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.     

Lattice Boltzmann modeling of microchannel flows in the transition flow regime

Owing to its kinetic nature and distinctive computational features, the lattice Boltzmann method for simulating rarefied gas flows has attracted significant research interest in recent years. In this article, a lattice Boltzmann (LB) model is presented to study microchannel flows in the transition flow regime, which have gained much attention because of fundamental scientific issues and technological applications in various micro-electro-mechanical system (MEMS) devices. In the model, a Bosanquet-type effective viscosity is used to account for the rarefaction effect on gas viscosity. To match the introduced effective viscosity and to gain an accurate simulation, a modified second-order slip boundary condition with a new set of slip coefficients is proposed. Numerical investigations demonstrate that the results, including the velocity profile, the non-linear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the direct simulation Monte Carlo (DSMC) results, and the experimental results over a broad range of Knudsen numbers. It is shown that taking the rarefaction effect on gas viscosity into consideration and employing an appropriate slip boundary condition can lead to a significant improvement in the modeling of rarefied gas flows with moderate Knudsen numbers in the transition flow regime.     

Numerical simulation of gas flow in rough microchannels: hybrid kinetic–continuum approach versus Navier–Stokes

The flow field in a rough microchannel is numerically analyzed using a hybrid solver, dynamically coupling kinetic and Navier–Stokes solutions computed in rarefied and continuum subareas of the flow field, respectively, and a full Navier–Stokes solver. The rough surface is configured with triangular roughness elements, with a maximum relative roughness of 5 % of the channel height. The effects of Mach number, Knudsen number (or Reynolds number) and roughness height are investigated and discussed in terms of Poiseuille number and mass flow rate. Discrepancies between full Navier–Stokes and hybrid solutions are analyzed, assessing the range of validity of Navier–Stokes equations provided with first-order slip boundary conditions for modeling gas flow along a rough surface. Results indicate that the roughness increases Poiseuille number and decreases mass flux in comparison with those for the smooth microchannel. Increasing rarefaction results in further enhancement of roughness effect. At the same time, the compressibility effect is more noticeable than the roughness one, although the compressibility effect is alleviated by increase in the rarefaction. It was found that, although the Navier–Stokes solution of the flow in a smooth channel is accurate up to Kn = 0.1, when relative roughness height is higher than 1.25 % significant errors already appear at Kn = 0.02.     

Buoyancy effects on gaseous slip flow in a vertical rectangular microchannel

Consideration is given to the buoyancy effects on the fully developed gaseous slip flow in a vertical rectangular microduct. Two different cases of the thermal boundary conditions are considered, namely uniform temperature at two facing duct walls with different temperatures and adiabatic other walls (case A) and uniform heat flux at two walls and uniform temperature at other walls (case B). The rarefaction effects are treated using the first-order slip boundary conditions. By means of finite Fourier transform method, analytical solutions are obtained for the velocity and temperature distributions as well as the Poiseuille number. Furthermore, the threshold value of the mixed convection parameter to start the flow reversal is evaluated. The results show that the Poiseuille number of case A is an increasing function of the mixed convection parameter and a decreasing function of the channel aspect ratio, whereas its functionality on the Knudsen number is not monotonic. For case B, the Poiseuille number is decreased by increasing each of the mixed convection parameter, the Knudsen number, and the channel aspect ratio.     

Thermohydrodynamic analysis of micro spherical spiral groove gas bearings under slip flow and surface roughness coupling effect

The effects of gas rarefaction and surface roughness are temperature-related and always neglected in macroscale. These effects were considered in the analysis of the thermohydrodynamic performance of micro spherical spiral groove bearings. The Reynolds equation and the energy equation incorporated with Wu’s slip model and the Weierstrass–Mandelbrot function to analysis the coupling effect of slip flow and surface roughness. The effects of spherical grooves on computational accuracy were reduced through parameter transformation and oblique coordinates. To describe the temperature boundary condition at the grooved region, a simple gas-mixing model was presented for the grooves. Prediction results showed that temperature reduced bearing forces and friction torque through the slip flow effect. Surface roughness increased not only temperature significantly but also bearing forces and friction torque through a high eccentricity ratio and a low bearing clearance.

     

Analytical solution of mixed electromagnetic/pressure driven gaseous flows in microchannels

The influences of wall-slip/jump conditions on the fluid flow and heat transfer for hydrodynamically and thermally fully developed electrically conducting gaseous flow subject to an electromagnetic field inside a parallel plate microchannel with constant heat flux at walls are studied under the assumptions of a low-magnetic Reynolds number. The governing equations are non-dimensionalized and then analytical solutions are derived for the friction and the heat transfer coefficients. The fluid flow and the heat transfer characteristics obtained in the analytical solutions are discussed in detail for different parameters such as the Knudsen, Hartmann, and Brinkman numbers. The velocity profiles verify that even with a constant Knudsen number, applying a stronger electromagnetic field gives rise to an increase in the slip velocity. The results also reveal that on increasing the Hartmann number, the heat transfer rate as well as the friction factor is enhanced, whereas it tends to suppress the movement of the fluid. Further, it is found that the Nusselt and the Poiseuille numbers are less sensitive to the electromagnetic field effects with increase in rarefaction.     

Gaseous slip flow forced convection through ordered microcylinders

This is a theoretical study dealing with longitudinal gaseous slip flow forced convection between a periodic bunch of microcylinders arranged in regular array. The selected geometry has applications in microscale pin fin heat sinks used for cooling of microchips. The flow is considered to be hydrodynamically and thermally fully developed. The two axially constant heat flux boundary conditions of H1 and H2 are considered in the analysis. The velocity and temperature discontinuities at the boundary are incorporated into the solutions using the first order slip boundary conditions. The method considered is mainly analytical in which the governing equations and three of the boundary conditions are exactly satisfied. The remaining symmetry condition on the right-hand boundary of the typical element is applied to the solution through the point matching technique. The results show that both the Poiseuille number and the Nusselt number are decreasing functions of the degree of rarefaction characterized by the Knudsen number. While an increase in the blockage ratio leads to a higher Poiseuille number, the functionality of the Nusselt number on this parameter is not monotonic. At small and moderate values of the blockage ratio, the Nusselt number is higher for a higher blockage ratio, whereas the opposite may be right for higher values of this parameter. It is also observed that the angular variations of the parameters are reduced at smaller blockage ratios. Accordingly, the H1 and H2 Nusselt numbers are the same for small and moderate blockage ratios.     

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.     

Theoretical analysis of the frictional losses in magnetohydrodynamic microflows considering slippage

In this article, we study the frictional losses in magnetohydrodynamic (MHD) microflows by analyzing the Poiseuille number defined through the Darcy–Weisbach friction factor. We consider two-dimensional fully developed flow models characteristic of MHD micropumps including the Hartmann braking effect and the existence of slippage. Unlike the purely hydrodynamic case, in MHD flows the Poiseuille number depends not only on the aspect ratio but also on the physical properties of the fluid and the externally applied magnetic field. Three different combinations of boundary conditions (slip and no-slip) are investigated. Calculations show that the Poiseuille number is considerably reduced as the dimensionless slip length is increased, while it increases as Hartmann number does. The obtained results are consistent with previous models and are helpful for the design of magnetohydrodynamic microflow devices.

     

A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows

A lattice Boltzmann model for simulating isothermal micro flows has been proposed by us recently [Niu XD, Chew YT, Shu C. A lattice Boltzmann BGK model for simulation of micro flows. Europhys Lett 2004;67(4):600]. In this paper, we extend the model to simulate the micro thermal flows. In particular, the thermal lattice Boltzmann equation (TLBE) [He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmann method in incompressible limit. J Comput Phys 1998;146:282] is used with modification of the relaxation times linking to the Knudsen number. The diffuse scattering boundary condition (DSBC) derived in our early model is extended to consider temperature jump at wall boundaries. Simple theoretical analyses of the DSBC are presented and the results are found to be consistent with the conventional velocity slip and temperature jump boundary conditions. Numerical validations are carried out by simulating two-dimensional thermal Couette flows and developing thermal flows in a microchannel, and the obtained results are found to be in good agreement with those given from the direct simulation Monte Carlo (DSMC), the molecular dynamics (MD) approaches and the Maxwell theoretical prediction.     

Two-dimensional molecular dynamic simulations on accommodation coefficients in nanochannels with various wall configurations

The accommodation coefficients are often utilized in slip boundary conditions to characterize gas-wall interactions. Due to the insufficient transport of momentum and energy in nanochannels, the accommodation coefficients are always less than unity and greatly influenced by temperature and surface structures. In the present paper, a statistical algorithm of the accommodation coefficients was described in molecular dynamic method. The accommodation coefficients were calculated for various wall configurations in two-dimensional nanochannels. The channels were constituted by several layers of platinum atoms, which vibrated and attached to face centered cubic (FCC) lattice sites. The results revealed that the NMAC and EAC are sensitive with the spring constant and wall atom layers. Subtle distinctions in FCC lattice and nanoscale roughness had strong effects. On FCC (1 1 1) lattice plane, the TMAC in isothermal flows was larger, while the NMAC and EAC in thermal conductions are smaller, than those on FCC (1 1 0) lattice plane. Moreover, larger roughness induced more normal momentum transferred into tangential momentum so that the NMAC decreases while the TMAC and EAC increases for larger roughness. In addition, the accommodation coefficients are also affected by rarefaction that the TMAC and EAC decrease as the Knudsen number increases.     

A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method

This work proposed a unified approach to impose both nonslip and slip boundary conditions for the lattice Boltzmann method (LBM). By introducing the tangential momentum accommodation coefficient (TMAC), the present implementation can determine the change of the tangential momentum on the wall and then impose the correct boundary conditions for LBM. The simulation results demonstrate that this implementation is equivalent to the first-order slip model.     

A lattice Boltzmann approach for the non-Newtonian effect in the blood flow

A new lattice Boltzmann approach within the framework of D2Q9 lattice for simulating shear-thinning non-Newtonian blood flows described by the power-law, Carreau-Yasuda and Casson rheology models is proposed in this study. The essence of this method lies in splitting the complete non-Newtonian effect up into two portions: one as the Newtonian result and the other as an effective external source. This arrangement takes the advantage in remaining fixed relaxation time during the whole course of numerical simulation that can avoid the potential numerical instability caused by the relaxation time approaches to 1/2, an inherent difficulty in the conventional lattice Boltzmann methods using varying relaxation times for the non-Newtonian effect. Macroscopically, consistency of the proposed model with the equations of motion for the three target non-Newtonian models is demonstrated through the technique of Chapman-Enskog multi-scale expansion. The feasibility and accuracy of the method are examined by comparing with the analytical solutions of the two-dimensional Poiseuille flows based on the power-law and Casson models. The results show that the velocity profiles agree very well with those of analytical solutions and the error analyses demonstrate that the proposed scheme is with second-order accuracy. The present approach also demonstrates its superiority over the conventional lattice Boltzmann method in the extent of numerical stability for simulating the power-law-based shear-thinning flows. The straightforwardness in scheme derivation and implementation renders the present approach as a potential method for the complex non-Newtonian flows.     

Direct numerical simulation of rarefied turbulent microchannel flow

Direct numerical simulation (DNS) has been carried out to investigate the effect of weak rarefaction on turbulent gas flow and heat transfer characteristics in microchannel. The Reynolds number based on the friction velocity and the channel half width is 150. Grid number is 64 × 128 × 64. Fractional time-step method is employed for the unsteady Navier–Stokes equations, and the governing equations are discretized with finite difference method. Statistical quantities such as turbulent intensity, Reynolds shear stress, turbulent heat flux and temperature variance are obtained under various Knudsen number from 0 to 0.05. The results show that rarefaction can influence the turbulent flow and heat transfer statistics. The streamwise mean velocity and temperature increase with increase of Kn number. In the near-wall-region rarefaction can increase the turbulent intensities and temperature variance. The effects of rarefaction on Reynolds shear stress and wall-normal heat flux are presented. The instantaneous velocity fluctuations in the vicinity of the wall are visualized and the influence of Kn number on the flow structure is discussed.     

Couette–Poiseuille flow of a gas in long microchannels

The flow of a compressible, isothermal gas under slightly rarefied conditions in a 2D planar geometry is considered. The gas is shear driven and is also subject to an applied pressure gradient, which is also known as Couette–Poiseuille (CP) flow. In this paper, the full Navier-Stokes (NS) equations are solved using a perturbation expansion up to the first order. The pressure profile is solved numerically. On the basis of the solutions, effects of rarefaction and compressibility on the flow characteristics are investigated in detail. The results show the parallel flow assumption to be invalid for cases with slight rarefaction. The axial and vertical velocity components are found to depend on the degree of rarefaction, applied pressure gradient and wall velocity. The effects of rarefaction on the occurrence of back flow are also discussed. In addition, the results for the Poiseuille and CP flow with and without rarefaction can be easily obtained from our results.     

A lattice Boltzmann study of gas flows in a long micro-channel

In this paper, the pressure-driven flow in a long micro-channel is studied via a lattice Boltzmann equation (LBE) method. With the inclusion of the gas–wall collision effects, the LBE is able to capture the flow behaviors in the transition regime. The numerical results are compared with available data of other methods. Furthermore, the effects of rarefaction and compressibility on the deviation of the pressure distribution from the linear one are also investigated.     

Pressure condition for lattice Boltzmann methods on domains with curved boundaries

We propose a lattice Boltzmann algorithm for an average pressure boundary condition at outlets in pipe flow systems. The advantage of this boundary condition is that only the average pressure is used to recover the non-trivial flow fields. The asymptotic analysis shows that this algorithm works for general curved boundaries and renders a second order accurate velocity and a first order accurate pressure approximation of the incompressible Navier–Stokes solution. Here, we verify the accuracy by numerical simulations of a Poiseuille flow and a less symmetric flow with non-trivial pressure field in channels inclined with arbitrary angle, and flows in a pipe with three outlets.     

Thermal boundary conditions for thermal lattice Boltzmann simulations

Consistent 2D and 3D thermal boundary conditions for thermal lattice Boltzmann simulations are proposed. The boundary unknown energy distribution functions are made functions of known energy distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definition of internal energy density. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. The discrete macroscopic energy equation is also derived for a steady and fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous energy distribution across the boundary, provided the boundary unknown energy distribution functions satisfy the macroscopic energy level. Four different local known energy distribution functions are experimented with to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating the 2D thermal Poiseuille flow, thermal Couette flow, thermal Couette flow with wall injection, natural convection in a square cavity, and 3D thermal Poiseuille flow in a square duct. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known energy distribution functions is, as expected, negligible, which are consistent with the results from the derived discrete macroscopic energy equation.     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

The effects of wall roughness on the methane flow in nano-channels using non-equilibrium multiscale molecular dynamics simulation

This paper presents a non-equilibrium multiscale molecular dynamics simulation method to investigate the effects of periodic wall surface roughness on the structure and mass transfer of methane fluid through the silicon nano-channels. In order to accurately capture the trajectories and microstructure of methane nano-fluidics, the present modification of OPLS fully atomic model is employed. Meanwhile, we introduce the corresponding coarse-grained model to solve the problem of wall–fluid interaction for methane Poiseuille flow within silicon atomic walls using the classical Lorentz–Berthelot mixing rules. The geometries of the upper wall roughness are modeled by rectangular waves with different amplitudes and wavelengths. The three-dimensional number densities of C (H) atom and kinetic energy distribution plots give a clear observation of the impacts of surface roughness on the localization micro-information of methane fluid. Moreover, the slip length of fluid over rough surface decreases with the increase in amplitude. The diffusion coefficients appear anisotropic, and the radial distribution functions decrease with the increase in the amplitude. These properties should be taken into account in the design of energy-saving emission reduction nano-fluidic devices. All numerical results also indicate that the presented method not only can well solve the issue of wall–fluid interactions, but also could accurately predict the micro-information and dynamic properties of methane Poiseuille flow.