We derive the pressure tensor and the heat flux to accompany the new macroscopic conservation equations that we developed previously in a volume-based kinetic framework for gas flows. This kinetic description allows for expansion or compression of a fluid element, and is characterized by a flux of volume that conventional modelling does not account for. A new convective form of transport arises from this new approach, which is due purely to macroscopic expansion or compression. It supplements the conventional transport processes, which are transport due purely to mass motion (classical convective transport) and transport due purely to random motion (diffusive transport). In addition, we show that the thermodynamic parameters of the fluid (temperature and pressure) appear with new non-equilibrium contributions that depend on density variations in the gas. Our new model may be useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluids and flows.
We develop constitutive equations for multi-component, multi-phase, macro-scale flow in a porous medium exposed to temperature-, composition-, and pressure -gradients. The porous medium is non-deformable. We define the pressure and the composition of the representative elementary volume (REV) in terms of the volume and surface averaged pressure and the saturation, and the respective driving forces from these variables. New contributions due to varying porosity or surface tension offer explanations for non-Darcy behavior. The interaction of a thermal and mechanical driving forces give thermal o…
Non-equilibrium effects play a vital role in high-speed and rarefied gas flows and the accurate simulation of these flow regimes are far beyond the capability of near-local-equilibrium Navier-Stokes-Fourier equations. Eu proposed generalized hydrodynamic equations which are consistent with the laws of irreversible thermodynamics to solve this problem. Based on Eu's generalized hydrodynamics equations, a computational model, namely the nonlinear coupled constitutive relations(NCCR),was developed by R.S.Myong and applied successfully to one-dimensional shock wave structure and two-dimensional ra…
We propose a constitutive model to predict the viscosity of fiber suspensions, which undergoes shear thinning, at various volume fractions, aspect ratios, and shear stresses/rates. We calibrate the model using the data from direct numerical simulation and prove the accuracy by predicting experimental measurements from the literature. We use a friction coefficient decreasing with the normal load between the fibers to quantitatively reproduce the experimentally observed shear thinning in fiber suspensions. In this model, the effective normal contact force, which is directly proportional to the b…
Southern University of Science and Technology（南方科技大学）；Hong Kong University of Science and Technology（香港科技大学）
Finding extended hydrodynamics equations valid from the dense gas region to the rarefied gas region remains a great challenge. The key to success is to obtain accurate constitutive relations for stress and heat flux. Data-driven models offer a new phenomenological approach to learning constitutive relations from data. Such models enable complex constitutive relations that extend Newton's law of viscosity and Fourier's law of heat conduction by regression on higher derivatives. However, the choices of derivatives in these models are ad-hoc without a clear physical explanation. We investigated d…
We develop direct inversion methods for inferring the rheology of a fluid from observations of its shallow flow. First, the evolution equation for the free-surface flow of an inertia-less current with general constitutive law is derived. The relationship between the volume flux of fluid and the basal stress, $τ_b$ is encapsulated by a single function $F(τ_b)$, which depends only on the constitutive law. The inversion method consists of (i) determining the flux and basal stress from the free-surface evolution, (ii) comparing the flux to the basal stress to constrain $F$ and (iii) inferring the …
The linear Navier-Stokes-Fourier (NSF) constitutive relations are capable of simulating the near-continuum flows, but fail in description of those flows which are removed far away from local equilibrium. In this paper, a diatomic nonlinear model named as nonlinear coupled constitutive relations (NCCR), derived from Eu's generalized hydrodynamics and proposed by Myong, is presented as an alternative for simulating these hypersonic gas flows with a goal of recovering NSF's solutions in continuum regime and being superior in transition regime. To guarantee stable computation, a reliable and effic…
High temperature gases, for instance in hypersonic reentry flows, show complex phenomena like excitation of rotational and vibrational energy modes, and even chemical reactions. For flows in the continuous regime, simulation codes use analytic or tabulated constitutive laws for pressure and temperature. In this paper, we propose a BGK model which is consistent with any arbitrary constitutive laws, and which is designed to make high temperature gas flow simulations in the rarefied regime. A Chapman-Enskog analysis gives the corresponding transport coefficients. Our approach is illustrated by a …
The constitutive equations for the heat flux and the Navier tensor are established for a high temperature dilute gas in two spatial dimensions. The Chapman-Enskog procedure to first order in the gradients is applied in order to obtain the dissipative energy and momentum fluxes from the relativistic Boltzmann equation. The solution for such equation is written in terms of three sets of orthogonal polynomials which are explicitly obtained for this calculation. As in the three dimensional scenario, the heat flux is shown to be driven by the density, or pressure, gradient additionally to the usual…
