Bounds and asymptotic results for the effective viscosity of a nondilute suspension of oriented fibres in longitudinal flow

Abstract

The medium considered in this paper is made of an assembly of rigid cylindrical parallel fibres suspended in a viscous incompressible fluid, flowing along the fibres in a Poiseuille-like configuration. An expression for the effective viscosity of the suspension, valid for any solid concentration, is obtained from the homogenization technique. When this special two-dimensional medium is isotropic on the macroscopic scale, upper and lower bounds are derived for its effective viscosity coefficient. These bounds depend on the solid concentration as well as on the geometry of the microstructure. They are calculated for fibres with a circular or a square shaped cross-section. Numerical results emphasize the good approximation of the effective viscosity coefficient provided by the upper or lower bounds. First order asymptotic expansions are also derived for the viscosity coefficient (i) for low concentrations of fibres with a circular cross-section (ii) for large concentrations of fibres with a square shaped cross-section, close to packing conditions. In the first case, it is shown that preceding results by Byron Pipes et al. [J. Compos. Mater. 25 (1991) 1204] disagree with these bounds.