Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid
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2025
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| author | Singeetham, Pavan Kumar Madival, Deepak Garg, Piyush Subramanian, Ganesh |
| author_facet | Singeetham, Pavan Kumar Madival, Deepak Garg, Piyush Subramanian, Ganesh |
| contents | We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ($κ$), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number ($De$), and the suspending fluid rheology is therefore modeled as a second-order fluid, with the constitutive equation involving a material parameter $ε$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to $ε\in[-0.7,-0.5]$. 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 $O(De)$. In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant $C$, although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For $De$ well below a threshold, $De_c(κ)$, weak viscoelasticity transforms the closed-trajectory topology into a tightly spiralling one. A multiple-scales analysis is used to interpret the resulting orientation dynamics in terms of an $O(De)$ orbital drift. The drift in orbit constant over a Jeffery period $ΔC$, when plotted as a function of $C$, identifies four different orientation dynamics regimes on the $κ-ε$ plane. For $ε$ in the polymeric range, prolate spheroids always drift towards the spinning mode. Oblate spheroids drift towards the tumbling mode for $κ> κ_c(ε)$, but towards an intermediate kayaking mode for $κ< κ_c(ε)$. The rotation of spheroids of extreme aspect ratios, either slender prolate spheroids ($κ\gg 1$) or thin oblate ones ($κ\ll 1$), about the vorticity axis, is arrested for $De \geq De_c(κ)$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_23361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid Singeetham, Pavan Kumar Madival, Deepak Garg, Piyush Subramanian, Ganesh Fluid Dynamics We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ($κ$), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number ($De$), and the suspending fluid rheology is therefore modeled as a second-order fluid, with the constitutive equation involving a material parameter $ε$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to $ε\in[-0.7,-0.5]$. 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 $O(De)$. In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant $C$, although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For $De$ well below a threshold, $De_c(κ)$, weak viscoelasticity transforms the closed-trajectory topology into a tightly spiralling one. A multiple-scales analysis is used to interpret the resulting orientation dynamics in terms of an $O(De)$ orbital drift. The drift in orbit constant over a Jeffery period $ΔC$, when plotted as a function of $C$, identifies four different orientation dynamics regimes on the $κ-ε$ plane. For $ε$ in the polymeric range, prolate spheroids always drift towards the spinning mode. Oblate spheroids drift towards the tumbling mode for $κ> κ_c(ε)$, but towards an intermediate kayaking mode for $κ< κ_c(ε)$. The rotation of spheroids of extreme aspect ratios, either slender prolate spheroids ($κ\gg 1$) or thin oblate ones ($κ\ll 1$), about the vorticity axis, is arrested for $De \geq De_c(κ)$ |
| title | Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2505.23361 |