The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

Abstract

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number Rc, the critical wave number ~c, and the critical wave speed cc are obtained for wide ranges of the magnetic Prandtl number Pm and the Hartmann number M. It is found that except for the case when Pm is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as Pm increases.

A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

“…[8]. We also present a series of critical-parameter calculations (see §5.4), confirming that results obtained via the fixed-boundary variants of our schemes are in close agreement with the corresponding ones by Takashima [19]. In free-surface problems, when Pm is increased from 10 −8 to 10 −4 the critical Reynolds number is seen to drop by a factor of five, while the corresponding relative variation in fixed-boundary problems is less than 0.003.…”

“…In what follows we consider the magnetic-field configuration 11) where (A −1 x , A −1 z ) is a uniform, externally imposed magnetic field, quantified in terms of the streamwise and flow-normal Alfvén numbers A x and A z , and B ∈ C 2 (Ω) is a function representing the magnetic field induced by the fluid motion U (z) within the background field (B is equal to the corresponding functionB in [19]). For the test calculations presented in §5 we employ the Hartmann profiles [1] 12) where…”

“…Moreover, they modify the velocity distribution of the perturbations, changing in turn the Reynolds stress and/or viscous dissipation. It is generally known, both on theoretical grounds [54,55], as well as from numerical calculations [16,19], that in channel problems the combined outcome of these effects is strongly stabilizing. In film problems, however, the existence of a resonance between the velocity and surface degrees of freedom may cause the F mode to deviate from that behavior [8].…”

Transition to Turbulence in the Hartmann Boundary Layer

“…For the flow considered here, i.e. for the case of insulating walls and vertical magnetic field, they found R c = 48250, which differs only slightly from the results of Takashima [9]. In summary, the critical Reynolds number for the linear instability of the Hartmann flow is much higher than the observed one.…”

“…More recently, the stability analysis using numerical techniques for modified plane Poiseuille flow and modified plane Couette flow in the presence of a transverse magnetic field ( [9], [10]) produced a critical Reynolds number of R c = 48311.016 for sufficiently high Hartmann number. An isolated Hartmann layer was investigated numerically by Lingwood & Alboussière [11] who studied the cases of electrically insulating and conducting walls with normal and arbitrarily oriented magnetic field.…”