Coupled constitutive relations: a second law based higher-order closure for hydrodynamics

In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

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A Derivation of the Main Relations of Nonequilibrium Thermodynamics

ISRN Thermodynamics ◽

10.1155/2013/906136 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Vladimir N. Pokrovskii

Keyword(s):

Nonequilibrium Thermodynamics ◽

Second Law Of Thermodynamics ◽

Thermodynamic System ◽

Internal Variables ◽

Second Law ◽

Basic Principles ◽

First Law Of Thermodynamics ◽

Sum Of Products ◽

Thermodynamic Forces ◽

Production Of Entropy

The principles of nonequilibrium thermodynamics are discussed, using the concept of internal variables that describe deviations of a thermodynamic system from the equilibrium state. While considering the first law of thermodynamics, work of internal variables is taken into account. It is shown that the requirement that the thermodynamic system cannot fulfil any work via internal variables is equivalent to the conventional formulation of the second law of thermodynamics. These statements, in line with the axioms introducing internal variables can be considered as basic principles of nonequilibrium thermodynamics. While considering stationary nonequilibrium situations close to equilibrium, it is shown that known linear parities between thermodynamic forces and fluxes and also the production of entropy, as a sum of products of thermodynamic forces and fluxes, are consequences of fundamental principles of thermodynamics.

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A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/0045-7825(86)90127-1 ◽

1986 ◽

Vol 54 (2) ◽

pp. 223-234 ◽

Cited By ~ 312

Author(s):

T.J.R. Hughes ◽

L.P. Franca ◽

M. Mallet

Keyword(s):

Fluid Dynamics ◽

Computational Fluid Dynamics ◽

Finite Element ◽

Stokes Equations ◽

Second Law Of Thermodynamics ◽

Navier Stokes ◽

Second Law ◽

Element Formulation ◽

Navier Stokes Equations ◽

Compressible Euler

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On the modelling of isothermal gas flows at the microscale

Journal of Fluid Mechanics ◽

10.1017/s0022112008001158 ◽

2008 ◽

Vol 604 ◽

pp. 235-261 ◽

Cited By ~ 60

Author(s):

DUNCAN A. LOCKERBY ◽

JASON M. REESE

Keyword(s):

Boundary Conditions ◽

Constitutive Relations ◽

Navier Stokes ◽

Test Case ◽

Gas Flows ◽

Hydrodynamic Models ◽

Slip Boundary ◽

Knudsen Numbers ◽

Isothermal Gas ◽

Non Equilibrium

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case.Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier--Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.

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The 2011 Koiter Lecture: The Simple Logic of Classical Nonlinear Thermodynamic Shell Theory

Journal of Applied Mechanics ◽

10.1115/1.4005536 ◽

2012 ◽

Vol 79 (4) ◽

Cited By ~ 2

Author(s):

J. G. Simmonds

Keyword(s):

Energy Density ◽

Constitutive Relations ◽

Equations Of Motion ◽

Three Dimensional ◽

Second Law Of Thermodynamics ◽

Second Law ◽

Virtual Power ◽

Ill Conditioning ◽

Simple Logic ◽

Reduced Forms

A classical nonlinear thermodynamic theory of elastic shells is derived by specializing the three-dimensional equations of motion and the second law of thermodynamics to a very general, shell-like body. No assumptions are made on how unknowns vary through the thickness. Extensional and bending strains are derived from the equations of motion via the principle of virtual power. The Coleman-Noll procedure plus the second law applied to an assumed form of the first law leads to constitutive relations plus reduced forms of the first and second laws. To avoid potential ill conditioning, a Legendre-Fenchel transformation is used to define a mixed-energy density, the logical place to impose the constitutive Kirchhoff hypothesis, if desired, because such an energy density rests, ultimately, on experiments. The Ladevèze-Pécastaings treatment of three-dimensional edge effects to obtain accurate two-dimensional solutions is discussed.

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On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

Journal of Scientific Computing ◽

10.1007/s10915-020-01388-6 ◽

2021 ◽

Vol 86 (3) ◽

Author(s):

Vladislav Balashov ◽

Alexander Zlotnik

Keyword(s):

Boundary Conditions ◽

Navier Stokes ◽

System Of Equations ◽

Spatial Discretization

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Is the Theory of Natural Selection a Statistical Theory?

Canadian Journal of Philosophy Supplementary Volume ◽

10.1080/00455091.1988.10715949 ◽

1988 ◽

Vol 14 ◽

pp. 187-207 ◽

Cited By ~ 7

Author(s):

Alexander Rosenberg

Keyword(s):

Statistical Theory ◽

Natural Selection ◽

Boundary Conditions ◽

Second Law Of Thermodynamics ◽

Second Law ◽

Local Boundary ◽

Long Run ◽

Equilibrium Level ◽

Statistical Version ◽

Average Fitness

In The Structure of Biological Science (Rosenberg [1985]) I argued that the theory of natural selection is a statistical theory for reasons much like those which makes thermodynamics a statistical theory. In particular, the theory claims that fitness differences are large enough and the life span of species long enough for increases in average fitness always to appear in the long run; and this claim, I held, is of the same form as the statistical version of the second law of thermodynamics.For the latter law also makes a claim about the long run, and its statistical character is due to this claim: thermodynamic systems must in the long run approach an equilibrium level of organization that maximizes entropy. Over finite times, given local boundary conditions, an isolated mechanical system, like the molecules in a container of gas, may sometimes interact so as to move the entropy of the system further from, instead of closer to the equilbrium level. But given enough interacting bodies, and enough time, the system will always eventually move in the direction prescribed by the law. Thus, we can attach much higher probabilities to the prediction that non-equilibrium systems will reflect greater entropy in future periods than we can to predictions that they will move in the opposite direction. And as we increase the amount of time and the number of bodies interacting, the strength of the probability we can attach to the prediction becomes greater and greater.

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Non-Reflective Boundary Conditions for a Consistent Model of Axisymmetric Electro-Magneto-Hydrodynamic Flows

10.1115/imece1999-1174 ◽

1999 ◽

Author(s):

Hyung-Jong Ko ◽

George S. Dulikravich

Keyword(s):

Magnetic Fields ◽

Flow Field ◽

Boundary Conditions ◽

Field Model ◽

Stokes Equations ◽

Constitutive Relations ◽

Axisymmetric Flow ◽

Navier Stokes ◽

Electro Magnetic Field ◽

Electro Magnetic

Abstract In this paper, the non-reflective boundary conditions for the axisymmetric electro-magneto-hydrodynamic (EMHD) flows have been derived. The electro-magneto-hydrodynamics (EMHD) deals with the motion of electrically conducting incompressible fluids under the combined influence of externally applied and internally generated electric and magnetic fields. A consistent axisymmetric EMHD flow model with linear constitutive relations and artificial compressibility was expressed in cylindrical coordinates. After some simplifications, the resulting EMHD system comprised of modified Maxwell equations for the electro-magnetic fields and modified Navier-Stokes equations for the flow-field, was transformed to a characteristic form, and the non-reflective boundary conditions were derived. The results show the strong mutual interactions between the axisymmetric flow-field and the electro-magnetic fields. The limiting cases, including the conventional axisymmetric flow-field model and the electro-magnetic field model in vacuum, are recoverable from these results.

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