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.     

Improved modified Reynolds equation for thin-film gas lubrication from an extended slip velocity boundary condition

An extended slip velocity boundary condition is derived from the regularized 13 moment equations firstly. Different from the existing slip velocity boundary condition, the slip coefficients of the extended one are not fixed, which will change with the wall accommodation coefficient and the Knudsen number of the gas flow. Using the extended slip velocity condition, an improved modified Reynolds equation for thin-film gas lubrication is established. From solving the improved modified Reynolds equation, the pressure distribution of the slider gas bearing is obtained and has a better agreement with that from the direct simulation Monte Carlo method under different pitch angles and wall velocities. It is found that the improved modified Reynolds equation can predict a more accurate pressure distribution of the slider gas bearing than the Fukui and Kaneko’s lubrication model from the linearized Boltzmann equation in the near transition regime.     

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.     

Electromechanical Model of Electrically Actuated Narrow Microbeams

A consistent one-dimensional distributed electromechanical model of an electrically actuated narrow microbeam with width/height between 0.5–2.0 is derived, and the needed pull-in parameters are extracted with different methods. The model accounts for the position-dependent electrostatic loading, the fringing field effects due to both the finite width and the finite thickness of a microbeam, the mid-plane stretching, the mechanical distributed stiffness, and the residual axial load. Both clamped–clamped and clamped-free (cantilever) microbeams are considered. The method of moments is used to estimate the electrostatic load. The resulting nonlinear fourth-order differential equation under appropriate boundary conditions is solved by two methods. Initially, a one-degree-of-freedom model is proposed to find an approximate solution of the problem. Subsequently, the meshless local Petrov–Galerkin (MLPG) and the finite-element (FE) methods are used, and results from the three methods are compared. For the MLPG method, the kinematic boundary conditions are enforced by introducing a set of Lagrange multipliers, and the trial and the test functions are constructed using the generalized moving least-squares approximation. The nonlinear system of algebraic equations arising from the MLPG and the FE methods are solved by using the displacement iteration pull-in extraction (DIPIE) algorithm. Three-dimensional FE simulations of narrow cantilever and clamped–clamped microbeams are also performed with the commercial code ANSYS. Furthermore, computed results are compared with those arising from other distributed models available in the literature, and it is shown that improper fringing fields give inaccurate estimations of the pull-in voltages and of the pull-in deflections. 1641     

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.     

Multi-scale study of liquid flow in micro/nanochannels: effects of surface wettability and topology

The paper reports parametric study, using a molecular dynamics–continuum hybrid simulation method, of liquid flow in micro/nanochannels with surface nanostructures. The effects of channel height, shape of roughening element, ratio of pitch to length of roughening element and liquid–solid bonding strengths (representing surface wettability) on the velocity and temperature boundary conditions are investigated. The velocity boundary condition is found to shift from significant slip to locking due to the blocking of the surface nanostructure. The blocking appears weak for small pitch ratio and weak liquid–solid bonding. Distorted streamlines, small random eddies and appreciable density oscillations are seen in the vicinity of the wall for small pitch ratio and strong liquid–solid bonding. On the other hand, smooth streamlines and weak density oscillations are seen for large pitch ratio and weak liquid–solid bonding. Results also reveals that: relative slip length, relative Kapitza length and minus pressure gradient vary with channel height and pitch ratio in functions of power law and approximately linear, respectively; relative slip and Kapitza lengths vary with liquid–solid bonding strength as approximately decreasing power functions (except for the strongest case), whereas minus pressure gradient varies with liquid–solid bonding strength as approximately a logarithm-like function. The effect of shape of roughening element is found to be much less significant compared with the other factors studied.     

