Statistical Mechanical Expressions of Slip Length

We provide general derivations of the partial slip boundary condition from microscopic dynamics and linearized fluctuating hydrodynamics. The derivations are based on the assumption of separation of scales between microscopic behavior, such as collision of particles, and macroscopic behavior, such as relaxation of fluid to global equilibrium. These derivations lead to several statistical mechanical expressions of the slip length, which are classified into two types. The expression in the first type is given as a local transport coefficient, which is related to the linear response theory that describes the relaxation process of the fluid. The second type is related to the linear response theory that describes the non-equilibrium steady state and the slip length is given as combination of global transport coefficients, which are dependent on macroscopic lengths such as a system size. Our derivations clarify that the separation of scales must be seriously considered in order to distinguish the expressions belonging to two types. Based on these linear response theories, we organize the relationship among the statistical mechanical expressions of the slip length suggested in previous studies.

     

Electrokinetic and hydrodynamic properties of charged-particles systems

Dynamic processes in dispersions of charged spherical particles are of importance both in fundamental science, and in technical and bio-medical applications. There exists a large variety of charged-particles systems, ranging from nanometer-sized electrolyte ions to micron-sized charge-stabilized colloids. We review recent advances in theoretical methods for the calculation of linear transport coefficients in concentrated particulate systems, with the focus on hydrodynamic interactions and electrokinetic effects. Considered transport properties are the dispersion viscosity, self- and collective diffusion coefficients, sedimentation coefficients, and electrophoretic mobilities and conductivities of ionic particle species in an external electric field. Advances by our group are also discussed, including a novel mode-coupling-theory method for conduction-diffusion and viscoelastic properties of strong electrolyte solutions. Furthermore, results are presented for dispersions of solvent-permeable particles, and particles with non-zero hydrodynamic surface slip. The concentration-dependent swelling of ionic microgels is discussed, as well as a far-reaching dynamic scaling behavior relating colloidal long- to short-time dynamics.     

Synchronization of oscillating reactions in an extended fluid system

We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.     

Increasing thermoelectric efficiency: a dynamical systems approach

Inspired by the kinetic theory of ergodic gases and chaotic billiards, we propose a simple microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, where we find exact expressions for all transport coefficients, the thermoelectric efficiency of ideal ergodic gases can approach the Carnot efficiency for sufficiently complex charge carrier molecules. Our results are clearly demonstrated with a simple numerical simulation of a Lorentz gas of particles with internal rotational degrees of freedom.     

Electro-osmotically actuated oscillatory flow of a physiological fluid on a porous microchannel subject to an external AC electric field having dissimilar frequencies

Electro-osmotic flow of a physiological fluid with prominent micropolar characteristics, flowing over a microchannel has been analyzed for a situation, where the system is subject to the action of an external AC electric field. In order to account for the rotation of the micro-particles suspended in the physiological fluid, the fluid has been treated as a micropolar fluid. The microchannel is considered to be bounded by two porous plates executing oscillatory motion. Such motion of the plates will normally induce oscillatory flow of the fluid. The governing equations of the fluid include a second-order partial differential equation depicting Gauss’s law of electrical charge distributions and two other partial differential equations of second order that arise out of the laws of conservation of linear and angular momenta. These equations have been solved under the sole influence of electrokinetic forces, by using appropriate boundary conditions. This enabled us to determine explicit analytical expressions for the electro-osmotic velocity of the fluid and the microrotation of the suspended micro-particles. These expressions have been used to obtain numerical estimates of important physical variables associated with the oscillatory electro-osmotic flow of a blood sample inside a micro-bio-fluidic device. The numerical results presented in graphical form clearly indicate that the formation of an electrical double layer near the vicinity of the wall causes linear momentum to reduce. In contrast, the angular momentum increases with the enhancement of microrotation of the suspended microparticles. The study will find important applications in the validation of results of further experimental and numerical models pertaining to flow in micro-bio-fluidic devices. It will also be useful in the improvement of the design and construction of various micro-bio-fluidic devices.     

