Global strong solutions to the Shliomis system for ferrofluids in a bounded domain

We consider the partial differential equations proposed by Shliomis to model the dynamics of an incompressible viscous ferrofluid submitted to an external magnetic field. The Shliomis system consists of the incompressible Navier‐Stokes equations, the magnetization equations, and the magnetostatic equations. The magnetization equations is of Bloch type, and no regularizing term is added. We prove the global existence of unique strong solution to the initial boundary value problem for the system in a bounded domain, with the small initial data and external magnetic field but without any restrictions on the physical parameters. The novelty of the analysis is to introduce a linear combination of magnetic fields.     

Intrinsic symmetries in constitutive modeling of magneto–elastic materials

We consider a class of non-dissipative materials whose constitutive equations are derived from a suitably constructed thermodynamic potential function. The Gibbs energy relation is introduced as a function of stress, strain, magnetic field, magnetization, and temperature. By minimization with respect to the chosen subset of independent variables we derive the corresponding set of constitutive equations. The chosen form of the free energy function leads to the linear elastic and nonlinear ferromagnetic coupling where non-linearity emerges in terms associated with the strength of magnetization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Markov brothers type inequality in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> on a closed interval

A recent modification of a classic Landau-Lifshitz equation that includes the socalled spin-transfer torque is widely recognized in physics community as a model of magnetization dynamics in certain nanodevices. Motivated by some experimental evidence, we introduce a generalization of this model, coupled Landau-Lifshitz equations with spin-transfer torque terms, and analyze it from dynamical systems standpoint. An explicit stability criterion for the critical points in terms of all parameters of the system is derived and illustrated with stability diagrams. Our analysis provides certain guidelines for the design of magnetic nanodevices with optimized response to control parameters.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

Nonlinear oscillations in a MEMS energy scavenger

The dynamics of an electret-based, capacitive, vibration-to-electric micro-converter (energy scavenger) is described by a set of ODEs where a second-order equation is coupled to two first-order equations through strongly-nonlinear terms. The nonlinear regimes of forced oscillations are analyzed with a semi-analytical approach, finding that the system exhibits features typical of Duffing-like nonlinear oscillators, such as jumps and multivalued frequency-response curves, with both stable and unstable periodic solutions. It is also proved that, for appropriate combinations of parameters, the system acts as a linear, damped oscillator, independently of the oscillation amplitude: in this case, the nonlinear coupling term reduces to a viscous-like term, physically interpretable as electromechanical damping.     

Coherent Swing Instability of Power Grids

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations??local plant, inter-machine, and inter-area modes??interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.     

Engineering magnetic polariton system with distributed coefficients: Applications to soliton management

In the wake of the recent design of a powerful method for generating higher-dimensional evolution systems with distributed coefficients Kuetche (2014) [15] illustrated on the dynamics of the current-fed membrane of zero Young’s modulus, we construct the general Lax-representation of a new higher-dimensional coupled evolution equations with varying coefficients. Discussing the physical meanings of these equations, we show that the coupled system above describes the propagation of magnetic polaritons within saturated ferrites, resulting structurally from the fast-near adiabatic magnetization dynamics combined to the Maxwell’s equations. Accordingly, we address some practical issues of the nonautonomous soliton managements underlying in the fast remagnetization process of data inputs within magnetic memory devices.     

Mixed Finite Element Methods for the Ferrofluid Model with Magnetization Paralleled to the Magnetic Field

Mixed finite element methods are considered for a ferrofluid flow model with magnetization paralleled to the magnetic field. The ferrofluid model is a coupled system of the Maxwell equations and the incompressible Navier-Stokes equations. By skillfully introducing some new variables, the model is rewritten as several decoupled subsystems that can be solved independently. Mixed finite element formulations are given to discretize the decoupled systems with proper finite element spaces. Existence and uniqueness of the mixed finite element solutions are shown, and optimal order error estimates are obtained under some reasonable assumptions. Numerical experiments confirm the theoretical results.     

