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.      

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

Nonlinear Stability Problem of a Rotating Doubly Diffusive Porous Layer

Lyapunov direct method is applied to study the non-linear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer. For a Darcy number greater than or equal to 1000, and for any Prandtl number, Taylor number, and solute Rayleigh number it is found that the non-linear stability bound coincides with linear instability bound. For a Darcy number less than 1000, for a Prandtl number greater than or equal to one, and for a certain range of Taylor number, a coincidence between the linear and nonlinear (energy) stability thermal Rayleigh number values is still maintained. However, it is noted that for a Darcy number less than 1000, as the value of the solute Rayleigh number or the Taylor number increases, the coincidence domain between the two theories decreases quickly.     

Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.     

Energy stability in the Bénard problem for a fluid of second grade

The critical Rayleigh number for stability of convection in a fluid of second grade is obtained by means of the energy method. If the thermodynamic restrictions derived by Dunn and Fosdick [1] are adopted the nonlinear energy boundary is shown to be the same as that of linear theory, and the nonexistence of subcritical instabilities is deduced.     

On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > k _c (for some critical wave number k _c) implies the stability of the mode for a class of basic flows. Furthermore, if $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} , where U ₀ is the basic velocity, T ₀ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) , where the flow domain is 0 ≤ z ≤ D.     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

Global stability of centrifugal filtration convection

A nonlinear stability analysis is performed to study the onset of convection in a fluid saturated porous layer subject to alternating direction of the centrifugal body force. By introducing a suitable energy functional, the analysis is carried out for the Darcy and the Brinkman models of flow through porous media. The nonlinear result is unconditional and its sharpest limit is determined and is compared with the corresponding linear limit. The failure of linear theory in describing the instability is established in a certain region of the parameter space where possible subcritical instabilities may arise. The stability boundaries are discussed graphically for various values of the Darcy number and comparison is made with the available known results.     

Finite-element method simulation of rotating disk flow: effect of the transport of a chemical species

Electrochemical cells containing an iron rotating disk electrode which is dissolved in the electrolyte, an 1 M H₂SO₄ solution present a current instability in the plateau region, where the current is controlled by the mass transport. Dissolution of the electrode gives rise to a thin concentration boundary layer, due to a Schmidt number Sc = 2000. This boundary layer, together with the potential applied to the electrode, leads to an increase in the fluid viscosity and in a decrease in the diffusion coefficient, coupling the concentration and the chemical species field. Since the current is proportional to the concentration gradient at the interface, an instability of the coupled fields at Reynolds numbers attained in experimental conditions could be responsible for the current instability. Mangiavacchi [1] performed a linear stability analysis of the problem and showed that this is indeed the case. In this paper we review the main results of the stability analysis and present the main features of the FEM code recently developed in our group, to proceed with the investigation of the current instability observed in electrochemical cells. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions

Stability analysis of the rotating Bénard problem gives a spectral instability threshold of the purely conducting solution that can be expressed as a critical Rayleigh number R ² depending on the Taylor number T ². The definition of a functional which can be used to prove Lyapunov stability up to the threshold of spectral instability (optimal Lyapunov function) is an important step forward both, for a proof of nonlinear stability and for the investigation of the basin of attraction of the equilibrium. In previous works a Lyapunov function was found, but its optimality could be proven only for small T ². In this work we describe the reason why this happens, and provide a weaker definition of Lyapunov function which allows to prove that, for the linearized system, the spectral instability threshold is also the Lyapunov stability threshold for every value of T ².     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k _s ?>?0; it is also shown that instability is possible only for 0?<?k?<?k _s and not for k?>?k _s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c _i, the imaginary part of the phase velocity as the wave number k →k _s ? to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.     

On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P_n(μ) of degree n is stable to infinitesimal perturbations of every invariant set I_m with |m| ≥ n. For each zonal number m, I_m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

Suppression of instability in rotatory hydromagnetic convection

Recently discovered hydrodynamic instability [1], in a simple Bénard configuration in the parameter regime under the action of a nonadverse temperature gradient, is shown to be suppressed by the simultaneous action of a uniform rotation and a uniform magnetic field both acting parallel to gravity for oscillatory perturbations whenever (Qσ ₁/π² + J/π⁴) > 1 and the effective Rayleigh numberR(1 -T ₀α₂) is dominated by either 27π⁴(1 + l/σ₁/4 or 27π⁴/2 according as σ₁ ≥1 or σ₁ ≤ 1 respectively. HereT ₀is the temperature of the lower boundary while α₂ is the coefficient of specific heat at constant volume due to temperature variation and σ₁,R,Q andJ respectively denote the magnetic Prandtl number, the Rayleigh number, the Chandrasekhar number and the Taylor number.     

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.     

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.