio $U_s/U_\tau$, where $U_s$ and $U_\tau$ are the suction and friction velocities, respectively. For example the suction impact on the mean velocity and the turbulence intensity profiles, which is felt across the boundary layer, decreases as $U_s/U_\tau$ decreases. Interestingly, the velocity spectra contour maps reveal that suction reduces the energy at all scales of motion across the boundary layer. Further, measurements of the velocity skewness indicate that the TBL undergoes a structural change when the ratio $U_s/U_\tau$ is relatively important. However, the measurements also reveal that TBL does not show a relaminarisation behaviour as it can be observed in a smooth wall TBL with similar localised wall suction. This lack of relaminarisation is due to the development of a growing internal boundary layer which evolves on a rough surface within the existing TBL.

Gravity-driven size segregation is important in mountain streams where a wide range of grain sizes are transported as bedload. More particularly, vertical size segregation is a multi-scale process that originates in interactions at the scale of particles with important morphological consequences for the river bed. To address this issue, a volume-averaged multi-phase flow model for immersed bi-disperse granular flows was developed based on an interparticle segregation force (Guillard et al., J. Fluid Mech., vol. 807, 2016, R1) and a granular Stokesian drag force (Tripathi & Khakhar, J. Fluid Mech., vol. 717, 2013, pp. 643–669). An advection–diffusion model was derived from this model yielding parametrisations for the advection and diffusion coefficients based on the interparticle interactions. This approach makes it possible to bridge the gap between grain-scale physics and continuum modelling. Both models were successfully tested against existing discrete element model (DEM) simulations of size segregation in bedload transport (Chassagne et al., J. Fluid Mech., vol. 895, 2020, A30). Through a detailed investigation of the granular forces, it is demonstrated that the observed scaling of the advection and diffusion coefficients with the inertial number can be explained by the granular drag force dependency on the viscosity. The drag coefficient is shown to be linearly dependent on the small particle concentration. A new scaling relationship for the segregation force including the small particle concentration and the pressure is proposed. Lastly, adding a size-ratio dependency in the segregation force fairly reproduces the DEM results for a large range of small particle concentrations and size ratios.

We conduct a theoretical study of a two-phase fluid–structure interaction problem in which air is driven at constant volume flux into a liquid-filled Hele-Shaw channel whose upper boundary is an elastic sheet. A depth-averaged model in the frame of reference of the advancing air–liquid interface is used to investigate the steady and unsteady interface propagation modes via numerical simulation. In slightly collapsed channels, the steadily propagating interface adopts a shape that is similar to the classic Saffman–Taylor finger in rigid Hele-Shaw cells. As the level of initial collapse increases, the induced gradients in channel depth alter the morphology of the propagating finger and promote a variety of instabilities from tip splitting to small-scale fingering on the curved interface, in qualitative agreement with experiments. The model has a complex solution structure with a wide range of stable and unstable, steady and time-periodic modes, many of which have similar driving pressures. We find good quantitative agreement between our model and the experimental data of Ducloué et al. (J. Fluid Mech., vol. 819, 2017, p. 121) for the finger width, sheet profile and finger pressure, provided that corrections to account for the presence of liquid films on the upper and lower walls of the channel are included in the model.

We study numerically the melting of a horizontal layer of a pure solid above a convecting layer of its fluid rotating about the vertical axis. In the rotating regime studied here, with Rayleigh numbers of order $10^{7}$, convection takes the form of columnar vortices, the number and size of which depend upon the Ekman and Prandtl numbers, as well as the geometry – periodic or confined. As the Ekman and Rayleigh numbers vary, the number and average area of vortices vary in inverse proportion, becoming thinner and more numerous with decreasing Ekman number. The vortices transport heat to the phase boundary, thereby controlling its morphology characterized by the number and size of the voids formed in the solid, and the overall melt rate, which increases when the lower boundary is governed by a no-slip rather than a stress-free velocity boundary condition. Moreover, the number and size of voids formed are relatively insensitive to the Stefan number, here inversely proportional to the latent heat of fusion. For small values of the Stefan number, the convection in the fluid reaches a slowly evolving geostrophic state wherein columnar vortices transport nearly all the heat from the lower boundary to melt the solid at an approximately constant rate. In this quasi-steady state, we find that the Nusselt number, characterizing the heat flux, co-varies with the interfacial roughness, for all the flow parameters and Stefan numbers considered here. This confluence of processes should influence the treatment of moving boundary problems, particularly those in astrophysical and geophysical problems where rotational effects are important.

A direct numerical simulation of turbulent heat transfer in a channel flow has been conducted for a Reynolds number of $5000$ and the Prandtl number of air, $0.71$. The mixed boundary condition has been used as the boundary condition of the thermal field. The computational domain has been set to $3.2 {\rm \pi}h$, $2h$ and $1.6 {\rm \pi}h$ in the $x$, $y$ and $z$ directions, respectively. This domain is large enough to accurately compute the statistics of the flow. Mean values and intensities of the temperature have been obtained. Derived parameters from the average thermal field, such as the von Kármán constant and the Nusselt number have been calculated. An asymptotic behaviour of the von Kármán constant is observed when ${\textit {Re}}_\tau$ is increased. A correlation for the Nusselt number is proposed. Also, the turbulent Prandtl number has been calculated and it does not present significant changes when ${\textit {Re}}_\tau$ is increased. Finally, the turbulent budgets are presented. A relation between the increment of the inner peak of the temperature intensities and the scaling failure of the dissipation and viscous diffusion terms is provided. The statistics of all simulations can be downloaded from the web page of our group: http://personales.upv.es/serhocal/.

