Rayleigh–Bénard magnetoconvection with temperature modulation

Floquet analysis of modulated magnetoconvection in Rayleigh–Bénard geometry is performed. A sinusoidally varying temperature is imposed on the lower plate. As Rayleigh number Ra is increased above a critical value Ra o , the oscillatory magnetoconvection begins. The flow at the onset of magnetoconvection may oscillate either subhar- monically or harmonically with the external modulation. The critical Rayleigh number Ra o varies non-monotonically with the modulation frequency ω for appreciable value of the modulation amplitude a . The temperature modulation may either postpone or prepone the appearance of magnetoconvection. The magnetoconvective flow always oscillates harmonically at larger values of ω . The threshold Ra o and the corresponding wavenumber k o approach to their values for the stationary magnetoconvection in the absence of modulation ( a = 0), as ω → ∞. Two different zones of harmonic instability merge to form a single instability zone with two local minima for higher values of Chandrasekhar’s number Q , which is qualitatively new. We have also observed a new type of bicritical point, which involves two different sets of harmonic oscillations. The effects of variation of Q and Pr on the threshold Ra o and critical wavenumber k o are also investigated.

On the porosity influence on stability of flow over porous medium

The stability of incompressible fluid plane-parallel flow over a layer of a saturated porous medium is studied. The results of a linear stability analysis are described at different porosity values. The considered system is bounded by solid wall from the porous layer bottom. Top fluid surface is free and rigid. A linear stability analysis of plane-parallel stationary flow is presented. It is realized for parameter area where the neutral stability curves are bimodal. The porosity variation effect on flow stability is considered. It is shown that there is a transition between two main instability modes: long-wave and short-wave. The long-wave instability mechanism is determined by inflection points within the velocity profile. The short-wave instability is due to the large transverse gradient of flow velocity near the interface between liquid and porous medium. Porosity decrease stabilizes the long wave perturbations without significant shift of the critical wavenumber. Simultaneously, the short-wave perturbations destabilize, and their critical wavenumber changes in wide range. When the porosity is less than 0.7, the inertial terms in filtration equation and magnitude of the viscous stress near the interface increase to such an extent that the Kelvin-Helmholtz analogue of instability becomes the dominant mechanism for instability development. The stability band realizes in narrow porosity area. It separates the two branches of the neutral curve.

The Horton–Rogers–Lapwood problem for an inclined porous layer with permeable boundaries

A formulation of the Horton–Rogers–Lapwood problem for a porous layer inclined with respect to the horizontal and characterized by permeable (isobaric) boundary conditions is presented. This formulation allows one to recover the results reported in the literature for the limiting cases of horizontal and vertical layer. It is shown that a threshold inclination angle exists which yields an upper bound to a parametric domain where the critical wavenumber is zero. Within this domain, the critical Darcy–Rayleigh number can be determined analytically. The stability analysis is performed for linear perturbations. The solution is found numerically, for the inclination angles above the threshold, by employing a Runge–Kutta method coupled with the shooting method.

Higher-order moment theories for dilute granular gases of smooth hard spheres

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

Modulations of viscous fluid conduit periodic waves

Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.

A numerical study on the energy transfer from surface waves to interfacial waves in a two-layer fluid system

Interaction between surface and interfacial waves with continuous energy spectra in a two-layer density stratified fluid system is investigated numerically. For an initial wave field which consists only of the surface waves all propagating in the same direction, it is confirmed that the spectra $S_{s}(k)$ of the surface waves and $S_{i}(k)$ of the interfacial waves change significantly due to the recently found class 3 triad resonance. When the bulk of the surface wave spectrum $S_{s}(k)$ is initially located well above the critical wavenumber $k_{crit}$, below which the class 3 triad resonance is prohibited, $S_{s}(k)$ downshifts gradually toward the lower wavenumber during the initial stage of evolution. However, this downshift halts when the peak of $S_{s}(k)$ reaches around $k_{crit}$, and after that a steep peak forms in $S_{s}(k)$ around $k_{crit}$. It is confirmed that the timescale of the spectral evolution is of $O(1/{\it\epsilon}^{2})$ (${\it\epsilon}$ is a characteristic non-dimensional wave amplitude) in most of the $k$ space, consistent with the prediction of the wave turbulence theory for a system with a decay-type dispersion relation. However, it is also found that the timescale of the formation and growth of the sharp peak in $S_{s}(k)$ around $k_{crit}$ is of $O(1/{\it\epsilon})$, i.e. the timescale of the deterministic three-wave resonance.

Resonance patterns in spatially forced Rayleigh–Bénard convection

We report on the influence of a quasi-one-dimensional periodic forcing on the pattern selection process in Rayleigh–Bénard convection (RBC). The forcing was introduced by a lithographically fabricated periodic texture on the bottom plate. We study the convection patterns as a function of the Rayleigh number ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$) and the dimensionless forcing wavenumber ($q_f$). For small $\mathit{Ra}$, convection takes the form of straight parallel rolls that are locked to the underlying forcing pattern. With increasing $\mathit{Ra}$, these rolls give way to more complex patterns, due to a secondary instability. The forcing wavenumber $q_f$ was varied in the experiment over the range of $0.6q_c<q_f<1.4q_c$, with $q_c$ being the critical wavenumber of the unforced system. We investigate the stability of straight rolls as a function of $q_f$ and report patterns that arise due to a secondary instability.

Super rogue waves in ultracold neutral nonextensive plasmas

The generation of ion-acoustic rogue waves in ultracold neutral plasmas (UNPs) composed of ion fluids and nonextensive electron distribution is investigated. For this purpose, basic equations are reduced to a nonlinear Schrödinger equation (NLSE) using a reductive perturbation technique. The existence region for the rogue waves defined precisely in terms of the critical wavenumber threshold kc. It is found that increasing the nonextensive parameter q would lead to a decrease of kc until q approaches to its critical value qc, then further increase of q beyond qc enhances kc; however, kc shrinks with the increase of the ions effective temperature ratio σ∗. The dependence of the first- and second-order rational solutions profile on the UNP parameters is numerically examined. It is noticed that near to the critical nonextensive parameter qc, the rogue wave amplitude becomes smaller, but it enhances whenever we stepped away from qc. However, the enhancement of the temperature ratio σ∗ and the wavenumber k reduces the envelope rogue wave amplitudes.

Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wavenumber and time. The method is applied to transient Rayleigh–Bénard–Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tension-driven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

Effect Of Dust Particles On Thermal Convection In A Ferromagnetic Fluid

This paper deals with the theoretical investigation of the effect of dust particles on the thermal convection in a ferromagnetic fluid subjected to a transverse uniform magnetic field. For a flat ferromagnetic fluid layer contained between two free boundaries, the exact solution is obtained, using a linear stability analysis. For the case of stationary convection, dust particles and non-buoyancy magnetization have always a destabilizing effect. The critical wavenumber and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of the buoyancy magnetization parameter M1. The results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because the heat capacity of the clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles. The oscillatory modes are introduced by the dust particles. A sufficient condition for the non-existence of overstability is also obtained.