On three simple experiments to determine slip lengths

It is now well established that for fluid flow at the micro- and nano-scales the standard no-slip boundary condition of fluid mechanics at fluid–solid interfaces is not applicable and must be replaced by a boundary condition that allows some degree of tangential fluid slip. Although molecular dynamics studies support this notion, an experimental verification of a slip boundary condition remains lacking, primarily due to the difficulty of performing accurate experimental observations at small scales. In this article, three simple fluid problems are studied in detail, namely a fluid near a solid wall that is suddenly set in motion (Stokes’ first problem), the long-time behavior of a fluid near an oscillating solid wall (Stokes’ second problem), and the long-time behavior of a fluid between two parallel walls one of which is oscillating (oscillatory Couette flow). The no-slip boundary condition is replaced with the Navier boundary condition, which allows a certain degree of tangential fluid slip via a constant slip length. The aim is to obtain analytical expressions, which may be used in an experimental determination of the constant slip length for any fluid–solid combination.     

Hybrid atomistic�Ccontinuum simulations of fluid flows involving interfaces

In this work, we use an hybrid atomistic–continuum (HAC) simulation method to study transient and steady isothermal flows of Lennard-Jones fluids near interfaces. Our hybrid method is based on a domain decomposition algorithm. The flow domain is composed of two overlapping regions: an atomistic region described by molecular dynamics, and a continuum region described by a finite volume discretization of the incompressible Navier–Stokes equations. To show the interest of such an hybrid method to compute flows near fluid/solid interface, we first applied our hybrid scheme to the classical Couette flow, where the moving wall is modelled at the atomistic scale. In addition, we also studied an oscillatory shear flow. Then, to compute flows near fluid/fluid interface, we applied our method to a two-phase Couette flow (liquid/gas), where the interface is modelled at the molecular scale. We show that hybrid results can sometimes differ from those provided by analytical solutions deduced from continuum mechanics equations combined with usual boundary/interface relations. For the Couette and oscillatory shear flows, a good agreement is found between hybrid simulations and macroscopic analytical solutions, however, we noticed that the fluid in contact with the wall can be more entailed than what expected. For the liquid/gas Couette flow, the hybrid simulation exhibits an unexpected jump of the velocity in the interfacial region, corresponding to a partial slip between the two fluid phases. Those interesting results highlight the interest of using an HAC method to deal with systems for which surfaces/interfaces effects are important.     

Effects of surface roughness and interface wettability on nanoscale flow in a nanochannel

Non-equilibrium molecular dynamics simulations have been carried out to investigate the effect of surface roughness and interface wettability on the nanorheology and slip boundary condition of simple fluids in a nanochannel of several atomic diameters width. The solid surfaces decorated with periodic nanostrips are considered as the rough surface in this study. The simulation results showed that the interface wettability and the surface roughness are important in determining the nanorheology of the nanochannel and fluid slip at solid–fluid interface. It is observed that the presence of surface roughness always suppresses the fluid slip for hydrophilic and hydrophobic surface nanochannels. For fluids over smooth and hydrophobic surfaces, the snapshots of fluid molecules show that an air gap or nanobubble exists at the fluid–solid interface, resulting in the apparent slip velocity. For a given surface with fixed interface wettability, the fluid velocities increase by increasing the driving force, while the driving force has no significant influence on the density structure of fluid molecules. The fluid slip and the flow rate are measured for hydrophilic and hydrophobic nanochannels. The flow rates in rough surface nanochannels are smaller than those of smooth surface walls due to the increase of drag resistance at the solid–fluid interface. The dependence between fluid slip and flow rate showed that the slip length increases approximately linearly with the flow rate for both the hydrophobic and hydrophilic surface nanochannels.     

