Weakly nonlinear stability analysis of MHD channel flow using an efficient numerical approach

Abstract

We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as ${\mu }_1\sim (0.814-\mathrm{i}19.8)\times 10^{-3}\text{Ha}$ and ${\mu }_2\sim (2.73-\mathrm{i}1.50)\times 10^{-5}{\text{Ha}}^{-4}$. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude $|A{|}^2=\Re [{\mu }_1]/\Re [{\mu }_2]({\text{Re}}_c-\text{Re})\sim 29.8{\text{Ha}}^5({\text{Re}}_c-\text{Re})$ which exists at Reynolds numbers below the linear stability threshold ${\text{Re}}_c\sim 4.83\times 10^4\text{Ha}.$ We find that the flow remains subcritically unstable regardless of the magnetic field strength. Our method for computing Landau coefficients differs from the standard one by the application of the solvability condition to the discretized rather than continuous problem. This allows us to bypass both the solution of the adjoint problem and the subsequent evaluation of the integrals defining the inner products, which results in a significant simplification of the method.

16 pages, 10 figures, revised version (to appear in Phys Fluids)

References in corpus (3)

• Quasi-two dimensional perturbations in duct flows under transverse magnetic field
• Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct
• Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method

Cited by in corpus (3)

• Subcritical transition to turbulence in quasi-two-dimensional shear flows
• Monotone energy stability of magnetohydrodynamics Couette and Hartmann flows
• Two-dimensional nonlinear travelling waves in magnetohydrodynamic channel flow