We investigate the removal of a dense bottom layer by a gravity current, via Navier–Stokes simulations in the Boussinesq limit. The problem is governed by a dimensionless thickness parameter for the bottom layer, and by the ratio of the density differences between bottom layer, gravity current and ambient fluids. A quasisteady gravity current forms that propagates along the interface and displaces some of the dense bottom fluid, which accumulates ahead of the gravity current and forms an undular bore or a series of internal gravity waves. Depending on the ratio of the gravity current front velocity to the linear shallow-water wave velocity, we observe the existence of different regimes, characterized by small-amplitude waves or by a train of steep, nonlinear internal waves. We develop a semiempirical model that provides reasonable estimates of several important flow properties. We also formulate a more sophisticated, self-contained model based on the conservation principles for mass and vorticity that does not require empirical closure assumptions. This model is able to predict such quantities as the gravity current height and the internal wave or bore velocity as a function of the governing dimensionless parameters, generally to within approximately a 10 $\%$ accuracy. An energy budget analysis provides information on the rates at which potential energy is converted into kinetic energy and then dissipated, and on the processes by which energy is transferred from the gravity current fluid to the dense and ambient fluids. We observe that the energy content of thicker and denser bottom layers grows more rapidly.

It is increasingly common for models of shallow-layer overland flows to include equations for the evolution of the underlying bed (morphodynamics) and the motion of an associated sedimentary phase. We investigate the linear stability properties of these systems in considerable generality. Naive formulations of the morphodynamics, featuring exchange of sediment between a well-mixed suspended load and the bed, lead to mathematically ill-posed governing equations. This is traced to a singularity in the linearised system at Froude number ${\textit {Fr}} = 1$ that causes unbounded unstable growth of short-wavelength disturbances. The inclusion of neglected physical processes can restore well posedness. Turbulent momentum diffusion (eddy viscosity) and a suitably parametrised bed load sediment transport are shown separately to be sufficient in this regard. However, we demonstrate that such models typically inherit an associated instability that is absent from non-morphodynamic settings. Implications of our analyses are considered for simple generic closures, including a drag law that switches between fluid and granular behaviour, depending on the sediment concentration. Steady morphodynamic flows bifurcate into two states: dilute flows, which are stable at low ${\textit {Fr}}$, and concentrated flows which are always unstable to disturbances in concentration. By computing the growth rates of linear modes across a wide region of parameter space, we examine in detail the effects of specific model parameters including the choices of sediment erodibility, eddy viscosity and bed load flux. These analyses may be used to inform the ongoing development of operational models in engineering and geosciences.

A rigorous bound on the energy dissipation in the flow around a sphere is derived analytically. The bound is allowed to depend on the viscosity of the fluid, the radius of the sphere, the free-stream velocity and the amplitude of the difference between the actual flow field and the irrotational flow around the sphere. The bound becomes asymptotically independent of viscosity as the Reynolds number tends to infinity. Provided that the velocity fluctuations are of the same order as the free-stream velocity, the scaling of the bound is consistent with a Reynolds-number-independent drag coefficient at high Reynolds numbers.

The motion of glaciers over their bedrock or drops of fluid along a solid surface can vary dramatically when these substrates are lubricated. We investigate the coupled flow of a gravity current (GC) of a strain-rate softening fluid that is lubricated by a denser, lower-viscosity Newtonian fluid. We present a set of experiments in which such GCs are discharged axisymmetrically and at constant flux over a flat surface. Using imaging techniques, we follow the evolution of the front and thickness field of each fluid. We find that, unlike purely Newtonian lubricated GCs, the fronts of the non-Newtonian and lubricating Newtonian fluids have a power-law time evolution with different exponents. Each of these exponents is similar to that of a non-lubricated GC of the same fluid. Nevertheless, the fronts of our lubricated GCs evolve faster than those of the corresponding non-lubricated GCs owing to larger intercepts. In addition, in contrast with the monotonically declining thickness of non-lubricated GCs, the thickness of the lubricated, non-Newtonian fluid is nearly uniform, and that of the lubricating fluid is non-monotonic with localised spikes. Despite these complex thickness patterns, lubricated GCs remain axisymmetric as long as the flux of the lubricating fluid is sufficiently smaller than that of the non-Newtonian fluid.

Modelling the flow of dry granular materials entering water is crucial for optimizing blending processes in industry and for natural hazard assessment when describing tsunami waves induced by landslides. In this study, we experimentally investigate the case of a jet of grains entering from the air into a water bath. After an initial transient state, a stationary impregnation front appears between the dry and the wet grains. The wet grains are then dispersed in the liquid. To describe this dry-to-wet transition, we focus on the first step of the process, when the liquid invades the dense granular medium. In this regime, the granular jet is modelled as a translating porous material, and we systematically characterize the impregnation process using a combination of experiments, analytical modelling, and numerical tools. We then compare this process to the situation of a confined granular jet entering a water bath. Our approach is a first step toward describing the interplay between dry grains and a liquid and the resulting dispersion of particles.

To isolate the multiscale dynamics of the logarithmic layer of wall-bounded turbulent flows, a novel numerical experiment is conducted in which the mean tangential Reynolds stress is eliminated except in a subregion corresponding to the typical location of the logarithmic layer in channels. Various statistical comparisons against channel flow databases show that, despite some differences, this modified flow system reproduces the kinematics and dynamics of natural logarithmic layers well, even in the absence of a buffer and an outer zone. This supports the previous idea that the logarithmic layer has its own autonomous dynamics. In particular, the results suggest that the mean velocity gradient and the wall-parallel scale of the largest eddies are determined by the height of the tallest momentum-transferring motions, which implies that the very large-scale motions of wall-bounded flows are not an intrinsic part of the logarithmic-layer dynamics. Using a similar set-up, an isolated layer with a constant total stress, which represents the logarithmic layer without a driving force, is simulated and examined.