Incompressible slip flow past a semi-infinite flat plate

1965 ◽  
Vol 22 (3) ◽  
pp. 463-469 ◽  
Author(s):  
J. D. Murray
Keyword(s):  
Boundary Condition ◽  
Shear Stress ◽  
Viscous Fluid ◽  
Flat Plate ◽  
Slip Velocity ◽  
Slip Flow ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Slip Boundary

An asymptotic solution to the Navier-Stokes equation is obtained for the incompressible flow of a viscous fluid past a semi-infinite flat plate when a slip boundary condition is applied at the plate. The results for the shear stress (and hence the slip velocity) on the plate differ basically from those obtained by previous authors who considered the same problem using some form of the Oseen equations.

ELECTROKINETIC SLIP FLOW OF MICROFLUIDICS IN TERMS OF STREAMING POTENTIAL BY A LATTICE BOLTZMANN METHOD: A BOTTOM-UP APPROACH

2007 ◽  
Vol 18 (04) ◽  
pp. 693-700 ◽  
Author(s):  
XIN FU ◽  
BAOMING LI ◽  
JUNFENG ZHANG ◽  
FUZHI TIAN ◽  
DANIEL Y. KWOK
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Slip Flow ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Slip Boundary ◽  
Surface Energetics ◽  
Boltzmann Method

In traditional computational fluid dynamics, the effect of surface energetics on fluid flow is often ignored or translated into an arbitrary selected slip boundary condition in solving the Navier-Stokes equation. Using a bottom-up approach, we show in this paper that variation of surface energetics through intermolecular theory can be employed in a lattice Boltzmann method to investigate both slip and non-slip phenomena in microfluidics in conjunction with the description of electrokinetic phenomena for electrokinetic slip flow. Rather than using the conventional Navier-Stokes equation with a slip boundary condition, the description of electrokinetic slip flow in microfluidics is manifested by the more physical solid-liquid energy parameters, electrical double layer and contact angle in the mean-field description of the lattice Boltzmann method.

MEMS-Scale Turbomachinery Based Vacuum Roughing Pump

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Anthony J. Gannon ◽  
Garth V. Hobson ◽  
Michael J. Shea ◽  
Christopher S. Clay ◽  
Knox T. Millsaps
Keyword(s):  
Boundary Condition ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Boundary Conditions ◽  
Knudsen Number ◽  
Slip Flow ◽  
Slip Boundary Condition ◽  
3D Simulations ◽  
Slip Boundary ◽  
Wall Shear

This study forms part of a program to develop a micro-electro-mechanical systems (MEMS) scale turbomachinery based vacuum pump and investigates the roughing portion of such a system. Such a machine would have many radial stages with the exhaust stages operating near atmospheric conditions while the inlet stages operate at near vacuum conditions. In low vacuum such as those to the inlet of a roughing pump, the flow can still be treated as a continuum; however, the no-slip boundary condition is not accurate. The Knudsen number becomes a dominant nondimensional parameter in these machines due to their small size and low pressures. As the Knudsen number increases, slip-flow becomes present at the walls. The study begins with a basic overview on implementing the slip wall boundary condition in a commercial code by specifying the wall shear stress based on the mean-free-path of the gas molecules. This is validated against an available micro-Poiseuille classical solution at Knudsen numbers between 0.001 and 0.1 with reasonable agreement found. The method of specifying the wall shear stress is then applied to a generic MEMS scale roughing pump stage that consists of two stators and a rotor operating at a nominal absolute pressure of 500 Pa. The zero flow case was simulated in all cases as the pump down time for these machines is small due to the small volume being evacuated. Initial transient two-dimensional (2D) simulations are used to evaluate three boundary conditions, classical no-slip, specified-shear, and slip-flow. It is found that the stage pressure rise increased as the flow began to slip at the walls. In addition, it was found that at lower pressures the pure slip boundary condition resulted in very similar predictions to the specified-shear simulations. As the specified-shear simulations are computationally expensive it is reasonable to use slip-flow boundary conditions. This approach was used to perform three-dimensional (3D) simulations of the stage. Again the stage pressure increased when slip-flow was present compared with the classical no-slip boundaries. A characteristic of MEMS scale turbomachinery are the large relative tip gaps requiring 3D simulations. A tip gap sensitivity study was performed and it was found that when no-slip boundaries were present the pressure ratio increased significantly with decreasing tip gap. When slip-flow boundaries were present, this relationship was far weaker.

