Fundamental aspects of inverse kinetic theories for the incompressible Navier-Stokes equations [Ellero and Tessarotto, 2004, 2005] include the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. Purpose of this work is to prove that under suitable prescriptions the inverse kinetic theory can be determined to satisfy such requirements.
The Brownian diffusion of micron-scale inclusions in freely suspended smectic A liquid crystal films a few nanometers thick and several millimeters in diameter depends strongly on the air surrounding the film. Near atmospheric pressure, the three-dimensionally coupled film/gas system is well described by Hughes/Pailthorpe/White hydrodynamic theory but at lower pressure (p < 70 torr), the diffusion coefficient increases substantially, tending in high vacuum toward the two-dimensional limit where it is determined by film size. In the absence of air, the films are found to be a nearly ideal physi…
A basic aspect of the kinetic descriptions of incompressible fluids based on an inverse kinetic approaches is the possibility of satisfying an H-theorem. This property is in fact related to the identification of the kinetic distribution function with a probability density in an appropriate phase space. Goal of this investigation is to analyze the conditions of validity of an H-theorem for the inverse kinetic theory recently proposed by Ellero and Tessarotto [2004, 2005]. It is found that the time-dependent contribution to the kinetic pressure, characteristic of such a kinetic model, can always…
A remarkable feature of fluid dynamics is its relationship with classical dynamics and statistical mechanics. This has motivated in the past mathematical investigations concerning, in a special way, the "derivation" based on kinetic theory, and in particular the Boltzmann equation, of the incompressible Navier-Stokes equations (INSE). However, the connection determined in this way is usually merely asymptotic (i.e., it can be reached only for suitable limit functions) and therefore presents difficulties of its own. This feature has suggested the search of an alternative approach, based on the …
In this work the issue of whether key energetic properties (nonlinear, exponential-type dissipation in the abscence of forcing and long-term stability under conditions of time dependent loading) are automatically inherited by deforming reference descriptions is resolved. These properties are intrinsic to real flows and the conventional Navier-Stokes equations. A completely general reference description of an incompressible, Newtonian fluid, which reconciles the differences between opposing schools of thought in the literature is derived for the purposes of this investigation. The work subseque…
Blood flow, dam or ship construction and numerous other problems in biomedical and general engineering involve incompressible flows interacting with elastic structures. Such interactions heavily influence the deformation and stress states which, in turn, affect the engineering design process. Therefore, any reliable model of such physical processes must consider the coupling of fluids and solids. However, complexity increases for non-Newtonian fluid models, as used, e.g., for blood or polymer flows. In these fluids, subtle differences in the local shear rate can have a drastic impact on the fl…
The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) method and unified gas kinetic scheme (UGKS) method, such as the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, DUGKS is proposed for near incompressible fluid flows, the existing compressible effect may cause some serious errors in simulating incompressible problems. To diminish the compressible effect, in this paper a novel DUGKS model with …
We investigate the two dimensional incompressible Navier-Stokes(NS) and the continuity equations in Cartesian coordinates and Eulerian description for non-Newtonian fluids. As a non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multi-dimensional generalization of the well-known self-similar Ansatz, which was already used for non-compressible and compressible viscous flow studies. The geometrical interpretations of the trial function is also discussed. We compare our recent results to the former Newtonian ones.
