Molecular dynamics–continuum hybrid simulation for condensation of gas flow in a microchannel

A molecular dynamics-continuum coupling method combining fluid flow and heat transfer is developed to study the condensation process of gas flow in a microchannel. The computational domain is decomposed into particle (P), continuum (C) and overlap (O) regions with solving approaches of molecular dynamics simulation, finite volume method and the developed coupling method, respectively. Continuities of momentum and energy in O region are ensured by constraint dynamics and the Langevin method. The validity of the developed method is confirmed by a good agreement between hybrid results and analytical solutions from two cases including the unsteady dynamical and thermal problems. For the condensation process of gas flow, the hybrid transient velocity and temperature fields indicate that the process does not progress smoothly but wavily with noticeable fluctuation, leading to oscillation in temperature field and recirculation flow in velocity field. Analysis based on heat and mass transfer is carried out in P region, and the Kapitza resistance and the thermal conductivity in liquid are obtained with the satisfying agreement with experimental data, which shows the availability of the developed model for the investigation on the thermal boundary resistance. The good performance had demonstrated that the developed coupling method and computational model are available to provide a multiscale overview in dynamical and thermal problems including phase-transition from nanoscale to microscale, which will show significantly potential in micro fluidics and thermal engineering.     

Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.     

Hydrodynamics of gas–liquid Taylor flow in rectangular microchannels

The effect of fluid properties and operating conditions on the generation of gas–liquid Taylor flow in microchannels has been investigated experimentally and numerically. Visualisation experiments and 2D numerical simulations have been performed to study bubble and slug lengths, liquid film hold-up and bubble velocities. The results show that the bubble and slug lengths increase as a function of the gas and liquid flow rate ratios. The bubble and slug lengths follow the model developed by Garstecki et?al. (Lab chip 6:437–446, 2006) and van Steijn et?al. (Chem Eng Sci 62:7505–7514, 2007), however, the model coefficients appear to be dependent on the liquid properties and flow conditions in some cases. The ratio of the bubble velocity to superficial two-phase velocity is close to unity, which confirms a thin liquid film under the assumption of a stagnant liquid film. Numerical simulations confirm the hypothesis of a stagnant liquid film and provide information on the thickness of the liquid film.     

Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels

In this paper we present a model for the calculation of pressure drop of three-phase liquid–liquid–gas slug flow in microcapillaries of a circular cross section. Introduced models consist of terms attributing for frictional and interfacial pressure drop, incorporating the presence of a stagnant thin film at the wall of the channel. Different formulations of the interfacial pressure drop equation were employed, using expressions developed by Bretherton (J Fluid Mech 10:166–188, 1961), Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989). Models were validated experimentally using oleic acid–water–nitrogen and heptane–water–nitrogen three-phase flows in round Teflon or Radel R microchannels of 254- and 508-µm nominal inner diameter, for capillary numbers Ca _b between 10^?4 and 4.9 × 10^?1 and Reynolds numbers Re between 0.095 and 300. Best agreement between measured and calculated values of pressure drop, with relative error between ?22 and 19 % or ?20 and 16 %, is reached for Warnier’s or Ratulowski and Chang’s interfacial pressure drop equation, respectively. The results prove that three-phase slug flow pressure drop can be successfully predicted by extending existing two-phase slug flow correlations. Good agreement of Bretherton’s equation was reached only at lower Ca numbers, indicating that an extension of the interfacial pressure drop equation as performed by Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989) for higher capillary numbers is necessary. Additionally it was demonstrated that pressure drop increases substantially if dry slug flow occurs or if microchannels with significant surface roughness are employed. Those influences were not accounted for in the models presented.     

Numerical simulation of mass transfer in gas–liquid hollow fiber membrane contactors for laminar flow conditions

This study presents a numerical simulation using computational fluid dynamics (CFD) of momentum and mass transfer in a hollow fiber membrane contactor for laminar flow conditions. Axial and radial diffusion inside the fiber, through the membrane, and within the shell side of the membrane contactor were considered in the mass transfer equations. The simulation results were compared with the experimental data obtained from literature for CO₂ absorption in pure water. The simulation results indicated that the removal of CO₂ increased with increasing liquid flow rate in the shell side. On the other hand, increasing temperature and gas flow rate in the tube side have an opposite effect.     

Numerical simulations of solutions of a two-level averaged Navier–Stokes system

An averaging procedure for the Navier–Stokes equations has been proposed in an earlier article [I. Moise, R.M. Temam, Renormalization group method. Application to Navier–Stokes Equation, Discrete Contin. Dyn. Syst. 6 (1) (2000) 191–210]. This averaging procedure is based on a two-level decomposition of the solution into low and high frequencies. The aim of the present article is to investigate, with the help of numerical simulations, the behavior of the small scales of the corresponding system. Space-periodic solutions with a non-resonant period are considered. The time evolution of the averaged and standard (non-averaged) small scales are compared at different Reynolds numbers and for different values of the cut-off level used to separate large and small scales of the flow variables. The numerical results illustrate the efficiency of the proposed averaging procedure for the Navier–Stokes equations. The averaged small scales provide an accurate prediction of the time-averaged small scales of the Navier–Stokes solutions. As the computational cost is reduced for the averaged equations, long time integrations on more than 50 eddy-turnover times have been performed for cut-off levels ensuring a proper resolution of the large scales. In these cases, development of instabilities in the averaged small scale equation is observed.     

Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

On the Numerical Controllability of the Two-Dimensional Heat,Stokes and Navier–Stokes Equations

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov–Imanuvilov formulation, see Fursikov and Imanuvilov (Controllability of Evolutions Equations, Lectures Notes Series, vol 34, Seoul National University, 1996); this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables \(x_1\), \(x_2\) and t that is second-order in time and fourth-order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.     

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.     

Numerical solution of the incompressible Navier–Stokes equations with exponential type schemes

A new exponential type finite-difference scheme of second-order accuracy for solving the unsteady incompressible Navier–Stokes equation is presented. The driven flow in a square cavity is used as the model problem. Numerical results for various Reynolds numbers are given, and are in good agreement with those presented by Ghia et al. (Ghia, U., Ghia, K.N. and Shin, C.T., 1982, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multi-grid method. Journal of Computational Physics, 48, 387–411.).     

Investigation of multiscale fluid flow characteristics based on a hybrid atomistic–continuum method

A computer program based on a molecular dynamics–continuum hybrid method has been developed in which the Navier–Stokes equations are solved in the continuum region and the molecular dynamics in the atomistic region. The coupling between the atomistic and continuum is constructed through constrained dynamics within an overlap region where both molecular and continuum equations are solved simultaneously. The simulation geometries are solved in three dimensions and an overlap region is introduced in two directions to improve the choice of using the molecular region in smaller areas. The proposed method is used to simulate steady and start-up Couette flow showing quantitative agreement with results from analytical solutions and full molecular dynamics simulations. The prepared algorithm and the computer code are capable of modeling fluid flows in micro and nano-scale geometries.     

CFD simulation of gas–liquid flow behaviour in an air-lift reactor: determination of the optimum distance of the draft tube

CFD simulation in an air-lift reactor containing a draft tube was employed. Three different layouts between the draft tube and the wall were used to determine the optimum distance between them. Appropriate distance of the draft tube and the wall was determined for the designing purposes, and the best distance was calculated. The simulation results were compared with the experimental results of the Menzel et al. Also it was also proven that the optimum distance results in the best possible effective mixing, higher rotary movements of the liquid and gas, and consequently improves the reactor performance.     

Detection of phishing attacks in Iranian e-banking using a fuzzy–rough hybrid system

Phishing is a method of stealing electronic identity in which social engineering and website forging methods are used in order to mislead users and reveal confidential information having economic value. Destroying the trust between users in business network, phishing has a negative effect on the budding area of e-commerce. Developing countries such as Iran have been recently facing Internet threats like phishing, whose methods, regarding the social differences, may be different from other experiences. Thus, it is necessary to design a suitable detection method for these deceits. The aim of current paper is to provide a phishing detection system to be used in e-banking system in Iran. Identifying the outstanding features of phishing is one of the important prerequisites in design of an accurate system; therefore, in first step, to identify the influential features of phishing that best fit the Iranian bank sites, a list of 28 phishing indicators was prepared. Using feature selection algorithm based on rough sets theory, six main indicators were identified as the most effective factors. The fuzzy expert system was designed using these indicators, afterwards. The results show that the proposed system is able to determine the Iranian phishing sites with a reasonable speed and precision, having an accuracy of 88%.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.     

Numerical modeling of heat exchange and unsaturated–saturated flow in porous media

We discuss the numerical modeling of heat and mass transport in unsaturated–saturated porous media. The heat is transported by infiltrated water underlying capillary and gravitation driven forces. Heat energy is governed by molecular diffusion, convection, dispersion and exchange between the infiltrated water and porous media matrix. An unsaturated–saturated flow is considered with boundary conditions reflecting the external driven forces. The presented mathematical model is motivated by analysis of hygrothermal isolation properties of facades. The main contribution is focused on the determination of model parameters including soil parameters, dispersion coefficients, thermal transmission coefficient, thermal conductivity of porous media matrix and external transmission coefficients. The used mathematical model does not include the vapor transport and its phase exchange with water due to vaporization and condensation. It will be the next step of our research. Thus, practical applications of our model are limited. The developed numerical method is a good candidate for solving corresponding inverse problems. Numerical experiments support our method.     

Numerical aspects in the finite element simulation of the Portevin–Le Chatelier effect

Serrated yielding and propagation of localized bands of plastic strain rate are the apparent phenomena of the Portevin–Le Chatelier (PLC) effect. The finite element modeling of this effect is investigated, using a model proposed by Zhang et al. [74] describing dynamic strain aging, and material parameters for a Nickel based superalloy at 500 °C. This work presents: (1) an efficient implicit integration scheme of the constitutive equations in the presence of instabilities; (2) a numerical tool to determine the type of plastic strain rate localization bands based on results of simulations; and (3) a mesh and time discretization sensitivity analysis of the model regarding the characteristics of PLC bands. This analysis is carried out in 2D and 3D for axisymmetric smooth and notched specimens.