Rheology of a suspension of deformable spheres in a weakly viscoelastic fluid

In this work, we consider a suspension of weakly deformable solid particles within a weakly viscoelastic fluid. The fluid phase is modelled as a second-order fluid, and particles within the suspended phase are assumed linearly elastic and relatively dilute. We apply a cell model as a proxy for mean field flow, and solve analytically within a cellular fluid layer and its enclosed particle. We use an ensemble averaging process to derive analytical results for the bulk stress in suspension, and evaluate the macroscopic properties in both shear and extensional flow. Our viscometric functions align with existing literature over a surprisingly broad range of fluid and solid elasticities. The suspension behaves macroscopically as a second-order fluid, and we give simple formulae by which the reader can calculate the parameters of this effective fluid, for use in more complex simulations. We additionally calculate the particle shape and orientation, and in simple shear flow show that the leading-order modifications to the angle of inclination ζ act to align the particle towards the flow direction, giving ζ=π/4−3Cae/4+α0Wi/2α1 where Cae is the elastic capillary number, Wi is the Weissenberg number, and αi are material properties of the suspending second-order fluid, for which the ratio α0/α1 is negative.