Isom H. Herron

On the Principle of Exchange of Stabilities in Rayleigh--Bénard Convection

The problem of Rayleigh--Benard convection with internal heat sources and a variable gravity field is treated. For the case of stress-free boundary conditions, it is proved that the principle of exchange of stabilities holds as long as the product of gravity field and the integral of the heat sourcesis nonnegative throughout the layer. The proof is based on the idea of a positive operator, and uses the positivity properties of Greens function.

The Orr-sommerfeld equation on infinite intervals

Numerical and asymptotic methods have been developed the last several years to solve the Orr–Sommerfeld (OS) stability equation for boundary layers and other unbounded domain flows. These are surveyed and some of the abstract mathematical questions they raise are mentioned.There has been a conventional wisdom among fluid dynamicists that it has not been “proved” that the eigenvalues of the OS equations are complete on an unbounded domain. There is something to this, since without the continuous spectrum, which occurs for boundary layer flows, the eigenvalues are not complete. However, much has been proved concerning the nature of the spectrum and the expansion properties of the generalized eigenfunctions. The results of Miklavcic and Williams, and of Bakenko and Herron, form the core of the survey.

A completeness observation on the stability equations for stratified viscous shear flows

The DiPrima–Habetler completeness theorem applies to the system of equations governing the linearized stability of stratified viscous shear flows.

Expansion problems in the linear stability of boundary layer flows

Several problems in the linearized stability of boundary layers are examined. They are all treated as perturbations of constant coefficient differential operators. Spectral theory and spectral expansions are developed. Possible anomalies, which might arise for nonparallel boundary layer flows with nonzero transverse component at infinity are also handled.

A Theoretical Model for Describing Iris Dynamics

We present a theoretical approach using what we know about tissue dynamics to explore the nonlinear dynamics of iris deformation. Current iris recognition algorithms assume a simple transformation to approximate the deformation of the iris tissue. Furthermore, current research work on iris deformation does not take into account the mechanical properties of the iris tissue nor the cause of deformation from the iris muscle activity. By looking at the tissue dynamics, we are able to gain a more comprehensive understanding of this deformation process. The results of this research work can potentially be leveraged into existing iris recognition systems.

The stability of Couette flow in a toroidal magnetic field

The stability of the hydromagnetic Couette flow is investigated when a constant current is applied along the axis of the cylinders. It is shown that if the resulting toroidal magnetic field depends only on this current, no linear instability to axisymmetric disturbances is possible.

ONSET OF INSTABILITY IN HYDROMAGNETIC COUETTE FLOW

The stability of viscous flow between rotating cylinders in the presence of a constant axial magnetic field is considered. The boundary conditions for general conductivities are examined. It is proved that the Principle of Exchange of Stabilities holds at zero magnetic Prandtl number, for all Chandrasekhar numbers, when the cylinders rotate in the same direction, the circulation decreases outwards, and the cylinders have insulating walls. The result holds for both the finite gap and the narrow gap approximation.

Exchange of Stabilities for Görtler Flow

A general approach to the principle of exchange of stabilities (PES), formulated by H. F. Weinberger for Couette flow is shown to apply to Gortler flows with a free surface boundary condition. PES has always been assumed for Gortler flow, but this is the first mathematical justification of it for any unbounded flow.

Instabilities in the Görtler model for wall bounded flows

Abstract The original Gortler model is analyzed with no-slip boundary conditions on the wall. These conditions have historically been the most difficult to treat. It is proved that the principle of exchange of stabilities holds, that is, the first unstable eigenvalue has imaginary part equal to zero. The techniques used involve factoring positive operators.

Stability criteria for flow along a convex wall

Flow along a curved wall is susceptible to centrifugal instability. The criterion of Synge [Proc. R. Soc. London Ser. A 167, 250 (1938)] for the stability of flow between rotating cylinders is generalized to the case of an arbitrary flow along a convex wall. Sufficient conditions for the linear stability of the flow are given, based on the behavior of a function similar to the Rayleigh discriminant.