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.     

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.     

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.     

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.     

Dynamisches Verhalten einer einfach besetzten rotierenden Welle mit Kardangelenken

Summary If a rotating, massless, elastic shaft carrying a disk is supported at the ends by Cardan links, the motion of the disk depends on the angles at the joints and the torques transmitted by the joints. The system is considered for constant angular velocity and constant torques of the driving shafts. The investigation of this nonstationary system leads to two second order differential equations with periodic coefficients. In order to establish conditions for instability the characteristics exponents are calculated by means of generalized Hills determinants. It is found that there exist critical intervals for the angular velocity.     

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)     

Parametric resonance in the problem of a heavy rigid body rolling along a straight line on a plane

The problem of the stability of a heavy rigid body, bounded by the surface of an ellipsoid and with a cavity in the form of a coaxial ellipsoid, rolling along a straight line on a horizontal rough plane is investigated. It is shown that in the case of a body that is close to being dynamically symmetrical, parametric resonance always occurs leading to instability of the rolling. Each ellipsoid has its own “individual” resonance angular velocity. In the general case, regions in which the necessary stability conditions are satisfied can be distinguished in parameter space. The problem of calculating the resonance coefficient corresponding to instability for parametric resonance in a reversible third-order system is solved.     

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.     

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)     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

Synchronization effects in rotors partially filled with fluid-influence on propulsion

In fluid-filled Rotors occur self-excited vibrations induced by surface waves of the fluid. A characteristic property is the instability over an interval of angular velocity above the natural frequency of the system. One explanation is the occurrence of synchronization effects between fluid waves and the critical rotor speed. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable for observer-based real-time control. This paper analyses a model based on a laval rotor and shows synchronization effects between fluid waves and rotor model and its influence on the rotor propulsion. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Distribution- Theoretic Approach to a Stability Problem of Orr

In a classic paper [1] of 1907, W. M'Farr Orr discovered, among other things, the “infinitesimal” instability of inviscid plane Couette flow. Surprisingly, although Orr's paper remains a standard reference in the field, later investigators [2, 3] have been able to call inviscid plane Couette flow stable without finding it necessary to controvert Orr's result. What has happened is that, at least in problems governed by linear (or linearized) equations with time-independent coefficients, the term “instability” has come to be identified with the presence of solutions exhibiting exponential time-growth. Orr found instability indeed: a class of solutions certain members of which grow in time by more than each preassigned factor. Unlike the exponential instabilities, however, Orr's solutions die away like 1/t after achieving their greatest growth. This ephemerality probably accounts for the discounting of Orr's result. Orr did not look into the general initial value problem. This is done in the sequel, with the result that the situation becomes clear. Under general disturbances, Couette flow turns out to be neither stable nor quasi-asymptotically stable*. The rate of growth depends on the smoothness of the initial data: classical solutions grow no faster than t, but sufficiently rough distribution-valued initial data leads to growth matching any power of t. Before presenting detailed results, we briefly review Orr's fundamental work on the problem.     

Unstable gap solitons in inhomogeneous nonlinear Schrödinger equations

A periodically inhomogeneous Schrödinger equation is considered. The inhomogeneity is reflected through a non-uniform coefficient of the linear and nonlinear term in the equation. Due to the periodic inhomogeneity of the linear term, the system may admit spectral bands. When the oscillation frequency of a localized solution resides in one of the finite band gaps, the solution is a gap soliton, characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, how to construct such gap solitons through a composite phase portrait is shown. By exploiting the phase-space method and combining it with the application of a topological argument, it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Surface gap solitons at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed. Numerical calculations are presented accompanying the analytical results.     

