The Stability of Plane Poiseuille Flow in a Finite-Length Channel

2021 ◽  
Author(s):  
Lei Xu ◽  
Zvi Rusak
Keyword(s):  
Kinetic Energy ◽  
Boundary Conditions ◽  
Linear Stability ◽  
Finite Length ◽  
Poiseuille Flow ◽  
Weak Form ◽  
Plane Poiseuille Flow ◽  
Various Boundary Conditions ◽  
New Perspective ◽  
The Stability

Abstract The linear stability of plane Poiseuille flow through a finite-length channel is studied. A weakly-divergence-free basis finite element method with SUPG stabilization is used to formulate the weak form of the problem. The linear stability characteristics are studied under three possible inlet-outlet boundary conditions and the corresponding perturbation kinetic energy transfer mechanisms are investigated. Active transfer of perturbation kinetic energy at the channel inlet and outlet, energy production due to convection and dissipation at the flow bulk provide a new perspective in understanding the distinct stability characteristics of plane Poiseuille flow under various boundary conditions.

Linear growth in two-fluid plane Poiseuille flow

1999 ◽  
Vol 381 ◽  
pp. 121-139 ◽  
Author(s):  
M. J. SOUTH ◽  
A. P. HOOPER
Keyword(s):  
Kinetic Energy ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Maximum Growth ◽  
Two Dimensions ◽  
Linear Stability Theory ◽  
Plane Poiseuille Flow ◽  
Stable Mode ◽  
Two Fluid ◽  
Interfacial Mode

In recent years a new paradigm has emerged in linear stability theory due to the recognition of the importance of non-normality in the Orr–Sommerfeld equation as derived from the method of normal modes. For single-fluid flows it has been shown that it is possible for the kinetic energy of certain stable mode combinations to grow transiently before decaying to zero. We look again at the linear stability of two-fluid plane Poiseuille flow in two dimensions, concentrating on transient growth and its dependence on the viscosity and depth ratio. The procedure is to solve the stability equations numerically and consider disturbances defined as a sum of the least stable eigenmodes (not just the least stable interfacial mode). It is found that the variational method used to find maximum growth cannot be based upon the kinetic energy of the flow only and that interface deflection must be included in the formulation. We show which modes are necessary for inclusion in the disturbance expression and find that the interfacial mode does not make a significant contribution to possible energy growth. We examine the magnitude of maximum growth and the nature of the disturbances that lead to this growth. The linear energy rate equation shows that at moderate Reynolds numbers the mechanism responsible for the largest two-fluid growth is transfer of energy from the basic flow via the Reynolds stresses. The energy transfer is facilitated by streamline tilting that can be seen at the channel walls or at the interface. A similar effect has been found in single-fluid plane Poiseuille flow.

scholarly journals Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

Fluids ◽  
2021 ◽  
Vol 6 (12) ◽  
pp. 432
Author(s):  
Silvia C. Hirata ◽  
Mohamed Najib Ouarzazi
Keyword(s):  
Linear Stability ◽  
Viscous Dissipation ◽  
Viscoelastic Fluid ◽  
Poiseuille Flow ◽  
Hydrodynamic Instability ◽  
Plane Poiseuille Flow ◽  
Volumetric Heating ◽  
Fluid Elasticity ◽  
Different Temperatures ◽  
Elastic Fluids

The onset of thermal instabilities in the plane Poiseuille flow of weakly elastic fluids is examined through a linear stability analysis by taking into account the effects of viscous dissipation. The destabilizing thermal gradients may come from the different temperatures imposed on the external boundaries and/or from the volumetric heating induced by viscous dissipation. The rheological properties of the viscoelastic fluid are modeled using the Oldroyd-B constitutive equation. As in the Newtonian fluid case, the most unstable structures are found to be stationary longitudinal rolls (modes with axes aligned along the streamwise direction). For such structures, it is shown that the viscoelastic contribution to viscous dissipation may be reduced to one unique parameter: γ=λ1(1−Γ), where λ1 and Γ represent the relaxation time and the viscosity ratio of the viscoelastic fluid, respectively. It is found that the influence of the elasticity parameter γ on the linear stability characteristics is non-monotonic. The fluid elasticity stabilizes (destabilizes) the basic Poiseuille flow if γ<γ* (γ>γ*) where γ* is a particular value of γ that we have determined. It is also shown that when the temperature gradient imposed on the external boundaries is zero, the critical Reynolds number for the onset of such viscous dissipation/viscoelastic-induced instability may be well below the one needed to trigger the pure hydrodynamic instability in weakly elastic solutions.

Linear Stability of a Plasma Diode

1969 ◽  
Vol 24 (8) ◽  
pp. 1235-1243 ◽  
Author(s):  
M Dobrowolny ◽  
F Engelmann ◽  
A Sestero
Keyword(s):  
Boundary Conditions ◽  
Linear Stability ◽  
Plane Waves ◽  
The Other ◽  
Other Hand ◽  
Free Particles ◽  
Plasma Diode ◽  
The Stability

AbstractThe stability of a plasma diode with respect to longitudinal oscillations is investigated. If there are free particles emitted by the electrodes, the perturbations do not have the same dynamics as they would in an infinite plasma, contrary to the case where only particles trapped in the diode are present. This can be interpreted as due to a coupling of plane waves of different wave lengths, introduced by the boundary conditions at the electrodes. The occurrence of resonant-particle effects, on the other hand, is subjected to precisely the same conditions as in an infinite plasma.

scholarly journals On the stability of plane Couette–Poiseuille flow with uniform crossflow

2010 ◽  
Vol 656 ◽  
pp. 417-447 ◽  
Author(s):  
ANIRBAN GUHA ◽  
IAN A. FRIGAARD
Keyword(s):  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Base Flow ◽  
Two Dimensional ◽  
Wall Velocity ◽  
Resonant Interactions ◽  
Flow Stabilization ◽  
Tollmien Schlichting ◽  
The Stability

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

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