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| Format: | Preprint |
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2015
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| Online Access: | https://arxiv.org/abs/1509.03606 |
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| _version_ | 1866911070730321920 |
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| author | Ozer, Saadet Sengul, Taylan |
| author_facet | Ozer, Saadet Sengul, Taylan |
| contents | In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate.
We show that unlike the Newtonian ($ε=0$) case, in the second grade model ($ε\neq 0$ case), the time independent base flow exhibits transitions as the Reynolds number $R$ exceeds the critical threshold $R_c \approx 4.124 ε^{-1/4}$ where $ε$ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects.
At $R=R_c$, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as $R$ tends to $R_c$. Our numerical calculations suggest that for low $ε$ values, the system prefers a catastrophic transition where the bifurcation is subcritical.
We also find that there is a Reynolds number $R_E$ with $R_E < R_c$ such that for $R<R_E$, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that $R_E \approx 12.87$ at $ε=0$ and $R_E$ approaches $R_c$ quickly as $ε$ increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1509_03606 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Stability and transitions of the second grade Poiseuille flow Ozer, Saadet Sengul, Taylan Analysis of PDEs In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian ($ε=0$) case, in the second grade model ($ε\neq 0$ case), the time independent base flow exhibits transitions as the Reynolds number $R$ exceeds the critical threshold $R_c \approx 4.124 ε^{-1/4}$ where $ε$ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At $R=R_c$, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as $R$ tends to $R_c$. Our numerical calculations suggest that for low $ε$ values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also find that there is a Reynolds number $R_E$ with $R_E < R_c$ such that for $R<R_E$, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that $R_E \approx 12.87$ at $ε=0$ and $R_E$ approaches $R_c$ quickly as $ε$ increases. |
| title | Stability and transitions of the second grade Poiseuille flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1509.03606 |