Inertial effects in dilute suspensions

Finite element approximations to the steady Navier-Stokes equations are used to compute the local flow experience by rigid particles in a sheared dilute suspension with an emphasis on inertial effects. A comparison of the basic flow features for the flow past a freely rotating sphere and past a freely rotating cylinder are presented, and the three dimensional features of the sphere problem are presented for the first time. Inertia causes asymmetric flow patterns, a collapse of the closed streamline region, and the emergence of flow reversal zones. Boundary layer separation for both particles, previously unreported, is also demonstrated. Steady simple shear flow past an elliptical cylinder, an oblate spheroid, and a prolate spheroid is computed. For all anisotropic particles studied, flow reversal regions are a basic flow pattern trait for inertial flows.; Numerical computations for simple shear flow past a sphere have been used to determine the effective stress of a dilute sheared suspension of rigid spheres with inertia. Inertia causes a dilute suspension to shear thicken, possess normal stress differences, and induces a suspension pressure. The numerical results agree quantitatively with previously published weak inertia asymptotic stress theories and provide the first verification of those theories. The rheology of a two dimensional dilute suspension is computed from numerical results of a freely rotating cylinder in simple shear flow and qualitatively predicts the same inertial effects present in the rheology of a dilute suspension of spheres.; A new approach to adaptive error control for the numerical approximation of elliptic partial differential equations is described. The approach rests on an a posteriori error estimate for a finite element solution of a nonlinear elliptic equation that takes into account both the local production of error, through the residual, and the global propagation of error, through a variational analysis using a dual problem. The error estimate provides a quantitative measure of solution quality and thus can be used to adaptively control the discretization of a computed solution.