System Of Hydrodynamic Equations Research Articles

Numerical Study of the Influence of Boundary Conditions on Calculations of the Dynamics of Polydisperse Gas Suspension

The work numerically simulates the flow of a polydisperse gas suspension in a channel. The carrier medium was described as a viscous, compressible, heat-conducting gas. The mathematical model implemented a continuum technique for the dynamics of multiphase media, taking into account the interaction of the carrier medium and the dispersed phase. For each component of the mixture, a complete hydrodynamic system of equations of motion for the carrier phase and dispersed phase fractions was solved. The dispersed phase consisted of particles with different sizes of dispersed inclusions. For the carrier medium, homogeneous Dirichlet boundary conditions were specified on the side surfaces of the channel. For fractions of the dispersed phase, boundary conditions for slippage. The influence of the boundary conditions of the flow of the carrier medium on the dynamics of gas suspension fractions has been revealed.

One-dimensional Carrollian fluids. Part I. Carroll-Galilei duality

Galilean and Carrollian algebras acting on two-dimensional Newton-Cartan and Carrollian manifolds are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. We describe the dynamics of these systems as they emerge from the relevant limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. We investigate the action of local hydrodynamic-frame transformations in the Galilean and Carrollian configurations, i.e. dual Galilean and Carrollian local boosts, and comment on their potential breaking. Emphasis is laid on the additional geometric elements that are necessary to attain complete systems of hydrodynamic equations in Newton-Cartan and Carroll spacetimes. Our analysis is conducted in general Cartan frames as well as in more explicit coordinates, specifically suited to Galilean or Carrollian use.

Numerical Modeling of the Interaction of a Monodisperse Gas Suspension with a Shock Wave Moving at an Angle to the Separation Boundary of a Homogeneous Gas and a Gas Suspension

The work is devoted to the study of the influence of the dispersed phase on the dynamics of gas suspensions during numerical modeling of the dynamics of gas suspensions. In this study, based on the continuum technique of dynamics of inhomogeneous media, the interaction of a shock wave propagating from a homogeneous gas with a gas suspension was numerically simulated. For each of the components of the mixture, a complete hydrodynamic system of equations of motion was solved, which included the equations of conservation of density, the equations of conservation of the spatial components of the momentum of the mixture components, and the equations of conservation of energy of the components. The carrier medium was described as a viscous, compressible heat-conducting gas. The mathematical model took into account interfacial heat transfer. The mathematical model also took into account the interphase exchange of momentum, which included the force of aerodynamic drag, the dynamic force of Archimedes and the force of added masses. The system of equations of the mathematical model was integrated using the finite difference method. To suppress numerical oscillations, a nonlinear correction scheme was used. Large volumetric contents of the dispersed phase were considered. The influence of interfacial interaction on the process of shock wave propagation has been studied.

Features of director reorientation in a thin nematic film under the influence of crossed electric and magnetic fields.

The theory based on numerical study of the system of hydrodynamic equations, which includes the director motion, shows that under the influence of crossed electric E and magnetic B fields, the director reorients in such a way that the transient quasiperiodic patterns may arise in microsized nematic volumes if the corresponding distortion mode has the fastest response and thus suppresses all other modes, including uniform ones. It has been shown that there is a threshold value of the amplitude of the thermal fluctuations of the director over the microsized nematic film which provides the nonuniform rotation mode rather than the uniform one, whereas the lower values of the amplitude dominate the uniform mode.

Визуализация течений вязкой несжимаемой жидкости, соответствующих точным решениям уравнений Навье-Стокса

The work visualizes flows corresponding to the exact solutions of the system of hydrodynamic equations previously published by the authors, consisting of the vector Navier-Stokes equation and the law of conservation of mass for an incompressible fluid. This work uses the MathGL library for the C/C++ language and ParaView for scientific visualization of the results of numerical and analytical calculations. Without the use of such means, it would be impossible to see that the fluid flow is stratified into invariant subregions, and the trajectories of motion of fluid particles are wound on torus-shaped surfaces. Most of the scientific works on the study of hydrodynamic equations cover the results of calculations and do not address the questions of the existence of exact analytical solutions. At the same time, these calculations are performed with a specially selected set of fitting parameters unique to the equipment used and the computer software used. Questions about trust in the results of such calculations, their verification with exact solutions and the creation of a bank of test examples of applied problems in order to certify the applicability of the calculation results in practice become relevant.

