Linear stability of viscoelastic Walters' liquid B in a channel in the presence of a magnetic field

The present study investigates the stability of viscoelastic Poiseuille flow of Walters' liquid B, placed within a transverse magnetic field and restricted between two infinite parallel plates. Stability is performed within the framework of modal and non-modal analyses to discuss the impact of the Reynolds number, fluid elasticity, and magnetic field strength. The governing fourth-order linearized disturbance equation, modified by the transverse magnetic field and the viscoelasticity of the fluid, is solved numerically using the Chebyshev spectral and shooting methods. A modal analysis examines the eigenspectrum, continuous spectrum, and neutral stability curves to characterize long-term stability behavior, revealing that increased fluid elasticity destabilizes the flow, whereas the transverse magnetic field tends to stabilize it. Analysis of ε-pseudospectrum and transient energy growth for optimal two-dimensional perturbations involving the non-normal Orr–Sommerfeld operator is investigated using a non-modal analysis. The ε-pseudospectral contours protrude into the unstable region, signaling flow instability, while the transient growth function shows that disturbances initially surge exponentially before decaying and subsequently evolving into sustained growth or further decay based on the magnetic effect and the elasticity number. Energy budget analysis of the perturbations further elucidates these dynamics by identifying regions of negative energy production due to Reynolds stress, alongside positive contributions from viscous dissipation and transverse magnetic field effects. The study provides a detailed discussion of these complex flow phenomena and their implications for flow stability.