In the above paper by Bechtel, Cai, Rooney and Wang, Physics of Fluids, 2004, 16, 3955-3974 six different theories of a Newtonian viscous fluid are investigated and compared, namely, the theory of a compressible Newtonian fluid, and five constitutive limits of this theory: the incompressible theory, the limit where density changes only due to changes in temperature, the limit where density changes only with changes in entropy, the limit where pressure is a function only of temperature, and the limit of pressure a function only of entropy. The six theories are compared through their ability to …
Model of incompressible turbulent flows via a kinetic theory
State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology（华中科技大学能源与动力工程学院煤燃烧国家重点实验室）；Institute of Interdisciplinary Research for Mathematics and Applied Science, Huazhong University of Science and Technology（华中科技大学数学与交叉科学研究院）；College of Energy Engineering, Zhejiang University（浙江大学能源工程学院）
Based on the recently-established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the…
Non-linear effects of the Navier-Stokes equations disappear under the Stokes regime of Newtonian fluid flows disallowing the fluid flow rectification. Here we show mathematically and experimentally that passive flow rectification of Newtonian fluids is obtainable under the Stokes regime of both compressible and incompressible flows by introducing nonlinearity into the otherwise linear Stokes equations. Asymmetric flow resistances arise in shallow nozzle/diffuser microchannels with deformable ceiling, in which the fluid flow is governed by a non-linear coupled fluid-solid mechanics equation. Fl…
Fundamental aspects of inverse kinetic theories for incompressible Navier-Stokes equations concern the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. In this paper we intend to prove that the non-uniqueness feature can be res…
Extending the statistical approach proposed in a parallel paper \cite% {Tessarotto2008-aa}, purpose of this work is to propose a stochastic inverse kinetic theory for small-scale hydrodynamic turbulence based on the introduction of a suitable \textit{local phase-space probability density function} (pdf). In particular, we pose the problem of the construction of Fokker-Planck kinetic models of hydrodynamic turbulence. The approach here adopted is based on the so-called IKT approach (inverse kinetic theory), developed by Ellero et al. (2004-2008) which permits an exact phase-space description of…
An interesting issue in fluid dynamics is represented by the possible existence of inverse kinetic theories (IKT) which are able to deliver, in a suitable sense, the complete set of fluid equations which are associated to a prescribed fluid. From the mathematical viewpoint this involves the formal description of a fluid by means of a classical dynamical system which advances in time the relevant fluid fields. The possibility of defining an IKT for the 3D incompressible Navier-Stokes equations (INSE), recently investigated (Ellero \textit{et al}, 2004-2007) raises the interesting question wheth…
In this article, we present a detailed asymptotic analysis of the lattice Boltzmann method with two different collision mechanisms of BGK-type on the D2Q9-lattice for generalized Newtonian fluids. Unlike that based on the Chapman-Enskog expansion leading to the compressible Navier-Stokes equations, our analysis gives the incompressible ones directly and exposes certain important features of the lattice Boltzmann solutions. Moreover, our analysis provides a theoretical basis for using the iteration to compute the rate-of-strain tensor, which makes sense specially for generalized Newtonian fluid…
The results studying various laminar flow regimes in diverging and converging plain channels (diffuser and confusor) with a small opening angle of channels (diverging and converging angles) are presented. The results are obtained for a viscous incompressible fluid by numerical simulation based on solving the Navier-Stokes equations. The paper presents the results concerning the change in the nature of flows from stationary - symmetric to stationary - asymmetric and to non-stationary in the diffuser and confusor in dependence on the Reynolds number. The ranges of existence of these flow regimes…
Despite the abundant literature on the subject appeared in the last few years, the lattice Boltzmann method (LBM) is probably the one for which a complete understanding is not yet available. As an example, an unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper aims at investigating discrete inverse kinetic theories (IKT) for incompressible fluids. We intend to show that the discrete IKT can be defined in su…
Pressure conditions in incompressible Navier-Stokes equations give rise to conservation of total energy. The energy rate getting into a volume is the same energy rate that gets out from it. Suitable choice of pressure counteracts energy disipation of viscosity term in such a way that total energy is preserved. As consequence, this prevents kinetic energy blow-up in a given volume of the fluid.
In spite of the large number of papers appeared in the past which are devoted to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain unchallenged. An unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper is theoretical and aims at developing an inverse kinetic approach of this type. In principle infinite solutions exist to this problem but the freedom can be exploited in order to meet…
Providing evidence of finite-time singularities of the incompressible Euler equations in three space dimensions is still an unsolved problem. Likewise, the zeroth law of turbulence has not been proven to date by numerical experiments. We address this issue by high-resolution numerical simulations of the inviscid three-dimensional Taylor-Green vortex problem using a novel high-order discontinuous Galerkin discretization approach. Our main finding is that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but in…
We discuss a pure hyperbolic alternative to the Navier-Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is …
We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are refor…
We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Grün for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.