Numerical Method for Calculating Plane Flows of Viscous Gas in a Cylinder

To solve scientific, technological and environmental problems in the field of road transport, it is necessary to develop new mathematical models. At this stage of development of internal combustion engines and their parts, it is necessary to model complex hydrodynamic technologies. This necessities revealing different forms of liquid and gas flows with arbitrary initial and boundary conditions for the region under consideration.The work is devoted to modelling a turbulent flow over a plate located in a cylinder. Reynolds-averaged Navier-Stokes equations were used as a mathematical model of the flow. Turbulent viscosity was calculated using the one-equation Spalart-Allmaras turbulence model. To solve the system of hydrodynamic equations, the finite difference method was used. A numerical method for calculating plane flows of viscous gas in a cylinder using turbulence models is presented. Suitable options were selected based on physical assumptions used to develop the models.Recommendations have been developed on the possibility of using the considered turbulence models.

Dirac Hydrodynamics in 19 Forms

We consider the relativistic spinor field theory re-formulated in polar variables to allow for the interpretation given in terms of fluid variables. After that, the dynamics of spinor fields are converted as dynamics of a special type of spin fluid. We demonstrate that such conversion into dynamical spin fluid is not unique, but it can be obtained through 19 different rearrangements, by explicitly showing the 19 minimal systems of hydrodynamic equations that are equivalent to the Dirac equations.

On the Existence of Optimal Control in the Problem of Optimizing the Lowest Coefficient of a Semilinear Evolutionary Equation

The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov’s hydrodynamic system of equations is considered.

SPECIFIC ACOUSTIC-GRAVITY WAVE MODES IN ISOTHERMAL ATMOSPHERE

In the paper, we show that the spectrum of acoustic-gravity waves in an isothermal atmosphere includes four specific evanescent modes. These modes are the solutions of the system of hydrodynamic equations for small atmospheric disturbances under the assumption that one of the quantities (horizontal or vertical components of particle velocity, density fluctuations, or temperature) is equal to zero. Three of the four specific modes (the Lamb wave, the Brunt-Väisälä oscillation, and the f-mode) are well known, but they were previously obtained as independent solutions. The recent discovery by the authors of the evanescent γ-mode made it possible to show that all four specified modes form a certain family of special modes of the isothermal atmosphere. On the spectral diagram of the frequency and the wave vector, there are four dispersion curves of these special modes in which one of the perturbed quantities is equal to zero. These curves belong to the evanescent region of the acoustic-gravity wave spectrum. They intersect each other at five points. It is shown that the specific modes cannot interact at the intersection points. The polarization ratios between two perturbed quantities have a different sign on either side of a particular curve if one of the quantities on this curve is zero. These properties can be used as indicators of the specific modes in experimental studies of the evanescent spectrum of AGWs. By using polarization relations, the possibility of observing these modes in the Earth’s atmosphere and on the Sun is also analyzed.

Heat-driven vortical flow in microsized liquid crystal film

ABSTRACTThe theoretical description of vortex dynamics in a hybrid aligned nematic (HAN) microvolume with an orientational defect excited by a temperature gradient, ∇T, is given based on the appropriate nonlinear extension of the classical Ericksen–Leslie theory. We have carried out a numerical study of the system of hydrodynamic equations including director reorientation, fluid flow v and the temperature redistribution across the HAN microvolume under the influence of ∇T, when the HAN microvolume is heated from above. Calculations show that, due to the interaction between the gradient of the director field ∇nˆ and ∇T, the HAN microvolume settles down to a stationary complex-vortex flow regime.PACS numbers: 61.30.Cz, 65.40.De

Is relativistic hydrodynamics always symmetric-hyperbolic in the linear regime?

Close to equilibrium, the kinetic coefficients of a thermodynamic system must satisfy a set of symmetry conditions, which follow from the Onsager-Casimir principle. Here, we show that, if a system of hydrodynamic equations is analyzed from the perspective of the Onsager-Casimir principle, then it is possible to impose very strong symmetry conditions also on the principal part of such equations (the part with highest derivatives). In particular, we find that, in the absence of macroscopic magnetic fields and spins, relativistic hydrodynamics should always be symmetric-hyperbolic, when linearized about equilibrium. We use these results to prove that Carter's multifluid theory and the Israel-Stewart theory in the pressure frame are both symmetric-hyperbolic in the linear regime. Connections with the GENERIC formalism are also explored.

