The Burnett equations in cylindrical coordinates and their solution for flow in a microtube

2014 ◽  
Vol 751 ◽  
pp. 121-141 ◽  
Author(s):  
Narendra Singh ◽  
Amit Agrawal
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Higher Order ◽  
Navier Stokes ◽  
Burnett Equations ◽  
Cylindrical Coordinates ◽  
Navier Stokes Equations

AbstractThe Burnett equations constitute a set of higher-order continuum equations. These equations are obtained from the Chapman–Enskog series solution of the Boltzmann equation while retaining second-order-accurate terms in the Knudsen number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Kn}$. The set of higher-order continuum models is expected to be applicable to flows in the slip and transition regimes where the Navier–Stokes equations perform poorly. However, obtaining analytical or numerical solutions of these equations has been noted to be particularly difficult. In the first part of this work, we present the full set of Burnett equations in cylindrical coordinates in three-dimensional form. The equations are reported in a generalized way for gas molecules that are assumed to be Maxwellian molecules or hard spheres. In the second part, a closed-form solution of these equations for isothermal Poiseuille flow in a microtube is derived. The solution of the equations is shown to satisfy the full Burnett equations up to $\mathit{Kn} \leq 1.3$ within an error norm of ${\pm }1.0\, \%$. The mass flow rate obtained analytically is shown to compare well with available experimental and numerical results. Comparison of the stress terms in the Burnett and Navier–Stokes equations is presented. The significance of the Burnett normal stress and its role in diffusion of momentum is brought out by the analysis. An order-of-magnitude analysis of various terms in the equations is presented, based on which a reduced model of the Burnett equations is provided for flow in a microtube. The Burnett equations in full three-dimensional form in cylindrical coordinates and their solution are not previously available.

Simulation-Based Analysis of a Biologically-Inspired Micropump With a Rotating Spiral Inside a Microchannel

2008 ◽  
Author(s):  
Mustafa Koz ◽  
Serhat Yesilyurt
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Asymptotic Results ◽  
Navier Stokes Equations ◽  
Rate Maximum ◽  
Finite Element Package ◽  
Resistive Force Theory

Microorganisms such as bacteria use their rotating helical flagella for propulsion speeds up to tens of tail lengths per second. The mechanism can be utilized for controlled pumping of liquids in microchannels. In this study, we aim to analyze the effects of control parameters such as axial span between helical rounds (wavelength), angular velocity of rotations (frequency), and the radius of the helix (amplitude) on the maximum time-averaged flow rate, maximum head, rate of energy transfer, and efficiency of the micropump. The analysis is based on simulations obtained from the three-dimensional time-dependent numerical model of the flow induced by the rotating spiral inside a rectangular-prism channel. The flow is governed by Navier-Stokes equations subject to continuity in time-varying domain due to moving boundaries of the spiral. Numerical solutions are obtained using a commercial finite-element package which uses arbitrary Lagrangian-Eulerian method for mesh deformations. Results are compared with asymptotic results obtained from the resistive-force-theory available in the literature.

Fluid motion inside a spinning nutating cylinder

1985 ◽  
Vol 150 ◽  
pp. 121-138 ◽  
Author(s):  
Harold R. Vaughn ◽  
William L. Oberkampf ◽  
Walter P. Wolfe
Keyword(s):  
Steady State ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
Internal Fluid

The incompressible three-dimensional Navier–Stokes equations are solved numerically for a fluid-filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin-stabilized artillery projectiles. Equations for the internal fluid motion are derived in a non-inertial aeroballistic coordinate system. Steady-state numerical solutions are obtained by an iterative finite-difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 × 104−1 × 109 cSt. The nature of the three-dimensional fluid motion inside the cylinder is discussed, and the moments generated by the fluid are explained. The calculated moments generally agree with experimental measurements.

Numerical Simulations of Three-Dimensional Flows in a Cubic Cavity With an Oscillating Lid

1993 ◽  
Vol 115 (4) ◽  
pp. 680-686 ◽  
Author(s):  
Reima Iwatsu ◽  
Jae Min Hyun ◽  
Kunio Kuwahara
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Time Dependent ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Sinusoidal Oscillations ◽  
Dependent Flow

Numerical studies are made of three-dimensional flow of a viscous fluid in a cubical container. The flow is driven by the top sliding wall, which executes sinusoidal oscillations. Numerical solutions are acquired by solving the time-dependent, three-dimensional incompressible Navier-Stokes equations by employing very fine meshes. Results are presented for wide ranges of two principal physical parameters, i.e., the Reynolds number, Re ≤ 2000 and the frequency parameter of the lid oscillation, ω′ ≤ 10.0. Comprehensive details of the flow structure are analyzed. Attention is focused on the three-dimensionality of the flow field. Extensive numerical flow visualizations have been performed. These yield sequential plots of the main flows as well as the secondary flow patterns. It is found that the previous two-dimensional computational results are adequate in describing the main flow characteristics in the bulk of interior when ω′ is reasonably high. For the cases of high-Re flows, however, the three-dimensional motions exhibit additional complexities especially when ω′ is low. It is asserted that, thanks to the recent development of the supercomputers, calculation of three-dimensional, time-dependent flow problems appears to be feasible at least over limited ranges of Re.

Low Reynolds-Number Pipe Flow in the Vicinity of Three-Dimensional Obstacles

1979 ◽  
Vol 21 (5) ◽  
pp. 335-343 ◽  
Author(s):  
A. D. Gosman ◽  
N. S. Vlachos ◽  
J. H. Whitelaw
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Recirculating Flow ◽  
Local Shear ◽  
Sector Angle

Numerical solutions of the three-dimensional Navier-Stokes equations are presented for boundary conditions corresponding to the laminar flow of Newtonian and non-Newtonian fluids in a round pipe with truncated sector-shaped obstacles. The influences of Reynolds number and sector angle on the velocity distributions, local shear stress and pressure drop are quantified and shown to be large. The results are complementary to those previously reported by Vlachos and Whitelaw (1)§ for axisymmetric obstacles, where related two-dimensional effects were quantified. They provide new information on three-dimensional, recirculating flow in ducts and form a basis for future calculations of corresponding turbulent flows.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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