scholarly journals The stability of stratified spatially periodic shear flows at low Péclet number

2015 ◽  
Vol 27 (8) ◽  
pp. 084104 ◽  
Author(s):  
Pascale Garaud ◽  
Basile Gallet ◽  
Tobias Bischoff
Keyword(s):  
Peclet Number ◽  
Shear Flows ◽  
Péclet Number ◽  
Spatially Periodic ◽  
The Stability

THE STABILITY OF SPATIALLY NONHOMOGENEOUS STEADY STATES IN A TUBULAR CHEMICAL REACTOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1477-1486
Author(s):  
JAMES M. ROTENBERRY ◽  
ANTONMARIA A. MINZONI
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Chemical Reactor ◽  
Higher Order ◽  
Steady States ◽  
Complex Conjugate ◽  
Heat Of Reaction ◽  
Péclet Number ◽  
The Stability ◽  
Steady State Solutions

We study the axial heat and mass transfer in a highly diffusive tubular chemical reactor in which a simple reaction is occurring. The steady state solutions of the governing equations are studied using matched asymptotic expansions, the theory of dynamical systems, and by calculating the solutions numerically. In particular, the effect of varying the Peclet and Damköhler numbers (P and D) is investigated. A simple expression for the approximate location of the transition layer for large Peclet number is derived and its accuracy tested against the numerical solution. The stability of the steady states is examined by calculating the eigenvalues and eigenfunctions of the linearized equations. It is shown that a Hopf bifurcation of the CSTR model (i.e., the limit as the P approaches zero) can be continued up to order 1 in the Peclet number. Furthermore, it is shown numerically that for appropriate values of the Peclet number, the Damköhler number, and B (the heat of reaction) these Hopf bifurcations merge with the limit points of an "S–shaped" bifurcation curve in a higher order singularity controlled by the Bogdanov–Takens normal form. Consequently, there must exist a finite amplitude, nonuniform, stable periodic solution for parameter values near this singularity. The existence of higher order degeneracies is also explored. In particular, it is shown for D ≪ 1 that no value of P exists where two pairs of complex conjugate eigenvalues of the steady state solutions can cross the imaginary axis simultaneously.

Miscible porous media displacements in the quarter five-spot configuration. Part 3. Non-monotonic viscosity profiles

1999 ◽  
Vol 388 ◽  
pp. 171-195 ◽  
Author(s):  
CHRISTIAN PANKIEWITZ ◽  
ECKART MEIBURG
Keyword(s):  
Porous Media ◽  
Linear Stability ◽  
Peclet Number ◽  
Base Flow ◽  
Péclet Number ◽  
Miscible Displacements ◽  
Flow Vorticity ◽  
Small Times ◽  
The Stability ◽  
Stability Results

The influence of a non-monotonic viscosity–concentration relationship on miscible displacements in porous media is studied for radial source flows and the quarter five-spot configuration. Based on linear stability results, a parametric study is presented that demonstrates the dependence of the dispersion relations on both the Péclet number and the parameters of the viscosity profile. The stability analysis suggests that any displacement can become unstable provided only that the Péclet number is sufficiently high. In contrast to rectilinear flows, for a given end-point viscosity ratio an increase of the maximum viscosity generally has a destabilizing effect on the flow. The physical mechanisms behind this behaviour are examined by inspecting the eigensolutions to the linear stability problem. Nonlinear simulations of quarter five-spot displacements, which for small times correspond to radial source flows, confirm the linear stability results. Surprisingly, displacements characterized by the largest instability growth rates, and consequently by vigorous viscous fingering, lead to the highest breakthrough recoveries, which can even exceed that of a unit mobility ratio flow. It can be concluded that, for non-monotonic viscosity profiles, the interaction of viscous fingers with the base-flow vorticity can result in improved recovery rates.

NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

2006 ◽  
Vol 17 (11) ◽  
pp. 1563-1577 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Peclet Number ◽  
Eigenvalue Problems ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Péclet Number ◽  
Velocity Models ◽  
Advection Diffusion ◽  
The Stability

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

Convective diffusion toward the sphere in laminar flow around the sphere

1979 ◽  
Vol 44 (4) ◽  
pp. 1218-1238
Author(s):  
Arnošt Kimla ◽  
Jiří Míčka
Keyword(s):  
Laminar Flow ◽  
Velocity Profile ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Péclet Number ◽  
Stability Of The Solution

The problem of convective diffusion toward the sphere in laminar flow around the sphere is solved by combination of the analytical and net methods for the region of Peclet number λ ≥ 1. The problem was also studied for very small values λ. Stability of the solution has been proved in relation to changes of the velocity profile.

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