Critical perturbations in a monostable active medium

An analysis is made of critical perturbations which initiate the evolution of instability in a monostable active medium described by a reaction-diffusion equation. A group-theoretical analysis of the problem yields an analytic expression for the energy of the critical perturbations. The results may be important for analyses of stability with respect to external perturbations of a wide range of active media. Pis’ma Zh. Tekh. Fiz. 24, 76–80 (November 26, 1998)     

The evolution of spatiotemporal chaos in a discrete-continuous active medium

A special approach to calculation of the spectrum of Lyapunov exponents has been developed and applied to diagnostics of the degree of regularity of wave structures in a model neural system. A transition between the regimes of regular wave dynamics and developed spatiotemporal chaos is demonstrated and quantitatively characterized.

     

Spectral evolution of an optical pattern generated by spatial modulation instability in a reorientational Kerr nonlinear medium

An experimental investigation is reported on the role of molecular reorientational nonlinearity in the spectral evolution of multifilamentation patterns induced by spatial modulation instability (MI) in a nonlinear Kerr medium. The influence of molecular reorientational Kerr nonlinear response on the spatial MI is analyzed theoretically by the standard stability analysis. The spatial MI gain spectra obtained by experimental measurements were found to agree with the theoretical derivation. In addition, by changing the launch average power, we also analyze the spatial spectral behavior of the optical pattern in such media. It is shown that, for higher launch average power, there will be secondary spatial stripes, manifesting as the original spatial modulation frequency doubling in the spatial frequency domain, which originated from the molecular reorientational Kerr nonlinearity response.     

Thermal instability in a porous medium with random vibrations

Summary Onset of thermal convection in a layer of saturated porous medium, heated from below, is examined when the layer is subjected to random vibrations. It is shown that when the vibrations are characterized by a white noise process, they hasten the onset of convection. Further, decrease in permeability tends to stabilize the flow field.With 4 Figures     

Stokes-vector evolution in a weakly anisotropic inhomogeneous medium

The equation for evolution of the four-component Stokes vector in weakly anisotropic and smoothly inhomogeneous media is derived on the basis of a quasi-isotropic approximation of the geometrical optics method, which provides the consequent asymptotic solution of Maxwell's equations. Our equation generalizes previous results obtained for the normal propagation of electromagnetic waves in stratified media. It is valid for curvilinear rays with torsion and is capable of describing normal mode conversion in inhomogeneous media. Remarkably, evolution of the four-component Stokes vector is described by the Bargmann-Michel-Telegdi equation for relativistic spin precession, whereas the equation for the three-component Stokes vector resembles the Landau-Lifshitz equation describing spin precession in ferromagnetic systems. The general theory is applied for analysis of polarization evolution in a magnetized plasma. We also emphasize fundamental features of the non-Abelian polarization evolution in anisotropic inhomogeneous media and illustrate them by simple examples.     

Autooscillation instability of the free surface of a viscoelastic medium

Oscillation instability of a layer of viscoelastic liquid is revealed on the basis of the numerical analysis of the dispersion equation. The instability is observed in the case of time-independent uniform external force action on the free surface of the liquid and may lead to the formation of a regular wavelike relief of a finite amplitude.     

Nonlinear radial oscillations of micro-bubbles in a compressible UCM medium

The nonlinear oscillations of acoustically forced spherical gas bubbles in an upper-convected Maxwell (UCM) compressible fluid are investigated. The nonlinear viscoelastic model used is suitable for large-amplitude excitation of bubbles that cannot be captured by linear models. The effects of acoustic excitation are studied for compressible nonlinear viscoelastic media, which increases the complexity and nonlinearity of the behavior. The Keller–Miksis equation is used to model the dynamics of a single bubble. The constitutive equations of compressible UCM are used for viscoelastic media. These governing equations are non-dimensionalized and coupled to determine the bubble dynamic behavior. The set of derived non-dimensionalized integro-differential equations developed are numerically solved simultaneously using the fourth-order Runge–Kutta method featured by the automatic variable time step-size module. The combined effects of compressibility and viscoelasticity of the fluid on bubble radius are investigated. The results show that the combination of compressibility and nonlinear viscoelasticity for bubble radial oscillations makes forced bubble dynamics more applicable for human needs, especially for large deformations in highly non-Newtonian fluids like industrial polymers or even tissue-like media. It can be seen that compressibility controls the oscillations at higher forcing amplitudes. The relevance and importance of these bubble dynamics to biomedical ultrasound applications and light emissions by sonoluminescence and other industries are evident.     

Convective instability in a gravity modulated anisotropic thermally stable porous medium

The effect of gravity modulation on the onset of convection in a horizontal fluid saturated porous layer in which the applied temperature gradient is opposite to that of gravity is investigated. The flow through the porous layer is governed by an extended form of Darcy’s law incorporating Brinkman’s boundary layer correction. The permeability and thermal conductivity of the medium are assumed to be transversely anisotropic. A stability analysis based on the method of small perturbations is performed using normal mode assumption. The study is focussed on low amplitude gravity modulation and the thresholds are found using Mathieu’s functions. The emergence of instability via the synchronous and subharmonic modes and the transition between them are discussed as a function of the physical parameters of interest.     