Явное представление сокращенных в размерности уравнений Эйлера сжимаемой жидкости и полной системы уравнений гидродинамики в интегральной форме

Various ways of reducing the complete system of hydrodynamic equations in volume to a system of equations on the surface are of great scientific interest. In this article, the unsteady Euler equations of a compressible fluid and the complete system of hydrodynamic equations are reduced to “stationary” integral-differential equations where time derivatives are absent. Such a procedure (3D → 2D, 2D → 1D) makes it possible to reduce the dimension of the problem by one, which significantly reduces the necessary computing power when modeling them. If to set the correct problem, then one can determine the entire unsteady flow in the volume without solving the unsteady problem. It is enough to set time-varying data only on some surface of this stream, which must be determined independently. The peculiarity of this work is that all equations reduced in dimension are obtained in explicit form, unlike previous works of the authors, where up to 200–500 equations with reduced dimension were proposed, which are very difficult to study and model. In previous authors’ work, dimensionality was reduced by reducing overdetermined systems of differential equations. In this method, in the case of a successful choice of an additional constraint equation for overdetermined systems of differential equations, there is a reduction to PDE systems of dimension less than that of the original PDE systems. The method that is used to reduce the dimension of the Euler equations and the complete system of hydrodynamic equations in this paper is actually a special case of the above. The technique of dimensional reduction in overdetermined systems of PDEs is obviously generalized to the case of integral-differential equations. The article also obtained integral equations in new variables (Lagrangian and pseudo-Lagrangian), which determine the evolution of the flow. Also, a new method is proposed for overdetermination of any system of PDEs using the general integral space relation, which is consequence of the Helmholtz decomposition theorem.

Vortex Dynamics in a Hybrid Aligned Nematic Microvolume with an Orientational Defect

The aim of this theoretical paper is to investigate the physical mechanism responsible for the appearance of vortex flow in a hybrid aligned nematic (HAN) microvolume with an orientational defect, excited by a temperature gradient ∇T. This was done in the framework of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation. We have carried out a numerical study of the system of hydrodynamic equations including director reorientation, fluid flow v, and the temperature redistribution across the HAN microvolume under the influence of ∇T, when the HAN microvolume is heated from above. Calculations show that, due to the interaction between the gradient of the director field ∇n^ and ∇T, the HAN microvolume settles down to a stationary complex vortex flow regime.

Численное моделирование турбулентных потоков на основе двухжидкостной модели турбулентности

This paper discusses the capabilities of a two-fluid turbulence model for solving complex physical problems such as separated flow around a square cylinder and laminar-turbulent flow in a suddenly expanding channel. The numerical solution to the system of hydrodynamic equations is implemented using a finite-difference scheme. At each time step, the velocities are corrected through pressure calculations according to the SIMPLE algorithm. For verification purposes, the obtained numerical results are compared with available experimental data. A comparison of numerical results has shown that the two-fluid model is easy to implement, requires less computational resources and is capable of predicting laminar and turbulent flows with high accuracy.

Quaternion equations for hydrodynamic two-fluid model of vortex plasma

We discuss the application of quaternionic space-time algebra for the generalization of self-consistent equations describing the hydrodynamic two-fluid model of vortex plasma. It is shown that quaternionic formalism allows one to write the system of hydrodynamic equations in a compact form as one quaternion equation, which can be easy generalized to the case of damping plasma in an external electromagnetic field. As an illustration, we apply the proposed equations for the description of sound waves in electron–ion and electron–positron plasmas.

From Boltzmann Equation for Granular Gases to a Modified Navier–Stokes–Fourier System

In this paper, we give an overview of the results established in Alonso (http://arxiv.org/org/abs/2008.05173, 2020) which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier–Stokes–Fourier system.

Сопоставление математических моделей динамики электрически заряженных газовзвесей для различных концентраций дисперсной компоненты

This work is devoted to mathematical modeling of the dynamics of inhomogeneous electrically charged media. A dusty environment - solid particles suspended in a gas – was considered as an inhomogeneous medium. The mathematical model implemented a continuous approach to modeling the dynamics of inhomogeneous media. The complete hydrodynamic system of equations was solved for each component. The system of equations for the dynamics of each component included the equations of mass continuity, momentum components, and the energy conservation equation for the mixture component. Intercomponent interaction took into account momentum exchange and intercomponent heat transfer. The carrier medium was described as a viscous compressible heat-conducting gas. The flow was described as a flow with a two- dimensional geometry. The equations of the mathematical model were supplemented with initial and boundary conditions. The mathematical model took into account the wall viscosity in the channel. The system of equations of the mathematical model was integrated by McCormack's explicit finite-difference method. To obtain a monotonic grid function, a nonlinear scheme for correcting the numerical solution was used. The mathematical model was supplemented by the Poisson equation describing the electric field formed by charged dispersed particles. Poisson's equation was integrated by finite-difference methods on a gas-dynamic grid. Such a choice of the computational grid was necessary to calculate the concentration of particles required both for solving the electric field equation and for calculating the physical fields of the dynamics of inhomogeneous media. The reciprocal motion of a gas suspension caused by the movement of dispersed particles under the action of the Coulomb force was numerically investigated. The values of the surface and mass densities are determined, at which the models of the surface and mass densities of charges in the simulation of such a process are the same. It is revealed that the surface and mass models of charges are identical with respect to the volumetric content.

