Kinetic model for uniform shear flow

The so-called ellipsoidal statistical (ES) kinetic model is used to study the uniform shear flow problem in a dilute gas. This model is an extension of the well-known BGK kinetic to account for the correct Prandtl number. The velocity moments and the velocity distribution function are obtained in terms of the shear rate and a parameter Pr which plays the role of the Prandtl number. It is shown that, independently of the numerical value of Pr, the expressions of the second-degree velocity moments (which are related to the pressure tensor) coincide with the ones derived from the Boltzmann equation for Maxwell molecules. A comparison with previous results obtained from the Boltzmann equation for the fourth-degree velocity moments and for the velocity distribution function is carried out. Surprisingly enough, the comparison shows a superiority of the BGK model (Pr = 1) over the ES model ( ) in this problem. If one chooses values of Pr larger than one, the ES predictions are improved significantly.