A parallel 3D Poisson solver for space charge simulation in cylindrical coordinates

This paper presents the development of a parallel three-dimensional Poisson solver in cylindrical coordinate system for the electrostatic potential of a charged particle beam in a circular tube. The Poisson solver uses Fourier expansions in the longitudinal and azimuthal directions, and Spectral Element discretization in the radial direction. A Dirichlet boundary condition is used on the cylinder wall, a natural boundary condition is used on the cylinder axis and a Dirichlet or periodic boundary condition is used in the longitudinal direction. A parallel 2D domain decomposition was implemented in the (r,θ) plane. This solver was incorporated into the parallel code PTRACK for beam dynamics simulations. Detailed benchmark results for the parallel solver and a beam dynamics simulation in a high-intensity proton LINAC are presented. When the transverse beam size is small relative to the aperture of the accelerator line, using the Poisson solver in a Cartesian coordinate system and a Cylindrical coordinate system produced similar results. When the transverse beam size is large or beam center located off-axis, the result from Poisson solver in Cartesian coordinate system is not accurate because different boundary condition used. While using the new solver, we can apply circular boundary condition easily and accurately for beam dynamic simulations in accelerator devices.     

Analytical solution of micro-/nanoscale convective liquid flows in tubes and slits

Analytical solutions examining heat transport in micro-/nanoscale liquid flows were developed. Using the energy equation coupled with fully developed velocity, we solved developing temperature profiles with axial conduction and viscous dissipation terms. A comprehensive literature review provided the published range of velocity slip and temperature jump conditions. While molecular simulations and experiments present constant slip and jump values for a specific liquid/surface couple independent of confinement size, non-dimensional forms of these boundary conditions were found appropriate to calculate non-equilibrium as a function of flow height. Although slip and jump conditions are specific for each liquid/surface couple and hard to obtain, we proposed modeling of the slip and jump as a function of the surface wetting, in order to create a general, easy to measure methodology. We further developed possible correlations to calculate jump using the slip value of the corresponding surface and tested in the results. Fully developed Nu showed strong dependence on slip and jump. Heat transfer stopped when slip and jump coefficients became higher than a certain value. Strong variation of Nu in the thermal development length was observed for low slip and jump cases, while an almost constant Nu in the flow direction was found for high slip and jump coefficients. Variation of temperature profiles was found to dominate the heat transfer through the constant temperature surface while surface and liquid temperatures became equal at heat transfer lengths comparable with confinement sizes for no-dissipation cases. In case of non-negligible heat dissipation, viscous heating dominated the Nu value by enhancing the heating while decreasing the heat removal in cooling cases. Implementation of proposed procedure on a micro-channel convection problem from a micro-fluidics application showed the dominant effect of the model defining the slip and jump relationship. Direct use of kinetic gas theory resulted in an increase of Nu by an increase in non-equilibrium, while models developed from published liquid slip and jump values produced an opposite behavior.     

Effect of human link length determination on posture reconstruction

Motion capture experiment results are often used as a means of validation for digital human simulations. Motion capture results are marker positions and joint centers in Cartesian space. However, joint angles are more intuitive and easy to understand compared to marker or joint center positions. Posture reconstruction algorithms are used to map Cartesian space to joint space by re-creating experimental postures with simulation models. This allows for direct comparison between the experimental results and digital human simulations. Besides the inherent experimental errors from motion capture system, one source of simulation error is the determination of the link lengths to be used in the simulation model. The link length errors can propagate through all simulation results. Therefore, it is critical to eliminate the link length errors. The objective of this study is to determine the best method of determining link lengths for the simulation model to best match the model to the experiment results containing errors. Specifically, the way that the link lengths are calculated in the posture reconstruction process from motion capture data has a significant effect on the recreated posture for the simulation model. Three link length calculation methods (experimental-average method, trial-specific method, and T-pose method) are developed and compared to a benchmark method (frame-specific method) for calculating link lengths. The results indicate that using the trial-specific method is the most accurate method when referring to calculating frame-specific link lengths.     

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.     

