Global stability in the Bénard problem for a mixture with superimposed plane parallel shear flows

The Lyapunov direct method is used to study the non‐linear stability of parallel convective shear flows of a mixture heated and salted from below for any Schmidt and Prandtl numbers. Global non‐linear exponential stability for small values of Reynolds number R is found and conditional stability results up to the criticality which are independent of R are given for rigid and stress‐free boundaries. Copyright © 2000 John Wiley & Sons, Ltd.     

Non‐linear stability for convection with quadratic temperature dependent viscosity

In this paper, we study the non‐linear stability of convection for a Newtonian fluid heated from below, where the viscosity of the fluid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Ra_c, the rest state of the fluid and the steady temperature distribution remains non‐linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16 :347–352). The central contribution of this paper lies in a simpler proof of non‐linear stability, than the ones in the current literature, by use of a suitable maximum principle argument. Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐linear stability in the Bénard problem for a double‐diffusive mixture in a porous medium

The linear and non‐linear stability of a horizontal layer of a binary fluid mixture in a porous medium heated and salted from below is studied, in the Oberbeck–Boussinesq–Darcy scheme, through the Lyapunov direct method. This is an interesting geophysical case because the salt gradient is stabilizing while heating from below provides a destabilizing effect. The competing effects make an instability analysis difficult. Unconditional non‐linear exponential stability is found in the case where the normalized porosity ? is equal to one. For other values of ? a conditional stability theorem is proved. In both cases we demonstrate the optimum result that the linear and non‐linear critical stability parameters are the same whenever the Principle of Exchange of Stabilities holds. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability of a layer of dipolar fluid heated from below

The equations of Bleustein and Green [2] are formulated in a way suitable to describe the convective instability which occurs when a layer of dipolar fluid is heated from below. The linear instability boundary is shown to coincide with the nonlinear stability curve and the critical Rayleigh numbers describing this boundary are found; in particular, the non-dimensional micro-length is found to always stabilize.     

Steady convection in the presence of an external magnetic field

The problem of the equilibrium of a liquid enclosed in a vessel heated from below has been considered by Sorokin [1], Iudovich and Ukhovskii [2] and Velt [3]. It has been established that if the Rayleigh number λ exceeds a certain critical value λ₀, then secondary steady flows arise in the liquid.

The stability of a conductive liquid heated from below has been studied by many authors. The most complete and general studies are those of Sorokin and Sushkin [4], whose paper contains the appropriate bibliography, and that of Shliomis [5]. The results of [4 and 5] make clear the physical picture of the phenomena associated with the heating of a conductive fluid and indicate the possible existence of secondary steady and periodic flows.

The existence of steady convective flows in a conductive liquid are proved below. Our study is based on the procedure set forth in [2].     

On stability of steady-state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid

A problem on linear stability of stationary plane–parallel shearing flows in a homogeneous in density inviscid incompressible fluid between two immovable impermeable solid parallel infinite plates is studied. With the use of the direct Lyapunov method it is shown that all sufficient conditions (by Rayleigh, Fjørtoft, Arnol’d) known to this moment for stability of these flows with respect to small plane perturbations are the necessary ones as well. An a priori lower estimate is constructed; the estimate displays exponential in time growth of the considered perturbations if these conditions are not affected. An analytical example of steady-state plane–parallel shearing flows and superimposed small plane perturbations growing in time in accordance with the constructed estimate is given.     

Blowup of solutions for a class of non‐linear evolution equations with non‐linear damping and source terms

We consider the blowup of solutions of the initial boundary value problem for a class of non‐linear evolution equations with non‐linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, α}, where m, α and p are non‐negative real numbers and m+1, α+1, p+1 are, respectively, the growth orders of the non‐linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above‐mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non‐linear terms, the states of the initial energy and the existence and non‐existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Experimental investigation of natural convection in rotating and non rotating wide spherical annuli

The axisymmetric and stationary basic flow due to the unstable stratification in the lower region of a spherical annulus cooled from inside has already been investigated in some detail ([5], [4]). The same applies to the columnar cell flow regime of high Taylor numbers dominated by the force component perpendicular to the rotation axis ([2] and [3]). Yet the question of the initial mechanism of symmetry breaking at medium Taylor numbers has still not been investigated. Therefore the present paper investigates the onset of natural convection in a differentially heated wide spherical gap. The stability curves for radius ratio η = 0.4 and a silicon oil fluid with Pr = 39 are determined by varying Rayleigh and Taylor number. The m = 2 vortex is investigated in detail using quasi 3D Particle Image Velocimetry (PIV). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.     

Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales

We study the Rayleigh–Bénard convection in a 2D rectangular domain with no‐slip boundary conditions for the velocity. The main mathematical challenge is due to the no‐slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free‐slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold R_c, the system bifurcates to an attractor, which is an (m ? 1)‐dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses R_c. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one‐dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, T_A1T_{A_1}, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T_A1T_{A_1} , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.      

Unconditional non‐linear stability for a fluid of third grade

The thermal convection in a layer of a third grade fluid is investigated, with viscosity being a general function of temperature. We develop a non‐linear stability analysis and prove that unconditional non‐linear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a fluid of third grade are preferable to those for a fluid of second grade or a dipolar fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

Double diffusive convection in porous media under the action of a magnetic field

The onset of thermal convection in an electrically conducting fluid saturating a porous medium, uniformly heated from below, salted by one chemical and embedded in an external transverse magnetic field is analyzed. The critical Rayleigh thermal numbers at which steady and Hopf convection can occur, are determined. Sufficient conditions guaranteeing the effective onset of convection via steady or oscillatory state are provided.

     

Effect of dust particles on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

The problem of the effect of dust particles on the thermal convection in micropolar ferromagnetic fluid saturating a porous medium subject to a transverse uniform magnetic field has been investigated theoretically. Linear stability analysis and normal mode analysis methods are used to find an exact solution for a flat micropolar ferromagnetic fluid layer contained between two free boundaries. In case of stationary convection, the effect of various parameters like medium permeability, dust particles, non-buoyancy magnetization, coupling parameter, spin-diffusion parameter and micropolar heat conduction parameter are analyzed. For sufficiently large values of magnetic parameter M₁, the critical magnetic thermal Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. It is also observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and dust particles.     

Zur Stabilität der Strömung in einem horizontalen Rohr bei ungleichmässig erwärmter Wand

Summary The linear fluid motion in a horizontal pipe heated from below is considered, when the temperature field outside the pipe is linear. By the method of small perturbations it is shown that the fluid motion becomes unstable for a critical value of the Rayleigh number, i.e., there is an additional convective motion, the streamlines of which are concentric curves in the plane of a cross-section of the pipe.

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Die Untersuchung wurde vom Wirtschaftsministerium des Landes Baden-Württemberg gefördert.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Hamel‐isomorphic images of the unit ball

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ?² and ?. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non‐measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non‐measurable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of hydromagnetic laminar flows in an inclined heated layer

Laminar flows of conducting fluids with an imposed magnetic field play an important role in many applications, for instance in geophysics, astrophysics, e.g. when dealing with solar winds, industry, biology, in metallurgy, in biofilms, etc. Also many engineering applications require heating at the boundaries. The inclination has been examined by some authors mainly in theoretical applications, geophysical studies, and materials processing. In Falsaperla et al. (Laminar hydromagnetic flows in an inclined heated layer, 2016) we have investigated analytical solutions of stationary laminar flows of an inclined layer filled with a hydromagnetic fluid heated from below and subject to the gravity field. In this article we study linear instability and nonlinear stability of some of the above solutions and investigate the critical stability/instability thresholds.     

On the Rayleigh wave on a curvilinear boundary between elastic and fluid media

Along the boundary between elastic and fluid media, the surface Rayleigh wave propagates. The velocity of this wave v _R0 in the case of a plane boundary is less than the velocity of the Rayleigh wave v _R on a free plane boundary of an elastic medium and less than the velocity v _P0 in a fluid medium. To investigate the velocity v _R0 in the case of curvilinear boundaries, the propagation of Rayleigh waves under consideration along cylindrical and spherical surfaces is studied. The velocity of the Rayleigh wave depends on the curvature of the wave trajectory and the curvature in the direction perpendicular to the trajectory. Furthermore this velocity depends on the presence or absence of a fluid medium. Bibliography: 5 titles.