Complex systems often have features that can be modeled by advanced mathematical tools [1]. Of special interests are the features of complex systems that have a network structure as such systems are important for modeling technological and social processes [3, 4]. In our previous research we have discussed the flow of a single substance in a channel of network. It may happen however that two substances flow in the same channel of network. In addition the substances may react and then the question arises about the distribution of the amounts of the substances in the segments of the channel. A study of the dynamics of the flow of the substances as well as a study of the distribution of the substances is presented in this paper on the base of a discrete - time model of flow of substances in the nodes of a channel of a network.
We investigate the flow of a nano-scale incompressible ridge of low-volatility liquid along a chemical channel: a long, straight, and completely wetting stripe embedded in a planar substrate, and sandwiched between two extended less wetting solid regions. Molecular dynamics simulations, a simple long-wavelength approximation, and a full stability analysis based on the Stokes equations are used, and give qualitatively consistent results. While thin liquid ridges are stable both statically and during flow, a (linear) pearling instability develops if the thickness of the ridge exceeds half of the width of the channel. In the flowing case periodic bulges propagate along the channel and subsequently merge due to nonlinear effects. However, the ridge does not break up even when the flow is unstable, and the qualitative behavior is unchanged even when the fluid can spill over onto a partially wetting exterior solid region.
The two-fluid (ions and electrons) plasma Richtmyer-Meshkov instability of a cylindrical light/heavy density interface is numerically investigated without an initial magnetic field. Varying the Debye length scale, we examine the effects of the coupling between the electron and ion fluids. When the coupling becomes strong, the electrons are restricted to co-move with the ions and the resulting evolution is similar to the hydrodynamic neutral fluid case. The charge separation that occurs between the electrons and ions results in self-generated electromagnetic fields. We show that the Biermann battery effect dominates the generation of magnetic field when the coupling between the electrons and ions is weak. In addition to the Rayleigh-Tayler stabilization effect during flow deceleration, the interfaces are accelerated by the induced spatio-temporally varying Lorentz force. As a consequence, the perturbations develop into the Rayleigh-Taylor instability, leading to an enhancement of the perturbation amplitude compared with the hydrodynamic case.
The previous study regarding the stabilization of a magnetized constant temperature plasma by shear flow with vorticity is extended to a plasma of non-constant temperature, where in the presence of heat source or sinks the thermomagnetic Nernst effect becomes important. Of special interest is what this effect has on the stabilization of a linear z-pinch discharge for which exact solutions are given. Solutions which are unstable for subsonic shear flow become stable if the flow is supersonic.
We propose an explanation for the onset of oscillations seen in numerical simulations of dense, inclined flows of inelastic, frictional spheres. It is based on a phase transition between disordered and ordered collisional states that may be interrupted by the formation of force chains. Low frequency oscillations between ordered and disordered states take place over weakly bumpy bases; higher-frequency oscillations over strongly bumpy bases involve the formation of particle chains that extend to the base and interrupt the phase change. The predicted frequency and amplitude of the oscillations induced by the unstable part of the equation of state are similar to those seen in the simulations and they depend upon the contact stiffness in the same way. Such oscillations could be the source of sound produced by flowing sand.
The role of turbulence in current generation and self-excitation of magnetic fields has been studied in the geometry of a mechanically driven, spherical dynamo experiment, using a three dimensional numerical computation. A simple impeller model drives a flow which can generate a growing magnetic field, depending upon the magnetic Reynolds number, Rm, and the fluid Reynolds number. When the flow is laminar, the dynamo transition is governed by a simple threshold in Rm, above which a growing magnetic eigenmode is observed. The eigenmode is primarily a dipole field tranverse to axis of symmetry of the flow. In saturation the Lorentz force slows the flow such that the magnetic eigenmode becomes marginally stable. For turbulent flow, the dynamo eigenmode is suppressed. The mechanism of suppression is due to a combination of a time varying large-scale field and the presence of fluctuation driven currents which effectively enhance the magnetic diffusivity. For higher Rm a dynamo reappears, however the structure of the magnetic field is often different from the laminar dynamo; it is dominated by a dipolar magnetic field which is aligned with the axis of symmetry of the mean-flow, apparently generated by fluctuation-driven currents. The fluctuation-driven currents have been studied by applying a weak magnetic field to laminar and turbulent flows. The magnetic fields generated by the fluctuations are significant: a dipole moment aligned with the symmetry axis of the mean-flow is generated similar to those observed in the experiment, and both toroidal and poloidal flux expulsion are observed.
Nikolay K. Vitanov
,Kaloyan N. Vitanov
,Zlatinka I. Dimitrova
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(2019)
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"Mathematical model of a flow of reacting substances in a channel of network"
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Nikolay K Vitanov
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