Correct definition of internal energy at the phenomenological construction of model of multicomponent continuous medium by methods of nonequilibrium thermodynamics

Taking into account the methods of thermodynamics of irreversible processes using the Onsager principle, a model of a multicomponent continuous medium is constructed, the internal energy of which is "free" of the kinetic energy of diffusion. The model is designed for an imperfect continuous medium with chemical reactions in the field of conservative external forces. Generalized StefanMaxwell relations are obtained, which represent a system of hydrodynamic equations of motion of a mixture with true inertial forces. The proposed thermodynamic technique made it possible to obtain a number of algebraic relations known from the kinetic theory of gases for the transfer coefficients, relating, in particular, the coefficients of multicomponent diffusion with binary diffusivitys, thermal diffusion ratios with thermal diffusion coefficients and multicomponent diffusion coefficients, true (molecular) thermal conductivity coefficient for multicomponent mixture with partial thermal conductivity coefficient, which indicates their versatility. The results obtained are intended for modeling not only liquid imperfect solutions, but also gas-dust mixtures with a finely dispersed dust component.

An explicit iterative scheme for 3D multicomponent heat conducting flow simulation

This paper presents the development and testing of time integration scheme for the system of hydrodynamic equations of gas mixture. These equations take into account the phenomena of multicomponent diffusion and heat transfer. The governing equation system is discretized by the finite volume approach on three-dimensional unstructured grids. According to the algorithm, computation of any single time step is split into the sequence of hyperbolic and parabolic stages. The hyperbolic subtask is solved using the Godunov-type scheme. The parabolic subtask is solved by the explicit iterative Chebyshev scheme, which is algorithmically simple and does not involve tuning parameters. This stage addresses dissipative fluxes (viscosity, multicomponent diffusion and thermal conductivity). The number of the explicit iterations is determined by the convective time step and by the upper bound for the discrete diffusion operator. The resulting scheme ensures the fulfillment of the conservation laws at the discrete level. The computer code can be used in highly parallel computing for large-scale simulation. The proposed approach is recommended for applied problems in which convective and dissipative processes are in close interaction, particularly for problems of plasma physics and astrophysics.

Post-Newtonian spherically symmetrical accretion

The objective of this work is to investigate the influence of the corrections to the spherical symmetrical accretion of an infinity gas cloud characterized by a polytropic equation into a massive object due to the post-Newtonian approximation. Starting with the steady state post-Newtonian hydrodynamic equations for the mass, mass-energy, and momentum densities, the post-Newtonian Bernoulli equation is derived. The post-Newtonian corrections to the critical values of the flow velocity, sound velocity and radial distance are obtained from the system of hydrodynamics equations in spherical coordinates. It was considered that the ratio of the sound velocity far the massive body and the speed of light was of order ${a}_{\ensuremath{\infty}}/c={10}^{\ensuremath{-}2}$. The analysis of the solution led to following results: the Mach number for the Newtonian and post-Newtonian accretion have practically the same values for radial distances of order of the critical radial distance; by decreasing the radial distance the Mach number for the Newtonian accretion is bigger than the one for the post-Newtonian accretion; the difference between the Newtonian and post-Newtonian Mach numbers when the ratio ${a}_{\ensuremath{\infty}}/c\ensuremath{\ll}{10}^{\ensuremath{-}2}$ is insignificant; the effect of the correction terms in post-Newtonian Bernoulli equation is more perceptive for the lowest values of the radial distance; the solutions for ${a}_{\ensuremath{\infty}}/c>{10}^{\ensuremath{-}2}$ does not lead to a continuous inflow and outflow velocity at the critical point; the comparison of the solutions with those that follow from a relativistic Bernoulli equation shows that the dependence of the Mach number with the radial distance of the former is bigger than the Newtonian and post-Newtonian Mach numbers.