Harmonic and anharmonic oscillations of a cylindrical plasma in the presence of a uniform axial magnetic field and a steady axial current

In the present note we have studied the harmonic and anharmonic oscillations of cylindrical plasma using Lagrangian formalism. In order to study the harmonic oscillations, the equations are linearized and the resulting equation for the displacement has been numerically solved. For situations present in thermonuclear reactors, the presence of axial magnetic field is found necessary to make the periods of oscillation to become comparable with the time required for the thermonuclear reactions to set in. A detailed analysis of the anharmonic oscillations reveals that the significant interaction is between the first and the second mode. The fundamental period of anharmonic oscillation is more than the corresponding period of harmonic oscillations by 9·2%. Graphs have been drawn for the amplitudes of relative variations in density and magnetic field and of the time-varying part of anharmonic oscillation.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

Biomagnetic viscoelastic fluid flow over a stretching sheet

Of concern in this paper is an investigation of biomagnetic flow of a non-Newtonian viscoelastic fluid over a stretching sheet under the influence of an applied magnetic field generated owing to the presence of a magnetic dipole. The viscoelasticity of the fluid is characterised by Walter’s B fluid model. The applied magnetic field has been considered to be sufficiently strong to saturate the ferrofluid. The magnetization of the fluid is considered to vary linearly with temperature as well as the magnetic field intensity. The theoretical treatment of the physical problem consists of reducing it to solving a system of non-linear coupled differential equations that involve six parameters, which are solved by developing a finite difference technique. The velocity profile, the skin-friction, the wall pressure and the rate of heat transfer at the sheet are computed for a specific situation. The study shows that the fluid velocity increases as the rate of heat transfer decreases, while the local skin-friction and the wall pressure increase as the magnetic field strength is increased. It is also revealed that fluid viscoelasticity has an enhancing effect on the local skin-friction. The study will have an important bearing on magnetic drug targeting and separation of red cells as well as on the control of blood flow during surgery.     

Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics

A range of representative models of intracellular calcium dynamics are surveyed, with the aim of determining which model characteristics are qualitatively unchanged by changes to details of the model components. Techniques from geometric singular perturbation theory are used to investigate the role of separation of timescales in determining model dynamics, with particular emphasis on identifying parameter regimes in which mixed mode oscillations are present as a result of the separation of timescales. We find that the number of distinct timescales and the number of variables evolving on each timescale varies between models and depends on both the model assumptions and on the parameter regime of interest within the model, but in all cases, the presence of canards and associated mixed mode oscillations provides a mechanism by which the models can robustly exhibit complex oscillations, with the frequency of oscillation depending sensitively on parameter values. We find that analysis of the number and nature of the distinct timescales in a model allows us to make useful predictions about the dynamics associated with the model, and that this may give us more information about the model dynamics than a classification according to the modelling assumptions made about different cellular mechanisms in deriving the models.     

Mathematical simulation of nonlinear oscillations of viscoelastic pipelines conveying fluid

A mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the paper. The Boltzmann–Volterra integral model with weakly singular kernels of heredity is used to describe the processes of pipeline strain. Using the Bubnov–Galerkin method, the mathematical model of the problem is reduced to the study of a system of ordinary integro-differential equations, where time is an independent variable. The solution of integro-differential equations is determined by a numerical method based on the elimination of the singularity in the relaxation kernel of the integral operator. Using the numerical method for unknowns, a system of algebraic equations is obtained. To solve a system of algebraic equations, the Gauss method is used. A computational algorithm is developed to solve the problems of the dynamics of viscoelastic pipelines with a flowing fluid. The algorithm of the proposed method makes it possible to investigate in detail the effect of rheological parameters on the character of vibrational strength of viscoelastic pipelines with a fluid flow, in particular, in the study of free oscillations of pipelines based on the theory of ideally elastic shells. On the basis of the computational algorithm developed, a set of applied computer programs has been created, which makes it possible to carry out numerical studies of pipeline oscillations. The influence of singularity in the heredity kernels and the geometric parameters of the pipeline on the vibrations of structures possessing viscoelastic properties is numerically investigated. It is shown that an account of viscoelastic properties of pipeline material leads decrease in the amplitude and frequency of oscillation. It is established that to reveal the influence of viscoelastic properties of structure material on the pipeline oscillations, it is necessary to use the Abel-type weakly singular kernels of heredity. The obtained results of numerical simulation can be used in the enterprises of oil and gas industries, as well as in design organizations.     

