Weakly nonlinear behavior of periodic disturbances in two‐layer Couette–Poiseuille flow

1989 ◽  
Vol 1 (10) ◽  
pp. 1666-1676 ◽  
Author(s):  
Yuriko Renardy
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Behavior ◽  
Weakly Nonlinear ◽  
Periodic Disturbances

Review on the Dynamics of Isothermal Liquid Bridges

2019 ◽  
Vol 72 (1) ◽  
Author(s):  
José M. Montanero ◽  
Alberto Ponce-Torres
Keyword(s):  
Nonlinear Behavior ◽  
Forced Oscillations ◽  
Liquid Bridges ◽  
Contact Lines ◽  
Surfactant Monolayers ◽  
Weakly Nonlinear ◽  
Nonlinear Motion ◽  
Bridge Problem ◽  
Equilibrium Shapes ◽  
Streaming Flows

Abstract In this review, we describe both theoretical and experimental results on the dynamics of liquid bridges under isothermal conditions with fixed triple contact lines. These two major restrictions allow us to focus on a well-defined body of literature, which has not as yet been reviewed in a comprehensive way. Attention is mainly paid to liquid bridges suspended in air, although studies about the liquid–liquid configuration are also taken into account. We travel the path from equilibrium to nonlinear dynamics of both Newtonian liquid bridges and those made of complex fluids. Specifically, we consider equilibrium shapes and their stability, linear dynamics in free and forced oscillations under varied conditions, weakly nonlinear behavior leading to streaming flows, fully nonlinear motion arising during stretching and breakup of liquid bridges, and problems related to rheological effects and the presence of surfactant monolayers. Although attention is mainly paid to fundamental aspects of these problems, some applications derived from the results are also mentioned. In this way, we intend to connect the two approaches to the liquid bridge problem, something that both theoreticians and experimentalists may find interesting.

On the nonlinear theory of stability of plane Poiseuille flow in the subcritical range

1982 ◽  
Vol 381 (1781) ◽  
pp. 407-418 ◽  
Keyword(s):  
Reynolds Number ◽  
Nonlinear Theory ◽  
Poiseuille Flow ◽  
Plane Poiseuille Flow ◽  
New Approach ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

A new approach is proposed in the application of the weakly nonlinear theory, initiated by J. T. Stuart, to the problem of stability of plane Poiseuille flow in the subcritical range. Some results of computation are presented. The calculated threshold amplitudes for Reynolds number R = 5000 and 4000 with different frequencies are compared with Nishioka’s experimental observations. The results are encouraging.

Effect of axial flow on viscoelastic Taylor–Couette instability

1998 ◽  
Vol 360 ◽  
pp. 341-374 ◽  
Author(s):  
M. D. GRAHAM
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Axial Flow ◽  
Weissenberg Number ◽  
Scaling Analysis ◽  
Plane Shear ◽  
Weakly Nonlinear ◽  
Elastic Stresses ◽  
Circular Couette Flow ◽  
The Stability

Viscoelastic flow instabilities can arise from gradients in elastic stresses in flows with curved streamlines. Circular Couette flow displays the prototypical instability of this type, when the azimuthal Weissenberg number Weθ is O(ε−1/2), where ε measures the streamline curvature. We consider here the effect of superimposed steady axial Couette or Poiseuille flow on this instability. For inertialess flow of an upper-convected Maxwell or Oldroyd-B fluid in the narrow gap limit (ε[Lt ]1), the analysis predicts that the addition of a relatively weak steady axial Couette flow (axial Weissenberg number Wez=O(1)) can delay the onset of instability until Weθ is significantly higher than without axial flow. Weakly nonlinear analysis shows that these bifurcations are subcritical. The numerical results are consistent with a scaling analysis for Wez[Gt ]1, which shows that the critical azimuthal Weissenberg number for instability increases linearly with Wez. Non-axisymmetric disturbances are very strongly suppressed, becoming unstable only when ε1/2Weθ= O(We2z). A similar, but smaller, stabilizing effect occurs if steady axial Poiseuille flow is added. In this case, however, the bifurcations are converted from subcritical to supercritical as Wez increases. The observed stabilization is due to the axial stresses introduced by the axial flow, which overshadow the destabilizing hoop stress. If only a weak (Wez[les ]1) steady axial flow is added, the flow is actually slightly destabilized. The analysis also elucidates new aspects of the stability problems for plane shear flows, including the exact structure of the modes in the continuous spectrum, and illustrates the connection between these problems and the viscoelastic circular Couette flow.

Degenerate bifurcation in stably stratified plane Poiseuille flow

1997 ◽  
Vol 331 ◽  
pp. 261-282 ◽  
Author(s):  
K. FUJIMURA ◽  
R. E. KELLY
Keyword(s):  
Prandtl Number ◽  
Poiseuille Flow ◽  
Liquid Metals ◽  
Landau Equation ◽  
Nonlinear Coefficient ◽  
Stable Stratification ◽  
Plane Poiseuille Flow ◽  
Weakly Nonlinear ◽  
Supercritical Bifurcation ◽  
Higher Order Terms

Bifurcation characteristics of stably stratified plane Poiseuille flow have been investigated on a weakly nonlinear basis. It is found that the results are sensitive to the value of the Prandtl number, in that subcritical bifurcation persists for most values of the Prandtl number but is replaced by supercritical bifurcation over a range of small values of the Prandtl number. This range includes values characteristic of some liquid metals. The bifurcation becomes degenerate at a particular parameter set where the real part of the cubic nonlinear coefficient in the Stuart–Landau equation vanishes at criticality, and the situation is discussed by including higher-order terms in the manner of Eckhaus & Iooss (1989). An exact hyper-degenerate situation is also found to be possible for which the cubic and the quintic nonlinear coefficients lose their real parts simultaneously; this case is also analysed. For large values of the Prandtl number, stable stratification tends to promote subcritical instability.

Weakly nonlinear behavior of a plate thickness-mode piezoelectric transformer

2007 ◽  
Vol 54 (4) ◽  
pp. 877-881 ◽  
Author(s):  
Jiashi Yang ◽  
Ziguang Chen ◽  
Yuantai Hu ◽  
Shunong Jiang ◽  
Shaohua Guo
Keyword(s):  
Plate Thickness ◽  
Nonlinear Behavior ◽  
Piezoelectric Transformer ◽  
Weakly Nonlinear ◽  
Thickness Mode

Weakly Nonlinear Analysis of a Localized Disturbance in Poiseuille Flow

2000 ◽  
Vol 105 (2) ◽  
pp. 121-141 ◽  
Author(s):  
D. Ponziani ◽  
C. M. Casciola ◽  
F. Zirilli ◽  
R. Piva
Keyword(s):  
Nonlinear Analysis ◽  
Poiseuille Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis

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.

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