The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field

The relationship between the ensemble average stress in a dilute suspension of spheres and the imposed rate of strain and rotation is derived for a general linear flow of a suspension in a second-order fluid. In a Newtonian fluid, the particulate phase only contributes to the stress via the shear viscosity; the contribution takes the form of a stresslet, the symmetric first moment of the force distribution on the surface of a suspended particle. In a second-order fluid, the interactions of the particles and polymers contribute to the stress in three ways: (1) the particle-induced fluid velocity disturbance alters the polymer stress in the fluid; (2) the polymer stresses exerted on the particle contribute to the particle’s stresslet; (3) the non-Newtonian nature of the fluid changes the pressure and velocity field, thereby modifying the Newtonian contributions to the particle stresslet. The particle contributions Ψ1P and Ψ2P to the first and second normal stress differences are related to the corresponding stress differences (Ψ10 and Ψ20) for the suspending fluid by Ψ1P=(5/2)ϕΨ10 and Ψ2P=(75/28)ϕΨ20−(5/28)ϕΨ10, where ϕϕ is the particle volume fraction.