Application of the homogenization method to a suspension of fibres

Abstract

The homogenization method is applied to the determination of a continuum model for an assembly of cylindrical parallel fibres suspended in the longitudinal flow of a viscous incompressible fluid. Whatever the solids concentration is, the homogenization gives the variations of the velocity and the pressure at the level of the microstructure and furnishes a rigorous deductive procedure for obtaining the governing equations of the homogenized bulk medium. The effective constitutive relation is that of a Newtonian incompressible anisotropic fluid. The viscosity is expressed by a second order tensor completely determined by the microstructure. Lower bounds of the viscosity coefficients are found when the microstructure exhibits symmetries. Numerical results are obtained for two cases of macroscopic isotropy. They emphasize the good approximation arising from the lower bounds and the importance of the microstructure geometry for nondilute suspensions. As a matter of fact the solids concentration is not able to describe correctly the influence of the microstructure on the bulk behaviour of concentrated suspensions.