Nonlinear Hydrodynamic Equations Research Articles

Rings, Shells, and Arc Structures Around B[e] Supergiants. I. Classical Tools of Nonlinear Hydrodynamics

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

Cauchy invariants and exact solutions of nonlinear equations of hydrodynamics

Cauchy invariants and exact solutions of nonlinear equations of hydrodynamics

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

Analog model for the BTZ black hole

We present an analog model for the Ba\~nados, Teitelboim, Zanelli (BTZ) black hole based on a hydrodynamical flow. We numerically solve the fully nonlinear hydrodynamic equations of motion and observe the excitation and decay of the analog BTZ quasinormal modes in the process. We consider both a small perturbation in the steady state configuration of the fluid and a large perturbation; the latter could be regarded as an example of formation of the analog (acoustic) BTZ black hole.

Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The nonlinear hydrodynamic equations of the WNT are solved analytically through a connection to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a nonvanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.

Hidden quasiconservation laws in fracton hydrodynamics.

We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden quasiconservation laws, in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. First, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of harmonic multipole charges within linear response; we calculate the decay or growth of these charges due to dangerously irrelevant nonlinearities. Second, we consider a model with xy and x^{2}-y^{2} quadrupole conservation, in addition to dipole conservation, which is described by isotropic fourth-order subdiffusion, yet has dangerously irrelevant sixth-order corrections necessary to relax the harmonic multipole charges. We confirm our predictions for the anomalously slow decay of the harmonic conserved charges in each setting by using numerical simulations, both of the nonlinear hydrodynamic differential equations, and in quantum automaton circuits on a square lattice.

Three-dimensional modulational instability of dust acoustic waves in the presence of generalized (r, q) distributed electrons

A three-dimensional (3D) modulational instability (MI) of dust acoustic waves (DAWs) in a three-component magnetized dusty plasma system consisting of a negatively charged fluid, inertialess generalized (r, q) distributed electrons and Boltzmann distributed ions, is investigated. The basic system of the nonlinear hydrodynamic equations is reduced to a 3D nonlinear Schrödinger equation (NLS) which is valid for small but finite amplitude DAWs using a reductive perturbation technique. The domain of the stability and instability regions is investigated that is strongly affected by the spectral parameters of the generalized (r, q) distribution and the electron-to-ion temperature ratio (T e /T i ). The existence domains for observing the first-and second-order solutions of the dust acoustic rogue waves (DARWs) are determined and the basic features (viz the width and amplitude) for the first-order solution are found to be significantly dependent on the system physical parameters changes such as T e /T i , number density ratio [n e0/(n d0 z d0)] and the dust cyclotron frequency (ω cd ) as well as the spectral indexes r and q. A comparison between the first-and second-order DARW amplitudes is presented. Moreover, another comparison between the first-order DARW amplitudes obtained by generalized (r, q) distributed electrons and those corresponds to Maxwellian is provided. Finally, implication of our consequences in specific plasma situations are briefly discussed.

An Extended Theoretical Scenario for Classical Cepheids. I. Modeling Galactic Cepheids in the Gaia Photometric System

We present a new extended and detailed set of models for Classical Cepheid pulsators at solar chemical composition ($Z=0.02$, $Y=0.28$) based on a well tested nonlinear hydrodynamical approach. In order to model the possible dependence on crucial assumptions such as the Mass-Luminosity relation of central Helium burning intermediate-mass stars or the efficiency of superadiabatic convection, the model set was computed by varying not only the pulsation mode and the stellar mass but also the Mass-Luminosity relation and the mixing length parameter that is used to close the system of nonlinear hydrodynamical and convective equations. The dependence of the predicted boundaries of the instability strip as well as of both light and radial velocity curves on the assumed Mass-Luminosity and the efficiency of superadiabatic convection is discussed. Nonlinear Period-Mass-Luminosity-Temperature, Period-Radius and Period-Mass-Radius relations are also computed. The theoretical atlas of bolometric light curves for both the fundamental and first overtone mode has been converted in the Gaia filters $G$, $G_{BP}$ and $G_{BR}$ and the corresponding mean magnitudes have been derived. Finally the first theoretical Period-Luminosity-Color and Period-Wesenheit relations in the Gaia filters are provided and the resulting theoretical parallaxes are compared with Gaia Data Release 2 results for both fundamental and first overtone Galactic Cepheids.

An alternative view on the Bateman–Luke variational principle

A new derivation of the Bernoulli equation for water waves in three-dimensional rotating and translating coordinate systems is given. An alternative view on the Bateman–Luke variational principle is presented. The variational principle recovers the boundary value problem governing the motion of potential water waves in a container undergoing prescribed rigid-body motion in three dimensions. A mathematical theory is presented for the problem of three-dimensional interactions between potential surface waves and a floating structure with interior potential fluid sloshing. The complete set of equations of motion for the exterior gravity-driven water waves, and the exact nonlinear hydrodynamic equations of motion for the linear momentum and angular momentum of the floating structure containing fluid, are derived from a second variational principle.