The effect of magnetic field on dynamics of gas bubbles in visco-elastic fluids

The effect of magnetic field on nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear visco-elastic media is studied. The constitutive equation UCM used for modeling the rheological behaviors of the fluid. By starting from the momentum equations for bubbles considering the magnetic force and considering some simplifying assumptions, the modified bubble dynamics equation (the modified Rayleigh–Plesset equation) has been achieved. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of visco-elastic constitutive equations into modified bubble dynamics equations. The governing equations are non-dimesionalized and numerically solved by using 4th order Runge–Kutta method. The accuracy of the calculations and the formulation is compared with the previous works done for models without the presence of magnetic field. Furthermore, the bubble size variations due to acoustic motivations and stress tensor components variations in presence of different magnitudes of magnetic fields are studied. Also, the bubble size dependence on fluid conductivity variations is declared. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that magnetic field may be an important consideration for the risk assessment of potential cavitations and also it could be possible to damp the bubble oscillations by using magnetic fields or in opposite case amplify the oscillations which could result in higher level light emissions in sonoluminescence approach.     

Development of a high order discretization scheme for solving fully nonlinear magnetohydrodynamic equations

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell''s equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of $64 \times 64$ is sufficient for weakly nonlinear problems and higher grids up to $512 \times 512$ are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for $Re=73$ which is in agreement with literature. When the magnetic Reynolds number $Rm<1$ separation length decreases linearly with strength of magnetic field on a log-log scale whereas if $Rm>1$, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

Asymptotic analysis of a model of nuclear magnetic autoresonance

We study the system of three first-order differential equations arising when averaging the Bloch equations in the theory of nuclear magnetic resonance. For the averaged system, we construct an asymptotic series for the stable solution with an infinitely increasing amplitude. This result gives a key to understanding the autoresonance in weakly dissipative magnetic systems as a phenomenon of significant growth of the magnetization initiated by a small external pumping.     

Adiabatic approximations for Landau-Lifshitz equations

The asymptotics with respect to a small parameter for solutions of a system of Landau-Lifshitz equations with slowly varying coefficients and small dissipative terms is investigated. These equations are a mathematical model of a uniaxial ferromagnet in a time-dependent magnetic field. The asymptotics constructed make it possible to describe the magnetization reversal effect and to reveal the influence of the parameters of the external magnetic field and dissipation on the stability of this process.     

Biomagnetic fluid flow over a stretching sheet with non linear temperature dependent magnetization

The flow of a heated ferrofluid over a linearly stretching sheet is studied in the pres- ence of an applied magnetic field due to a magnetic dipole. It is assumed that the applied magnetic field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a non linear function of temperature difference. By introducing appropriate non dimensional variables the problem is described by a coupled and non linear system of ordinary differential equations with its boundary conditions which is solved numerically by applying an efficient numerical technique based on the common finite difference method. The obtained results are presented graphically for different values of the parameters entering into the problem under consideration and the dependence of the flow field from these parameters is discussed. A comparative study, with a similar problem which has already been solved and documented in literature, is also made wherever necessary, emphasizing the impor- tance of the non-linear variation of magnetization with temperature. Emphasis is also given in the obtained results for Prandtl number equal to 21 and critical exponent = 0.368 which are important and interesting in Biomagnetic Fluid Dynamics.