Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid

Abstract

Abstract

We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ( kappa κ $\kappa$ ), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number ( italic De De ${\textit{De}}$ ), and the suspending fluid rheology is therefore modelled as a second-order fluid, with the constitutive equation involving a material parameter epsilon ϵ $\epsilon$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to epsilon element of left bracket negative 0.7 comma negative 0.5 right bracket ϵ ∈ [ − 0.7 , − 0.5 ] $\epsilon \in [-0.7,-0.5]$ . Employing a reciprocal theorem formulation, along with expressions for the relevant disturbance fields in terms of vector spheroidal harmonics, we obtain the spheroid angular velocity to upper O left parenthesis upper D e right parenthesis O ( D e ) $O(De)$ . In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant upper C C $C$ , although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For italic De De ${\textit{De}}$ well below a threshold, italic De Subscript c Baseline left parenthesis kappa right parenthesis <mml:msu