Mesoscopic hydrodynamics

Thermal fluctuations can be included into the hydrodynamic equations via a Gaussian fluctuating stress tensor S(r, The stochastic version of the Navier-Stokes equation then reads 

For nanofluidic systems the left-hand side of Eq. 3.16 can be neglected because in general the Reynolds number is tiny. This leads to the linear Stokes equation 

For incompressible fluids the conservation law for mass reduces to the condition  

Pol3mieric liquids play an important role in nanofluidics. However, they tend to be non-Newtonian. In this respect Eq. 3.16 can be generalized to take into account viscoelasticity however, this is beyond the scope of the present discussion. 

Together with the disjoining pressure n, the noise tensor S appears also in the boundary conditions for the pressure p and the viscous stress tensor Ty = rj j at the liquid-vapor 

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The main message of these lectures is the need to amend the classical hydrodynamic theory by direct inclusion of intermolecular interactions. This is necessary not only in the theory of contact line motion outlined here, but in all mesoscopic hydrodynamic problems, e.g. in fluid mechanics of microdevices, which attracts lately a lot of attention. The specific feature of the contact line problem is the connection between microscopic and macroscopic. The motion in the precursor film can and should be treated more precisely, on the statistical level with due account for fluctuations or directly through molecular dynamics simulations. A challenging problem is matching the microscopic theory with classical hydrodynamics applicable in macroscopic domains away from the immediate vicinity of the contact line. 

The main theoretical challenge for nonequilibrium systems is to bridge the gap in length and time scales between the molecular motion and the collective motion of the fluid, such as the translation of droplets. In top-down approaches macroscopic hydrodynamics is combined with equilibrium statistical physics. The resulting mesoscopic hydrodynamic equations partially include the effects of boundary slip, thermal fluctuations, and the long range of molecular interactions. So far, bottom-up approaches for nonequilibrium systems are only available for a small class of systems with purely diffusive dynamics. For the other liquids one has to resort to numerical simulations. 

Solvent-free models, triangulated surfaces and other discretized curvature models have the disadvantage that they do not contain a solvent, and therefore do not describe the hydrodynamic behavior correctly. However, this apparent disadvantage can be turned into an advantage by combining these models with a mesoscopic hydrodynamics technique. This approach has been employed for dynamically triangulated surfaces [37,180] and for meshless membrane models in combination with MPC [188], as well as for fixed membrane triangulations in combination with both MPC [187] and the LB method [189]. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

These results show that hydrodynamic interactions and the spatial dependence of the friction tensor can be investigated in regimes where continuum descriptions are questionable. One of the main advantages of MPC dynamics studies of hydrodynamic interactions is that the spatial dependence of the friction tensor need not be specified a priori as in Langevin dynamics. Instead, these interactions automatically enter the dynamics from the mesoscopic particle-based description of the bath molecules. 

Hybrid MPC-MD schemes are an appropriate way to describe bead-spring polymer motions in solution because they combine a mesoscopic treatment of the polymer chain with a mesoscopic treatment of the solvent in a way that accounts for all hydrodynamic effects. These methods also allow one to treat polymer dynamics in fluid flows. 

Since hydrodynamic interactions are included in MPC dynamics, the collective motion of many self-propelled objects can be studied using this mesoscopic simulation method. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Remark. Instability and bistability are defined as properties of the macroscopic equation. The effect of the fluctuations is merely to make the system decide to go to one or the other macroscopically stable point. Similarly the Taylor instability and the Benard cells are consequences of the macroscopic hydrodynamic equations. ) Fluctuations merely make the choice between different, equally possible macrostates, and, in these examples, determine the location of the vortices or of the cells in space. (In practice they are often overruled by extraneous influences, such as the presence of a boundary.) Statements that fluctuations shift or destroy the bistability are obscure, because on the mesoscopic level there is no sharp separation between stable and unstable systems. Some authors call a mesostate (i.e., a probability distribution P) bistable when P has two maxima, however flat. This does not correspond to any observable fact, however, unless the maxima are well-separated peaks, which can each be related to separate macrostates, as in (1.1). 

Yet another promising line of research lies in creating mesoscopic representations whose fundamental scale is somewhere within the 1010 range of distance scales and for which one defines closed (or fully consistent) equations of motion. At the macroscopic limit, hydrodynamic models are a very successful and standard example. More recent approaches include the Cahn-Hillard coarse-grained models and phase-field models. In some cases, one aims to ascertain the degree to which the systems exhibit self-similarity at various length scales hence the lack of a specific parameterization—which would be necessary using reduceddimensional models—is not of much importance. 

The hydrodynamics of a circulating fluidized bed can be analyzed from both the macroscopic and mesoscopic points of view. The nonuniformity of the solids concentration in the radial and axial directions represents macroscopic behavior. The existence of solid clusters characterizes mesoscopic behavior (see 10.5). The hydrodynamic behavior in a macroscale is discussed in the following. 

Since the theories developed by Novotny [53] (conventional diffusion model) and Mate [74] (hydrodynamic model) fail to describe the L oc t behavior exhibited in regime I, the constmction of a simple mesoscopic model (prior to investigating the molecular simulation see Section III) has been suggested as described below to explain overall L-t behavior. 

There is also another way the mesoscopic time evolution Equation (55) can be introduced. We collect a list of well-established (i.e., well tested with experimental observations) time evolution equations on many different levels of description and try to identify their common features. This is indeed the way the time evolution Equation (55) has been first introduced. The Hamiltonian structure of the nondissipative part has been discovered first in the context of hydrodynamics by Clebsch (1895). Equations of the type (55) have started to appear in Dzyaloshinskii and Volovick (1980) and later in... 

The hydrodynamic study at a mesoscopic scale requires the understanding of instantaneous local solids flow structure. The time-variant flow behavior is complex. Analyses of the instantaneous flow structure require recognizing the following factors. 

Roby and Joanny [27] improved the model of Benmouna, et al. [23] by incorporating interchain hydrodynamic interactions. At elevated concentrations, reptation dynamics were assumed, approximating the solution as a polymer melt in which mesoscopic polymer-solvent blobs are effective monomers. Hammouda [28] repeated the calculation, removing the restriction that the system contained equal amounts of two species having the same molecular weight, and analyzing tagged-tracer experiments. 

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