Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

1996 ◽  
Vol 106 (3) ◽  
pp. 281-287 ◽  
Author(s):  
Mihir B Banerjee ◽  
R G Shandil ◽  
Balraj Singh Bandral
Keyword(s):  
Eigenvalue Bounds ◽  
Backward Wave ◽  
Sommerfeld Equation

Eigenvalue Bounds for the Orr-Sommerfeld Equation and Their Relevance to the Existence of Backward Wave Motion

1999 ◽  
Vol 103 (1) ◽  
pp. 43-50 ◽  
Author(s):  
Mihir B. Banerjee ◽  
R. G. Shandil ◽  
M. G. Gourla ◽  
S. S. Chauhan
Keyword(s):  
Wave Motion ◽  
Eigenvalue Bounds ◽  
Backward Wave ◽  
Sommerfeld Equation

Eigenvalue Bounds for the Orr—Sommerfeld Equation and their Relevance to the Existence of Backward Wave Motion—II

2000 ◽  
Vol 105 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Mihir B. Banerjee ◽  
R. G. Shandil ◽  
S. S. Chauhan ◽  
Daleep Sharma
Keyword(s):  
Wave Motion ◽  
Eigenvalue Bounds ◽  
Backward Wave ◽  
Sommerfeld Equation

Note on eigenvalue bounds for the Orr-Sommerfeld equation

1969 ◽  
Vol 38 (2) ◽  
pp. 273-278 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Boundary Conditions ◽  
Wave Velocity ◽  
Channel Flow ◽  
Eigenvalue Bounds ◽  
Complex Wave ◽  
Sommerfeld Equation

Bounds for the complex wave velocity c, determined by the Orr-Sommerfeld equation and the boundary conditions for channel flow, have been given by Joseph (1968a,b). In these notes it is shown how two of Joseph's theorems can be uniformly improved.

Eigenvalue bounds for the Orr—Sommerfeld equation. Part 2

1969 ◽  
Vol 36 (4) ◽  
pp. 721-734 ◽  
Author(s):  
Daniel D. Joseph
Keyword(s):  
Boundary Layer ◽  
Linear Stability ◽  
Channel Flow ◽  
Sufficient Conditions ◽  
Neutral Stability ◽  
Eigenvalue Bounds ◽  
Wave Speeds ◽  
Parallel Motion ◽  
Flow Part ◽  
Sommerfeld Equation

Rigorous estimates of amplification rates, wave speeds and sufficient conditions for linear stability are derived for the manifold of solutions of the Orr—Sommerfeld problem governing parallel motion in the boundary layer and in round pipes. The estimates for channel flow (part I) are improved and compared with numerical results for the neutral stability of Jeffery—Hamel flow.

Eigenvalue bounds for the Orr-Sommerfeld equation

1968 ◽  
Vol 33 (3) ◽  
pp. 617-621 ◽  
Author(s):  
Daniel D. Joseph
Keyword(s):  
Linear Stability ◽  
Isoperimetric Inequalities ◽  
Considerable Improvement ◽  
Eigenvalue Bounds ◽  
Sommerfeld Equation

Estimates of the eigenvalues C belonging to the manifold of solutions of the Orr-Sommerfeld equation are constructed by application of elementary isoperimetric inequalities. The inequalities also lead to a considerable improvement on the estimate of (αR) regions of linear stability given by Synge.

Stability and eigenvalue bounds of the flow of a dipolar fluid between two parallel plates

2005 ◽  
Vol 461 (2057) ◽  
pp. 1401-1421 ◽  
Author(s):  
Pratap Puri
Keyword(s):  
Newtonian Fluids ◽  
Sufficient Condition ◽  
Parallel Plates ◽  
Eigenvalue Bounds ◽  
Dipolar Fluids ◽  
Parallel Flows ◽  
Dipolar Fluid ◽  
The Stability ◽  
Sommerfeld Equation

In this article, we derive the Orr–Sommerfeld equation for the stability of parallel flows of a dipolar fluid. The classical results found by Squire, for viscous Newtonian fluids, are generalized to the case of dipolar fluids. A sufficient condition for stability is obtained for dipolar fluids and eigenvalue bounds for the Orr–Sommerfeld equation are found.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

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