Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Abstract

Abstract The unstable spectra of plane Poiseuille flow (PF) in the presence of a longitudinal magnetic field are numerically investigated using an eigenvalue solver of incompressible magnetohydrodynamic equations. It is found that the strength of the magnetic field and the dissipative effect of the magnetic perturbation have played different roles in different parameter regions. The magnetic field has a strong suppression effect on the classical plane PF instability with a large Reynolds number in the region with the magnetic Prandtl number or the magnetic Reynolds number . Here, the Reynolds number and the magnetic Reynolds number are defined as and , where a, V 0, ν and η are the typical length, velocity, viscosity and resistivity, respectively. The magnetic Prandtl number is defined as , which is proportional to the ratio of the viscosity and the resistivity of the fluid medium. As the strength of the magnetic field increases, the PF instability can be completely stabilized in the limit of or/and . It is interestingly found that a new instability branch is excited in the small magnetic Prandtl number ( ) or moderate magnetic Reynolds number ( ) and large Reynolds number ( ) regions. This new type of instability is verified to be driven by the magnetic Reynolds stress and modulated by the dissipative effect of the magnetic perturbation. The wavelength of the original PF instability gradually shifts to the long wavelength region, but the wavelength of the new branch is almost unchanged, as increases with fixed . However, the wavelength of the original instability branch is almost unchanged, but the wavelength of the new instability branch shifts to the long wavelength region, as increases with fixed .