Slippage of binary fluid mixtures in a nanopore

This work focuses on the slip phenomenon at the fluid–solid interface accompanying Poiseuille flow of simple binary miscible fluids in a slit nanopore. To explore such flows, molecular dynamics simulations are used on Lennard–Jones binary mixtures composed of species of varying affinities with the walls. The results have shown that the apparent slip magnitude at the fluid–solid interface depends largely on the species that is dominant in contact with the walls. In addition, it has been shown that the velocity profiles of each species (of different “wettability”) does not superpose with the velocity profile of the mixture and such a result points out the limitations of the classical approaches based on a single momentum conservation equation to deal with mixtures flow in nanochannels.     

Slip in nanoscale shear flow: mechanisms of interfacial friction

The atomistic mechanism of fluid–solid interfacial friction as the basis of slip is still not fully understood. This study explores the interfacial friction mechanisms and their interplay with the nanoscale slip behavior using non-equilibrium molecular dynamics simulations. Our results show that there is an abrupt jump of slip length at a critical shear rate, corresponding to the transition from “defect slip” at low shear rates to “collective slip” at high shear rates. Here, we identified two mechanisms of interfacial friction: surface potential and collision mechanisms. Their impacts on slip are elaborated through a quantitative scaling estimation and our results show that both mechanisms contribute to the defect slip at low shear rates, while the collision mechanism dominates the collective slip at high shear rates. We also verify the importance of the bulk viscous heating via a comparison among different thermostat strategies.     

Molecular dynamics-based prediction of boundary slip of fluids in nanochannels

Computational modeling and simulation can provide an effective predictive capability for flow properties of the confined fluids in micro/nanoscales. In this paper, considering the boundary slip at the fluid–solid interface, the motion property of fluids confined in parallel-plate nanochannels are investigated to couple the atomistic regime to continuum. The corrected second-order slip boundary condition is used to solve the Navier–Stokes equations for confined fluids. Molecular dynamics simulations for Poiseuille flows are performed to study the influences of the strength of the solid–fluid coupling, the fluid temperature, and the density of the solid wall on the velocity slip at the fluid boundary. For weak solid–fluid coupling strength, high temperature of the confined fluid and high density of the solid wall, the large velocity slip at the fluid boundary can be obviously observed. The effectiveness of the corrected second-order slip boundary condition is demonstrated by comparing the velocity profiles of Poiseuille flows from MD simulations with that from continuum.     

Scale effects in nano-channel liquid flows

Force-driven liquid argon flows both in nanoscale periodic domains and in gold nano-channels are simulated using non-equilibrium molecular dynamics to investigate the scale and wall force field effects. We examined variations in liquid density, viscosity, velocity profile, slip length, shear stress and mass flow rate in different sized periodic domains and nano-channels at a fixed thermodynamic state. In the absence of walls, liquid argon obeys Newton’s law of viscosity with the desired absolute viscosity in domains as small as 4 molecular diameters in height. Results prove that deviations from continuum solution are solely due to wall effects. Simulations in nano-channels with heights varying from 3.26 to 36 nm exhibit parabolic velocity profiles with constant slip length modeled by Navier-type slip boundary condition. Both channel averaged density and “apparent viscosity” decrease with reduced channel height, which has competing effects in determination of the mass flow rate. Density layering and wall force field induce deviations from Newton’s law of viscosity in the near-wall region, while constant “apparent viscosity” with the deformation rate from a parabolic velocity profile successfully predicts shear stress in the bulk flow region.     

Near-wall effects in micro scale Couette flow and heat transfer in the Maxwell-slip regimes

The effects of modified transport characteristics within an extremely thin layer adjacent to the fluid–solid interfaces are investigated for fully developed laminar micro-scale Couette flows with slip boundary conditions. The wall-adjacent layer effects are incorporated into the continuum-based mathematical model by imposing variable viscosity and thermal conductivity values close to the channel walls, for solving the momentum and energy conservation equations. Analytical expressions for the velocity profiles are derived and are subsequently utilized to obtain the temperature variations within the parallel plate channel, as a function of the significant system parameters. It is revealed that the variations in effective viscosity and thermal conductivity values within the wall-adjacent layer have profound influences on the fluid flow and the heat transfer characteristics within the channel, with an interesting interplay with the wall slip boundary conditions. These effects cannot otherwise be accurately captured by employing classical continuum based models for microscale Couette flows that do not take into account the alterations in effective transport properties within the wall adjacent layers.     

Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves

Superhydrophobic surfaces have been demonstrated to be capable of reducing fluid resistance in micro- and nanofluidic applications. The objective of this paper is to present analytical solutions for the Stokes flow through microchannels employing superhydrophobic surfaces with alternating micro-grooves and ribs. Results are presented for both cases where the micro-grooves are aligned parallel and perpendicular to the flow direction. The effects of patterning the grooves on one or both channel walls are also analyzed. The reduction in fluid resistance has been quantified in terms of a dimensionless effective slip length, which is found to increase monotonically with the shear-free fraction and the periodic extent of each groove–rib combination normalized by the channel half-height. Asymptotic relationships have been derived for the normalized effective slip length corresponding to large and small limiting values of the shear-free fraction and the normalized groove–rib period. A detailed comparison has been made between transverse and longitudinal grooves, patterned on one or both channel walls, to assess their effectiveness in terms of enhancing the effective slip length. These comparisons have been carried out for small and large limiting values, as well as finite values of the shear-free fraction and normalized groove–rib period. Results for the normalized effective slip length corresponding to transverse and longitudinal grooves are further applied to model the Stokes flow through microchannels employing superhydrophobic surfaces containing a periodic array of micro-grooves inclined at an angle to the direction of the applied pressure gradient. Results are presented for the normalized effective slip lengths parallel to the direction of the applied pressure gradient and the normalized cross flow rate perpendicular to the direction of the applied pressure gradient.     

On the influence of the wall shear stress vector form on hemodynamic indicators

Hemodynamic indicators such as the averaged wall shear stress (AWSS) and the oscillatory shear index (OSI) are well established to characterize areas of arterial walls with respect to the formation and progression of aneurysms. Here, we study two different forms for the wall shear stress vector from which AWSS and OSI are computed. One is commonly used as a generalization from the two-dimensional setting, the latter is derived from the full decomposition of the wall traction force given by the Cauchy stress tensor. We compare the influence of both approaches on hemodynamic indicators by numerical simulations under different computational settings. Namely, different (real and artificial) vessel geometries, and the influence of a physiological periodic inflow profile. The blood is modeled either as a Newtonian fluid or as a generalized Newtonian fluid with a shear rate dependent viscosity. Numerical results are obtained by using a stabilized finite element method. We observe profound differences in hemodynamic indicators computed by these two approaches, mainly at critical areas of the arterial wall.

     

Effects of shear rate and small periodic corrugation on the slip velocity in microscopic domain

We obtain analytically the slip velocity (up to the second order) of shear-thinning fluids inside the periodically corrugated microtube by using the verified fluid model and boundary perturbation method. Our results show that, even the slip length being zero, there is a slip velocity which is proportional to the small amplitude of periodic corrugation, the (referenced) shear rate, the applied forcing, and the orientation or the angle. Our results could be applied to the flow control in microfluidics as well as biofluidics.     

Numerical analysis on electroosmotic flows in a microchannel with rectangle-waved surface roughness using the Poisson–Nernst–Planck model

The present study has numerically investigated two-dimensional electroosmotic flows in a microchannel with dielectric walls of rectangle-waved surface roughness to understand the roughness effect. For the study, numerical simulations are performed by employing the Nernst–Planck equation for the ionic species and the Poisson equation for the electric potential, together with the traditional Navier–Stokes equation. Results show that the steady electroosmotic flow and ionic-species transport in a microscale channel are well predicted by the Poisson–Nernst–Planck model and depend significantly on the shape of surface roughness such as the amplitude and periodic length of wall wave. It is found that the fluid flows along the surface of waved wall without involving any flow separation because of the very strong normal component of EDL (electric double layer) electric field. The flow rate decreases exponentially with the amplitude of wall wave, whereas it increases linearly with the periodic length. It is mainly due to the fact that the external electric-potential distribution plays a crucial role in driving the electroosmotic flow through a microscale channel with surface roughness. Finally, the present results using the Poisson–Nernst–Planck model are compared with those using the traditional Poisson–Boltzmann model which may be valid in these scales.     

