An asymptotic solution is derived for the motion of inertial particles exposed to Stokes drag in an unsteady random flow. This solution provides an estimate for the sum of Lyapunov exponents as a function of the Stokes number and Lagrangian strain- and rotation-rate autocovariance functions. The sum of exponents in a Lagrangian framework is the rate of contraction of clouds of particles, and in an Eulerian framework, it is the concentration-weighted divergence of the particle velocity field. Previous literature offers an estimate of the divergence of the particle velocity field, which is applicable only in the limit of small Stokes numbers [Robinson, Comm. Pure Appl. Math. 9, 69 (1956) and Maxey, J. Fluid Mech. 174, 441 (1987)] (R-M). In addition to reproducing R-M at this limit, our analysis provides a first-order correction to R-M at larger Stokes numbers. Our analysis is validated by a directly computed rate of contraction of clouds of particles from simulations of particles in homogeneous isotropic turbulence over a broad range of Stokes numbers. Our analysis and R-M predictions agree well with the direct computations at the limit of small Stokes numbers. At large Stokes numbers, in contrast to R-M, our model predictions remain bounded. In spite of an improvement over R-M, our analysis fails to predict the expansion of high Stokes clouds observed in the direct computations. Consistent with the general trend of particle segregation versus Stokes number, our analysis shows a maximum rate of contraction at an intermediate Stokes number of $O\left(1\right)$ and minimal rates of contraction at small and large Stokes numbers.

We analyze the response of a gas in a micro-slab, set at an initial pure-conduction state, to instantaneous thermal insulation of its boundaries. In line with ongoing efforts in generating gas flows at the microscale, thermal insulation is suggested as a means for flow excitation with no moving parts. The problem is analyzed in the entire range of gas rarefaction rates and for arbitrary initial temperature differences between the walls. Analytical solutions are obtained in the linearized limit of small temperature differences for large (collisionless) and small (continuum) Knudsen numbers. These solutions are supported by direct simulation Monte Carlo calculations, which are then used to investigate the nonlinear problem with large initial temperature differences. Followed by the system's initial state, boundary insulation results in a series of time-decaying waves, propagating across the slab, and transferring the system between its conductive and adiabatic equilibrium states. While larger initial temperature differences result in higher flow rates, it is found that nonlinear effects reduce the efficiency of flow excitation through boundaries insulation. At high Knudsen numbers, this is rationalized through the system's initial state, in which the gas uniform temperature is lower than the arithmetic mean of walls temperatures. At low Knudsen numbers, the dominant effect of molecular collisions causes thermal dissipation, which in turn results in kinetic energy losses. The analysis may be readily applied to calculate the gas response to arbitrary time variations of the boundary-imposed heat flux.

We investigate the dynamics of freely expanding liquid sheets prepared with fluids with different rheological properties: (i) Viscous fluids with a zero-shear viscosity ${\eta }_0$ in the range $1–1000\mathrm{mPa}\mathrm{s}$ and (ii) viscoelastic fluids whose linear viscoelastic behavior in the frequency range $0.1–100$ rad/s can be accounted for by a Maxwell fluid model, whose characteristic elastic modulus $G_0$, relaxation time $\tau$, and zero-shear viscosity ${\eta }_0=G_0\tau$, can be tuned over several orders of magnitude. The sheets are produced by a drop of fluid impacting a small cylindrical solid target. For viscoelastic fluids, we show that, when $\tau$ is shorter than the typical lifetime of the sheet ($\sim 10$ ms), the dynamics of the sheet is similar to that of Newtonian viscous liquids with equal zero-shear viscosity. In that case, for little viscous samples (${\eta }_0\lesssim 30$ mPa s), the maximal expansion of the sheet $d_{\max }$ is independent of ${\eta }_0$, whereas for more viscous samples, $d_{\max }$ decreases as ${\eta }_0$ increases. We provide a simple model for the dependence of the maximal expansion of the sheet with the viscosity that accounts well for our experimental data. By contrast, when $\tau$ is longer than the typical lifetime of the sheet, the behavior drastically differs. The sheet expansion is strongly enhanced as compared to that of viscous samples with comparable zero-shear viscosity, but is heterogeneous with the occurrence of cracks, revealing the elastic nature of the viscoelastic fluid.

