An efficient method for simulation of the dynamics of a large number of deformable droplets in the stokes regime

Direct simulations of the interaction of a large number of deformable droplets are necessary for more accurate predictions of rheological properties and the microstructure of liquid-liquid systems. In the present study, a mathematical model of a three-dimensional flow of a mixture of two Newtonian liquids of a droplet structure in an unbounded domain at low Reynolds numbers is considered. An efficient computational method for simulation of the dynamics of a large number of deformable drops is developed and tested. This approach is based on the boundary element method for three-dimensional problems accelerated both via an advanced scalable algorithm (FMM), and via utilization of a heterogeneous computing architecture (multicore CPUs and graphics processors). This enables direct simulations of systems of tens of thousands of deformable droplets on PCs, which is confirmed by test and demo computations. The developed method can be used for solution of a wide range of problems related to the emulsion flow in micro- and nanoscales. Also it can be used to establish the closing relations for simulation of two-phase liquid-liquid flow based on the continuum approach in macroscales.     

Conservation laws and exact solutions for nonlinear diffusion in anisotropic media

Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.     

Geometric Proof of Lie's Linearization Theorem

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.     

Self-adjoint sub-classes of generalized thin film equations

In this work we consider a class of fourth-order nonlinear partial differential equation containing several un-specified coefficient functions of the dependent variable which encapsulates various mathematical models used, e.g. for describing the dynamics of thin liquid films. We determine the subclasses of these equations which are self-adjoint. By using a general theorem on conservation laws proved by one of the authors (NHI) we find conservation laws for some of these partial differential equations without classical Lagrangians.     

Quantized separations of phase-locked soliton pairs in fiber lasers

We report the discovery of a quantization of the separation between phase-locked soliton pairs that is related to the radiation waves known as Kelly sidebands, in a passively mode-locked fiber ring laser. Our numerical simulations that predict this phenomenon have been confirmed by our experimental results.     

Investigations of polymer dynamics in nanoporous media by field cycling NMR relaxometry and the dipolar correlation effect

The chain dynamics of short-chain perfluoropolyether melts confined in Vycor nanoporous media has been characterized by field cycling nuclear magnetic resonance relaxometry and the dipolar correlation effect. The slowdown of motions under confinement, leading to larger residual dipolar couplings, has been probed by looking at the quotient of stimulated and primary echoes. Using field cycling relaxometry, it has been shown that there is strong evidence of reptation-like motion, even for such short-chain polymers as shown by the frequency and molecular weight dependences of the spin-lattice relaxation time.     

Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes

Nonlinear dissipative systems display the full (3+1)D spatiotemporal dynamics of stable optical solitons. We review recent results that were obtained within the complex cubic-quintic Ginzburg-Landau equation model. Numerical simulations reveal the existence of stationary bell-shaped (3+1)D solitons for both anomalous and normal chromatic dispersion regimes, as well as the formation of double soliton complexes. We provide additional insight concerning the possible dynamics of these soliton complexes, consider collision cases between two solitons, and discuss the ways nonstationary evolution can lead to optical pattern formation.     

Deuteron and proton spin-lattice relaxation dispersion of polymer melts: intrasegment, intrachain, and interchain contributions

Proton and deuteron field-cycling NMR relaxometry was applied to deuterated and undeuterated bulk polyethyleneoxide and polybutadiene melts and mixtures thereof with molecular weights above the critical value. Spin-lattice relaxation data due to intrasegment (quadrupolar) couplings and intra- and interchain (dipolar) interactions were evaluated. Diverse dynamic limits are identified both with the proton and deuteron frequency dispersion data. The comparison between the intrachain and the interchain contributions leads to the conclusion that only model theories based on largely isotropic chain dynamics can account for the experimental findings. The extremely anisotropic character of the well-known tube/reptation model is too restrictive in this respect.