Fast nanofluidics by travelling surface waves

In this paper, we investigate the fast flow in nanochannels, which is induced by the travelling surface waves. The nanoscale fluid mechanism in nanochannels has been influenced by both amplitude and frequency of travelling surface waves, and the hydrodynamic characteristics have been obtained by molecular dynamics simulations. It has been found that the flow rate is an increasing function of the amplitude of travelling surface waves and can be up to a sevenfold increase. However, the flow rate is only enhanced in the limited range of frequency of travelling surface waves such as low frequencies, and a maximum fivefold increase in flow rate is pronounced. In addition, the fluid–wall interaction (surface wettability) plays an important role in the nanoscale transport phenomena, and the flow rate is significantly increased under a strong fluid–wall interaction (hydrophilicity) in the presence of travelling surface waves. Moreover, the friction coefficient on the wall of nanochannels is decreased obviously due to the large slip length, and the shear viscosity of fluid on the hydrophobic surface is increased by travelling surface waves. It can be concluded that the travelling surface wave has a potential function to facilitate the flow in nanochannels with respect to the decrease in surface friction on the walls. Our results allow to define better strategies for the fast nanofluidics by travelling surface waves.     

Analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with slip-dependent zeta potential

The present study is an analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with emphasis on the slip effects under coupling of interfacial electric and fluid slippage phenomena. Commonly used linear model with slip-independent zeta potential and the nonlinear model at limiting (high-K) condition with slip-dependent zeta potential are solved analytically. Then, numerical solutions of the electrokinetic flow model with zeta potential varying with slip length are analyzed. Different from the general notion of “the more hydrophobic the channel wall, the higher the flowrate,” the results with slip-independent and slip-dependent zeta potentials both disclose that flowrate becomes insensitive to the wall hydrophobicity or fluid slippage at sufficiently large slip lengths. Boundary slip not only assists fluid motion but also enhances counter-ions transport in EDL and, thus, results in strong streaming potential as well as electrokinetic retardation. With slip-dependent zeta potential considered, flowrate varies non-monotonically with increasing slip length due to competition of the favorable and adverse effects with more complicated interactions. The influence of the slip on the electrokinetic flow is eventually nullified at large slip lengths for balance of the counter effects, and the flowrate becomes insensitive to further hydrophobicity of the microchannel. The occurrence of maximum, minimum, and insensitivity on the flowrate-slip curves can be premature at a higher zeta potential and/or larger electrokinetic separation distance.     

Hydrodynamic boundary condition of polymer melts at simple and complex surfaces

Tailoring surface interactions or grafting of polymers onto surfaces is a versatile tool for controlling wettability, lubrication, adhesion, and interactions between surfaces. Many of those properties - e.g., excess free energy and friction at the surface - are dictated by the local structure. Using molecular dynamics simulation of a coarse-grained, bead-spring model, we study the equilibrium structure and near-surface flow of a polymer melt. Two prototypical surfaces are considered: (i) a hard substrate comprised of the first two layers of an FCC solid and (ii) a soft substrate that consists of a polymer brush. We show that the slip length strongly depends on temperature and surface structure. At high temperatures and low grafting densities, we find small slippage. At low temperatures, in the immediate vicinity of the glass transition temperature of the polymer melt, we observe very large slip lengths. At strongly attractive, hard substrates and polymer brushes of intermediate grafting density, we find that the Navier slip condition fails to describe Couette and Poiseuille flows simultaneously. This failure is rationalized within a schematic, two-layer model, which demonstrates that the failure of the Navier slip condition will occur if the fluid at the surface exhibits a higher viscosity than the bulk liquid.     