Within the past decade, the separation of particles via continuous flow through microfluidic devices has been developed largely through an Edisonian approach whereby devices have been developed based on observation and intuition. This is particularly true in the development of vector chromatography at vanishingly small Reynolds number for non-Brownian particles. Note that this latter phenomenon has its origins in the irreversible forces that are at work in the device, since Stokes flow reversibility typically prohibits their function otherwise. We present a numerical simulation of the vector separation of non-Brownian particles of different sizes and deformabilities in the Stokes flow through channels whose lower surface is composed of slanted cavities. The simulations are designed to understand the physical principles behind the separation as well as to provide design criteria for devices for separating particles in a given size and flexibility range. The numerical simulations are Stokes flow boundary element simulations using techniques defined elsewhere in the literature, but including a close-range repulsive force between the particles and the slanted cavities. We demonstrate that over a range of repulsive force that is comparable to the roughness in the experimental devices, the separation data (particularly in particle size) are predicted quantitatively and are a very weak function of the range of the force. We then vary the geometric parameters of the simulated devices to demonstrate the sensitivity of the separation efficiency to these parameters, thus making design predictions as to which devices are appropriate for separating particles in different size, shape, and deformability ranges.

The comparison of the results of direct numerical simulations of isotropic turbulence of Newtonian and viscoelastic fluid provides evidence that viscoelasticity modifies qualitatively the behavior of the smallest scales: we observe a power law in the far dissipation range of the fluid kinetic energy spectrum and we show that it is a robust feature, roughly independent of the large scale dynamics. A detailed analysis of the energy transfer shows that at these scales energy is injected into the fluid flow through polymer relaxation. It is further shown that a part of the total energy is transferred among scales through an interaction of the velocity field with the polymer field.

Eukaryotic cilia and flagella are chemo-mechanical oscillators capable of generating long-range coordinated motions known as metachronal waves. Pair synchronization is a fundamental requirement for these collective dynamics, but it is generally not sufficient for collective phase-locking, chiefly due to the effect of long-range interactions. Here we explore experimentally and numerically a minimal model for a ciliated surface: hydrodynamically coupled oscillators rotating above a no-slip plane. Increasing their distance from the wall profoundly affects the global dynamics, due to variations in hydrodynamic interaction range. The array undergoes a transition from a traveling wave to either a steady chevron pattern or one punctuated by periodic phase defects. Within the transition between these regimes the system displays behavior reminiscent of chimera states.

When deposited on a plate moving quickly enough, any liquid can levitate as it does when it is volatile on a very hot solid (Leidenfrost effect). In the aerodynamic Leidenfrost situation, air gets inserted between the liquid and the moving solid, a situation that we analyze. We observe two types of entrainment. (i) The thickness of the air gap is found to increase with the plate speed, which is interpreted in the Landau-Levich-Derjaguin frame: Air is dynamically dragged along the surface and its thickness results from a balance between capillary and viscous effects. (ii) Air set in motion by the plate exerts a force on the levitating liquid. We discuss the magnitude of this aerodynamic force and show that it can be exploited to control the liquid and even to drive it against gravity.

We investigate the impact of nonideal diffusion on the structure of supercritical cryogenic binary mixing layers. This situation is typical of liquid fuel injection in high-pressure rocket engines. Nonideal diffusion has a dramatic impact in the neighborhood of chemical thermodynamic stability limits where the components become quasi-immiscible and ultimately form a nonmixing layer. Numerical simulations are performed for mixing layers of ${\mathrm{H}}_2$ and ${\mathrm{N}}_2$ at a pressure of 100 atm and temperature around 120–150 K near chemical thermodynamic stability limits.

