On the spectral problem in the linear stability study of flows on a sphere

A normal mode instability study of a steady nondivergent flow on a rotating sphere is considered. A real-order derivative and family of the Hilbert spaces of smooth functions on the unit sphere are introduced, and some embedding theorems are given. It is shown that in a viscous fluid on a sphere, the operator linearized about a steady flow has a compact resolvent, that is, a discrete spectrum with the only possible accumulation point at infinity, and hence, the dimension of the unstable manifold of a steady flow is finite. Peculiarities of the operator spectrum in the case of an ideal flow on a rotating sphere are also considered. Finally, as examples, we consider the normal mode stability of polynomial (zonal) basic flows and discuss the role of the linear drag, turbulent diffusion and sphere rotation in the normal mode stability study.     

Spectral approximation in the numerical stability study of nondivergent viscous flows on a sphere

The accuracy of calculating the normal modes in the numerical linear stability study of two-dimensional nondivergent viscous flows on a rotating sphere is analyzed. Discrete spectral problems are obtained by truncating Fourier's series of the spherical harmonics for both the basic flow and the disturbances to spherical polynomials of degrees K and N, respectively. The spectral theory for the closed operators [1], and embedding theorems for the Hilbert and Banach spaces of smooth functions on a sphere are used to estimate the rate of convergence of the eigenvalues and eigenvectors. It is shown that the convergence takes place if the basic state is sufficiently smooth, and the truncation numbers K and N of Fourier's series for the basic flow and disturbances tend to infinity keeping the ratio N/K fixed. The convergence rate increases with the smoothness of the basic flow and with the power s of the Laplace operator in the vorticity equation diffusion term. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:143–157, 1998     

On the search for singularities in incompressible flows

In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.     

On uniform exponential stability of linear switching system

We prove that the linear switching system , where is bounded valued square matrices and ?:[0,1,2,…)→Ω is an arbitrary switching signals, is uniformly exponentially stable if the sequence is bounded, where s(k) is bounded valued sequence.     

Bounds on the phase velocity in the linear instability of viscous shear flow problem in the β-plane

Results obtained by Joseph(J. Fluid Mech. 33 (1968) 617) for the viscous parallel shear flow problem are extended to the problem of viscous parallel, shear flow problem in the beta plane and a sufficient condition for stability has also been derived.     

Effect of slip on the linear stability of flow through a tube

A theoretical investigation of the linear stability of the flow of a Newtonian fluid through a tube is presented using an alternative boundary condition to the standard no-slip condition. The linear stability analysis is based on the classical method of infinitesimal axially symmetric harmonic perturbations super-imposed on the steady state solution. In this analysis the standard no-slip boundary condition is replaced with the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter, which may be regarded as a positive slip length. The aim of this analysis is to assess the effect of a nonzero slip length on the behavior of infinitesimal disturbances in the flow through a tube. It is demonstrated that for positive slip the rate of decay of the least damped disturbance is reduced, although the flow still remains stable to all infinitesimal disturbances of the type considered, as it does for the no-slip boundary condition.     

Non‐linear stability and convection for laminar flows in a porous medium with Brinkman law

The non‐linear stability of plane parallel shear flows in an incompressible homogeneous fluid heated from below and saturating a porous medium is studied by the Lyapunov direct method.In the Oberbeck–Boussinesq–Brinkman (OBB) scheme, if the inertial terms are negligible, as it is widely assumed in literature, we find global non‐linear exponential stability (GNES) independent of the Reynolds number R. However, if these terms are retained, we find a restriction on R (depending on the inertial convective coefficient) both for a homogeneous fluid and a mixture heated and salted from below. In the case of a mixture, when the normalized porosity ε is equal to one, the laminar flows are GNES for small R and for heat Rayleigh numbers less than the critical Rayleigh numbers obtained for the motionless state. Copyright © 2003 John Wiley & Sons, Ltd.     

Global existence of three dimensional incompressible MHD flows

We investigate the initial value problem for the three‐dimensional incompressible magnetohydrodynamics flows. Global existence and uniqueness of flows are established in the function space , provided that the norm of the initial data is less than the minimal value of the viscosity coefficients of the flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value ReE depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, ReE is shown to be at least 174.     

Description of stability for linear time-invariant systems based on the first curvature

This paper focuses on using the first curvature κ(t) of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems (Wang, Sun, Song et al, arXiv:1808.00290) to n-dimensional systems. We prove that for a system $\text{◂=▸}\dot{r}(t)=\text{◂⋅▸}Ar(t)$, (a) if there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂≠▸}\underset{t\to +\infty }{\lim }\kappa (t)\ne 0$ or $\underset{t\to +\infty }{\lim }\kappa (t)$ does not exist for any initial value in this set, then the zero solution of the system is stable; (b) if the matrix A is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂=▸}\underset{t\to +\infty }{\lim }\kappa (t)=+\infty$ for any initial value in this set, then the zero solution of the system is asymptotically stable.     

On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.     

Decay results of solutions to the incompressible Navier-Stokes flows in a half space

In a half space, we consider the asymptotic behavior of the strong solution for the non-stationary Navier-Stokes equations. In particular, the decay rates of the second order derivatives of the Navier-Stokes flows in (n?2) with 1?r?∞ are derived by using Lq−Lr estimates and a clever analysis on the fractional powers of the Stokes operator. In addition, we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space.     

On a necessary condition for stability in perturbed linear and convex programming

We show that a necessary condition for stable perturbations in linear and convex programming is valid on an arbitrary region of stability. Using point-to-set mappings, two new regions of stability are identified.Contribution of this author is part of his M.Sc. thesis in Applied Mathematics at McGill University.     

On the stability of infinite-dimensional linear inequality systems

The principal aim of this paper is to study the stability of the solution set mapping of a system composed by an arbitrary set of linear inequalities in an infinite-dimensional space. The unknowns space is assumed to be metrizable, which allows us to measure the size of any possible perturbation. Conditions guaranteeing the closedness, the lower semicontinuity and the upper semicontmuity of this mapping, at a particular nominal system, are given in the paper. The more significant differences with respect to the finite dimensional case, previously approached in the context of the so-called semi-infinite optimization, are illustrated by means of convenient examples.     

On the linear stability of hyperbolic PDEs and viscoelastic flows

The issue addressed in this paper is whether linear stability can be determined from the spectrum. We present a counterexample for a hyperbolic PDE in two dimensions and a positive result for parallel shear flows of a class of viscoelastic fluids.     

On Howard’s conjecture in heterogeneous shear flow problem

Howard’s conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy forcegΒ ≪ 1 (Miles J W,J. Fluid Mech. 10 (1961) 496–508), where Β is the basic heterogeneity distribution function). An erratum to this article is available at .