The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n, with application to plane poiseuille flow

1989 ◽  
Vol 106 (4) ◽  
pp. 335-376 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Hopf Bifurcation ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations

scholarly journals Subcritical transition in channel flows

2002 ◽  
Vol 451 ◽  
pp. 35-97 ◽  
Author(s):  
S. JONATHAN CHAPMAN
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Domain Of Attraction ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Laminar State

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

Wave patterns in plane Poiseuille flow created by concentrated disturbances

1989 ◽  
Vol 208 ◽  
pp. 639-656 ◽  
Author(s):  
Fei Li ◽  
Sheila E. Widnall
Keyword(s):  
Laminar Flow ◽  
Reynolds Stress ◽  
Poiseuille Flow ◽  
Fourier Transforms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Turbulent Spot ◽  
Navier Stokes Equations ◽  
Wave Patterns

A model is constructed for perturbations created in the surrounding laminar flow by a turbulent spot in plane Poiseuille flow. The turbulent spot is represented as a distribution of increased Reynolds stress, which travels steadily through the surrounding laminar flow. The Navier-Stokes equations are linearized and are solved by using Fourier transforms in the plane parallel to the channel walls and a finite-difference method in the direction perpendicular to the walls. The travelling Reynolds stress distribution acts as a forcing term in the equations.Numerical results show that a packet of oblique waves are generated around the disturbance when the force is antisymmetric with respect to the channel centreline, whereas no identifiable wave crests are found when the forcing is symmetric. Furthermore, wavelengths of the typical waves composing the packet are insensitive to the size of the region of Reynolds stress. The dependencies of the flow field on Reynolds number and spot speed are investigated. In the case of symmetric forcing, the flow is forced around the disturbance, causing distortions to the basic velocity profiles. These results are in qualitative agreement with experimental observations.

On spatially-growing finite disturbances in plane Poiseuille flow

1962 ◽  
Vol 14 (2) ◽  
pp. 211-221 ◽  
Author(s):  
J. Watson
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations

Possible solutions of the Navier-Stokes equations are given representing certain finite disturbances in plane Poiseuille flow which vary with distance parallel to the bounding walls. These solutions are based on infinitesimal disturbances which vary exponentially with distance (upstream or downstream) instead of with time, and they are more closely related to the disturbances investigated experimentally than the corresponding ‘time-dependent’ solutions.

Non-linear theory of unstable plane Poiseuille flow

1969 ◽  
Vol 38 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Linear Theory ◽  
Poiseuille Flow ◽  
Theoretical Study ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Mathematical Technique ◽  
Navier Stokes Equations ◽  
Fourier Decomposition ◽  
Non Linear

A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

Internal laminar flow

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Concentric Cylinder ◽  
Navier Stokes Equations ◽  
Concentric Cylinders ◽  
Constant Viscosity ◽  
Pressure Driven Flow

In this chapter it is shown that solutions to the Navier-Stokes equations can be derived for steady, fully developed flow of a constant-viscosity Newtonian fluid through a cylindrical duct. Such a flow is known as a Poiseuille flow. For a pipe of circular cross section, the term Hagen-Poiseuille flow is used. Solutions are also derived for shear-driven flow within the annular space between two concentric cylinders or in the space between two parallel plates when there is relative tangential movement between the wetted surfaces, termed Couette flows. The concepts of wetted perimeter and hydraulic diameter are introduced. It is shown how the viscometer equations result from the concentric-cylinder solutions. The pressure-driven flow of generalised Newtonian fluids is also discussed.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

Flow Through Rough Microchannels: A Lubrication Perspective

2003 ◽  
Author(s):  
S. Boedo
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Upper And Lower Bounds ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Test Setup ◽  
One Dimensional ◽  
Reference Velocity ◽  
Sample Application ◽  
Flow Through

This paper provides concise specifications where idealized Poiseuille flow is applicable in representing one-dimensional flow through wide, thin, rough microchannels subjected to prescribed pressures at the channel ends. Starting with the general (compressible) form of the Navier-Stokes equations, new expressions which discuss the effect of body forces on flow through thin channels are first presented, leading to upper and lower bounds on channel reference velocity where idealized Poiseuille flow dominates. These results are combined with previously published studies related to the predicability of flow through stochastically rough surfaces. An arbitrarily chosen microchannel model based loosely on a previously published experimental test setup is used as a sample application.

scholarly journals Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel

2019 ◽  
Vol 25 ◽  
pp. 66
Author(s):  
Sourav Mitra
Keyword(s):  
Feedback Control ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Boundary Stabilization ◽  
Navier Stokes Equations ◽  
Local Boundary ◽  
Hyperbolic Dynamics ◽  
Finite Dimensional ◽  
Dimensional Range

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

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