A slip model for micro/nano gas flows induced by body forces

A slip model for gas flows in micro/nano-channels induced by external body forces is derived based on Maxwell’s collision theory between gas molecules and the wall. The model modifies the relationship between slip velocity and velocity gradient at the walls by introducing a new parameter in addition to the classic Tangential Momentum Accommodation Coefficient. Three-dimensional Molecular Dynamics simulations of helium gas flows under uniform body force field between copper flat walls with different channel height are used to validate the model and to determine this new parameter.     

Magnetohydrodynamic flow with slippage in an annular duct for microfluidic applications

In this paper, we investigate theoretically the 3D laminar flow of an electrolyte in an annular duct driven by a Lorentz force. The duct is formed by two concentric electrically conducting cylinders limited by insulating bottom and top walls. A uniform magnetic field acts along the axial direction, while a potential difference is applied between the cylinders so that a radial electric current traverses the fluid. The interaction of the current and the magnetic field produces a Lorentz force that drives an azimuthal flow. The steady flow is solved using a Galerkin method with Bessel–Fourier series in the radial direction and trigonometric series along the vertical direction, allowing different combinations of slip conditions at the walls. The orthogonality of both series with the general boundary conditions of the third kind is used to find an analytic approximation. Velocity patterns and flow rates are explored by varying the aspect ratio of the duct and the gap between the cylinders, as well as the slippage at the walls. Results can provide useful information for optimization and design of annular microfluidic devices.     

Apparent slip arising from Stokes shear flow over a bidimensional patterned surface

A mathematical model is presented for the problem of apparent slip arising from Stokes shear flow over a composite surface featuring mixed boundary conditions on the microscale. The surface can be composed of a bidimensional array of solid areas placed on an otherwise no-shear surface corresponding to an envelope over the tops of posts, or no-shear areas placed on an otherwise solid surface corresponding to an envelope over the tops of holes. Posts and holes of circular or square cross section, and solid areas of no-slip or partial-slip types are studied. Following some previously proposed scaling laws, the effective slip length is expressed as a certain function of the solid fraction for some specific cases. More refined equations based on linear regression of the computed results are obtained for these cases. Amounts of slippage arising from these bidimensional patterns are compared with those from the one-dimensional patterns of grooves/grates. It is also shown that a larger slip length can result from an arrangement where the pitch is larger in the spanwise direction than in the streamwise direction.     

Computation of physiological bifurcation flows using a patched grid

A finite volume method in a boundary-fitted coordinate system together with a zonal grid method is employed to compute the flow field of a real-shape two-dimensional aortic bifurcation. The steady terms in the governing equations are treated by a fully explicit scheme. The zonal gridding procedure is discussed in detail. The numerical method is first tested in a laminar backward facing step flow to demonstrate the features of the method. The effect of the interface treatment on the flow fields can be significant. A 90° T-junction is then computed. The results are in good agreement with the available experimental data. The method is then applied to simulate the flows of an atherosclerotic human aorta. Both the steady and pulsatile flows are considered. It is shown that the mean shear stresses in recirculation regions of a pulsatile flow cannot be adequately described by a corresponding steady flow with a mean Reynolds number. In pulsatile flows, a sinusoidal input pulse and a realistic input pulse are both used in the computations. It is found that the “averaged” flow behavior is similar in both cases. However, the details of the flow field are significantly different. During pulsatile flow, permanent eddies are not present. That is, for a certain period in a cycle, the entire wall is free from eddies. On the other hand, in another period, the overall wall is almost completely in the reversing flow near the walls. These phenomena have been observed by other authors experimentally. Distributions of wall shear stresses and locations of recirculation zones in a realistic flow are shown and discussed briefly.