A spectral solution of the Boltzmann equation for the infinitely strong shock

1990 ◽  
Vol 330 (1611) ◽  
pp. 217-252 ◽  
Keyword(s):  
Boltzmann Equation ◽  
Constitutive Relations ◽  
Local Equilibrium ◽  
Nonlinear Dynamical System ◽  
Strong Shock ◽  
Singular Part ◽  
Orthogonal Functions ◽  
The Boltzmann Equation ◽  
Highly Nonlinear ◽  
Background Gas

We formulate and implement a new spectral method for the solution of the Boltzmann equation, making extensive use of the theory of irreducible tensors together with the symbolic notation of Dirac. These tools are shown to provide a transparent organization of the algebra of the method and the efficient automation of the associated calculations. I he power of the proposed method is demonstrated by application to the highly nonlinear problem of the infinitely strong shock. It is shown that the distribution function can in this limit be decomposed into a singular part corresponding to the molecular beam, which represents the supersonic side of the shock, and a regular part, which provides the evolving ‘ background gas and covers the rest of velocity space. Separate governing equations for the singular and regular parts are derived, and solved by an expansion of the latter in an infinite series of orthogonal functions. The basis for this expansion is the same set that was used by Burnett ( Proc. Lond. math. Soc . 39, 385-430 (1935)), but is centred around the (fixed) downstream maxwellian. This basis, because of the presence of spherical harmonics which provide an irreducible representation of the group SO (3), lends itself to the utilization of powerful group-theoretic tools. The present expansion, not being about local equilibrium, does not imply any constitutive relations; instead it reduces the Boltzmann equation to an equivalent infinite-order nonlinear dynamical system. A solution with six modes shows encouraging convergence in the density profile, towards a shock thickness of about 6.7 hot-side mean free paths.

Thermodynamics of Continuous Systems

2003 ◽  
Author(s):  
E. S. Geskin
Keyword(s):  
Boltzmann Equation ◽  
Local Equilibrium ◽  
Continuous System ◽  
Intrinsic Property ◽  
Continuous Systems ◽  
Initial State ◽  
The Boltzmann Equation ◽  
Generalized Variables ◽  
Classical Thermodynamics ◽  
Non Equilibrium

The formalism of the classical thermodynamics, for example Gibbs equations, is routinely and successfully applied to the highly non-equilibrium processes in dynamic systems. Such applications are based on the local equilibrium hypothesis. The presented paper discusses the conditions of the application of this hypothesis. It is shown that the local equilibrium hypothesis is applicable with no limitations to continuous systems. This application is validated by the solution of the Boltzmann equation. This solution was obtained by Enskog, Chapman and Bogolubov. From the Boltzmann equation follows that regardless of the initial state of the system the duration of its approach to the local equilibrium conditions by far less than the time scale of the evolution of the macro properties of the continuous media. This result shows that the local equilibrium is the intrinsic property of this media. Thus it is possible to apply the formalism of the non-equilibrium thermodynamics (the generalized variables, forces and fluxes) to description of the continuous system with no limitations. The derivation of the Carnot theorem is presented to show the effectiveness of such an application.

Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave

1974 ◽  
Vol 65 (3) ◽  
pp. 603-624 ◽  
Author(s):  
J. P. Elliott ◽  
D. Baganoff
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Mach Number ◽  
Singular Point ◽  
Shock Waves ◽  
Strong Shock ◽  
Singular Points ◽  
The Boltzmann Equation ◽  
Closure Relations ◽  
Strong Shock Waves

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

On Nonlocal Constitutive Relations, Continued Fraction Expansion for the Wave Vector Dependent Diffusion Coefficient

1977 ◽  
Vol 32 (7) ◽  
pp. 678-684 ◽  
Author(s):  
Siegfried Hess
Keyword(s):  
Diffusion Coefficient ◽  
Boltzmann Equation ◽  
Wave Vector ◽  
Continued Fraction ◽  
Constitutive Relations ◽  
Transport Coefficients ◽  
Mean Free Path ◽  
Upper And Lower Bounds ◽  
Continued Fraction Expansion ◽  
The Boltzmann Equation

Abstract Nonlocal constitutive relations which involve wave vector dependent transport coefficients can be derived from the Boltzmann equation. Diffusion of a Lorentzian gas is treated as an illustrative example. Transport-relaxation equations obtained from the Boltzmann equation with the help of the moment method lead to a continued fraction expansion for the wave vector dependent diffusion coefficient D(k). Rapidly converging upper and lower bounds on D(k)/D(0) are found which are meaningful for all values of lk where l is a mean free path and k is the magnitude of the wave vektor k. Also some remarks on a frequency and wave vector dependent diffusion coefficient are made.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

Approach to Equilibrium, the H-Theorem and Irreversibility

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Detailed Knowledge ◽  
Approach To Equilibrium ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

Boltzmann’s Kinetic Theory

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Statistical Physics ◽  
Neutron Transport ◽  
Condensed Matter ◽  
Relativistic Fluids ◽  
Classical Statistical Mechanics ◽  
Dilute Gas ◽  
Equilibrium Processes ◽  
The Boltzmann Equation

Kinetic theory is the branch of statistical physics dealing with the dynamics of non-equilibrium processes and their relaxation to thermodynamic equilibrium. Established by Ludwig Boltzmann (1844–1906) in 1872, his eponymous equation stands as its mathematical cornerstone. Originally developed in the framework of dilute gas systems, the Boltzmann equation has spread its wings across many areas of modern statistical physics, including electron transport in semiconductors, neutron transport, quantum-relativistic fluids in condensed matter and even subnuclear plasmas. In this Chapter, a basic introduction to the Boltzmann equation in the context of classical statistical mechanics shall be provided.

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