We compute the first order correction of the effective viscosity for a suspension containing solid particles with arbitrary shapes. We rewrite the computation as an homogenization problem for the Stokes equations in a perforated domain. Then, we extend the method of reflections [14], [20] to approximate the solution to the Stokes problem with a fixed number of particles. By obtaining sharp estimates, we are able to prove that this method converges for small volume fraction of the solid phase whatever the number of particles. This allows to address the limit when the number of particles diverges while their radius tends to 0. We obtain a system of PDEs similar to the Stokes system with a supplementary term in the viscosity proportional to the volume fraction of the solid phase in the mixture.

我们计算包含任意形状的固体颗粒的悬浮液的有效粘度的一级修正。我们将计算重写为穿孔域中Stokes方程的均质化问题。然后，我们扩展了反射方法[14]，[20]，以近似的方法解决了具有固定数量粒子的斯托克斯问题。通过获得精确的估计，我们能够证明该方法收敛于固相的小体积分数，无论颗粒数量如何。当颗粒数趋于零而半径趋于0时，这可以解决极限问题。我们获得了一种类似于Stokes系统的PDE系统，其粘度的补充项与混合物中固相的体积分数成比例。