Linear stability of Navier–Stokes–Voigt fluid under the influence of uniform magnetic field in a porous medium

Abstract

We analyze the instability of plane Poiseuille flow for a viscoelastic Navier–Stokes–Voigt fluid influenced by a vertical magnetic field within a homogeneous porous medium. The viscoelastic fluid, confined between two parallel plates, is driven by a pressure gradient. To examine the effects of the Voigt parameter $\Lambda$, permeability $\Omega$, and Hartmann number $Ha$ on flow stability, we numerically solve the generalized eigenvalue problem using the Chebyshev spectral collocation method, which provides an accurate upper bound for stability. For the set of parameters, the plane Poiseuille flow of the Navier–Stokes–Voigt fluid stabilizes as $\Omega$ and $Ha$ increase, preserving the Newtonian eigenspectrum within a specific range of $\Lambda$, $\Omega$, and $Ha$. The eigenspectra display a characteristic $Y$-shaped structure with three distinct modes, depending on the phase speed. The instability consistently emerges in the wall mode for the physical parameters $\Lambda$, $\Omega$, and $Ha$. The transition from laminar flow to turbulence is identified at a Reynolds number $Re=5772.22$ and a disturbance wavenumber $a=1.02$ in the case of channel Poiseuille flow of a Newtonian fluid and this is termed as Tollmien–Schlichting instability. The neutral stability curves corresponding to different values of $\Lambda$, $\Omega$, and $Ha$ extend the Tollmien–Schlichting instability observed in Newtonian channel flow. Stability regions expand with increasing $\Lambda$, $\Omega$, and $Ha$, as demonstrated by the neutral stability diagrams obtained for various associated parameters. Furthermore, the energy budget method is employed to validate our findings by quantifying the physical mechanisms driving instability. While Reynolds stress leads to negative energy production, viscous dissipation and viscoelastic contributions positively influence energy production, ultimately reducing the kinetic energy growth rate and enhancing stability.