We study the long-time behaviour of a Poiseuille-type flow of a weakly conducting incompressible polymeric fluid governed by the Pokrovskii-Vinogradov model. Under the Poiseuille ansatz, the electrohydrodynamic system separates into velocity-stress and charge-potential subsystems. In the constant-relaxation regime with zero initial transverse normal stress, the velocity-shear subsystem reduces to a damped wave equation. If the pressure gradient has a finite limit and the residual forcing vanishes, the unique energy solution converges in $ H_0^1(0, 1)\times L_2(0, 1) $ to the corresponding stationary state. For the homogeneous deviation problem, a modified Lyapunov functional yields exponential stability, while modal analysis determines the supremal decay rate. The same representation provides a fading-memory estimate and a standard $ L_\infty $ input-to-state stability bound. Under additional regularity, the shear and streamwise normal stresses converge in $ L_2(0, 1) $, and the transverse normal stress remains zero. Staggered-grid computations support the predicted decay rates. The results concern streamwise-independent perturbations and constant coefficients.
Citation: Xuelin Guan. Stability of a Poiseuille-type velocity-stress subsystem for an electrohydrodynamic polymeric fluid[J]. AIMS Mathematics, 2026, 11(10): 35832-35854. doi: 10.3934/math.20261389
We study the long-time behaviour of a Poiseuille-type flow of a weakly conducting incompressible polymeric fluid governed by the Pokrovskii-Vinogradov model. Under the Poiseuille ansatz, the electrohydrodynamic system separates into velocity-stress and charge-potential subsystems. In the constant-relaxation regime with zero initial transverse normal stress, the velocity-shear subsystem reduces to a damped wave equation. If the pressure gradient has a finite limit and the residual forcing vanishes, the unique energy solution converges in $ H_0^1(0, 1)\times L_2(0, 1) $ to the corresponding stationary state. For the homogeneous deviation problem, a modified Lyapunov functional yields exponential stability, while modal analysis determines the supremal decay rate. The same representation provides a fading-memory estimate and a standard $ L_\infty $ input-to-state stability bound. Under additional regularity, the shear and streamwise normal stresses converge in $ L_2(0, 1) $, and the transverse normal stress remains zero. Staggered-grid computations support the predicted decay rates. The results concern streamwise-independent perturbations and constant coefficients.
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Figures(2) / Tables(3)
Xuelin Guan. Stability of a Poiseuille-type velocity-stress subsystem for an electrohydrodynamic polymeric fluid[J]. AIMS Mathematics, 2026, 11(10): 35832-35854. doi: 10.3934/math.20261389
Normalised $ H_0^1 $ velocity errors obtained from the independent staggered-grid solver: (a) Reynolds-number scan; (b) Weissenberg-number scan
Normalised velocity and stress errors for the nontrivially forced start-up: (a) $ \lambda = 0.5 $, for which the forcing controls the convergence; (b) $ \lambda = 2 $, for which the intrinsic exponent $ \sigma_\ast $ controls the convergence. The thick dotted curve is $ |\Theta(t)|/|\Theta(0)| = e^{-\lambda t} $, and the dashed reference segments have the rates predicted by (5.4)
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