Loi de Sutherland

La loi de Sutherland exprime la viscosité d'un gaz en fonction de la température en utilisant un potentiel interatomique particulier introduit par William Sutherland^[1], qui souhaitait une expression plus réaliste que celle obtenue à partir du potentiel des sphères élastiques infiniment dures.

Potentiel de Sutherland

Le potentiel utilisé, dit potentiel de Sutherland, prolonge celui des sphères dures par un terme en $r^{-\gamma }$ au-delà du rayon $r$ de non-pénétration $\sigma$ ^[2] :

E_{\mathrm {p} }=\left\{{\begin{array}{cl}\infty &\mathrm {si} \quad r\leq \sigma \\-\varepsilon \,\left({\dfrac {\sigma }{r}}\right)^{\gamma }&\mathrm {si} \quad r>\sigma \end{array}}\right.

où $\varepsilon$ est la valeur attractive maximale du potentiel.

La viscosité dynamique obtenue s'écrit :

\eta ={\frac {\eta _{0}}{1+{\frac {S}{T}}}}\,,\quad \eta _{0}={\frac {5}{16}}\left({\frac {k_{\mathrm {B} }}{\pi N_{\mathrm {A} }}}\right)^{\frac {1}{2}}{\frac {(MT)^{\frac {1}{2}}}{\sigma ^{2}}}\,,\quad S=f(\gamma ){\frac {\varepsilon }{k_{\mathrm {B} }}}

où $\eta _{0}$ est la viscosité obtenue avec un modèle de sphères dures ( $M$ est la masse molaire) et $f(\gamma )$ un coefficient calculable numériquement^[2]. Pour $\gamma =6$ (valeur typique pour un potentiel physiquement réaliste), $f(\gamma )=$ 0,1667.

Loi de Sutherland

L'expression précédente est à l'origine de la loi de Sutherland, une formule semi-empirique pour exprimer la viscosité dynamique d'un fluide :

\eta (T)\approx \eta (T_{\mathrm {ref} })\,\left({\frac {T}{T_{\mathrm {ref} }}}\right)^{\!3/2}\,{\frac {T_{\mathrm {ref} }+S}{T+S}}

$T_{\mathrm {ref} }$ est une température de référence, généralement 273,15 K, et $S$ une constante numérique qui dépend du gaz considéré. Cette expression se déduit de la précédente en remplaçant $T^{\frac {1}{2}}$ par $T^{\frac {3}{2}}$ .

Pour l'air par exemple, on prend $S$ = 110,4 K et $\eta (T_{\mathrm {ref} })$ = 1,715 × 10⁻⁵ Pa s, ce qui donne une bonne approximation sur une plage de température de l'ordre de 170 à 1 500 K environ.

Références

↑ (en) William Sutherland, « The viscosity of gases and molecular force », The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 5^e série, vol. 36,‎ 1893, p. 507-531 (DOI 10.1080/14786449308620508, lire en ligne [PDF])
↑ ^{a et b} (en) Joseph Oakland Hirschfelder, Charles Francis Curtiss et Robert Byron Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, 1966 (ISBN 978-0-471-40065-3).

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