Compressible Fluid Equations Research Articles

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

وجود حل محلي لنظام الجذب الكيميائي والتنافر مع الانتشار غير الخطي والمصدر اللوجستي

In this paper, we are concerned with a model arising from biology, which is a coupled system of chemotaxis equations and viscous compressible fluid equations through transport and external forcing. The local existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for an attraction-repulsion chemotaxis model system over three space dimensions, we obtain local existence and uniqueness of convergence on classical solutions near constant states. We prove local existence of unique solutions in three dimensions by using energy estimates.

The conduit equation: Hyperbolic approximation and generalized Riemann problem

The conduit equation is a dispersive non-integrable scalar equation modeling the flow of a low-viscous buoyant fluid embedded in a highly viscous fluid matrix. This equation can be written in a particular form reminiscent of the famous Godunov form proposed in 1961 for the Euler equations of compressible fluids. We propose a hyperbolic approximation of the conduit equation by retaining the Godunov-type structure. The comparison of solutions to the conduit equation and those to the approximate hyperbolic system is performed: the wave fission of a large initial perturbation of a rectangular or Gaussian form. The results are in good agreement. New generalized solutions to the conduit equation composed of a finite set of waves of the same period and linked with a constant solution by generalized Rankine-Hugoniot relations are discovered. Such multi-hump structures interact with each other almost as solitary waves: they collide, merge, and reconstruct after the interaction. This partly indicates the stability of such multi-hump solutions under small perturbations. The exact and approximate hyperbolic system describes such an interaction with good accuracy.

Roll waves in a predictive model for open-channel flows in the smooth turbulent case

A depth-averaged model for turbulent open-channel flows with breaking roll waves on a sloping smooth bottom is derived under an assumption of independence between the wall turbulence and the roller turbulence. The model includes four variables – the water depth, the average velocity, and two variables called enstrophy, the shear enstrophy and the roller enstrophy – which take into account the deviations of the velocity with respect to its depth-averaged value due to shear effect and roller turbulence, respectively. The four equations of the model are the mass, momentum, energy and shear enstrophy balance equations, with the mathematical structure of the Euler equations of compressible fluids, with an additional transport equation and with source terms. The system is hyperbolic. The roller enstrophy is created by shocks. A non-zero value of the roller enstrophy indicates a breaking wave and a turbulent roller. The model is solved by a fast and well-known numerical scheme, with an explicit finite-volume method in one step. The model is used to simulate periodic and natural roll waves with a good agreement with existing experimental results. There is no parameter to calibrate in the model, which gives it a predictive character.

Time decay of solutions for compressible isentropic non-Newtonian fluids

In this paper, we consider the Cauchy problem of a compressible Navier–Stokes system of Eills-type non-Newtonian fluids. We investigate the time decay properties of classical solutions for the compressible non-Newtonian fluid equations. More specifically, we construct a new linearized system in terms of a combination of the solutions, and then we investigate the long-time behavior of the Cauchy problem for the three-dimensional isentropic compressible Eills-type non-Newtonian fluids with an initial perturbation.

On spherically symmetric solutions for the equations of heat-conducting and compressible fluids

On spherically symmetric solutions for the equations of heat-conducting and compressible fluids

Global classical solutions to the 3D Cauchy problem of compressible magneto-micropolar fluid equations with far field vacuum

Global classical solutions to the 3D Cauchy problem of compressible magneto-micropolar fluid equations with far field vacuum

Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.

Low-frequency Internal Gravity Waves Are Pseudo-incompressible

Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequency ω is much less than that of surface gravity waves, i.e., ω≪gkh , where g is the gravitational acceleration and k h is the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency ε=ω/gkh . Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz–Braginsky–Roberts formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation—whereas, in the anelastic approximation, Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.

Dispersion of axisymmetric longitudinal waves propagating in an inhomogeneous prestressed hollow cylinder immersed in an inviscid compressible fluid

The paper investigates the dispersion of axisymmetric waves in a cylinder with inhomogeneous initial stresses caused by the hydrostatic pressure of the compressible inviscid fluid in which this cylinder is immersed. In the context of this study, the motion of the cylinder is described by the three-dimensional linearized equations for elastic waves in bodies with initial stresses, but the flow of the fluid is described by the linearized Navier-Stokes equations for compressible (barotropic) inviscid fluids. The initial stresses in the cylinder, due to the action of hydrostatic pressure on the outer surface of the cylinder, are determined according to the well-known Lame formulas of classical linear elasticity theory. The corresponding eigenvalue problem for the dispersion of axisymmetric waves propagating in the hydro-elastic system under consideration is solved using the discrete-analytic method. The dispersion equation is obtained and solved numerically. Concrete numerical investigations are carried out for three pairs of materials: Soft Rubber + Glycerin, Lucite + Water and Steel + Water. The dispersion curves obtained for these pairs of materials are presented and discussed for different values of the intensity of the external pressure. In particular, it is found that the inhomogeneous initial stresses in the cylinder cause zero group velocity (ZGV) points to appear on the dispersion curves of the lowest modes. However, the ZGV points (or modes) appear on the dispersion curves of the higher modes as a result of the contact of the hollow cylinder with the fluid.

Well‐posedness of reynolds averaged equations for compressible fluids with a vanishing pressure

We show that the Reynolds averaged equations for compressible fluids (neglecting third order correlations) are well‐posed in when the pressure vanishes in dimensions and 3. In order to do this, we show that the system is Friedrichs‐symmetrizable. This model belongs to the class of non‐conservative hyperbolic systems. Hence the usual symmetrisation method for conservation laws can not be used here.

