Transient rod-climbing in an Oldroyd-B fluid
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909446602489856 |
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| author | Ruangkriengsin, Tachin Brandão, Rodolfo Wu, Katie Hwang, Jonghyun Boyko, Evgeniy Stone, Howard A. |
| author_facet | Ruangkriengsin, Tachin Brandão, Rodolfo Wu, Katie Hwang, Jonghyun Boyko, Evgeniy Stone, Howard A. |
| contents | The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph \& Fosdick (\emph{Arch. Rat. Mech. Anal.}, vol. 49, 1973, pp. 321--380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Oldroyd-B model. We begin by neglecting surface tension and inertia to focus on the interaction between gravity and viscoelasticity. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface profile on this short time scale and recover the steady-state profile on a longer time scale. Subsequently, we reintroduce small but finite inertial effects to investigate their interplay with viscoelasticity and propose a criterion for determining the conditions under which rod-climbing occurs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transient rod-climbing in an Oldroyd-B fluid Ruangkriengsin, Tachin Brandão, Rodolfo Wu, Katie Hwang, Jonghyun Boyko, Evgeniy Stone, Howard A. Fluid Dynamics The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph \& Fosdick (\emph{Arch. Rat. Mech. Anal.}, vol. 49, 1973, pp. 321--380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Oldroyd-B model. We begin by neglecting surface tension and inertia to focus on the interaction between gravity and viscoelasticity. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface profile on this short time scale and recover the steady-state profile on a longer time scale. Subsequently, we reintroduce small but finite inertial effects to investigate their interplay with viscoelasticity and propose a criterion for determining the conditions under which rod-climbing occurs. |
| title | Transient rod-climbing in an Oldroyd-B fluid |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2501.01016 |