# Entropy in self-similar shock profiles

## Abstract

In this paper, we study the structure of a gaseous shock, and in particular the distribution of entropy within, in both a thermodynamics and a statistical mechanics context. The problem of shock structure has a long and distinguished history that we review. We employ the Navier–Stokes equations to construct a self–similar version of Becker’s solution for a shock assuming a particular (physically plausible) Prandtl number; that solution reproduces the well–known result of Morduchow & Libby that features a maximum of the equilibrium entropy inside the shock profile. We then construct an entropy profile, based on gas kinetic theory, that is smooth and monotonically increasing. The extension of equilibrium thermodynamics to irreversible processes is based in part on the assumption of local thermodynamic equilibrium. We show that this assumption is not valid except for the weakest shocks. Finally, we conclude by hypothesizing a thermodynamic nonequilibrium entropy and demonstrating that it closely estimates the gas kinetic nonequilibrium entropy within a shock.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- U.S. Naval Research Lab., Stennis Space Center, MS (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1372792

- Report Number(s):
- LA-UR-15-26684

Journal ID: ISSN 0020-7462

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal of Non-Linear Mechanics

- Additional Journal Information:
- Journal Volume: 95; Journal ID: ISSN 0020-7462

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; shock waves; entropy; Clausius Duhem

### Citation Formats

```
Margolin, Len G., Reisner, Jon Michael, and Jordan, Pedro M.
```*Entropy in self-similar shock profiles*. United States: N. p., 2017.
Web. doi:10.1016/j.ijnonlinmec.2017.07.003.

```
Margolin, Len G., Reisner, Jon Michael, & Jordan, Pedro M.
```*Entropy in self-similar shock profiles*. United States. doi:10.1016/j.ijnonlinmec.2017.07.003.

```
Margolin, Len G., Reisner, Jon Michael, and Jordan, Pedro M. Sun .
"Entropy in self-similar shock profiles". United States.
doi:10.1016/j.ijnonlinmec.2017.07.003. https://www.osti.gov/servlets/purl/1372792.
```

```
@article{osti_1372792,
```

title = {Entropy in self-similar shock profiles},

author = {Margolin, Len G. and Reisner, Jon Michael and Jordan, Pedro M.},

abstractNote = {In this paper, we study the structure of a gaseous shock, and in particular the distribution of entropy within, in both a thermodynamics and a statistical mechanics context. The problem of shock structure has a long and distinguished history that we review. We employ the Navier–Stokes equations to construct a self–similar version of Becker’s solution for a shock assuming a particular (physically plausible) Prandtl number; that solution reproduces the well–known result of Morduchow & Libby that features a maximum of the equilibrium entropy inside the shock profile. We then construct an entropy profile, based on gas kinetic theory, that is smooth and monotonically increasing. The extension of equilibrium thermodynamics to irreversible processes is based in part on the assumption of local thermodynamic equilibrium. We show that this assumption is not valid except for the weakest shocks. Finally, we conclude by hypothesizing a thermodynamic nonequilibrium entropy and demonstrating that it closely estimates the gas kinetic nonequilibrium entropy within a shock.},

doi = {10.1016/j.ijnonlinmec.2017.07.003},

journal = {International Journal of Non-Linear Mechanics},

number = ,

volume = 95,

place = {United States},

year = {Sun Jul 16 00:00:00 EDT 2017},

month = {Sun Jul 16 00:00:00 EDT 2017}

}

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