The problem of stability for classical Couette and Poiseuille flows (flows of a horizontal layer of fluid between plates in relative motion) is still open. Recently, Falsaperla et al. obtained values for critical linear and nonlinear energy Reynolds numbers which are in good agreement with the experiments of Prigent et al. and with the numerical simulation of Barkley and Tuckerman. Such critical values are computed for tilted perturbations which have the same inclination of the secondary motions appearing in the experiments.

In this paper we consider the same problem when the fluid is electrically conducting and subject to a magnetic field orthogonal to the layer. We investigate the stability of the stationary solution called magnetic Couette and Hartmann flows and we show that such flows are nonlinearly stable if the Reynolds number $\mathrm{Re}$ is less then ${\bar{R}}_{\theta }={\mathrm{Re}}_{Orr}^{\left(m\right)}(2\pi ∕(\lambda sin\theta ))∕sin\theta ,$when the perturbations are rolls inclined by an angle $\theta$ with respect to the direction of the fluid motion. In the expression above ${\mathrm{Re}}_{Orr}^{\left(m\right)}\left(\mu \right)$ is the magnetic Orr–Reynolds number evaluated at the wavenumber $\mu =2\pi ∕(\lambda sin\theta )$, where $\lambda$ is the wavelength of the perturbation. We conjecture that in an experiment instability will emerge with secondary states that are rolls with inclination and wavelength related to the critical Reynolds number by the above formula

经典库埃特流和泊厄流的稳定性（相对运动的两板之间的流体水平层流）的稳定性问题仍然存在。最近，Falsaperla等。获得的临界线性和非线性能量雷诺数的值与Prigent等人的实验非常吻合。并用Barkley和Tuckerman进行数值模拟。对于具有与实验中出现的次级运动相同的倾斜度的倾斜扰动，计算这些临界值。

在本文中，当流体导电并且受到垂直于该层的磁场时，我们考虑相同的问题。我们调查称为稳态解的稳定性磁库埃特和哈特曼流，我们表明，这种流动是非线性稳定的，如果雷诺数$\mathrm{回覆}$ 小于 ${\overline{\mathrm{[R}}}_{\theta }={\mathrm{回覆}}_{Ø\mathrm{[R}\mathrm{[R}}^{（米）}（2\pi ∕（\lambda 罪\theta ））∕罪\theta ，$当扰动是倾斜一定角度的辊子时 $\theta$关于流体运动的方向。在上面的表达式中${\mathrm{回覆}}_{Ø\mathrm{[R}\mathrm{[R}}^{（米）}（\mu ）$ 是在波数上评估的磁性奥尔-雷诺数 $\mu =2\pi ∕（\lambda 罪\theta ）$，在哪里 $\lambda$是扰动的波长。我们推测，在实验中，次级状态会出现不稳定状态，次级状态是具有与上式相关的临界雷诺数的倾角和波长的滚动