Sensitivity to solid volume fraction of gravitational instability in a granular medium

We report experimental results on incipient dynamics of gravitational destabilization in an immersed granular medium. Two protocols have been used to reach instability: a progressive loading and a dam-break situation. The instability triggering, analyzed by high-speed imaging and CIV post-processing, reveals distinct dynamical behaviours depending on the initial packing fraction of the medium. Two destabilisation modes have been observed: a surface avalanche and a deeper phenomenon that we called circular collapse.     

Nonlinear instability of mixed convection flow over a horizontal cylinder

Vortex shedding and instability of the mixed convection flow over a horizontal cylinder have been studied numerically for different flow directions with respect to the direction of buoyancy—with emphasis on assisting and opposing flows. Nonlinear instability and vortex shedding have been investigated with the help of the Landau equation—that is modified to identify critical Reynolds and Richardson numbers. Effects of flow direction are studied for a representative Richardson number. The average Nusselt number is estimated for all the cases to represent average heat transfer.     

Shape evolution of a light pulse in a linear birefringent medium

The spreading of a three-dimensional quasi-monochromatic progressive directional wave packet (such as a laser pulse) propagating freely in a linear and transparent birefringent medium is described geometrically by means of the ellipsoid representative of the pulse's second-order moments. The medium is characterized by a second-order expansion of its dispersion relation omega(K) about the mean wave vector Km of the pulse, i.e., by its Hessian matrix (H(Km)omega), which plays two important roles. Then, for some elements of(H(Km)omega), practical expressions are provided that are related to the curvature and dispersion properties of the normal surface of the medium. Then cases in which these properties determine completely the asymptotic transverse or axial spreadings of the wave packet are specified, and the results of an experiment are discussed.     

Onset of spatial instability in parametric four-wave in a medium with a diagonally bipolar response

A method of analyzing equations on the phase plane is used to solve a system of equations describing four-wave mixing in media with diagonally bipolar nonlinearity. It is shown theoretically that for certain parameters of the interacting waves, parametric instabilities may occur. It is established that the conversion of the mixed wave power depends strongly on their initial phase shift. Pis’ma Zh. Tekh. Fiz. 23, 61–65 (July 26, 1997)     

Nonlinear electrohydrodynamic instability of two liquid layers

The nonlinear instability of two superposed dielectric fluids is studied under the influence of an oscillating external electric field. In addition, a small constant and normal electric field is superimposed on the system. With the use of the method of multiple scaling, a generalized derivation of the amplitude evolution is obtained in marginally unstable regions of parameter space. A Melnikov function is formulated for such an instability and it is shown that there exist transverse homoclinic orbits leading to chaotic motions.     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

Time evolution of few-cycle pulse in a dense V-type three-level medium

Propagation of a few-cycle laser pulse in a dense V-type three-level atomic medium is investigated by the numerical solution of the full Maxwell–Bloch equations without the rotating wave and slowly varying envelope approximations, and the numerical solution is obtained by using the predictor–corrector method and the finite-difference time-domain method. It is shown that, due to the strength of the electric field induced by the macroscopic polarization in a dense medium being stronger than that in a dilute medium and the influence of the near dipole–dipole (NDD) interaction, the time evolution of a few-cycle pulse in the dense medium is remarkably different from that in the corresponding dilute medium. In the dilute medium, oscillation arises at the trailing edge of the pulse; while in the dense medium, it appears at both the leading and trailing edges of the pulse; moreover, the oscillation at the leading edge is more obvious with the pulse area decreasing. The carrier-envelope phase has an obvious difference in the two cases with and without NDD interaction. The ratio, γ, of the transition dipole moments has strong influence on the time evolution and split of the pulse. In the dense medium, when?γ?= 1, NDD interaction delays propagation and split of the pulse; while when?γ?> 1, NDD interaction accelerates propagation and split of the pulse, moreover, the phenomenon is more obvious with the input pulse area decreasing. In the dilute medium, the larger area pulse doesn't split when?γ?= 1 while it splits when?γ?> 1.     

Nonlinear waves in an active-dissipative disperse medium

Nonlinear waves in a medium involving dissipation, dispersion, and enhancement described by the generalized Kuramoto-Sivashinsky equation are discussed. Analytical solutions of the equation are obtained in the form of solitary waves. For numerical modeling of the nonlinear waves a difference scheme is suggested. Interaction of nonlinear waves described by the Kuramoto-Sivashinsky model is considered. It is shown that for specified values of the problem parameters there is one solitary wave described by the initial model. The dependences of the velocity and amplitude of this wave on the problem parameters are determined. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 1, pp. 149–154, January–February, 1998.     

Nonlinear parametric instability of antisymmetrically laminated angle-ply plates

Parametric instability of antisymmetrically laminated angle-ply rectangular plates is considered in this paper. The boundaries of instability domains are determined using linear theory. Then the frequency-amplitude relationship of the nonlinear parametric vibration within the instability domains is obtained using an asymptotic method.     

Nonlinear parametric instability of antisymmetrically laminated angle-ply plates

Parametric instability of antisymmetrically laminated angle-ply rectangular plates is considered in this paper. The boundaries of instability domains are determined using linear theory. Then the frequency-amplitude relationship of the nonlinear parametric vibration within the instability domains is obtained using an asymptotic method.     

Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration?

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum–fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape—no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems.