The influence of secondary flows induced by normal stress differences on the shear-induced migration of particles in concentrated suspensions

It was first demonstrated experimentally by H. Giesekus in 1965 that the second normal stress difference in polymers can induce a secondary flow within the cross-section of a non-axisymmetric conduit. In this paper, we show through simulations that the same may be true for suspensions of rigid non-colloidal particles that are known to exhibit a strong negative second normal stress difference. Typically, the magnitudes of the transverse velocity components are small compared to the average axial velocity of the suspension; but the ratio of this transverse convective velocity to the shear-induced migration velocity is characterized by the shear-induced migration Péclet number χ which scales as B2/a2, B being the characteristic length scale of the cross-section and a being the particle radius. Since this Péclet number is kept high in suspension experiments (typically 100 to 2500), the influence of the weak circulation currents on the concentration profile can be very strong, a result that has not been appreciated in previous work. The principal effect of secondary flows on the concentration distribution as determined from simulations using the suspension balance model of Nott & Brady (J. Fluid Mech. vol. 275, 1994, p. 157) and the constitutive equations of Zarraga et al. (J. Rheol. vol. 44, 2000, p. 185) is three-fold. First, the steady-state particle concentration distribution is no longer independent of particle size; rather, it depends on the aspect ratio B/a. Secondly, the direction of the secondary flow is such that particles are swept out of regions of high streamsurface curvature, e.g. particle concentrations in corners reach a minimum rather than the local maximum predicted in the absence of such flows. Finally, the second normal stress differences lead to instabilities even in such simple geometries as plane-Poiseuille flow.