Modal, Nonmodal, and Energy Budget Stability Analysis of Viscoelastic Plane Poiseuille Flow in the Presence of a Transverse Magnetic Field

Abstract

Abstract. The linear stability of plane Poiseuille flow of a viscoelastic Navier–Stokes–Voigt (Kelvin–Voigt-type) fluid is analyzed under a transverse magnetic field. High-accuracy spectral Chebyshev collocation and Galerkin methods are employed to solve the resulting modified Orr–Sommerfeld stability equations. Unlike earlier studies, this work links energy budget analysis with nonmodal growth suppression to reveal distinct viscoelastic and magnetic dissipation mechanisms across the channel. The modal stability analysis, including eigenspectrum and temporal growth rate evaluations, reveals that both viscoelastic effects (parameterized by Λ) and magnetic damping (via the Hartmann number M) exert strong stabilizing influences by suppressing Tollmien–Schlichting (TS) instabilities and shifting the critical Reynolds number to higher values. Neutral stability curves and growth-rate profiles confirm that increasing Λ and M enlarges the stable parameter space, delaying the onset of linear instability. Beyond modal analysis, transient energy growth calculations capture the potential for nonmodal amplification in linearly stable regimes. Results show that both Λ and M significantly reduce the peak transient energy growth and expedite its decay, thereby limiting the risk of subcritical transition. The ε-pseudospectrum analysis supports these findings, illustrating that the extent of spectral bulging into the unstable region diminishes with increasing damping. An in-depth energy budget analysis highlights spatially localized stabilization: viscoelasticity acts near the walls by attenuating shear production and enhancing Voigt-type regularization-induced dissipation, while magnetic damping dominates in the channel core via enhanced Lorentz force-induced dissipation. The results obtained using the Kelvin–Voigt-type model provide baseline insights that can guide future investigations employing more complex constitutive models such as Oldroyd-B or FENE-P.