Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.

A comparison of particle and fluid models for positive streamer discharges in air

Both fluid and particle models are commonly used to simulate streamer discharges. In this paper, we quantitatively study the agreement between these approaches for axisymmetric and 3D simulations of positive streamers in air. We use a drift-diffusion-reaction fluid model with the local field approximation and a PIC-MCC (particle-in-cell, Monte Carlo collision) particle model. The simulations are performed at 300 K and 1 bar in a 10 mm plate-plate gap with a 2 mm needle electrode. Applied voltages between 11.7 and 15.6 kV are used, which correspond to background fields of about 15 to 20 kV/cm. Streamer properties like maximal electric field, head position and velocity are compared as a function of time or space. Our results show good agreement between the particle and fluid simulations, in contrast to some earlier comparisons that were carried out in 1D or for negative streamers. To quantify discrepancies between the models, we mainly look at streamer velocities as a function of streamer length. For the test cases considered here, the mean deviation in streamer velocity between the particle and fluid simulations is less than 4\%. We study the effect of different types of transport data for the fluid model, and find that flux coefficients lead to good agreement whereas bulk coefficients do not. Furthermore, we find that with a two-term Boltzmann solver, data should be computed using a temporal growth model for the best agreement. The numerical convergence of the particle and fluid models is also studied. In fluid simulations the streamer velocity increases somewhat using finer grids, whereas the particle simulations are less sensitive to the grid. Photoionization is the dominant source of stochastic fluctuations in our simulations. When the same stochastic photoionization model is used, particle and fluid simulations exhibit similar fluctuations.

Fluid Energy Cascade Rate and Kinetic Damping: New Insight from 3D Landau-fluid Simulations

Using an exact law for incompressible Hall magnetohydrodynamics (HMHD) turbulence, the energy cascade rate is computed from three-dimensional HMHD-CGL (biadiabatic ions and isothermal electrons) and Landau-fluid numerical simulations that feature different intensities of Landau damping over a broad range of wavenumbers, typically 0.05 ≲ k ⊥ d i ≲ 100. Using three sets of cross-scale simulations where turbulence is initiated at large, medium, and small scales, the ability of the fluid energy cascade to “sense” the kinetic Landau damping at different scales is tested. The cascade rate estimated from the exact law and the dissipation calculated directly from the simulation are shown to reflect the role of Landau damping in dissipating energy at all scales, with an emphasis on the kinetic ones. This result provides new prospects on using exact laws for simplified fluid models to analyze dissipation in kinetic simulations and spacecraft observations, and new insights into theoretical description of collisionless magnetized plasmas.

Rogue and lump waves for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice

Two-layer fluid models are used to depict some nonlinear phenomena in fluid mechanics, medical science and thermodynamics. In this paper, we investigate a (3[Formula: see text]1)-dimensional Yu-Toda-Sasa-Fukuyama equation for the interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice. Via the Kadomtsev-Petviashvili hierarchy reduction, we derive the rational solutions in the determinant forms and semi-rational solutions. The [Formula: see text]th-order lump waves and multi-lump waves are obtained, where [Formula: see text] is a positive integer. We observe the second-order lump waves: Two-lump waves interact with each other and separate into two new lump waves. Two-lump waves are observed: Overtaking interaction takes place between the two-lump waves; After the interaction, the two-lump waves propagate with their original velocities and amplitudes. Studying the semi-rational solutions, we show the fusion between a lump wave and a bell-type soliton and fission of a bell-type soliton. Interaction between a line rogue wave and a bell-type soliton is shown.

Sheath collapse at critical shallow angle due to kinetic effects

The Debye sheath is known to vanish completely in magnetised plasmas for a sufficiently small electron gyroradius and small angle between the magnetic field and the wall. This angle depends on the current onto the wall. When the Debye sheath vanishes, there is still a potential drop between the wall and the plasma across the magnetic presheath. The magnetic field angle corresponding to the predicted sheath collapse is shown to be much smaller than previous estimates, scaling with the electron-ion mass ratio and not with the square root of the mass ratio. This is shown to be a consequence of the kinetic electron and finite ion orbit width effects, which are not captured by fluid models. The wall potential with respect to the bulk plasma at which the Debye sheath vanishes is calculated. Above this wall potential, it is possible that the Debye sheath will invert.