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The forced phase locking in a system of the oscillating electroconvective rolls that form in a nematic liquid crystal layer in a dc electric field is studied. As a result of the action of an additive ac electric field with a small amplitude, the system of oscillators is found to be divided into clusters, where oscillations are fully phase locked. The electroconvective rolls in neighboring clusters oscillate in antiphase and the clusters are separated by Ising walls. The phase locking is shown to be maximal at the forcing frequency that is close to the double frequency of the natural oscillations of rolls. A model is proposed to describe spatially distributed phase oscillators, and it takes into account the symmetries of a system of electroconvective rolls and external forcing. The results of numerical simulations agree well with the experimental data.

The actin cytoskeleton and its response to external chemical stimuli is fundamental to the mechano-biology of eukaryotic cells and their functions. One of the key players that governs the dynamics of the actin network is the motor protein myosin II. Based on a phase space embedding we have identified from experiments three phases in the cytoskeletal dynamics of starved Dictyostelium discoideum in response to a precisely controlled chemotactic stimulation. In the first two phases the dynamics of actin and myosin II in the cortex is uncoupled, while in the third phase the time scale for the recovery of cortical actin is determined by the myosin II dynamics. We report a theoretical model that captures the experimental observations quantitatively. The model predicts an increase in the optimal response time of actin with decreasing myosin II-actin coupling strength highlighting the role of myosin II in the robust control of cell contraction.

We investigate flexodomains, which are observed in planar layers of certain nematic liquid crystals, when a dc voltage $U$ above a critical value ${U}_{c}$ is applied across the layer. They are characterized by stationary stripelike spatial variations of the director in the layer plane with a wave number $p(U)$. Our experiments for different nematics demonstrate that $p(U)$ varies almost linearly with $U$ for $Ug{U}_{c}$. That is confirmed by a numerical analysis of the full nonlinear equations for the director field and the induced electric potential. Beyond this numerical study, we demonstrate that the linearity of $p(U)$ follows even analytically, when considering a special parameter set first used by Terent'ev and Pikin [Sov. Phys. JETP 56, 587 (1982)]. Their theoretical paper serves until now as the standard reference on the nonlinear analysis of flexodomains, since it has arrived at a linear variation of $p(U)$ for large $U\ensuremath{\gg}{U}_{c}$. Unfortunately, the corresponding analysis suffers from mistakes, which in a combination led to that result.

Many theoretical and experimental studies indicate that a propagation block represents an important factor in spiral wave initiation in excitable media. The analytical and numerical results we obtained for a generic two-component reaction-diffusion system demonstrate quantitative conditions for the propagation block in a one-dimensional and a two-dimensional medium due to a sharp spatial increase of the medium's excitability or the coupling strength above a certain critical value. Here, we prove that this critical value strongly depends on the medium parameters and the geometry of the inhomogeneity. For an exemplary two-dimensional medium, we show how the propagation block can be used to initiate spiral waves by a specific choice of the size and shape of the medium's inhomogeneity.

Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

We describe the dynamics of zigzag oscillations in a system of convective rolls in a nematic liquid crystal above the electroconvection threshold under the action of an ac voltage with a biased position of the mean value. It is found that an increase in the contribution from the constant component leads to a substantial increase in the spatiotemporal ordering of zigzag rolls and their synchronization with the homogeneous twist mode. The results confirm the flexoelectric mechanism of locking.

This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg-Landau theory as a unified language. In the first part, mean-field theory is presented, for both statics and dynamics, and its validity tested self-consistently. As is well known, the mean-field approximation breaks down below four spatial dimensions, where it can be replaced by a scaling phenomenology. The Ginzburg-Landau formalism can then be used to justify the phenomenological theory using the renormalization group, which elucidates the physical and mathematical mechanism for universality. In the second part of the paper it is shown how near pattern forming linear instabilities of dynamical systems, a formally similar Ginzburg-Landau theory can be derived for nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau equations thus obtained yield nontrivial solutions of the original dynamical system, valid near the linear instability. Examples of such solutions are plane waves, defects such as dislocations or spirals, and states of temporal or spatiotemporal (extensive) chaos.

This contribution deals with three types of orientational phenomena generated by oscillatory #ow in nematic liquid crystals: (1) Uniform orientational instabilities in the geometry where the director is initially oriented within the #ow plane. The instability occurs under rectilinear Poiseuille #ow and not under Couette #ow. (2) The "rst instability under rectilinear Couette #ow is a bifurcation to a stationary roll pattern with the roll axis perpendicular to the #ow direction. The threshold amplitude has been measured for homeotropic alignment and calculated by linear stability analysis. (3) An interesting slow director precession and nonlinear waves which arise in homeotropically aligned nematics exposed to an elliptical shear #ow in the bend FreH edericksz distorted state. The experimental data are compared with a theoretical analysis. The precession can also be generated by oscillatory compression. ( 2000 Elsevier Science B.V. All rights reserved.

