Singular behavior of the velocity moments of a dilute gas under uniform shear flow.

Abstract

The hierarchy of moment equations derived from the nonlinear Boltzmann equation describing uniform shear flow is analyzed. It is shown that all the moments of order k\ensuremath{\ge}4 diverge in time for shear rates larger than a critical value ${\mathit{a}}_{\mathit{c}}^{(\mathit{k})}$, which decreases as k increases. Furthermore, the results suggest an asymptotic behavior of the form ${\mathit{a}}_{\mathit{c}}^{(\mathit{k})}$\ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}}$ for large k. Consequently, even for very small shear rates, either a stationary solution fails to exist (which implies the absence of a normal solution) or a stationary solution exists but with only a finite number of convergent moments. Although the uniform shear flow may be experimentally unrealizable for large shear rates, the above conclusions can be of interest for more realistic flows. \textcopyright{} 1996 The American Physical Society.