We perform a thorough investigation of the drying dynamics of a charged colloidal dispersion drop in a confined geometry. We develop an original methodology based on Raman microspectroscopy to measure spatially resolved colloids concentration profiles during the drying of the drop. These measurements lead to estimates of the collective diffusion coefficient of the dispersion over a wide range of concentration. The collective diffusion coefficient is one order of magnitude higher than the Stokes-Einstein estimate, showing the importance of the electrostatic interactions for the relaxation of concentration gradients. At the same time, we also perform fluorescence imaging of tracers embedded within the dispersion during the drying of the drop, which reveals two distinct regimes. At early stages, concentration gradients along the drop lead to buoyancy-induced flows. Strikingly, these flows do not influence the colloidal concentration gradients that generate them, as the mass transport remains dominated by diffusion. At longer time scales, the tracer trajectories reveal the formation of a gel that dries quasihomogeneously. For such a gel, we show using linear poroelastic modeling that the drying dynamics is still described by the same transport equations as for the liquid dispersion. However, the collective diffusion coefficient follows a modified generalized Stokes-Einstein relation, as also demonstrated in the context of unidirectional consolidation by Style and Peppin [Style and Peppin, Crust formation in drying colloidal suspensions, Proc. R. Soc. A 467, 174 (2011)].

We report simultaneous measurements of the mean temperature profile $\theta \left(z\right)$ and temperature variance profile $\eta \left(z\right)$ near the lower conducting plate of a specially designed quasi-two-dimensional cell for turbulent Rayleigh-Bénard convection. The measured $\theta \left(z\right)$ is found to have a universal scaling form $\theta (z/\delta )$ with varying thermal boundary layer (BL) thickness $\delta$, and its functional form agrees well with the recently derived BL equation by Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)]. The measured $\eta \left(z\right)$, on the other hand, is found to have a scaling form $\eta (z/\delta )$ only in the near-wall region with $z/\delta \lesssim 2$. Based on the experimental findings, we derive a BL equation for $\eta (z/\delta )$, which is in good agreement with the experimental results. These BL equations thus provide a common framework for understanding the effect of BL fluctuations.

The Lagrangian dynamics of semiflexible particles in homogeneous and isotropic turbulent flows is studied by means of analytically solvable stochastic models and direct numerical simulations. The stationary statistics of the bending angle shows a strong dependence on the dimension of the flow. In two-dimensional turbulence, particles are found in either a fully extended or a fully folded configuration; in three dimensions, the predominant configuration is the fully extended one. Such a sensitivity of the bending statistics on the dimensionality of the flow is peculiar to fluctuating flows and is not observed in laminar stretching flows.

The generalized Chapman-Enskog (GCE) method for rapid and slow thermochemical processes is employed to formulate a set of continuum breakdown parameters for chemically reacting flows. These GCE breakdown parameters are derived for one-temperature, two-temperature, and three-temperature models, through classification of the relevant thermochemical time scales relative to fast elastic collisional processes and slow flow processes associated with changes in macroscopic observables. Continuum breakdown mechanisms owing to multicomponent diffusion, thermal diffusion, normal and shear stresses, Fourier-type heat fluxes based on translational, rotational, and vibrational temperatures, bulk viscosity, and relaxation pressure are presented for chemically reacting flows. The GCE breakdown parameters, derived from rigorous kinetic theory, capture the proper physical mechanism leading to continuum breakdown. These breakdown parameters are used to analyze continuum breakdown in a Mach 24 reacting air flow over a sphere and continuum breakdown is observed in the shock and close to the sphere surface. The flow field near the sphere surface is found to be characterized by sharp species concentration gradients due to gas-phase and surface reactions. Chemical reactions thus lead indirectly to the distortion of the velocity distribution function (VDF), providing a pathway to continuum breakdown that is captured by the GCE specieswise diffusion breakdown parameter.

