Based on the generalized kinetic equation for the one-particle distribution function with a small source, the transition from the kinetic to the hydrodynamic description of many-particle systems is performed. The basic feature of this new technique to obtain the hydrodynamic limit is that the latter has been partially incorporated into the kinetic equation itself. The hydrodynamic equations for capillary fluids are derived from the characteristic function for the local moments of the distribution function. The Fick's law appears as a consequence of the transformation law for the hydrodynamic quantities under time inversion.
Fluid dynamics video of an evaporating sessile drop of capillary size is presented. The corresponding simulation represents the description taking into account jointly time dependent hydrodynamics, vapor diffusion and thermal conduction in an evaporating sessile drop. The fluid convection in the drop is driven by Marangoni forces associated with the temperature dependence of the surface tension. For the first time the evolution of the vortex structure in the drop during an evaporation process is obtained.
Many familiar events feature a distinctive sound: paper crumpling or tearing, squeaking doors, drumming rain or boiling water. Such characteristic sounds actually carry a profusion of informations about the fleeting physical processes at the root of acoustic emission, which appears appealing especially in situations precluding direct or in-situ measurements, such as e.g. the rupture of micron-thick liquid sheet. Here we report on such a link between fast interfacial hydrodynamics and sound. The acoustic emission of a bursting soap bubble is captured by means of antennae and deciphered with the…
We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train from considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles.
We study numerically the hydrodynamics of dip coating from a suspension and report a mechanism for colloidal assembly and pattern formation on smooth and uniform substrates. Below a critical withdrawal speed of the substrate, capillary forces required to deform the meniscus prevent colloidal particles from entering the coating film. Capillary forces are overcome by hydrodynamic drag only after a minimum number of particles organize in a close-packed formation within the meniscus. Once within the film, the formed assembly moves at nearly the withdrawal speed and rapidly separates from the next …
Capillary waves are perhaps the simplest example to consider for an introduction to wave turbulence. Since the first paper by Zakharov and Filonenko (1967), capillary wave turbulence has been the subject of many studies but a didactic derivation of the kinetic equation is still lacking. It is the objective of this paper to present such a derivation in absence of gravity and in the approximation of deep water. We use the Eulerian method and a Taylor expansion around the equilibrium elevation for the velocity potential to derive the kinetic equation. The use of directional polarities for three-w…
We reported an electro-hydrodynamic shooting phenomenon of liquid metal stream. A small voltage direct current electric field would induce ejection of liquid metal inside capillary tube and then shooting into sodium hydroxide solution to form discrete droplets. The shooting velocity has positive relationship with the applied voltage while the droplet size is dominated by the aperture diameter of the capillary nozzle. Further, the motion of the liquid metal droplets can be flexibly manipulated by the electrodes. This effect suggests an easy going way to generate metal droplets in large quantity…
This paper presents the second-order perturbation theory of the Navier-Stokes equations for free surface flows, with the wave amplitude considered as the perturbation parameter. Gravity-capillary surface waves in incompressible viscous fluids are subjected. The results provide a systematic derivation of a nonlinear surface wave equation that fully takes into account dispersion, while nonlinearity is included in the leading order. However, the presence of infinitely many overdamped modes has been neglected, and only the two least damped modes are considered. Finally, to describe the elevations …
The hydrodynamics of droplets is significant in wide-ranging applications involving immiscible fluids and emulsions in food and pharmaceutical. The control and manipulation of droplets are primarily a function of flow governing and geometrical parameters. The finite element and level set approaches are used in this work to explore the influences of capillary number (Ca) and flow rate ratio (Qr) of dispersed and continuous phases on hydrodynamics of droplet generation in two-phase flow through T-junction cross-flow microfluidic device. A mathematical model based on a mass continuity, Navier-Sto…
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our deriv…
This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wavenumber of critical damping based on a harmonic oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers t…
Thin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, self-similarity of the first kind of the full evolution is demonstrated and a long-term expression for the excess free energy is derived. In addition, hydro…
We study the dynamics of the interface between two immiscible fluids in contact with a chemically homogeneous moving solid plate. We consider the generic case of two fluids with any viscosity ratio and of a plate moving in either directions (pulled or pushed in the bath). The problem is studied by a combination of two models, namely an extension to finite viscosity ratio of the lubrication theory and a Lattice Boltzmann method. Both methods allow to resolve, in different ways, the viscous singularity at the triple contact between the two fluids and the wall. We find a good agreement between th…
Capillary attraction between identical millimeter sized spheres floating at a liquid-air interface and the resulting aggregation is investigated at low Reynolds number. We show that the measured capillary forces between two spheres as a function of distance can be described by expressions obtained using the Nicolson approximation at low Bond numbers for far greater particle sizes than previously assumed. We find that viscous hydrodynamics interactions between the spheres needs to be included to describe the dynamics close to contact. We then consider the aggregates formed when a third sphere i…
We derive the second-order hydrodynamic equation for reactive multi-component systems from the relativistic Boltzmann equation. In the reactive system, particles can change their species under the restriction of the imposed conservation laws during the collision process. Our derivation is based on the renormalization group (RG) method, in which the Boltzmann equation is solved in an organized perturbation method as faithfully as possible and possible secular terms are resummed away by a suitable setting of the initial value of the distribution function. The microscopic formulae of the relaxati…
Stability criterion for the surface gravity capillary waves in a flowing two-layered fluid system with viscous dissipation is investigated. It is seen that the dissipative instability of negative energy waves is absent,- contrary to what earlier authors have concluded. Their error is identified to arise from an erroneous choice of the dissipation law, in which the wave profile velocity is wrongly equated to the particle velocity. Our corrected dissipation law is also shown to restore Galilean invariance to the stability condition of the system.