Double-disk rotating viscous micro-pump with slip flow

In this paper numerical solution was provided for the 2D, axisymmetric Navier-Stokes equations coupled with energy equation for gaseous slip flow between two micro rotating disks pump. A first-order slip boundary condition was applied to all internal solid walls. The objective is to study the effect of Knudsen number, rotational Reynolds number and gap height on pump head, flow rate, coefficient of moments and overall micro-pump efficiency. Pump head, flow rate, coefficient of moments and pump efficiency were calculated for various pump operating conditions when the mass flow rate is applied at the pump inlet port. Detailed investigations were performed for rotational Reynolds number equals to 10. Effect of gap height between the two disks was studied. Effect of rotational Reynolds number on maximum flow rate and maximum pressure rise was simulated. The present numerical results for no-slip were compared with previously published experimental and theoretical data and found to be in a very good agreement. Knudsen number Kn values were found to be major parameters that affect the performance of pump. Pump performance decreases with increasing Kn. Optimal pump performance occurs around middle point of pump operating range. Pump operating range decreases with increasing Kn numbers. Pump performance is found to experience a steep degradation for Kn approaching 0.1. Maximum flow rate increases with rotational speed almost linearly. Maximum pressure rise also increases with rotational speed. Reducing gap height results in increasing maximum pressure rise, while increasing gap height results in larger maximum flow rate.     

Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels

This paper significantly extends previous studies to the transition regime by employing the second-order slip boundary conditions. A simple analytical model with second-order slip boundary conditions for a normalized Poiseuille number is proposed. The model can be applied to either rarefied gas flows or apparent liquid slip flows. The developed simple models can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels and nanochannels by the research community for the practical engineering design of microchannels and nanochannels. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier–Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as the Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given.     

Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model

This article investigates vibration and instability analysis of a single-walled carbon nanotube (SWCNT) conveying viscous fluid flow. For this purpose, the first-order shear deformation shell model is developed in the framework of nonlocal strain gradient theory (NSGT) for the first time. The proposed model is a conveying viscous fluid in which the external force of fluid flow is applied by the modified Navier–Stokes relation and considering slip boundary condition and Knudsen number. The NSGT can be reduced to the nonlocal elasticity theory, strain gradient theory or the classical elasticity theory by inserting their specific nonlocal parameters and material length scale parameters into the governing equations. Comparison of above-mentioned theories suggests that the NSGT predicts the greatest critical fluid flow velocity and stability region. The governing equations of motion and corresponding boundary conditions are discretized using the generalized differential quadrature method. Furthermore, the effects of the material length scale, nonlocal parameter, Winkler elastic foundation and Pasternak elastic foundation on vibration behavior and instability of a SWCNT conveying viscous fluid flow with simply supported and clamped–clamped boundary conditions are investigated.     

Theoretical modeling of microfabricated beams with elasticallyrestrained supports

A mathematical model is developed to analyze the mechanical behavior of step-up supports which typically result from surface micromachining. This model accounts for the finite stiffness of the step-up support, which is more accurately represented by an elastically restrained boundary condition rather than a fixed or built-in boundary condition. Based on the model developed, the deflection of cantilever and doubly-supported beams under arbitrarily located concentrated and distributed loads is determined for a wide range of beam geometries. In addition, the buckling load of doubly-supported beams with elastically restrained boundary conditions is analyzed. Finite element analysis verifies the accuracy of the models developed. The models show that significant error in predicting the beam mechanical behavior may result if the finite stiffness of the step-up support is not correctly modeled     

A two-dimensional lattice Boltzmann model for uniform channel flows

In this paper a novel two-dimensional lattice Boltzmann model (LBM) is developed for uniform channel flows. The axial velocity is solved from a momentum diffusion equation over the cross-sectional plane. An extrapolation boundary condition is also introduced to enhance the no-slip boundary in the momentum equation. This boundary treatment can also be applied to LBM simulations of other diffusion processes. The algorithm and boundary treatment are validated by simulations of steady Poiseuille and pulsatile Womersley flows in circular pipes. The numerical convergence and accuracy are comparable to those of existing models. Moreover, comparison with general three-dimensional lattice Boltzmann simulations demonstrates the advantages of our two-dimensional model, including lower computational resource requirements (memory and time), easier boundary treatment for arbitrary cross-sectional shapes, and no velocity constraint. These features are attractive for practical applications with uniform channel flows.     

Molecular dynamics simulation of nanoscale liquid flows

Molecular dynamics (MD) simulation is a powerful tool to investigate the nanoscale fluid flow. In this article, we review the methods and the applications of MD simulation in liquid flows in nanochannels. For pressure-driven flows, we focus on the fundamental research and the rationality of the model hypotheses. For electrokinetic-driven flows and the thermal-driven flows, we concentrate on the principle of generating liquid motion. The slip boundary condition is one of the marked differences between the macro- and micro-scale flows and the nanoscale flows. In this article, we review the parameters controlling the degree of boundary slip and the new findings. MD simulation is based on the Newton's second law to simulate the particles' interactions and consists of several important processing methods, such as the thermal wall model, the cut-off radius, and the initial condition. Therefore, we also reviewed the recent improvement in these key methods to make the MD simulation more rational and efficient. Finally, we summarized the important discoveries in this research field and proposed some worthwhile future research directions.     

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.     

Robust adaptive boundary control of a perturbed hybrid Euler-Bernoulli beam with coupled rigid and flexible motion

In this paper, a robust adaptive boundary controller is proposed to stabilize the coupled rigid-flexible motion of an Euler-Bernoulli beam in presence of boundary and distributed perturbations. Applying Hamilton’s principle, the dynamics of the hybrid beam model, including the actuators hub and the payload at its ends, is represented through four nonhomogeneous nonlinear partial differential equations (PDEs) subject to ordinary differential equations (ODEs) of boundary conditions. Using a Lyapunov-based control synthesis procedure, a robust nonlinear boundary controller is established that asymptotically stabilizes the perturbed beam vibration while regulating the rigid motion coordinates. A redesign of the proposed control laws produces a robust adaptive boundary controller that achieves control objectives in the presence of both parametric and modelling uncertainties. Control design is directly based on system PDEs without truncating the model so that instabilities from spillover effects are mitigated. The control inputs to the beam consist of three forces/torque applied to the actuators hub and a transverse force applied to the tip payload. Simulation results are used to investigate the efficiency of the proposed approach.

     

Generating efficient dynamical models for microelectromechanicalsystems from a few finite-element simulation runs

In this paper, we demonstrate how efficient low-order dynamical models for micromechanical devices can be constructed using data from a few runs of fully meshed but slow numerical models such as those created by the finite-element method (FEM). These reduced-order macromodels are generated by extracting global basis functions from the fully meshed model runs in order to parameterize solutions with far fewer degrees of freedom. The macromodels may be used for subsequent simulations of the time-dependent behavior of nonlinear devices in order to rapidly explore the design space of the device. As an example, the method is used to capture the behavior of a pressure sensor based on the pull-in time of an electrostatically actuated microbeam, including the effects of squeeze-film damping due to ambient air under the beam. Results show that the reduced-order model decreases simulation time by at least a factor of 37 with less than 2% error. More complicated simulation problems show significantly higher speedup factors. The simulations also show good agreement with experimental data     

Fluid Effects in Vibrating Micromachined Structures

Squeeze film damping and hydrodynamic lift for a micromechanical perforated proof mass are calculated and measured. This paper has resulted in closed-form expressions that can be used to design accelerometers, tuning-fork gyroscopes (TFGs), and other micromechanical devices. The fluid damping and lift are determined using finite-element analyses of the normalized and linearized governing equations where the boundary condition of the pressure relief holes is derived using pipe flow analysis. The rarefaction of gas is incorporated in the governing equations based on slip flow condition. As a further check, a one-dimensional (1-D) network model is developed to account for the boundary condition of the holes on a tilted proof mass. Both closed-form and numerical solutions are compared against experimental data over a range of pressures.hfillhbox[1221]