Generalization of compressible fluid equations in terms of complexified octonions

Using the attractive properties of octonion algebra, an alternative formulation has been proposed for the Maxwell-type equations of compressible fluids. Although the origins of electromagnetic theory and fluid mechanics are completely different, a series of suitable and elegant 8-dimensional equations have been derived in a form similar to electromagnetic, gravitational counterparts previously given in relevant literature. Moreover, the corresponding matrix representations of derived expressions have also been presented after describing the connections between the algebraic properties of Cayley–Dickson octonions and some pseudo-real matrices.

Relative energy inequality and weak-strong uniqueness for an isothermal non-Newtonian compressible fluid

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a de­ri­va­tion of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the strong solution exists. For this purpose, a new assumption on the coercivity of the viscous stress tensor was introduced along with two natural examples satisfying it.

High-order discontinuous Galerkin hydrodynamics with sub-cell shock capturing on GPUs

ABSTRACT Hydrodynamical numerical methods that converge with high-order hold particular promise for astrophysical studies, as they can in principle reach prescribed accuracy goals with higher computational efficiency than standard second- or third-order approaches. Here we consider the performance and accuracy benefits of Discontinuous Galerkin (DG) methods, which offer a particularly straightforward approach to reach extremely high order. Also, their computational stencil maps well to modern GPU devices, further raising the attractiveness of this approach. However, a traditional weakness of this method lies in the treatment of physical discontinuities such as shocks. We address this by invoking an artificial viscosity field to supply required dissipation where needed, and which can be augmented, if desired, with physical viscosity and thermal conductivity, yielding a high-order treatment of the Navier–Stokes equations for compressible fluids. We show that our approach results in sub-cell shock capturing ability, unlike traditional limiting schemes that tend to defeat the benefits of going to high order in DG in problems featuring many shocks. We demonstrate exponential convergence of our solver as a function of order when applied to smooth flows, such as the Kelvin–Helmholtz reference problem of Lecoanet et al. We also demonstrate excellent scalability of our GPU implementation up to hundreds of GPUs distributed on different compute nodes. In a first application to driven, subsonic turbulence, we highlight the accuracy advantages of high-order DG compared to traditional second-order accurate methods, and we stress the importance of physical viscosity for obtaining accurate velocity power spectra.

The pointwise estimates of solutions for the 3D compressible viscoelastic fluids

In this paper, we investigate the Cauchy problem of the 3D compressible viscoelastic hydrodynamic model. By decomposing the compressible viscoelastic fluid equations, according to Duhamel's principle, we make full use of the decomposition of divergence and curl, and obtain the expression of the corresponding solution. By refined integral calculation and classification discussion, we analyze the interaction of different waves in detail, and finally obtain the pointwise time-space estimates of the solutions.

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids, which is different from the case studied by Mei et al. (2018) for the free boundary compressible Navier-Stokes system.

A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert’s sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.

Experimental methods in chemical engineering: Computational fluid dynamics/finite volume method—CFD/FVM

AbstractComputational fluid dynamics (CFD) applies numerical methods to solve transport phenomena problems. These include, for example, problems related to fluid flow comprising the Navier–Stokes transport equations for either compressible or incompressible fluids, together with turbulence models and continuity equations for single and multi‐component (reacting and inert) systems. The design space is first segmented into discrete volume elements (meshing). The finite volume method, the subject of this article, discretizes the equations in time and space to produce a set of non‐linear algebraic expressions that are assigned to each volume element—cell. The system of equations is solved iteratively with algorithms like the semi‐implicit method for pressure‐linked equations (SIMPLE) and the pressure implicit splitting of operators (PISO). CFD is especially useful for testing multiple design elements because it is often faster and cheaper than experiments. The downside is that this numerical method is based on models that require validation to check their accuracy. According to a bibliometric analysis, the broad research domains in chemical engineering include: (1) dynamics and CFD‐DEM, (2) fluid flow, heat transfer, and turbulence, (3) mass transfer and combustion, (4) ventilation and the environment, and (5) design and optimization. Here, we review the basic theoretical concepts of CFD and illustrate how to set up a problem in the open‐source software OpenFOAM to isomerize n‐butane to i‐butane in a notched reactor under turbulent conditions. We simulated the problem with 1000, 4000, and 16 000 cells. According to the Richardson extrapolation, the simulation underestimates the adiabatic temperature rise by 7% with 16 000 cells.

Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum

This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in [Formula: see text]. For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.

The acoustic streaming effects of sonic black hole

In this paper, the acoustic streaming effects of Sonic Black Hole (SBH) are deeply studied by using the numerical and simulation method. By solving Navier–Stokes equations of compressible fluid, we analyze the flow characteristics of acoustic medium in SBH excited by sound wave, and further discuss the acoustic streaming effects and sound wave capture mechanism of SBH. The phenomenon of simultaneous reduction of medium and high-frequency sound reflection and transmission is analyzed. Under the excitation of medium- and high-frequency sound waves, the interaction between fluid medium flow and SBH structure leads to the uneven internal velocity distribution in the sound propagation direction, which can lead to the phenomenon of sound wave capture and deceleration. For low-frequency sound waves, the velocity distribution of sound medium is uniform, and there is almost no change in velocity gradient, resulting in the uniform distribution of sound pressure without sound absorption. These theoretical and numerical results are in good agreement with the experimental results in the literature, and also verify our results. Finally, by improving the structure, increasing the complexity of sound medium flow and the gradient change in the direction of sound propagation, the sound absorption and insulation ability of the SBH can be further improved. This study reveals the sound transmission mechanism of SBH, which can provide a new idea for the suppression of low-frequency sound waves in SBH.