This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. The starting point is the kinetic model considered in earlier work of Degond & Motsch with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in Degond & Motsch, the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be s…
We introduce non-perturbative analytical techniques for the derivation of the hydrodynamic manifolds from kinetic equations. The new approach is analogous to the Schwinger-Dyson equation of quantum field theories, and its derivation is demonstrated with the construction of the exact diffusion manifold for a model kinetic equation. The novel approach is superior to the classical Chapman-Enskog method.
Immiscible fluid-fluid displacement in porous media is of great importance in many engineering applications, such as enhanced oil recovery, agricultural irrigation, and geologic CO2 storage. Fingering phenomena, induced by the interface instability, are commonly encountered during displacement processes and somehow detrimental since such hydrodynamic instabilities can significantly reduce displacement efficiency. In this study, we report a possible adjustment in pore geometry which aims to suppress the capillary fingering in porous media with hierarchical structures. Through pore-scale simulat…
In this paper we systematically derive a fourth-order continuum theory capable of reproducing mesoscale turbulence in a three-dimensional suspension of microswimmers. We start from overdamped Langevin equations for a generic microscopic model (pushers or pullers), which include hydrodynamic interactions on both, small length scales (polar alignment of neighboring swimmers) and large length scales, where the solvent flow interacts with the order parameter field. The flow field is determined via the Stokes equation supplemented by an ansatz for the stress tensor. In addition to hydrodynamic inte…
We study isothermal transport of a nonelectrolyte binary fluid mixture, which lies in the homogeneous phase near the demixing critical point, through a capillary tube connecting two reservoirs. Usually, one component is preferentially adsorbed onto the tube wall, and the adsorption becomes significant owing to large osmotic susceptibility. The mixture flowing out of the tube is rich in the preferred component when flow is driven by the pressure difference between the reservoirs. When flow is driven by the mass-fraction difference, the total mass flow occurs in the presence of the preferential …
The boundary conditions at the deformable interface between two contacting fluids are derived for the general case of the large-amplitude perturbations. The interface is modeled as perturbed free boundary that evolves in time, and the non-linear description is performed and analyzed in a wide range of physical situations. The differential equations of the interfacial motion thus obtained may be useful in research of the non-linear development of the classical hydrodynamic instabilities. They should play an important role in the understanding of the hydrodynamic phenomena associated with the fl…
The dynamics of particles attached to an interface separating two immiscible fluids are encountered in a wide variety of applications. Here we present a combined asymptotic and numerical investigation of the fluid motion past spherical particles attached to a deformable interface undergoing uniform creeping flows in the limit of small Capillary number and small deviation of the contact angle from 90 degrees. Under the assumption of a constant three-phase contact angle, we calculate the interfacial deformation around an isolated particle and a particle pair. Applying the Lorentz reciprocal theo…
From extensive molecular dynamics simulations on immiscible two-phase flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilli…
After a brief account of the derivation of the first-order relativistic hydrodynamic equation as a construction of the invariant manifold of relativistic Boltzmann equation, we give a sketch of derivation of the second-order hydrodynamic equation (extended thermodynamics) both in the nonrelativistic and relativistic cases. We show that the resultant equation suggests a novel ansatz for the functional form to be used in Grad moment method, which turns out to give the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for …
We derive generic relativistic hydrodynamical equations with dissipative effects from the underlying Boltzmann equation in a mechanical and systematic way on the basis of so called the renormalization-group (RG) method. A macroscopic frame vector is introduced to specify the frame on which the macroscopic dynamics is described. Our method is so mechanical with only few ansatz that our method give a microscopic foundation of the available hydrodynamical equations, and also can be applied to make a reduction of the kinetic equations other than the simple Boltzmann equation.
Theoretical description and numerical simulation of an evaporating sessile drop are developed. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. A shape of the rotationally symmetric drop is determined within the quasistationary approximation. Nonstationary effects in the diffusion of the vapor are also taken into account. Simulation results agree well with the data of evaporation rate measurements for the toluene drop. Marangoni forces associated with the temperature dependence of the…
Fluctuating hydrodynamics provides a model for fluids at mesoscopic scales where thermal fluctuations can have a significant impact on the behavior of the system. Here we investigate a model for fluctuating hydrodynamics of a single component, multiphase flow in the neighborhood of the critical point. The system is modeled using a compressible flow formulation with a van der Waals equation of state, incorporating a Korteweg stress term to treat interfacial tension. We present a numerical algorithm for modeling this system based on an extension of algorithms developed for fluctuating hydrodynam…
A geometric approach to derive the Nambu brackets for ideal two-dimensional (2D) hydrodynamics is suggested. The derivation is based on two-forms with vanishing integrals in a periodic domain, and with resulting dynamics constrained by an orthogonality condition. As a result, 2D hydrodynamics with vorticity as dynamic variable emerges as a generic model, with conservation laws which can be interpreted as enstrophy and energy functionals. Generalized forms like surface quasi-geostrophy and fractional Poisson equations for the stream-function are also included as results from the derivation. The…
We present four different ways of deriving the Oseen tensor which is the fundamental solution to the Stokes equations for an incompressible viscous fluid. This solution corresponds to a point force acting on an infinite fluid. The derivations follow the books of Kim & Karilla, Zapryanov & Tabakova, Dhont, and Pozrikidis.
