Kumaran V
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jan;79(1 Pt 1):011302. doi: 10.1103/PhysRevE.79.011302. Epub 2009 Jan 14.
The effect of correlations on the viscosity of a dilute sheared inelastic fluid is analyzed using the ring-kinetic equation for the two-particle correlation function. The leading-order contribution to the stress in an expansion in =(1-e);{12} is calculated, and it is shown that the leading-order viscosity is identical to that obtained from the Green-Kubo formula, provided the stress autocorrelation function in a sheared steady state is used in the Green-Kubo formula. A systemmatic extension of this to higher orders is also formulated, and the higher-order contributions to the stress from the ring-kinetic equation are determined in terms of the terms in the Chapman-Enskog solution for the Boltzmann equation. The series is resummed analytically to obtain a renormalized stress equation. The most dominant contributions to the two-particle correlation function are products of the eigenvectors of the conserved hydrodynamic modes of the two correlated particles. In Part I, it was shown that the long-time tails of the velocity autocorrelation function are not present in a sheared fluid. Using those results, we show that correlations do not cause a divergence in the transport coefficients; the viscosity is not divergent in two dimensions, and the Burnett coefficients are not divergent in three dimensions. The equations for three-particle and higher correlations are analyzed diagrammatically. It is found that the contributions due to the three-particle and higher correlation functions to the renormalized viscosity are smaller than those due to the two-particle distribution function in the limit -->0 . This implies that the most dominant correlation effects are due to the two-particle correlations.
利用两粒子关联函数的环动力学方程，分析了关联对稀薄剪切非弹性流体粘度的影响。计算了在(\epsilon=(1 - e);{12})展开中应力的主导阶贡献，结果表明，若在格林 - 库博公式中使用剪切稳态下的应力自相关函数，则主导阶粘度与从格林 - 库博公式得到的粘度相同。还给出了将此系统扩展到高阶的形式，并根据玻尔兹曼方程的查普曼 - 恩斯科格解中的项确定了环动力学方程对应力的高阶贡献。对该级数进行解析重整以得到重整化应力方程。对两粒子关联函数最主要的贡献是两个相关粒子守恒流体动力学模式的本征向量的乘积。在第一部分中，已表明剪切流体中不存在速度自相关函数的长时间尾项。利用这些结果，我们表明关联不会导致输运系数发散；二维中的粘度不会发散，三维中的伯内特系数也不会发散。对三粒子及更高阶关联的方程进行了图解分析。发现在(\epsilon\to0)的极限情况下，三粒子及更高阶关联函数对重整化粘度的贡献小于两粒子分布函数的贡献。这意味着最主要的关联效应是由两粒子关联引起的。
