scholarly journals On a model equation that reflects some of the shear flow hydrodynamic stability properties

2006 ◽  
Vol 133 (31) ◽  
pp. 29-40
Author(s):  
V.D. Djordjevic
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Hydrodynamic Stability ◽  
Model Equation ◽  
Landau Equation ◽  
Wave Number ◽  
Legendre Functions ◽  
Weakly Nonlinear ◽  
Stability Properties ◽  
Weakly Nonlinear Theory

A model equation is proposed in the paper that mimics some of the shear flow hydrodynamic stability properties. It contains the basic velocity profile which can be arbitrarily chosen, and a nonlinear term, whose form can be appropriately adjusted to any particular problem. Full linear and weakly nonlinear theories for the Bickley jet velocity profile are elaborated. The solution of the linear problem is obtained in terms of associated Legendre functions. Within the weakly nonlinear theory a Landau equation is derived that describes the evolution of the perturbations near the critical wave number. The conditions for supercritical stability and subcritical instability are revealed. AMS Mathematics Subject Classification (2000) 76E05, 76E30.

scholarly journals Weakly Nonlinear Hydrodynamic Stability of the Thin Newtonian Fluid Flowing on a Rotating Circular Disk

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Cha'o-Kuang Chen ◽  
Ming-Che Lin
Keyword(s):  
Stability Analysis ◽  
Newtonian Fluid ◽  
Rotation Number ◽  
Hydrodynamic Stability ◽  
Stability Region ◽  
Multiple Scales ◽  
Landau Equation ◽  
Circular Disk ◽  
Necessary Condition ◽  
Weakly Nonlinear

The main object of this paper is to study the weakly nonlinear hydrodynamic stability of the thin Newtonian fluid flowing on a rotating circular disk. A long-wave perturbation method is used to derive the nonlinear evolution equation for the film flow. The linear behaviors of the spreading wave are investigated by normal mode approach, and its weakly nonlinear behaviors are explored by the method of multiple scales. The Ginzburg-Landau equation is determined to discuss the necessary condition for the existence of such flow pattern. The results indicate that the superctitical instability region increases, and the subcritical stability region decreases with the increase of the rotation number or the radius of circular disk. It is found that the rotation number and the radius of circular disk not only play the significant roles in destabilizing the flow in the linear stability analysis but also shrink the area of supercritical stability region at high Reynolds number in the weakly nonlinear stability analysis.

Instability of spatially quasi-periodic states of the Ginzburg-Landau equation

1994 ◽  
Vol 444 (1921) ◽  
pp. 347-362 ◽  
Keyword(s):  
Linear Stability ◽  
Hydrodynamic Stability ◽  
Model Equation ◽  
Landau Equation ◽  
Geometric Interpretation ◽  
Sufficient Condition ◽  
Ginzburg Landau ◽  
Frequency Map ◽  
The Stability ◽  
Two Parameter

The Ginzburg-Landau (GL) equation with real coefficients is a model equation appearing in superconductor physics and near-critical hydrodynamic stability problems. The stationary GL equation has a two-parameter ( I 1 , I 2 ) family of spatially quasi-periodic (QP) states with frequencies ( ω 1 , ω 2 ) and frequency map with determinant ∆ K = ∂( ω 1 , ω 2 ) / ∂( I 1 , I 2 ). In this paper the linear stability of these QP states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of ∆ K : when ∆ K < 0 the spatially QP state is unstable and ∆ K > 0 is a necessary but not sufficient condition for linear stability. There is an interesting relation between ∆ K and the KAM persistence theorem for invariant toroids.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

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