Finite-Amplitude Elastic Instability of Plane-Poiseuille Flow of Viscoelastic Fluids

1999 ◽  
Vol 67 (4) ◽  
pp. 834-837 ◽  
Author(s):  
R. E. Khayat ◽  
N. Ashrafi
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Dynamical System ◽  
Elastic Stability ◽  
Finite Amplitude ◽  
Base Flow ◽  
Galerkin Projection ◽  
Plane Poiseuille Flow ◽  
Nonlinear Dynamical ◽  
Stability And Bifurcation ◽  
The Stability

The purely elastic stability and bifurcation of the one-dimensional plane Poiseuille flow is determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The problem is reduced to a nonlinear dynamical system using the Galerkin projection method. It is shown that elastic normal stress effects can be solely responsible for the destabilization of the base (Poiseuille) flow. It is found that the stability and bifurcation picture is dramatically influenced by the solvent-to-solute viscosity ratio, ε. As the flow deviates from the Newtonian limit and ε decreases below a critical value, the base flow loses its stability. Two static bifurcations emerge at two critical Weissenberg numbers, forming a closed diagram that widens as the level of elasticity increases. [S0021-8936(00)00703-0]

Numerical Computation of the Stability of Plane Poiseuille Flow

1978 ◽  
Vol 45 (1) ◽  
pp. 13-18 ◽  
Author(s):  
L. Wolf ◽  
Z. Lavan ◽  
H. J. Nielsen
Keyword(s):  
Stream Function ◽  
Poiseuille Flow ◽  
Numerical Technique ◽  
Finite Amplitude ◽  
Wave Number ◽  
Plane Poiseuille Flow ◽  
Governing Equations ◽  
The Stability ◽  
Successive Over Relaxation ◽  
Small Disturbances

The hydrodynamic stability of plane Poiseuille flow to infinitesimal and finite amplitude disturbances is investigated using a direct numerical technique. The governing equations are cast in terms of vorticity and stream function using second-order central differences in space. The vorticity equation is used to advance the vorticity values in time and successive over-relaxation is used to solve the stream function equation. Two programs were prepared, one for the linearized and the other for the complete disturbance equations. Results obtained by solving the linearized equations agree well with existing solutions for small disturbances. The nonlinear calculations reveal that the behavior of a disturbance depends on the amplitude and on the wave number. The behavior at wave numbers below and above the linear critical wave number is drastically different.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve

1967 ◽  
Vol 29 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Chaim L. Pekeris ◽  
Boris Shkoller
Keyword(s):  
Periodic Solutions ◽  
Linear Theory ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Lower Branch ◽  
Plane Poiseuille Flow ◽  
Periodic Disturbances ◽  
Existence Of Periodic Solutions ◽  
The Stability

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

1983 ◽  
Vol 133 ◽  
pp. 179-206 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu
Keyword(s):  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Plane Poiseuille Flow ◽  
Supercritical Region ◽  
Subcritical Region ◽  
Poor Convergence ◽  
The Stability ◽  
Good Agreement ◽  
False Problem

In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.

Stability of Viscoelastic Channel Flow

2007 ◽  
Author(s):  
Nariman Ashrafi
Keyword(s):  
Reynolds Number ◽  
Viscosity Ratio ◽  
Weissenberg Number ◽  
Base Flow ◽  
Galerkin Projection ◽  
Large Reynolds Number ◽  
One Dimensional ◽  
Stress Effects ◽  
The Stability ◽  
The One

The nonlinear stability and bifurcation of the one-dimensional channel (Poiseuille) flow is examined for a Johnson-Segalman fluid. The velocity and stress are represented by orthonormal functions in the transverse direction to the flow. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. The stability picture is dramatically influenced by the viscosity ratio. The range of shear rate or Weissenberg number for which the base flow is unstable increases from zero as the fluid deviates from the Newtonian limit as decreases. Typically, two turning points are observed near the critical Weissenberg numbers. The transient response is heavily influenced by the level of inertia. It is found that the flow responds oscillatorily. When the Reynolds number is small, and monotonically at large Reynolds number when elastic effects are dominated by inertia.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

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