Inflection point instability in Hartmann channel flow with variable electric conductivity

TL;DR

This study investigates the stability of a Hartmann-type channel flow with a wall-normal magnetic field when the fluid's electrical conductivity and viscosity vary across the channel, focusing on linear σ(z)=1+κ z and ν(z)=1+ς z. By formulating a modified Orr-Sommerfeld–Squire system that includes Lorentz forces and solving it with a spectral method (Dedalus), the authors identify two linear instabilities: Hartmann Layer Instability (HLI) and Inflection Point Instability (IPI). IPI arises from inflection points in the asymmetric base profile and is long-wave and core-localized, with its onset characterized by κ_cr(Ha) ≈ Ha e^{-Ha/2}; Re_cr^IPI decreases rapidly with κ and can be much smaller than Re_cr^HLI for κ ≳ 0.5–0.7. Direct numerical simulations corroborate the linear predictions and show a four-stage transition to turbulence driven by IPI, highlighting a new mechanism for destabilization in MHD duct flows with nonuniform properties and potential implications for liquid-metal technologies.

Abstract

The stability of a flow of an electrically conducting, incompressible fluid in a channel with an imposed uniform wall-normal magnetic field and electrically insulating walls is studied using linear stability analysis and direct numerical simulations. The novelty of the system, which differentiates it from the classical Hartmann channel flow, is that, as in some technological applications of liquid metals, the electric conductivity and viscosity of the fluid vary across the channel. This variation is found to have a strong influence on the stability characteristics of the flow. Specifically, a linear variation in electric conductivity significantly alters the base velocity profile, leading to pronounced asymmetry and the development of inflection points. When this transformation is sufficiently strong, the flow becomes linearly unstable at Reynolds numbers much lower than the threshold for the linear instability of the Hartmann channel flow. The instability exhibits distinct features: a large typical axial wavelength and the localization of perturbation growth in the channel core. The characteristics of this instability suggest a mechanism similar to the classical inviscid inflection-point instability of one-dimensional velocity profiles. The resulting transition to turbulence is demonstrated in direct numerical simulations.

Figures (13)

• Figure 1: Schematics of the flow. Variation of the electric conductivity of the fluid $\sigma$ and the corresponding profile of the base flow velocity $U(z)$ are illustrated.
• Figure 2: (a) Base velocity profiles at ${\textit{Ha}} = 50$, $\varsigma = 0$ under varying conductivity. Inflection points are indicated by dots. (b) Base velocity profiles within the boundary layer at ${\textit{Ha}}=50$, $\varkappa=0$ under varying viscosity. Red dashed line corresponds to the ordinary Hartmann flow profile.
• Figure 3: $d^2U/dz^2$ at ${\textit{Ha}}=8$ (red), ${\textit{Ha}}=10$ (green), ${\textit{Ha}}=20$ (blue), ${\textit{Ha}}=50$ (black) for $\varkappa=0.1$(a) and $\varkappa=0.7$(b). Solid lines correspond to uniform viscosity ($\varsigma=0$) and dotted lines correspond to $\varsigma=0.1$.
• Figure 4: Numerically calculated $\varkappa_{cr}$, such that inflection points do not appear at $\varkappa<\varkappa_{cr}$, but appear at $\varkappa>\varkappa_{cr}$, versus ${\textit{Ha}}$ for two values of $\varsigma$. Black dashed line corresponds to the asymptotic relation \ref{['eq:kap_crit']}.
• Figure 5: Dispersion curves at ${\textit{Ha}}=50$ for $\varkappa=0$ (a), $\varkappa = 0.5$ (b), $\varkappa = 0.7$ (c). Peaks at low wavelengths ($\alpha<3$) correspond to the inflection point instability. Peaks at moderate and high wavelengths ($\alpha>3$) correspond to the Hartmann layer instability.