تقرير
Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables
| العنوان: | Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables |
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| المؤلفون: | Dolce, Michele |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics Physics (Other) |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Physics - Fluid Dynamics, 35Q35, 76W05 |
| الوصف: | We consider a 2D incompressible and electrically conducting fluid in the domain $\mathbb{T}\times\mathbb{R}$. The aim is to quantify stability properties of the Couette flow $(y,0)$ with a constant homogenous magnetic field $(\beta,0)$ when $|\beta|>1/2$. The focus lies on the regime with small fluid viscosity $\nu$, magnetic resistivity $\mu$ and we assume that the magnetic Prandtl number satisfies $\mu^2\lesssim\mathrm{Pr}_{\mathrm{m}}=\nu/\mu\leq 1$. We establish that small perturbations around this steady state remain close to it, provided their size is of order $\varepsilon\ll\nu^{2/3}$ in $H^N$ with $N$ large enough. Additionally, the vorticity and current density experience a transient growth of order $\nu^{-1/3}$ while converging exponentially fast to an $x$-independent state after a time-scale of order $\nu^{-1/3}$. The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system's dynamic behavior. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2308.12589 |
| رقم الأكسشن: | edsarx.2308.12589 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |