A perturbation-based solution of Burnett equations for gaseous flow in a long microchannel

2018 ◽  
Vol 844 ◽  
pp. 1038-1051 ◽  
Author(s):  
Aishwarya Rath ◽  
Narendra Singh ◽  
Amit Agrawal
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Perturbation Expansion ◽  
Direct Simulation Monte Carlo ◽  
Analytical Investigation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Perturbation Approach ◽  
Burnett Equations ◽  
Gaseous Flow

In this paper, an analytical investigation of two-dimensional conventional Burnett equations has been undertaken for gaseous flow through a long microchannel. The analytical solution is obtained by using perturbation analysis around the classical Navier–Stokes solution with appropriate boundary conditions. The perturbation expansion is employed with the smallness parameter $\unicode[STIX]{x1D716}$, taken as the ratio of height to length of the microchannel. The solution for pressure is obtained by solving the cross-stream momentum equation while the velocity distribution is obtained from the streamwise momentum equation. The resulting ordinary differential equations in pressure and velocity are third-order and second-order, respectively. The required boundary conditions for pressure are obtained from direct simulation Monte Carlo (DSMC) data. The obtained analytical solution matches the available DSMC solution well. This is perhaps the first analytical solution of the Burnett equations using the perturbation approach.

The planar Couette flow with slip and jump boundary conditions in a microchannel

2016 ◽  
Vol 22 (4) ◽  
Author(s):  
Mohamed Hssikou ◽  
Jamal Baliti ◽  
Mohammed Alaoui
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Direct Simulation Monte Carlo ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Direct Simulation ◽  
Dilute Gas ◽  
Viscous Heat ◽  
Simulation Monte Carlo

AbstractThe steady state of a dilute gas enclosed within a rectangular cavity, whose upper and lower sides are in relative motion, is considered in the slip and early transition regimes. The DSMC (Direct simulation Monte Carlo) method is used to solve the Boltzmann equation for analysing a Newtonian viscous heat conducting ideal gas with the slip and jump boundary conditions (SJBC) in the vicinity of horizontal walls. The numerical results are compared with the Navier–Stokes solutions, with and without SJBC, through the velocity, temperature, and normal heat flux profiles. The parallel heat flux and shear stress are also evaluated as a function of rarefaction degree; estimated by the Knudsen number

A Numerical Study of the Impact of Wavy Walls on Steady Fluid Flow Through a Curved Tube

2013 ◽  
Vol 135 (7) ◽  
Author(s):  
Chekema Prince ◽  
Mingyao Gu ◽  
Sean D. Peterson
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Perturbation Expansion ◽  
Analytical Investigation ◽  
Navier Stokes ◽  
Radius Of Curvature ◽  
Navier Stokes Equations ◽  
Curved Tube ◽  
Small Tube ◽  
The Impact

In this paper, we discuss the impact of a wavy-walled pipe cross-section on steady flow in a curved tube at moderate Dean numbers and small tube radius-to-radius-of-curvature ratios. Parameters investigated include the protrusion height, the number of protrusions around the tube circumference, and the pipe curvature. This work extends a previous analytical investigation that employed a double perturbation expansion to elucidate the flow field as a function of these parameters. Due to the rapid growth in the solution complexity as the number of terms in each expansion increases, the analytical work is relegated to small wall perturbations and low Dean numbers. These barriers are removed in the present study by numerically solving the Navier–Stokes equations at Dean numbers up to 2500. The impact on the axial and secondary flow structures are emphasized, along with the resulting wall shear stress distributions.

scholarly journals Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 680 ◽  
Author(s):  
Alexander Beckmann ◽  
Anirudh Rana ◽  
Manuel Torrilhon ◽  
Henning Struchtrup
Keyword(s):  
Kinetic Theory ◽  
Boundary Conditions ◽  
Rarefied Gas ◽  
Continuum Hypothesis ◽  
Direct Simulation Monte Carlo ◽  
Transport Equations ◽  
Particle Methods ◽  
Navier Stokes ◽  
Knudsen Numbers ◽  
Constant Coefficients

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Analytical solution of the Burnett equations for gaseous flow in a long microchannel

2021 ◽  
Vol 912 ◽  
Author(s):  
Aishwarya Rath ◽  
Upendra Yadav ◽  
Amit Agrawal
Keyword(s):  
Analytical Solution ◽  
Burnett Equations ◽  
Gaseous Flow

Abstract

Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions

2001 ◽  
Vol 124 (2) ◽  
pp. 338-345 ◽  
Author(s):  
Yichuan Fang ◽  
William W. Liou
Keyword(s):  
Heat Transfer ◽  
Fluid Dynamics ◽  
Boundary Conditions ◽  
Microelectromechanical Systems ◽  
Stokes Equations ◽  
Direct Simulation Monte Carlo ◽  
Velocity Slip ◽  
Navier Stokes ◽  
Slip Boundary ◽  
Calculated Velocity

The heat transfer and the fluid dynamics characteristics of subsonic gas flows through microchannels are examined using the direct simulation Monte Carlo (DSMC) method. A simple implicit treatment for the low-speed inflow and outflow boundaries for the DSMC of the flows in microelectromechanical systems (MEMS) is used. Micro-Couette flows and micro-Poiseuille flows are simulated with the value of the Knudsen numbers ranging between 0.06 and 0.72. Where appropriate, the calculated velocity slip and temperature distribution are compared with analytical solutions derived from the Navier-Stokes equations with slip-boundary conditions. A patterned microstructure with nonuniform surface temperature is also simulated. The computational results show that the Knudsen number and the geometric complexity have significant effects on the heat transfer as well as the fluid dynamics properties of the microfluid flows studied.

Computation of Inviscid Incompressible Flow Using the Primitive Variable Formulation

1986 ◽  
Vol 108 (1) ◽  
pp. 68-75 ◽  
Author(s):  
S. Abdallah ◽  
H. G. Smith
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Type Equation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Fine Grid ◽  
Navier Stokes Equations ◽  
Pressure Poisson Equation ◽  
Fine Mesh

The primitive variable formulation originally developed for the incompressible Navier–Stokes equations is applied for the solution of the incompressible Euler equations. The unsteady momentum equation is solved for the velocity field and the continuity equation is satisfied indirectly in a Poisson-type equation for the pressure (divergence of the momentum equation). Solutions for the pressure Poisson equation with derivative boundary conditions exist only if a compatibility condition is satisfied (Green’s theorem). This condition is not automatically satisfied on nonstaggered grids. A new method for the solution of the pressure equation with derivative boundary conditions on a nonstaggered grid [25] is used here for the calculation of the pressure. Three-dimensional solutions for the inviscid rotational flow in a 90 deg curved duct are obtained on a very fine mesh (17 × 17 × 29). The use of a fine grid mesh allows for the accurate prediction of the development of the secondary flow. The computed results are in good agreement with the experimental data of Joy [15].

scholarly journals Analysis of natural in-plane vibration of rectangular plates using homotopy perturbation approach

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Igor V. Andrianov ◽  
Jan Awrejcewicz ◽  
Vladimir Chernetskyy
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Perturbation Approach ◽  
Rectangular Plates ◽  
Plane Vibration ◽  
Homotopy Perturbation

An analytical solution of the problem of free in-plane vibration of rectangular plates with complicated boundary conditions is proposed.

scholarly journals Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

2021 ◽  
pp. 1-21
Author(s):  
Claudia Gariboldi ◽  
Takéo Takahashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Slip ◽  
Slip Boundary ◽  
Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

Study on the Deposition Profile Characteristics in the Micron-Scale Trench Using Direct Simulation Monte Carlo Method

1998 ◽  
Vol 120 (2) ◽  
pp. 296-302 ◽  
Author(s):  
Masato Ikegawa ◽  
Jun’ichi Kobayashi ◽  
Morihisa Maruko
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Boundary Conditions ◽  
Gas Flow ◽  
Direct Simulation Monte Carlo ◽  
Low Pressure ◽  
Sputter Deposition ◽  
Direct Simulation ◽  
Simulation Monte Carlo ◽  
Profile Characteristics

As integrated circuits are advancing toward smaller device features, step-coverage in submicron trenches and holes in thin film deposition are becoming of concern. Deposition consists of gas flow in the vapor phase and film growth in the solid phase. A deposition profile simulator using the direct simulation Monte Carlo method has been developed to investigate deposition profile characteristics on small trenches which have nearly the same dimension as the mean free path of molecules. This simulator can be applied to several deposition processes such as sputter deposition, and atmospheric- or low-pressure chemical vapor deposition. In the case of low-pressure processes such as sputter deposition, upstream boundary conditions of the trenches can be calculated by means of rarefied gas flow analysis in the reactor. The effects of upstream boundary conditions, molecular collisions, sticking coefficients, and surface migration on deposition profiles in the trenches were clarified.

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