Unstable spectra of plane Poiseuille flow with a uniform magnetic field

Abstract

The unstable spectra of plane Poiseuille flow (PF) in the present of a longitudinal magnetic field are numerically investigated using an eigenvalue solver of an 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 strong suppression effect on the classical plane PF instability with large Reynolds number Rein the region with magnetic Prandtl number Pm = [0.1, 1] or magnetic Reynolds number Rm = [103, 106]. Here, the Reynolds number and the magnetic Reynolds number are defined as Re=aV0/ν and Rm=aV0μ/η with a, V0, ν and η are the typical length, velocity, viscosity and resistivity respectively. The magnetic Prandtl number is defined as Pm = Rm/Re∝ν/η, which is proportional to the ratio of the viscosity and the resistivity of the fluid medium. As the strength of magnetic field increases, the PF instability can be completely stabilized in the limit of Pm → ∞ or/and Rm → ∞. It is interestingly found that a new instability branch is excited in the small magnetic Prandtl number (Pm → 0) or moderate magnetic Reynolds number (Rm = 104∼ 106) and large Reynolds number (Re →∞) regions. The new type 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 Re increases with fixed Rm. 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 Rm increases with flexed Re.