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.     

The small magnetic Prandtl number approximation suppresses magnetorotational instability

Axisymmetric stability of viscous resistive magnetized Couette flow is re-examined, with emphasis on flows that would be hydrodynamically stable according Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. In this regime, mag- netorotational instability (MRI) may occur. The governing system in cylindrical coordinates is of tenth order. It is proved, by methods based on those of Synge and Chandrasekhar, that by dropping one term from the system, MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions. This term is often neglected because it has the magnetic Prandtl number, which is very small, as a factor; nevertheless it is crucially important. (Received: August 11, 2005; revised: January 3, 2006)     

Standard and Helical Magnetorotational Instability

The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable rotating shear flows, e.g., in accretion disks. What laws of differential rotation are susceptible to the destabilization by axial, azimuthal, or helical magnetic field? The answer to this question, which is vital for astrophysical and experimental applications, inevitably leads to the study of spectral and geometrical singularities on the instability threshold. The singularities provide a connection between seemingly discontinuous stability criteria and thus explain several paradoxes in the theory of MRI that were poorly understood since the 1950s.     

Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.     

Spurious Behaviour of Numerically Computed Fluid Flow

We investigate the stability to aliasing errors of numericalschemes for hydrodynamics, taking the viscous Burgers' equationas a model for systems with a term that is quadratic in thevelocity. Considering wavelengths equal to three times the mesh-spacing,and arbitrary mean flow, we are able to demonstrate explicitlyfor common schemes (a) a sufficient criterion for stabilityand (b) blow-up of solutions in a finite time when (a) is violated.Singular behaviour is shown to persist at all wavelengths: studiesof wavelengths up to thirty times mesh-spacing make it clearthat a profile with a single region of strong convergent flowis most conducive to instability. In contrast, spectral (Galerkin)and upwind schemes are shown to be stable for all flows andperiods.     

Singularities on the boundaries of magnetorotational instabilities and scaling laws

In the theory of magnetorotational instability and its modern extensions such as the helical MRI, non-trivial scaling laws between the critical parameters are observed. In case of the standard MRI it is well known that the Reynolds and Hartmann numbers are scaled as Re ∼ Ha² while for the helical MRI Re ∼ Ha³ . What is less known is that the thresholds of SMRI and HMRI plotted as surfaces in the space of parameters, possess singularities that determine the scaling laws. Moreover, the two paradoxes of SMRI and HMRI in the limits of infinite and zero magnetic Prandtl number (Pm), respectively, sharply correspond to the singularities on the instability thresholds. In either case, it is the local Plücker conoid structure that explains the non-uniqueness of the critical Rossby number, and its crucial dependence on the Lundquist number. For HMRI, we have found an extension of the former Liu limit Ro_c ≃ −0.828 (valid for Lu = 0 ) to a somewhat higher value Ro ≃ −0.802 at Lu = 0.618 which is, however, still below the Kepler value. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of Thermal Stratification on the Stability of Ekman-Couette-Flow

The Ekman-Couette-System consists of two infinitely extended plates which are sheared in opposite directions over a fluid and are additionally rotated about their normal axis. In the case of angular velocities which tend to zero, the system becomes the classical Couette-System, whereas for high angular velocities the boundary layers of the upper and lower plate are separated and represent Ekman boundary layers. For both limit cases the influence of thermal stratification on the stability of the base flow has been a subject of research for some time, but not so for moderate angular velocities. This was the motivation for doing a linear stability analysis for that case, including both stable and unstable stratification for a Prandtl number equal to unity. The results show, that as expected, stable stratification is suppressing the emergence of stationary as well as Type I- and Type II-shear-instabilities, while unstable stratification is supporting them. For unstable stratification, the system can also become unstable to a convection instability with all its properties known from other systems, except for that their orientation angle is not coincidental but determined due to the influence of the shear and Coriolis forces. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Onset of thermal convection and influence of rotation in wide spherical gap flow

The onset of thermal convection and the effect of rotation in a high Prandtl number fluid in a wide gap between two concentric spheres with an axial force field are investigated experimentally. Both spheres rotate along the vertical axis with the same angular velocity Ω while the inner one (r₁) is cooled and the outer one (r₂) is heated. The velocity field is investigated by different visualization techniques and Particle Image Velocimetry (PIV). The axisymmetric basic flow is disturbed by local instabilities. At a Rayleigh number of Ra = 6.97 · 10⁶, a pulsing vortex develops in the south polar region. A different, coexisting instability in the outer boundary layer appears at Ra = 1.79 · 10⁷. Rotating with Taylor numbers Ta > 1.4 · 10⁵, this instability vanishes. The instabilities occur mainly in the southern hemisphere where the thermal stratification is unstable.     

A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics(MHD)boundary layer system,it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by G′erard-Varet and Dormy(2010).This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.     

The null spaces of elliptic partial differential operators in Rn

Overstability in a horizontal layer of a viscoelastic fluid is considered in the presence of a uniform magnetic field. The equations of motion appropriate to hydromagnetics in a Maxwellian fluid have been established and the analysis has been carried out in terms of normal modes. The proper solutions have been obtained for the case of two free boundaries. The dispersion relation obtained is found to be quite complex and involves the Prandtl number p₁, magnetic Prandtl number p₂, a parameter Q characterizing the strength of the magnetic field, and a parameter Γ which characterizes the elasticity of the fluid. Numerical calculations have been performed for different values of the parameters involved and the values of critical Rayleigh numbers, wave numbers, and frequencies for the onset of instability as overstability have been obtained. It is found that the magnetic field has a stabilizing influence on the overstable mode of convection in a viscoelastic fluid. Elasticity is found to have a destabilizing influence as in the absence of a magnetic field. Thus the effect of a magnetic field is the same as that for an ordinary viscous fluid.     

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in {\mathbb{R}^{3}}

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015).     

Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate

A numerical study of magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate is presented. The onset position characterized by the local Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The Coriolis force and the Lonertz force have significant effects on the formation of longitudinal vortices and the associated instability. Positive rotation stabilizes the flow on the rotating flat surface. On the contrary, a negative rotation destabilizes the flow. The flow is found more stable as the value of Hartmann number increases. The numerical data show reasonable agreement with the experimental results with the case of thermal instability in natural convection over a flat plate heated from below.     

前后排列旋转圆柱体对流热交换的格子Boltzmann方法解

用格子Boltzmann方法,数值研究流过前后排列两旋转圆柱体的二维层流．用二阶精度的速度场和温度场,数值化涉及运动的曲线边界．在Reynolds数为100,Prandtl数为0.71时,研究旋转速度比的变化和不同间距的影响．在4种不同间距(3, 1.5, 0.7, 0.2)下,研究旋转速度比的不同范围．结果表明,当间距取大数值时,第1个圆柱体的升力和阻力系数,与单个圆柱体相类似;对所有间距(除间距3以外),第2个圆柱体的升力系数,随着角速度的增加而减小,而阻力系数反而增加．圆柱体表面平均周期Nusselt数的结果表明,当两圆柱体间距小且角速度又低时,热传导是主要的传热机理,而当间距大且角速度又高时,对流是主要的传热机理．     

Frequency-Domain Criterion for the Global Stability of Dynamical Systems with Prandtl Hysteresis Operator

In the present paper, dynamical systems with Prandtl hysteresis operator are considered. For the class of dynamical systems under consideration, a frequency-domain global stability criterion is formulated and proved. For a second-order dynamical system with Prandtl operator, we demonstrate the advantage of the obtained criterion as compared to the well-known criterion derived by Logemann and Ryan.     

An Instability Criterion for Nonlinear Standing Waves on Nonzero Backgrounds

A nonlinear Schrödinger equation with repulsive (defocusing) nonlinearity is considered. As an example, a system with a spatially varying coefficient of the nonlinear term is studied. The nonlinearity is chosen to be repelling except on a finite interval. Localized standing wave solutions on a non-zero background, e.g., dark solitons trapped by the inhomogeneity, are identified and studied. A novel instability criterion for such states is established through a topological argument. This allows instability to be determined quickly in many cases by considering simple geometric properties of the standing waves as viewed in the composite phase plane. Numerical calculations accompany the analytical results.     

Mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium under the influence of Ohmic heating,Soret and Dufour effects

A numerical model is developed to examine the combined effects of Soret and Dufour on mixed convection magnetohydrodynamic heat and mass transfer in micropolar fluid-saturated Darcian porous medium in the presence of thermal radiation, non-uniform heat source/sink and Ohmic dissipation. The governing boundary layer equations for momentum, angular momentum (microrotation), energy and species transfer are transformed to a set of non-linear ordinary differential equations by using similarity solutions which are then solved numerically based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters with appropriate boundary conditions. The influence of Darcy number, Prandtl number, Schmidt number, Soret number and Dufour number, magnetic parameter, local thermal Grashof number and local solutal Grashof number on velocity, temperature and concentration fields are studied graphically. Finally, the effects of related physical parameters on local Skin-friction, local Nusselt number and local Sherwood number are also studied. Results showed that the fields were influenced appreciably by the Soret and Dufour effects, thermal radiation and magnetic field, etc.     

Fundamental principle and a basis in invariant subspaces

In the paper, first-order complex sequences with finite maximal angular density are studied. A criterion for such a sequence to be a part of a regularly distributed set with a given angular density is obtained. Using this criterion, we present complete solutions of fundamental principle problems and basis for an invariant subspace of analytic functions in a bounded convex domain.     

Spatial parity violation and the turbulent magnetic Prandtl number

Using the field theory renormalization-group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the model of kinematic magnetohydrodynamic turbulence, where the magnetic field behaves as a passive vector quantity advected by the helical turbulent environment given by the stochastic Navier-Stokes equation. We show that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and that the two-loop helical contribution to the turbulent magnetic Prandtl number is up to 4.2% of its nonhelical value. This result demonstrates the strong stability of the properties of diffusion processes of the magnetic field in turbulent environments with spatial parity violation compared with the corresponding systems without the helicity.     

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.     

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.