Nonlinear Features of a Fluid Flow in an Elastic Pipeline

The process of a fluid flow in an elastic pipeline with fixed ends is considered. It is shown that in such a pipeline auto-oscillations of the wall and the fluid stream or flutter of the skin mode can take place. A solution of the system of nonlinear hydrodynamical equations describing these auto-oscillations in elementary functions is presented. It is shown that standing waves in an elastic pipeline arise not due to the summation of direct and returning waves.

Complex dynamics of a sheared nematic fluid

Nonlinearities in constitutive equations of extended objects in shear flow lead to novel phenomena, e.g. ‘rheochaos’ in solutions of wormlike micelles and ‘elastic turbulence’ in polymer solutions. Since both phenomena involve anisotropic objects, their contributions to the deviatoric stress are likely to be similar. However, these two fields have evolved rather independently and an attempt at connecting these fields is still lacking. We show that a minimal model in which the anisotropic nature of the constituting objects is taken into account by a nematic alignment tensor field reproduces several statistical features found in rheochaos and elastic turbulence. We numerically analyse the full non-linear hydrodynamic equations of a sheared nematic fluid under shear stress and strain rate controlled situations, incorporating spatial heterogeneity only in the gradient direction. For a certain range of imposed stress and strain rates, this extended dynamical system shows signatures of spatiotemporal chaos and transient shear banding. In the chaotic regime the power spectra of the order parameter stress, velocity fluctuations and the total injected power show power law behavior and the total injected power shows a non-gaussian, skewed probability distribution. These dynamical features bear resemblance to elastic turbulence phenomena observed in polymer solutions. The scaling behavior is independent of the choice of shear rate/stress controlled method.

Enhancement of valley polarization in graphene with an irradiating charged particle

We use a two-dimensional two-component nonlinear hydrodynamic method to study valley-dependent plasmons in bounded strained graphene in the presence of an irradiating proton, realizing electrostatic control of the valley-dependent plasmons in actual space. We use flux-corrected transport to solve the nonlinear hydrodynamic equations numerically and self-consistently. Our results answer the important question of whether full valley polarization can be obtained in a specific valley at the boundary or inside the graphene sheet in the presence of the injected proton. The electrons experience collective excitations due to the proton interaction and the strain-induced pseudomagnetic field. The electron density fluctuation can be much larger than the equilibrium electron density, leading to a strong nonlinear effect and thus full valley polarization. This demonstrates the nature of the nonlinear response of electrons in graphene to strong interactions, a response that originates from the strong nonperturbative interaction between the irradiating proton and the electrons. Thus, our method opens up the possibility of investigating the nonlinear behavior of valley-dependent plasmons in strong modulations. The effects of the proton on the valley polarization are examined. There is K-polarization inside the surface behind the proton, whereas there is K′-polarization at the edge which decays away from the edge, thereby switching the valley polarization. This work establishes a link between actual-space valley-dependent plasmons in graphene and the irradiating proton and provides an alternative way to realize full valley polarization with tunable polarity. Compared to the case with no proton, the valley polarization is enhanced considerably in the presence of a proton.

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Multiple harmonics control of edge pseudomagnetoplasmons in strained graphene

Valley-resolved edge plasmons are relevant to nano-optics at subwavelength scales. However, less attention has been paid to their tunable properties in time domain. In this work we investigate edge pseudomagnetoplasmons in a strained graphene modulated by multiple harmonics with frequency in the THz regime. The edge plasmon is described by a set of nonlinear hydrodynamic equations, which are self-consistently solved by the flux-corrected transport method. Without the applied voltage, there exist two unidirectional-propagating edge-plasmon modes with weak valley polarization P. It is demonstrated that by varying the amplitude of multiple harmonics one can alter both the amplitude and the polarity of the valley polarization in the edge plasmon. One can achieve a full valley polarization P=1 at the instant of half cycle of the multiple harmonics and P=-1 at the instant of one cycle. The edge-plasmon density and the transverse velocity vanish for the frozen valley.