Synchronization effects in rotors partly filled with fluid

In fluid-filled rotors self-excited vibrations occur induced by a surface wave of the fluid. A characteristic property is the instability over the full range of angular velocity above the Eigenfrequency of the system. A possible explanation is the occurrence of synchronization effects between fluid and rotor. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable due to the complexity of fluid modelling. This paper analyses a simple minimal model and shows synchronization effects between fluid and rotor model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The relativistic theory of chemical binding

Starting from Breit’s relativistic equation for a system of two electrons, it is shown that for a hydrogen molecule (or for a system of two electrons moving in a field of cylindrical symmetry) the component of the total angular momentum (J _x ) along the axis of the molecule (axis of symmetry) is a constant of motion. Thus every eigenstate of the system is simultaneously an eigenstate of J _x also, and a state of the system will specify, besides its energy, only the eigenvalue of the component of the angular momentum parallel to the axis of symmetry. The form of the four large components of the wave function relating to their dependence on the azimuthal co-ordinates has been given. The case of Russel-Saunders approximation has been considered in detail and the nature of the components of the wave function for the singlet and triplet states has been discussed. It is shown that the wave function for the ground state of the hydrogen molecule could be expressed as a sum of a set of symmetric functions of which the first term is the Heitler-London function, and that the wave function for a triplet state should be a superposition of anti-symmetric molecular orbitals. It is shown that relativistic theory brings about in a natural manner the facts relating to the ground state of the molecules C₂ and O₂. Finally, some remarks are made concerning the case of molecules for which the spinorbit and the spin-spin couplings are strong.     

The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure

In this paper we investigate the effect of structural flexibility on rocking motion of a system consisting of a free standing rigid block with an attached chain of uniaxially moving point masses. Motion is excited by random acceleration of the ground; instability is associated with overturning of the overall structure. The condition of instability is constructed by the stochastic Melnikov method. We demonstrate a twofold effect of structural flexibility on the rocking response. The attached structure may increase the critical angular displacement and velocity in comparison with the similar parameters of the single rigid block. At the same time, the enlargement of the domain of stability enhances the contribution of the random perturbation in the Melnikov process. As a result, a lower level of random forcing can result in overturning of the structure. As an example, an effect of a single-mass secondary structure on the dynamic behavior of the system is discussed. The paper is restricted to the consideration of seismic vulnerability of the structure. A similar approach can be applied to systems with wind or wave excitation.     

Instability and refinement of models in long term planning of large scale projects

Instability of models used in long term planning of large scale industrial projects is demonstrated. The uncertainty band around the balance set is introduced to account for unpredictable variations of important parameters or utilities created by instability in multi-objective optimization of large scale projects. Then, instability of dynamic models of growth is considered including population dynamics with saturation and long term optimal planning in social and economic spheres. A method of successive refinement in a synthetic multi-model system is proposed, and application of the sequence of refined models is illustrated on a real-life example of construction of a dam with yearly refinements of the initial model, based on past history of project realization.     

Investigating the Turing Conditions for Diffusion-driven Instability in Predator-prey System with Hunting Cooperation Functional Response

In this paper, we focus on stability analysis of steady-state solutions of a predator-prey system with hunting cooperation functional response. The results show that the Turing instability can be affected not only the existence of hunting cooperation, but also the diffusion coefficients: (1) in the absence of predator diffusion, diffusion-driven instability can be induced by hunting cooperation, but no stable patterns appear; (2) the system can occur diffusion-driven instability and Turing patterns, when both predator and prey have diffusion, and the diffusion coefficient of prey is greater than that of the predator. The numerical simulations of two cases are presented to verify the validity of our theoretical results.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

Order-Up-To policies in Information Exchange supply chains

Collaboration in Supply Chains (SC) is concerned with the alignment of the decision making process amongst SC partners. This is crucial in the planning and inventory management area where this alignment is enabled by the exchange of information. Several benefits deriving from such effective collaboration exist, such as: excess inventory elimination, lead times reduction, improved customer service, efficient product development, etc. Operations Management literature proclaims the virtues of collaboration and information sharing but academicians and practitioners have recently identified various gaps that still need further work. More specifically it has been shown that several deleterious phenomena as the bullwhip effect; inventory instability and intermittent orders are not completely eliminated in Information Exchange supply chains. The reason is mainly because companies adopt order policies that are prone to create instability along the SC. In this paper we show how the performance of an Information Exchange SC can be improved by shifting from a myopic periodic review Order-Up-To policy to a periodic review Order-Up-To with feedback gain. To do so, we model the SC structure through difference equations and study the system response in term of internal process efficiency and customer service level.