~Received 19 March 1998! The response of a homeotropically aligned nematic liquid crystal layer to oscillatory rectilinear shear ~Couette flow ! was investigated for frequencies f between 0.01 and 200 Hz and layer thickness d between 10 and 130 mm. Below the onset of instability the cell was placed between crossed polars and light transmission was studied using a parallel light beam. The experimental results for the transmitted light intensity agree quantitatively with numerical solutions of the nematodynamic equations for different cell thicknesses, oscillation frequencies, and amplitudes. For frequencies between 25 and 150 Hz the critical oscillation amplitude for the onset of a spatial pattern, observed in polarized white light, could be reached. The pattern consisted of stationary rolls perpendicular to the direction of the oscillation. The experimentally obtained frequency dependence of the critical shear amplitude for the roll instability for different cell thicknesses is compared with an existing theory and the results of numerical calculations. @S1063-651X~98!09411-2# Nematic liquid crystals ~NLCs! exhibit interesting flow phenomena due to the coupling between the local molecular orientation ~director n ˆ! and the velocity field v. The flow properties of NLCs are characterized by Leslie viscosity coefficients a 1 ,...,a6 , two of which a 2(,0) and a 3 are important in describing the coupling between flow and director orientation @1‐4#. In the case of steady, plane, parallel shear flow, e.g., along the x axis with a velocity field v x(z )( v y 5 v z 50) and in the absence of other torques, the director will tend to align in the flow plane ( x-z plane! at a fixed angle u fl 56arctan(a3 /a2) 1/2 with the x axis if a 3,0 ~the 6

Pattern forming instabilities induced by ultralow frequency sinusoidal voltages were studied in a rodlike nematic liquid crystal by microscopic observations and simultaneous electric current measurements. Two pattern morphologies, electroconvection (EC) and flexodomains (FD), were distinguished, both appearing as time separated flashes within each half period of driving. A correlation was found between the time instants of the EC flashes and those of the nonlinear current response. The voltage dependence of the pattern contrast $C(U)$ for EC has a different character than that for the FD. The flattening of $C(U)$ at reducing the frequency was described in terms of an imperfect bifurcation model. Analyzing the threshold characteristics of FD, the temperature dependence of the difference $|{e}_{1}\ensuremath{-}{e}_{3}|$ of the flexoelectric coefficients was also determined by considering elastic anisotropy.

Optical methods are most convenient to analyze spatially periodic patterns with wavevector $\bm q$ in a thin layer of a nematic liquid crystal. In the standard experimental setup a beam of parallel light with a 'short' wavelength $\lambda \ll 2 \pi/q$ passes the nematic layer. Recording the transmitted light the patterns are either directly visualized by shadowgraphy or characterized more indirectly by the diffraction fringes due to the optical grating effects of the pattern. In this work we present a systematic short-wavelength analysis of these methods for the commonly used planar orientation of the optical axis of liquid crystal at the confining surfaces. Our approach covers general 3D experimental geometries with respect to the relative orientation of $\bm q$ and of the wavevector $\bm k$ of the incident light. In particular the importance of phase grating effects is emphasized, which are not accessible in a pure geometric optics approach. Finally, as a byproduct we present also an optical analysis of convection rolls in Rayleigh-B\'enard convection, where the refraction index of the fluid is isotropic in contrast to its uniaxial symmetry in nematic liquid crystals. Our analysis is in excellent agreement with an earlier physical optics approach by Trainoff and Cannell [Physics of Fluids {\bf 14}, 1340 (2002)], which is restricted to a 2D geometry and technically much more demanding., Comment: 22 pages

The nonlinear response of a nematic slab subjected to a rectilinear low-frequency oscillatory Couette and Poiseuille flow is investigated theoretically. We find that under Poiseuille flow and with appropriate alignment conditions by surface anchoring and/or magnetic field a state with slow, spontaneous director rotation appears. This may provide a model for the slow director rotations observed in high-frequency Couette flow. \textcopyright{} 1996 The American Physical Society.

Generalizing a recently introduced approximation scheme valid when the director relaxation time is large compared to the inverse frequency of an oscillatory flow, we derive equations for the time-averaged torques on the director for a prescribed plane shear flow. Whereas for a linear flow field (simple Couette flow) the average torques vanish, one has for Poiseuille flow (and more general flow fields) torques that tend to orient the director essentially perpendicular to the flow plane. For flow-aligning materials the orientation parallel to the flow is also (weakly) stable. Including the effect of homeotropic surface alignment we estimate the threshold of the oscillation amplitude for the out-of-plane transition. The results are essentially recovered (and improved) by direct numerical simulations

It is demonstrated that the key findings of the paper by Y.G. Marinov and H.P. Hinov, Eur. Phys. J. E 31, 179-189 (2010), are in direct conflict with the general physical background of flexoelectric domains. This is caused by a methodological error in the theoretical analysis of the paper.

