Hydrodynamic boundary condition

In the preceding sections, we have presented simulation methodologies to study systems away from equilibrium. In particular, we have concentrated on the problem of shear flow in bulk systems. In the present section, we illustrate the effectiveness of the SLLOD dynamics coupled with the robustness of the extended system approach to initiate and sustain a shear flow in a fluid in the absence of moving boundary conditions. 

The physical problem we consider is the characterization of the boundary for a liquid flow over a stationary solid surface. The fluid flow can he studied hy means of the Navier-Stokes equation 

In the section on nonequilibrium molecular dynamics, the equations of motion for field-driven dynamics were introduced. We have also noted that for bulk fluids, the field must be accompanied by boundary conditions that are 

In the absence of shearing periodic boundary conditions (of the type introduced earlier) the system is totally isolated that is, all the degrees of freedom of the system are explicitly accounted for in the equations of motion. In this case, it is possible to obtain a conserved quantity for field-driven dynamics in general and SLLOD in particular. The approach we employ is similar to that introduced in the section on Molecular Dynamics and Equilibrium Statistical Mechanics. The SLLOD equations of motion are 

note that in Eq. [202] we have introduced a variable, I, which although inconsequential to the dynamics of the variables representing the fluid (i.e., p, q) is essential to obtain a conserved energy for SLLOD dynamics with time-independent boundary conditions. The conserved energy for the dynamics in Eqs. [202] is given by 

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Recent times have seen much discussion of the choice of hydrodynamic boundary conditions that can be employed in a description of the solid-liquid interface. For some time, the no-slip approximation was deemed acceptable and has constituted something of a dogma in many fields concerned with fluid mechanics. This assumption is based on observations made at a macroscopic level, where the mean free path of the hquid being considered is much smaller... 

In the following a semiquantitative argument is presented on the recovery of the hydrodynamic boundary condition from microscopic considerations. 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

Polymers can be confined one-dimensionally by an impenetrable surface besides the more familiar confinements of higher dimensions. Introduction of a planar surface to a bulk polymer breaks the translational symmetry and produces a pol-ymer/wall interface. Interfacial chain behavior of polymer solutions has been extensively studied both experimentally and theoretically [1-6]. In contrast, polymer melt/solid interfaces are one of the least understood subjects in polymer science. Many recent interfacial studies have begun to investigate effects of surface confinement on chain mobility and glass transition [7], Melt adsorption on and desorption off a solid surface pertain to dispersion and preparation of filled polymers containing a great deal of particle/matrix interfaces [8], The state of chain adsorption also determine the hydrodynamic boundary condition (HBC) at the interface between an extruded melt and wall of an extrusion die, where the HBC can directly influence the flow behavior in polymer processing. 

L. Bocquet and J. L. Barrat, Phys. Rev. Lett., 70, 2726 (1993). Hydrodynamic Boundary Conditions and Correlation Function of Confined Fluids. 

C. J. Mundy, S. Balasubramanian, and M. L. Klein, /. Chem. Phys., 105, 3211 (1996). Hydrodynamic Boundary Conditions for Confined Fluids Via a Nonequilibrium Molecular Dynamics Simulation. 

P. G. Wolynes, Phys. Rev. A, 13, 1235 (1976). Hydrodynamic Boundary Conditions and Mode-Mode Coupling Theory. 

The origin of the observed enhancement of D in hard-sphere fluids relative to = kT/ at intermediate densities has been the subject of many theoretical investigations. " The aim in many instances has been to show the evolution of D from the Enskog result at intermediate densities to the Stokes -Einstein result = kT/6nriR at liquid densities. For example, the analysis of Hynes et al., in which a modified hydrodynamic boundary condition was invoked, yields the simple result... 

It is shown that fluid flow and heat transfer at microscale differ greatly from those at macroscale. At macroscale, classical conservation equations are successfully coupled with the corresponding wall boundary conditions, usual no-slip for the hydrodynamic boundary condition and no-temperature-jump for the thermal boundary condition. These two boimdary conditions are valid only if the fluid flow adjacent to the surface is in thermal equilibrium. However, they are not valid for gas flow at microscale. For this case, the gas no longer reaches the velocity or the temperature of the surface and therefore a slip condition for the velocity and a jump condition for the temperature should be adopted. 

The following treatment, adapted from the work of Van Oene et al. is used to illustrate the assumptions and general conclusions of the surface chemical approach in which the no slip condition is ignored the hydrodynamic boundary condition that for a liquid moving over a solid surface there can be no motion of the liquid immediately adjacent to the liquid/solid interface. The work of Van Oene is similar to that of Schon-horn et al. and at the end of this section we will compare the results of the two investigations. 

The applicability of slip flow is not well accepted to date by the academic community. One of the problems is the small length scale of slip flow regime (if present) in comparison to the length scale of the flow or system. The hydrodynamic boundary condition appears to be one of no slip, unless the flow is examined on a length scale comparable to the slip length. Hence, very accurate techniques with high spatial resolution capable of interfacial flow measurements are required to detect the effects of slip. Some of the experimental techniques for quantification of liquid slip are presented in the following sections. 

The surface boundary conditions are critical to modeling electrokinetic phenomena in nanofluidics. For the hydrodynamic boundary condition, we use the nonslip model at the silica surfaces. Although the slip boundaries have been adopted and have shown significant effects to improve the energy-conversirai efficiency, a careful molecular study showed that the hydro-dynamic boundary conditirm, slip or not, depended on the molecular interactions between fluid and solid and the channel size. For the dilute solution in silica nanochannels considered in this work(/x 2 nm), the nonslip boundary condition is still valid very well. 

The boundary condition implementations play a very critical role in the accuracy of the numerical simulations. The hydrodynamic boundary conditions for the LBM have been smdied extensively. The conventional bounce-back rule is the most popular method used to treat the velocity boundary condition at the solid-fluid interface due to its easy implementation, where momentum from an incoming fluid particle is bounced back in the opposite direction as it hits the wall [20]. However, the conventional bounce-back rale has two main disadvantages. First, it requires the dimensionless relaxation time to be strictly within the range (0.5, 2) otherwise, the prediction will deviate from the correct result. Second, the nonslip boundary implemented by the conventional bounce-back rule is not located exactly on the boundary nodes, as mentioned before, which will lead to inconsistence when coupling with other partial differential equation (PDF) solvers on a same grid set [17]. 

At the boundary, the following hydrodynamic boundary condition holds ... 

Sokhan VP, Nicholson D, Quirke N (2001) Fluid flow in nanopores an examination of hydrodynamic boundary conditions. J ChemPhys 115(8) 3878... 

We have not been able to find Eq. (125b) in this form in the literature. (Cercignani gives expressions for the hydrodynamic boundary condition that are not entirely equivalent to ours.) It is derived in Ref. (78). For the case of stationary flow, where V o = 0, a boundary condition equivalent to (125b) has been derived. ... 

Botan A, Marry V, Rotenberg B, Turq P, Noetinger B (2013) How electrostatics influences hydrodynamic boundary conditions Poiseuille and electro-osmostic flows in clay nanopores. J Phys Chem C 117 978-985... 

Besides this confusion over v and F, it is further incoirect to confuse a thermodynamic quantity ifi) with a hydrodynamic one (F,). The quantity Fe was determined from the hydrodynamic theory of rigid, impermeable ellipsoids. However, the protein may not be ellipsoidal in shape, it may not be rigid in a hydrodynamic field, and it may not be impermeable to the flow of solvent. In addition, the hydrodynamic boundary condition of no slippage on the. surface of the particle may not be satisfied, and the... 

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