Most large-scale planetary magnetic fields are thought to be driven by low Rossby number convection of a low magnetic Prandtl number fluid. Here kinematic dynamo action is investigated with an asymptotic, rapidly rotating dynamo model for the plane layer geometry that is intrinsically low magnetic Prandtl number. The thermal Prandtl number and Rayleigh number are varied to illustrate fundamental changes in flow regime, ranging from laminar cellular convection to geostrophic turbulence in which an inverse energy cascade is present. A decrease in the efficiency of the convection to generate a dynamo, as determined by an increase in the critical magnetic Reynolds number, is observed as the buoyancy forcing is increased. This decreased efficiency may result from both the loss of correlations associated with the increasingly disordered states of flow that are generated, and boundary layer behavior that enhances magnetic diffusion locally. We find that the spatial characteristics of the large-scale magnetic field is dependent only weakly on changes in flow behavior. In contrast, the behavior of the small-scale magnetic field is directly dependent on, and therefore shows significant variations with, the small-scale convective flow field. However, our results are limited to the linear, kinematic dynamo regime; future simulations that include the Lorentz force are therefore necessary to assess the robustness of these results.

The scaling properties of the mean wall-normal velocity $V(x,y)$ in zero-pressure-gradient laminar and turbulent boundary-layer flows are investigated using numerical simulation data, physical experiment data, and integral analyses of the governing equations. The maximum mean wall-normal velocity $V_{\infty }$ and the boundary-layer thickness $\delta$ are evidenced to be the proper scaling for $V$ over most if not all of the boundary layer. This is different from the behavior of the mean streamwise velocity $U$ or the turbulent shear stress $T=-\rho \langle uv\rangle$, which depend on different characteristic length scales in the regions near and away from the surface, respectively. The reason for this apparent difference in scaling behaviors is described physically relative to the downstream development of the $U$ velocity profile and the mechanisms of boundary-layer growth. Insights pertaining to this are further surmised from an analytical relationship for the ratio of the displacement to momentum thickness, i.e., shape factor $H$. Integral analyses using the continuity and mean momentum equation show that $U_{\infty }{V}_{\infty }/u_{\tau }^2=H$, where $u_{\tau }$ is the friction velocity. Both the laminar similarity solution and direct numerical simulation data in post-transitional flows convincingly support this relation. Over the transitional regime, data of sufficiently high quality are lacking to check if this relation remains valid.

We investigate the stability of thin viscous films coated on the inside of a horizontal cylindrical substrate. In such a case, gravity acts both as a stabilizing force through the progressive drainage of the film and as a destabilizing force prone to form droplets via the Rayleigh-Taylor instability. The drainage solution, derived from lubrication equations, is found asymptotically stable with respect to infinitesimally small perturbations, although in reality, droplets often form. To resolve this paradox, we perform an optimal transient growth analysis for the first-order perturbations of the liquid's interface, generalizing the results of Trinh et al. [Phys. Fluids 26, 051704 (2014)]. We find that the system displays a linear transient growth potential that gives rise to two different scenarios depending on the value of the Bond number (prescribing the relative importance of gravity and surface tension forces). At low Bond numbers, the optimal perturbation of the interface does not generate droplets. In contrast, for higher Bond numbers, perturbations on the upper hemicircle yield gains large enough to potentially form droplets. The gain increases exponentially with the Bond number. In particular, depending on the amplitude of the initial perturbation, we find a critical Bond number above which the short-time linear growth is sufficient to trigger the nonlinear effects required to form dripping droplets. We conclude that the transition to droplets detaching from the substrate is noise and perturbation dependent.