COMPARISON OF GAS SLIP MODELS WITH SOLUTIONS OF LINEARIZED BOLTZMANN EQUATION AND DIRECT SIMULATION OF MONTE CARLO METHOD

2007 ◽  
Vol 18 (02) ◽  
pp. 203-216 ◽  
Author(s):  
G. H. TANG ◽  
Y. L. HE ◽  
W. Q. TAO
Keyword(s):  
Friction Factor ◽  
Flow Rate ◽  
Slip Flow ◽  
Flow Analysis ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Slip Boundary ◽  
Slip Regime ◽  
Linearized Boltzmann Equation

Analytical solutions of the Navier–Stokes equation based on a locally fully-developed flow assumption with various gas slip models are presented and comparisons for velocity profile, flow rate, friction factor, and pressure distribution are performed. The effect of the second-order coefficient in the slip boundary condition becomes significant as the Knudsen number increases. Most slip models are limited to slip regime or marginally transition regime and break down around Kn = 0.1 while Sreekanth's model, followed by Mitsuya's model, gives a good agreement with the linearized Boltzmann solutions from slip regime up to Kn = 2 for flow rate and friction factor predictions. These two models should be of great use for slip flow analysis in micro-electro-mechanical systems (MEMS) and, in particular, in situations where the flow rate and flow resistance are of interest.

scholarly journals On the dynamic slip boundary condition for Navier–Stokes-like problems

2021 ◽  
pp. 1-48
Author(s):  
Anna Abbatiello ◽  
Miroslav Bulíček ◽  
Erika Maringová
Keyword(s):  
Boundary Condition ◽  
Shear Stress ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Slip Boundary ◽  
Boundary Terms ◽  
The Moment ◽  
Dynamic Slip ◽  
Proper Generalization ◽  
Slip Phenomenon

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

On a New Approach to Slip Flow Using Rayleigh’s Problem

1967 ◽  
Vol 34 (4) ◽  
pp. 837-839 ◽  
Author(s):  
E. P. Russo ◽  
O. A. Arnas
Keyword(s):  
Boundary Condition ◽  
Drag Coefficient ◽  
Flat Plate ◽  
Slip Velocity ◽  
Stokes Equations ◽  
Traditional Approach ◽  
Slip Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
New Approach

A new approach, involving a perturbation of the Navier-Stokes equations, is used to analyze the phenomenon of slip flow over a flat plate. An expression for the coefficient of drag is derived and compared to the drag coefficient obtained by the traditional approach of solving the Navier-Stokes equations with a slip-velocity boundary condition.

A Mean-Field Lattice Boltzmann Model for Electrokinetic Slip Flow

2007 ◽  
Author(s):  
Xu Fu ◽  
Baoming Li ◽  
Junfeng Zhang ◽  
Fuzhi Tian ◽  
Daniel Y. Kwok
Keyword(s):  
Fluid Flow ◽  
Slip Flow ◽  
Mean Field ◽  
Slip Boundary Condition ◽  
Lattice Boltzmann Model ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Slip Boundary ◽  
Surface Energetics ◽  
Boltzmann Model

Traditional computational fluid dynamics does not normally consider surface energetic effect on fluid flow in microchannels. Even if it does, the effect is usually translated into an arbitrary selected slip boundary condition in solving the Navier-Stokes equation. Moreover, this treatment has neglected another important phenomenon: electrokinetics. In this paper, we consider both effects of electrokinetics and surface energetics on fluid flow in microfluidics which are normally considered independently.

MEMS-Scale Turbomachinery Based Vacuum Roughing Pump

2013 ◽  
Author(s):  
Anthony J. Gannon ◽  
Garth V. Hobson ◽  
Michael J. Shea ◽  
Christopher S. Clay ◽  
Knox T. Millsaps
Keyword(s):  
Boundary Condition ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Boundary Conditions ◽  
Knudsen Number ◽  
Slip Flow ◽  
Three Dimensional ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
Three Dimensional Simulations

This study forms part of a program to develop a micro-electro-mechanical-systems (MEMS) scale turbomachinery based vacuum pump and investigates the roughing portion of such a system. Such a machine would have many radial stages with the exhaust stages operating near atmospheric conditions while the inlet stages operate at near vacuum conditions. In low vacuum such as those to the inlet of a roughing pump the flow can still be treated as a continuum however the no-slip boundary condition is not accurate. The Knudsen number becomes a dominant non-dimensional parameter in these machines due to their small size and low pressures. As the Knudsen number increases slip flow becomes present at the walls. The study begins with a basic overview on implementing the slip wall boundary condition in a commercial code by specifying the wall shear stress based on the mean-free-path of the gas molecules. This is validated against an available micro-Poiseuille classical solution at Knudsen numbers between 0.001–0.1 with reasonable agreement found. The method of specifying the wall-shear stress is then applied to a generic MEMS scale roughing pump stage that consists of two stators and a rotor operating at a nominal absolute pressure of 500 Pa. The zero flow case was simulated in all cases as the pump down time for these machines is small due to the small volume being evacuated. Initial transient two-dimensional simulations are used to evaluate three boundary conditions, classical no-slip, specified-shear and slip-flow. It is found that the stage pressure rise increased as the flow began to slip at the walls. In addition it was found that at lower pressures the pure slip boundary condition resulted in very similar predictions to the specified shear simulations. As the specified-shear simulations are computationally expensive it is reasonable to use slip-flow boundary conditions. This approach was used to perform three-dimensional simulations of the stage. Again the stage pressure increased when slip-flow was present compared with the classical no-slip boundaries. A characteristic of MEMS scale turbomachinery are the large relative tip gaps requiring three-dimensional simulations. A tip gap sensitivity study was performed and it was found that when no-slip boundaries were present the pressure ratio increased significantly with decreasing tip gap. When slip-flow boundaries were present this relationship was far weaker.

Sign in / Sign up

Export Citation Format

Share Document