Anomalous diffusion is predicted for Brownian particles in inhomogeneous viscosity landscapes by means of scaling arguments, which are substantiated through numerical simulations. Analytical solutions of the related Fokker-Planck equation in limiting cases confirm our results. For an ensemble of particles starting at a spatial minimum (maximum) of the viscous damping we find subdiffusive (superdiffusive) motion. Superdiffusion occurs also for a monotonically varying viscosity profile. We suggest different substances for related experimental investigations., Comment: 15 pages

We present a theoretical analysis of phase separation in the presence of a spatially periodic forcing of wavenumber q traveling with a velocity v. By an analytical and numerical study of a suitably generalized 2d-Cahn-Hilliard model we find as a function of the forcing amplitude and the velocity three different regimes of phase separation. For a sufficiently large forcing amplitude a spatially periodic phase separation of the forcing wavenumber takes place, which is dragged by the forcing with some phase delay. These locked solutions are only stable in a subrange of their existence and beyond their existence range the solutions are dragged irregularly during the initial transient period and otherwise rather regular. In the range of unstable locked solutions a coarsening dynamics similar to the unforced case takes place. For small and large values of the forcing wavenumber analytical approximations of the nonlinear solutions as well as for the range of existence and stability have been derived.

For many years it has been commonly accepted that electroconvection (EC) as primary instability in nematic liquid crystals for the "classical" planar geometry requires a positive anisotropy of the electric conductivity, sigma(a), and a slightly negative dielectric anisotropy, epsilon(a). This firm belief was supported by many experimental and theoretical studies. Recent experiments, which have surprisingly revealed EC patterns at negative conduction anisotropy as well, have motivated the theoretical studies in this paper. It will be demonstrated that extending the common hydrodynamic description of nematics by the usually neglected flexoelectric effect allows for a simple explanation of EC in the "nonstandard" case sigma(a)

We study theoretically ferrofluid pipe flow in a wide range of the flow rate in the presence of an oscillating magnetic field applied along the pipe axis. As demonstrated, the field‐dependent part of ferrofluid viscosity (it can be either positive or negative) reveals significant dependence on the flow vorticity, i.e., ferrofluids exhibit non‐Newtonian behavior. This is manifested in an alteration of the velocity profile — it ceases to be parabolic — and in deviation of the flow rate from the value prescribed by the Poiseuille’s formula. The presented theoretical model is based on the conventional ferrohydrodynamic equations, an assumption of chainlike aggregates composed of magnetic grains, and some concepts of polymer physics. The model allows to describe self‐consistently how the microstructure influences the ferrofluid flow in an externally imposed magnetic field while the field and the flow alter the ferrofluid microstructure. Using the model, we succeeded fitting well recent experimental data on the ferrofluid pipe flow in an oscillating magnetic field.

We study theoretically the slow director precession and nonlinear waves observed in homeotropically oriented nematic liquid crystals subjected to circular or elliptical Couette and Poiseuille flow and an electric field. From a linear analysis of the nematodynamic equations it is found that in the presence of the flow the electric bend Fréedericksz transition is transformed into a Hopf-type bifurcation. In the framework of an approximate weakly nonlinear analysis we have calculated the coefficients of the modified complex Ginzburg-Landau equation, which slightly above onset describes nonlinear waves with strong nonlinear dispersion. We also derive the equation describing the precession and waves well above the Fréedericksz transition and for small flow amplitudes. Then the nonlinear waves are of diffusive nature. The results are compared with full numerical simulations and with experimental data.

We study the homogeneous and the spatially periodic instabilities in a nematic liquid crystal layer subjected to steady plane Couette or Poiseuille flow. The initial director orientation is perpendicular to the flow plane. Weak anchoring at the confining plates and the influence of the external electric and/or magnetic field are taken into account. Approximate expressions for the critical shear rate are presented and compared with semianalytical solutions in case of Couette flow and numerical solutions of the full set of nematodynamic equations for Poiseuille flow. In particular the dependence of the type of instability and the threshold on the azimuthal and the polar anchoring strength and external fields is analyzed.

Elliptic shear applied to a homeotropically oriented nematic above the electric bend Freedericks transition (FT) generates slow precession of the director. The character of the accompanying nonlinear waves changes from diffusive phase waves to dispersive ones exhibiting spirals and spatio-temporal chaos as the FT is approached from above. An exact solution of the flow alignment equations captures the observed precession and predicts its reversal for non-flow aligning materials. The FT transforms into a Hopf bifurcation opening the way to understand the wave phenomena., Comment: 4 pages, 4 figures, submitted to PRL

Ferrofluid pipe flow in an oscillating magnetic field along the pipe axis is studied theoretically in a wide range of the flow rate. The field-dependent part of viscosity (it can be positive or negative) reveals significant dependence on the flow vorticity, i.e., ferrofluids exhibit non-Newtonian behavior. This is manifested in an alteration of the velocity profile—it ceases to be parabolic—and deviation of the flow rate from the value prescribed by Poiseuille’s formula. The presented model based on the conventional ferrohydrodynamic equations and an assumption of the ferrofluid structure fits well experimental data recently obtained by Schumacher, Sellien, Konke, Cader, and Finlayson [“Experiment and simulation of laminar and turbulent ferrofluid pipe flow in an oscillating magnetic field,” Phys. Rev. E 67, 026308 (2003)].