Influence of hydrodynamic coupling on pair diffusion in a quasi-one-dimensional colloid system

The effect of hydrodynamic interaction on the separation dependence of the center of mass and relative pair diffusion coefficients of colloid particles in a quasi-one-dimensional system, including the influence of proximate walls, is calculated using the method of reflections. There is excellent agreement between the theoretical predictions and the experimental data. We show that the separation dependence of the relative pair diffusion coefficient has an oscillatory structure on the scale length of the correlation length in the system, and we directly relate that oscillatory structure to the pair correlation function of the system.     

Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles

We present experimental studies of interacting excitable and oscillatory catalytic particles in well-stirred and spatially distributed systems. A number of distinct paths to synchronized oscillatory behavior are described. We present an example of a Kuramoto type transition in a well-stirred system with a collective rhythm emerging on increasing the number density of oscillatory particles. Groups of spatially distributed oscillatory particles become entrained to a common frequency by organizing centers. Quorum sensing type transitions are found in populations of globally and locally coupled excitable particles, with a sharp transition from steady state to fully synchronized behavior at a critical density or group size.     

Evaluation of the translational and rotational diffusion coefficients of a cubic particle (for the application to Brownian dynamics simulations)

By estimating the force and torque acting on the cube for the two cases of a uniform flow field and a rotational flow field, we have discussed whether or not there is a coupling between the translational and the rotational motion. From the characteristics of the friction coefficients, we may understand that there is no coupling between the translation motion and the rotational motion in the situation of the Reynolds number being sufficiently smaller than unity. In contrast, in the case of a non-slow flow field with the Reynolds number larger than unity, the coupling characteristics of the motion of a cube is certainly recognised and therefore the interaction with the ambient fluid is characterised by a variety of friction coefficients including friction coefficients that relate the forces acting on the cube to the angular velocities of the rotational motion. Hence, the employment of these translational and rotational diffusion coefficients for a cube enables the implementation of Brownian dynamics simulations for a suspension composed of cubic particles in order to analyse the dynamic characteristics of a cubic particle suspension.

Highlights

1. We have considered a flow problem around a cube in order to numerically clarify the characteristics of the translational and rotational friction or diffusion coefficients.

2. In a slow flow field the motion of the cube need only to be characterised by two friction coefficients, i.e. the translational and rotational friction coefficients.

3. In the case of a non-slow flow field, the coupling characteristics between the translational motion and the rotational motion are recognised.

4. Employment of these diffusion coefficients enables the implementation of Brownian dynamics simulations for a suspension composed of cubic particles.

     

Note on the Kaplan–Yorke Dimension and Linear Transport Coefficients

A number of new relations between the Kaplan–Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small but non-zero external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the linear transport coefficients in terms of the Kaplan–Yorke dimension is derived. Alternatively, the Kaplan–Yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.     

Atomistic simulations of fluid structure and solvation forces in atomic force microscopy

We describe results of atomistic molecular dynamics simulations modelling an atomic force microscope (AFM) tip immersed in a fluid. Both the tip and the surface are modelled by rigid arrays of atoms. The tip is pyramidal and the surface is the (100) face of a fcc crystal. The focus is on the solvation forces acting on the tip and on the surface and their relation to the structural and dynamic properties of the fluid. Fluid particles in the neighborhood of the tip-surface junction are found to be highly ordered compared to the bulk, as shown by localized variations in the average fluid density. The atomistic nature of the model gives rise to several effects related to the discrete sizes of the fluid, tip, and surface particles which are not observed in continuum-based theories. A number of simulated force-distance curves are presented, along with an analysis of the effect of changing fluid particle size, tip (lateral) position, tip shape, and the lyocompatability of the tip and surface materials. The atomic-scale distribution of fluid-surface forces is examined for various positions of the tip, and the extent to which the fluid can act as a “cushion” by increasing the effective area of the tip-surface interaction is studied. The effect of a fluid on AFM imaging is investigated by generating “fluid images”, which are shown to be comparable in magnitude to the direct tip-surface interaction in the noncontact mode. We compare images generated by defective and defect-free surfaces, and analyse the fluid-tip forces acting in a lateral direction. An image formed from fluid forces acting in the direction of the surface normal does not show the presence of a vacancy, but an image formed from lateral fluid forces does.     

Losses and nonlinear steady-state particle distribution functions for fully ionized tokamak-plasmas in the collisional transport regimes

Fully ionized L-mode tokamak plasmas in the fully collisional (Pfirsch-Schlüter) and in the low-collisional (banana) nonlinear transport regimes are analyzed. We derive the expressions for particles and heat losses together with the steady-state particle distribution functions in the several collisional transport regimes. The validity of the nonlinear closure equations, previously derived, has been indirectly tested by checking that the obtained particle distribution functions are indeed solutions of the nonlinear, steady-state, Vlasov-Landau gyro-kinetic equations. A quite encouraging result is the fact that, for L-mode tokamak plasmas a dissymmetry appears between the ion and electron transport coefficients: the latter submits to a nonlinear correction, which makes the radial electron coefficients much larger than the former. In particular we show that when the L-mode JET plasma is out of the linear region, the Pfirsch-Schlüter electron transport coefficients are corrected by an amplification factor, which may reach values of order 10². Such a correction is absent for ions. On the contrary, in the banana regime, the ion transport coefficients are increased by a factor 2 and the nonlinear corrections for electrons are negligible. These results are in line with experiments.     

Numerical simulation for separation of multi-phase immiscible fluids in porous media

Based on a lattice Boltzmann method and general principles of porous flow, a numerical technique is presented for analysing the separation of multi-phase immiscible fluids in porous media. The total body force acting on fluid particles is modified by axiding relative permeability in Nithiarasu＇s expression with an axiditional surface tension term. As a test of this model, we simulate the phase separation for the case of two immiscible fluids. The numerical results show that the two coupling relative permeability coefficients K12 and K21 have the same magnitude, so the linear flux-forcing relationships satisfy Onsager reciprocity. Phase separation phenomenon is shown with the time evolution of density distribution and bears a strong similarity to the results obtained from other numerical models and the flows in sands. At the same time, the dynamical rules in this model are local, therefore it can be run on massively parallel computers with well computational efficiency.     

Low-viscosity lattice gases

A class of lattice gas models are studied which are variants of the FCHC model. The aim is to achieve the highest possible Reynolds coefficient (inverse dimensionless viscosity) for efficient simulations of the three-dimensional incompressible Navier-Stokes equations. The models include an arbitrary number of rest particles and violation of semi-detailed balance. Within the framework of the Boltzmann approximation exact expressions are obtained for the Reynolds coefficients. The minimization of the viscosity is done by solving a Hitchcock-type optimization problem for the fine tuning of the collision rules. When the number of rest particles exceeds one, there is a range of densities at which the viscosity takes negative values. Various optimal models with up to 26 bits per node have been implemented on a CRAY-2 and their true transport coefficients have been measured with good accuracy. Fairly large discrepancies with Boltzmann values are observed when semi-detailed balance is violated; in particular, no negative viscosity is obtained. Still, the best model has a Reynolds coefficient of 13.5, twice that of the best previously implemented model, and thus is about 16 times more efficient computationally. Suggestions are made for further improvements. It is proposed to use models with very high Reynolds coefficients for sub-grid-scale modeling of turbulent flows.     

Molecular derivation of the hydrodynamic equations for a binary fluid

In this paper, the hydrodynamic equations and the associated transport coefficients are derived for a simple binary fluid from molecular considerations. This is a generalization of the methods of Felderhof and Oppenheim and of Selwyn to multicomponent systems. A linear response formalism is used to describe the relaxation of the binary system from an initial nonequilibrium state. Explicit molecular expressions are given for the transport coefficients in terms of time correlation functions of generalized current densities. These densities have the useful property of not containing a conserved part. The correlation functions are then related to a set of phenomenological coefficients in the theory of nonequilibrium thermodynamics. This explicit identification enables one to relate the correlation functions to experimentally measured transport coefficients.Supported by the National Science Foundation.     

Interacting Brownian motion in a periodic potential

We study transport properties of a system composed of Brownian particles immersed in a periodic potential. Interaction among the Brownian particles is treated perturbationally in a framework of a generalized Fokker-Planck equation, which due to the interaction contains a renormalized periodic potential and an extra mean field term. We solve the kinetic equation numerically and discuss effects of the (repulsive) interaction on dynamic response functions and transport coefficients.     

A non-equilibrium molecular dynamics approach to fluid transfer through microporous membranes

The colour field non-equilibrium molecular dynamics method is applied to model fluid transfer between two fluid phases separated by microporous membranes (i.e. pore width ≤ 2 nm). The model is simple. All particles show short range spherical interactions, but reproduce the main features of the experimental phenomenology of molecular sieve membranes. An external force is applied to the fluid particles and, due to the presence of the membrane, a stationary flux across the system results. A range of mass transfer coefficients was investigated and the fluxes were found to be linear with force intensities. The density was found to have a small effect on molecular mobility, while temperature had a more significant effect, showing the features of an activated process. In addition, the changes in these fluxes with the membrane size, and the cross interaction parameters between the fluid and the membrane particles were examined.     

Stability of thermal convection of an Oldroyd-B fluid in a porous medium with Newtonian heating

A linear stability analysis determining the onset of oscillatory convection of an Oldroyd-B fluid in a bounded two-dimensional rectangular porous medium generated by Newtonian heating is conducted. Influences of viscoelastic parameters and Biot number on the onset of oscillatory convection, preferred modes and patterns of disturbed temperature contours are discussed.     

Free oscillations of magnetic fluid in strong magnetic field

The paper presents the esults of measuring the elastic parameters of an oscillatory system (coefficient of pondermotive elasticity, damping factor, and oscillation frequency) whose viscous inertial element is represented by a magnetic fluid confined in a tube by magnetic levitation in a strong magnetic field. The role of elasticity is played by the pondermotive force acting on thin layers at the upper and lower ends of the fluid column. It is shown that, by measuring the elastic oscillation frequencies of the magnetic fluid column, it is possible to develop a fundamentally new absolute method for determining the saturation magnetization of a magnetic colloid.     

Modified Eulerian–Lagrangian formulation for hydrodynamic modeling

We present the modified Eulerian–Lagrangian (MEL) formulation, based on non-divergent forms of partial differential balance equations, for simulating transport of extensive quantities in a porous medium. Hydrodynamic derivatives are written in terms of modified velocities for particles propagating phase and component quantities along their respective paths. The particles physically interpreted velocities also address the heterogeneity of the matrix and fluid properties. The MEL formulation is also implemented to parabolic Partial Differential Equations (PDE’s) as these are shown to be interchangeable with equivalent PDE’s having hyperbolic – parabolic characteristics, without violating the same physical concepts. We prove that the MEL schemes provide a convergent and monotone approximation also to PDE’s with discontinuous coefficients. An extension to the Peclet number is presented that also accounts for advective dominant PDE’s with no reference to the fluid velocity or even when this velocity is not introduced.In Sorek et al. [27], a mathematical analysis for a linear system of coupled PDE’s and an example of nonlinear PDE’s, proved that the finite difference MEL, unlike an Eulerian scheme, guaranties the absence of spurious oscillations. Currently, we present notions of monotone interpolation associated with the MEL particle tracking procedure and prove the convergence of the MEL schemes to the original balance equation also for discontinuous coefficients on the basis of difference schemes approximating PDE’s. We provide numerical examples, also with highly random fields of permeabilities and/or dispersivities, suggesting that the MEL scheme produces resolutions that are more consistent with the physical phenomenon in comparison to the Eulerian and the Eulerian–Lagrangian (EL) schemes.     

Study of mass transport in rotating couple stress liquid under concentration modulation

A weakly non-linear stability analysis has been performed to examine the effect of time-periodic concentration modulation on the mass transport. We consider an infinite horizontal fluid layer is rotating with angular velocity Ω₁ about Z-axis subjected to an imposed time-periodic boundary concentration (ITBC). The concentration gradient between the plates of the fluid layer consists of a steady part and a time-dependent oscillatory part. The concentration of both walls is modulated in this case. We have expanded the infinitesimal disturbances in terms of power series of amplitude of modulation, which is assumed to be small. Ginzburg–Landau equation is obtained to find the rate of mass transfer. It is found that, the effect of Taylor number is to stabilize the system. Effect of Schmidt number and Couple stress parameter on mass transfer are also discussed. Further, it is found that the mass transport can be controlled by suitably adjusting the frequency and amplitude of modulation .