Transition induced by stationary streaks undergoing transient growth in a supersonic flat plate boundary layer flow is studied using numerical computations. While the possibility of strong transient growth of small-amplitude stationary perturbations in supersonic boundary layer flows has been demonstrated in previous works, its relation to laminar-turbulent transition cannot be established within the framework of linear disturbances. Therefore, this paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite amplitude streaks in the downstream region, and then studies the modal instability of those streaks as a likely cause for the onset of bypass transition. The nonmodal evolution of linearly optimal stationary perturbations in a supersonic, Mach 3 flat plate boundary layer is computed via the nonlinear plane-marching parabolized stability equations (PSE) for stationary perturbations, or equivalently, the perturbation form of parabolized Navier-Stokes equations. To assess the effect of the nonlinear finite-amplitude streaks on transition, the linear form of plane-marching PSE is used to investigate the instability of the boundary layer flow modified by the spanwise periodic streaks. The onset of transition is estimated using an $N$-factor criterion based on modal amplification of the secondary instabilities of the streaks. In the absence of transient growth disturbances, first mode instabilities in a Mach 3, zero pressure gradient boundary layer reach $N=10$ at ${\text{Re}}_x\approx {10}^7$. However, secondary instability modes of the stationary streaks undergoing transient growth are able to achieve the same $N$-factor at ${\text{Re}}_x<2\times {10}^6$ when the initial streak amplitude is sufficiently large. In contrast to the streak instabilities in incompressible flows, subharmonic instability modes with twice the fundamental spanwise wavelength of the streaks are found to have higher amplification ratios than the streak instabilities at fundamental wavelength. This behavior is shown to be related to the destabilizing influence of the streaks on oblique first mode disturbances, which tend to have longer spanwise wavelengths than those of the optimal stationary disturbances.

Two vortex rings can form a localized configuration whereby they continually pass through one another in an alternating fashion. This phenomenon is called leapfrogging. Using parameters suitable for superfluid helium-4, we describe a recurrence phenomenon that is similar to leapfrogging, which occurs for two coaxial straight vortex filaments with the same Kelvin wave mode. For small-amplitude Kelvin waves we demonstrate that our full Biot-Savart simulations closely follow predictions obtained from a simplified model that provides an analytical approximation developed for nearly parallel vortices. Our results are also relevant to thin-cored helical vortices in classical fluids.

The distributions of flow topologies within the flames representing the corrugated flamelets, thin reaction zones, and broken reaction zone regimes of premixed turbulent combustion are investigated using direct numerical simulation data of statistically planar turbulent ${\mathrm{H}}_2$-air flames with an equivalence ratio $\varphi =0.7$. It was found that the diminishing influence of dilatation rate with increasing Karlovitz number has significant influences on the statistical behaviors of the first, second, and third invariants (i.e., $P,Q$, and $R$) of the velocity gradient tensor. These differences are reflected in the distributions of the flow topologies within the flames considered in this analysis. This has important consequences for those topologies that make dominant contributions to the scalar-turbulence interaction and vortex-stretching terms in the scalar dissipation rate and enstrophy transport equations, respectively. Detailed physical explanations are provided for the observed regime dependences of the flow topologies and their implications on the scalar dissipation rate and enstrophy transport.

We investigate experimentally the influence of the liquid viscosity on the problem of the generation of waves by a turbulent wind at the surface of a liquid, extending the results of Paquier et al. [A. Paquier et al., Phys. Fluids 27, 122103 (2015)] over nearly three decades of viscosity. The surface deformations are measured with micrometer accuracy using the free-surface synthetic schlieren method. We recover the two regimes of surface deformations previously identified: the wrinkle regime at small wind velocity, resulting from the viscous imprint on the liquid surface of the turbulent fluctuations in the boundary layer, and the regular wave regime at large wind velocity. Below the wave threshold, we find that the characteristic amplitude of the wrinkles scales as ${\nu }^{-1/2}{u}^{*3/2}$ over nearly the whole range of viscosities, whereas their size is essentially unchanged. We propose a simple model for this scaling, which compares well with the data. We show that the critical friction velocity $u^{*}$ for the onset of regular waves slowly increases with viscosity as ${\nu }^{0.2}$. Whereas the transition between wrinkles and waves is smooth at low viscosity, including for water, it becomes rather abrupt at high viscosity. A third wave regime is found at $\nu >(100–200)\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, characterized by a slow, nearly periodic emission of large-amplitude isolated fluid bumps.