We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move …
In this work, we extend the discrete unified gas-kinetic scheme (DUGKS) [Guo et al., Phys. Rev. E 88, 033305 (2013)] to continue two-phase flows. In the framework of DUGKS, two kinetic model equations are used to solve the quasi-incompressible phase-field governing equations [Yang et al., Phys. Rev.E 93, 043303 (2016)]. One is for the Chan-Hilliard (CH) equation and the other is for the Navier-Stokes equations. The DUGKS can correctly recover the quasi-incompressible phase-field governing equations through the Chapman-Enskog analysis. Unlike previous phase-field-based LB models, the Courant-Fr…
It is well-known that in fluid dynamics an alternative to customary direct solution methods (based on the discretization of the fluid fields) is provided by so-called \emph{particle simulation methods}. Particle simulation methods rely typically on appropriate \emph{kinetic models} for the fluid equations which permit the evaluation of the fluid fields in terms of suitable expectation values (or \emph{momenta}) of the kinetic distribution function $f(\mathbf{r,v},t),$ being respectively $\mathbf{r}$ and\textbf{\}$\mathbf{v}$ the position an velocity of a test particle with probability density …
A key aspect of fluid dynamics is the correct definition of the \textit{% phase-space} Lagrangian dynamics which characterizes arbitrary fluid elements of an incompressible fluid. Apart being an unsolved theoretical problem of fundamental importance, the issue is relevant to exhibit the connection between fluid dynamics and the classical dynamical systems underlying incompressible and non-isothermal fluid, typically founded either on: a) a \textit{configuration-space} Lagrangian description of the dynamics of fluid elements; b) a kinetic description of the molecular dynamics, based on a discre…
Phase-space Lagrangian dynamics in ideal fluids (i.e, continua) is usually related to the so-called {\it ideal tracer particles}. The latter, which can in principle be permitted to have arbitrary initial velocities, are understood as particles of infinitesimal size which do not produce significant perturbations of the fluid and do not interact among themselves. An unsolved theoretical problem is the correct definition of their dynamics in ideal fluids. The issue is relevant in order to exhibit the connection between fluid dynamics and the classical dynamical system, underlying a prescribed flu…
We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density, nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic $α$-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field f…
In this paper the issue of the determination of the fluid pressure in incompressible fluids is addressed, with particular reference to the search of algorithms which permit to advance in time the fluid pressure without actually solving numerically the Poisson equation. Based on an inverse kinetic approach recently proposed for the incompressible Navier-Stokes equations we intend to prove that an exact evolution equation can be obtained which advances in time self-consistently the fluid pressure. The new equation is susceptible of numerical implementation in Lagrangian CFD simulation codes.
We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence fre…
We showed that various expressions of stress in different models of fluids and in different forms of applying Newton's Second Law can be chosen, however there is always one Newton's Second Law which restricts and provides flexibility in these expressions. When fluids are regarded as being made of cells from a microscopic point of view, temperature force (gradient of internal kinetic energy) can also drive them flow.
In the present work a simple kinetic model based on the Enskog equation is solved to get the rheological properties of a hard-disk fluid under shear far from equilibrium, as functions of the density and shear rate. Comparison with Monte Carlo simulations of the Enskog equation show excellent agreement.
In this paper, a phase-field based model under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn (A-C) equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapmann-Ensko…
In this paper, we focus on investigating the influence on hydrodynamic factors of different coupled computational models describing the interaction between an incompressible fluid and two symmetric elastic or poroelastic structures. The fluid region is governed by time dependent Navier-Stokes equations; while for the structure region, we employ two different types of fully dynamic models to study the effects of elasticity and poroelasticity. It is known that blood flow shows a non- Newtonian property in small vessels and in situations of complex geometries. On one hand, we perform numerical ex…
We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illustration of the method, the equations of motion for both the elastic matrix and the fluid are derived in the spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems, such as sponges in water, where the elastic porous …
Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting 'sneezing' dynamics of a freshwater sponge, when the sponge contracts and expands to clear itself from surrounding polluted water. We derive the equations of motion for the dynamics of such an active porous media (i.e., a deformable porous media that is capable of applying a force to itself w…
A novel formulation of fluid dynamics as a kinetic theory with tailored, on-demand constructed particles removes any restrictions on Mach number and temperature as compared to its predecessors, the lattice Boltzmann methods and their modifications. In the new kinetic theory, discrete particles are determined by a rigorous limit process which avoids ad hoc assumptions about their velocities. Classical benchmarks for incompressible and compressible flows demonstrate that the proposed discrete-particles kinetic theory opens up an unprecedented wide domain of applications for computational fluid d…
Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to …
