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):011301. doi: 10.1103/PhysRevE.79.011301. Epub 2009 Jan 14.
The hydrodynamic modes and the velocity autocorrelation functions for a dilute sheared inelastic fluid are analyzed using an expansion in the parameter =(1-e);{12} , where e is the coefficient of restitution. It is shown that the hydrodynamic modes for a sheared inelastic fluid are very different from those for an elastic fluid in the long-wave limit, since energy is not a conserved variable when the wavelength of perturbations is larger than the "conduction length." In an inelastic fluid under shear, there are three coupled modes, the mass and the momenta in the plane of shear, which have a decay rate proportional to k;{23} in the limit k-->0 , if the wave vector has a component along the flow direction. When the wave vector is aligned along the gradient-vorticity plane, we find that the scaling of the growth rate is similar to that for an elastic fluid. The Fourier transforms of the velocity autocorrelation functions are calculated for a steady shear flow correct to leading order in an expansion in . The time dependence of the autocorrelation function in the long-time limit is obtained by estimating the integral of the Fourier transform over wave number space. It is found that the autocorrelation functions for the velocity in the flow and gradient directions decay proportional to t;{-52} in two dimensions and t;{-154} in three dimensions. In the vorticity direction, the decay of the autocorrelation function is proportional to t;{-3} in two dimensions and t;{-72} in three dimensions.
利用参数(\epsilon=(1 - e))（其中(e)为恢复系数）的展开式，分析了稀薄剪切非弹性流体的流体动力学模式和速度自相关函数。结果表明，在长波极限下，剪切非弹性流体的流体动力学模式与弹性流体的模式有很大不同，因为当扰动波长大于“传导长度”时，能量不是一个守恒变量。在剪切作用下的非弹性流体中，存在三种耦合模式，即剪切平面内的质量和动量，在(k\rightarrow0)的极限情况下，如果波矢沿流动方向有一个分量，则其衰减率与(k^{2/3})成正比。当波矢沿梯度 - 涡度平面排列时，我们发现增长率的标度与弹性流体的相似。计算了稳态剪切流速度自相关函数的傅里叶变换，其在(\epsilon)展开式中精确到主导阶。通过估计波数空间上傅里叶变换的积分，得到了自相关函数在长时间极限下的时间依赖性。结果发现，流动方向和梯度方向上速度的自相关函数在二维中按(t^{-5/2})衰减，在三维中按(t^{-15/4})衰减。在涡度方向上，自相关函数的衰减在二维中与(t^{-3})成正比，在三维中与(t^{-7/2})成正比。