Correct Boundary Conditions for the High-Resolution Model of Nonlinear Acoustic-Gravity Waves Forced by Atmospheric Pressure Variations

Currently, an international network of operating high-resolution microbarographs was established to record wave-induced pressure variations at the Earth’s surface. Based on these measurements, simulations are performed to analyze the characteristics of waves corresponding to the observed variations of atmospheric pressure. Such a mathematical problem involves a set of primitive nonlinear hydrodynamic equations considering lower boundary conditions in the form of pressure variations at the Earth’s surface. Selection of upward propagating acoustic-gravity waves (AGWs) generated or reflected at the Earth’s surface requires the Neumann boundary conditions involving the vertical gradients of vertical velocity at the lower boundary. To analyze the correctness of the mathematical problem, linearized equations are used for small-surface wave amplitudes excited near the ground. Using the relation for wave energy, it is proven that the solution of the boundary problem based on the nondissipative approximation is uniquely determined by the variable pressure field at the Earth’s surface. The respective dissipative problem has also a unique solution with the appropriate choice of lower boundary conditions for temperature and velocity components. To test the numerical algorithm, solutions of the linearized equations for AGW modes are used. Developed boundary conditions are implemented into the model describing acoustic-gravity wave propagation from the surface atmospheric pressure source. Atmospheric waves propagating from the observed surface pressure variations to the upper atmosphere are simulated using the obtained algorithms and the computer codes.

Viscous Overstability in Saturn’s Rings: Influence of Collective Self-gravity

Abstract We investigate the influence of collective self-gravity forces on the nonlinear, large-scale evolution of the viscous overstability in Saturn’s rings. We numerically solve the axisymmetric nonlinear hydrodynamic equations in the isothermal and non-isothermal approximation, including radial self-gravity and employing transport coefficients derived by Salo et al. We assume optical depths to model Saturn’s dense rings. Furthermore, local N-body simulations, incorporating vertical and radial collective self-gravity, are performed. Vertical self-gravity is mimicked through an increased frequency of vertical oscillations, while radial self-gravity is approximated by solving the Poisson equation for an axisymmetric thin disk with a Fourier method. Direct particle–particle forces are omitted, which prevents small-scale gravitational instabilities (self-gravity wakes) from forming, an approximation that allows us to study long radial scales and to compare directly the hydrodynamic model and the N-body simulations. Our isothermal and non-isothermal hydrodynamic model results with vanishing self-gravity compare very well with results of Latter & Ogilvie and Rein & Latter, respectively. In contrast, for rings with radial self-gravity we find that the wavelengths of saturated overstable waves settle close to the frequency minimum of the nonlinear dispersion relation, i.e., close to a state of vanishing group velocities of the waves. Good agreement is found between non-isothermal hydrodynamics and N-body simulations for moderate and strong radial self-gravity, while the largest deviations occur for weak self-gravity. The resulting saturation wavelengths of viscous overstability for moderate and strong self-gravity ( ) agree reasonably well with the length scales of axisymmetric periodic microstructure in Saturn’s inner A ring and the B ring, as found by Cassini.

Odd viscosity in chiral active fluids

We study the hydrodynamics of fluids composed of self-spinning objects such as chiral grains or colloidal particles subject to torques. These chiral active fluids break both parity and time-reversal symmetries in their non-equilibrium steady states. As a result, the constitutive relations of chiral active media display a dissipationless linear-response coefficient called odd (or equivalently, Hall) viscosity. This odd viscosity does not lead to energy dissipation, but gives rise to a flow perpendicular to applied pressure. We show how odd viscosity arises from non-linear equations of hydrodynamics with rotational degrees of freedom, once linearized around a non-equilibrium steady state characterized by large spinning speeds. Next, we explore odd viscosity in compressible fluids and suggest how our findings can be tested in the context of shock propagation experiments. Finally, we show how odd viscosity in weakly compressible chiral active fluids can lead to density and pressure excess within vortex cores.

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.

Axion as a Cold Dark Matter Candidate: Proof to Fully Nonlinear Order

Abstract We present proof of the axion as a cold dark matter (CDM) candidate to the fully nonlinear order perturbations based on Einstein’s gravity. We consider the axion as a coherently oscillating massive classical scalar field without interaction. We present the fully nonlinear and exact, except for ignoring the transverse-tracefree tensor-type perturbation, hydrodynamic equations for an axion fluid in Einstein’s gravity. We show that the axion has the characteristic pressure and anisotropic stress; the latter starts to appear from the second-order perturbation. But these terms do not directly affect the hydrodynamic equations in our axion treatment. Instead, what behaves as the effective pressure term in relativistic hydrodynamic equations is the perturbed lapse function and the relativistic result coincides exactly with the one known in the previous non-relativistic studies. The effective pressure term leads to a Jeans scale that is of the solar-system scale for conventional axion mass. As the fully nonlinear and relativistic hydrodynamic equations for an axion fluid coincide exactly with the ones of a zero-pressure fluid in the super-Jeans scale, we have proved the CDM nature of such an axion in that scale.