We reconsider some fundamental aspects of the fluid mechanics model, and the derivation of continuum flow equations from gas kinetic theory. Two topologies for fluid representation are presented, and a set of macroscopic equations are derived through a modified version of the classical Boltzmann kinetic equation for monatomic gases. The free volumes around the gaseous molecules are introduced into the set of kinetic microscopic parameters. Our new description comprises four, rather than three, conservation equations; the classical continuity equation, which conflates actual mass-density and nu…
The general synthetic iteration scheme (GSIS) is extended to find the steady-state solution of nonlinear gas kinetic equation, removing the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredients of GSIS are that the gas kinetic equation and macroscopic synthetic equations are tightly coupled, and the constitutive relations in macroscopic synthetic equations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated…
We investigate sound wave propagation in a monatomic gas using a volume-based hydrodynamic model. In Physica A vol 387(24) (2008) pp6079-6094, a microscopic volume-based kinetic approach was proposed by analyzing molecular spatial distributions; this led to a set of hydrodynamic equations incorporating a mass-density diffusion component. Here we find that these new mass-density diffusive flux and volume terms mean that our hydrodynamic model, uniquely, reproduces sound wave phase speed and damping measurements with excellent agreement over the full range of Knudsen number. In the high Knudsen …
We simulate complex fluids by means of an on-the-fly coupling of the bulk rheology to the underlying microstructure dynamics. In particular, a macroscopic continuum model of polymeric fluids is constructed without a pre-specified constitutive relation, but instead it is actively learned from mesoscopic simulations where the dynamics of polymer chains is explicitly computed. To couple the macroscopic rheology of polymeric fluids and the microscale dynamics of polymer chains, the continuum approach (based on the finite volume method) provides the transient flow field as inputs for the (mesoscopi…
Classical continuum-based liquid vapour phase-change models typically assume continuity of temperature at phase interfaces along with a relation which describes the rate of evaporation at the interface (Hertz-Knudsen-Schrage, for example). However, for phase transitions processes at small scales, such as the evaporation of nanodroplets, the assumption that the temperature is continuous across the liquid-vapour interface leads to significant inaccuracies (McGaughey & Ward 2002; Rana et al. 2019), as may the adoption of classical constitutive relations that lead to the Navier-Stokes-Fourier equa…
An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume-averaging of the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme…
We extend our recent work on the creeping flow of a Bingham fluid in a lid-driven cavity, to the study of inertial effects, using a finite volume method and the Papanastasiou regularisation of the Bingham constitutive model [J. Rheology 31 (1987) 385-404]. The finite volume method used belongs to a very popular class of methods for solving Newtonian flow problems, which use the SIMPLE algorithm to solve the discretised set of equations, and have matured over the years. By regularising the Bingham constitutive equation it is easy to extend such a solver to Bingham flows since all that this requ…
The role of thermodynamics in deriving constitutive equations is unique, and various approaches have been developed in the last decades. In the present paper, the frameworks of Extended Irreversible Thermodynamics (EIT) and Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) are discussed and compared to each other on the basis of a particular problem of rarefied gases. In this comparison, both theoretical and experimental aspects are taken into account. Eventually, an experiment by Meyer and Sessler covering a wide range of pressures and frequencies is investigated. Here, concentr…
We investigate the performance of the finite volume method in solving viscoplastic flows. The creeping square lid-driven cavity flow of a Bingham plastic is chosen as the test case and the constitutive equation is regularised as proposed by Papanastasiou [J. Rheol. 31 (1987) 385-404]. It is shown that the convergence rate of the standard SIMPLE pressure-correction algorithm, which is used to solve the algebraic equation system that is produced by the finite volume discretisation, severely deteriorates as the Bingham number increases, with a corresponding increase in the non-linearity of the eq…
Using the kinetic-theory-based two-fluid models as a starting point, we develop filtered two-fluid models for a gas-particle flow in the presence of an isothermal, first-order, solid-catalyzed reaction of a gaseous species. As a consequence of the filtering procedure, terms describing the filtered reaction rate and filtered reactant dispersion need to be constituted in order to close the filtered species balance equation. In this work, a constitutive relation for filtered reaction rate is developed by performing fine-grid, two-fluid model simulations of an isothermal, solid-catalyzed, first-or…
To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows over (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to the fourth-order GKS-based traditional finite volume (FV) method, the proposed method realizes compactness effectively by subdividing each cell into a set of subcells or control volumes (CVs) and selecting only face-neighboring cells for high-order co…
