Unsteady Poiseuille-Type Flow of a Vinogradov–Pokrovskii Polymer Fluid in a Flat Channel: An Explicit Modal Solution and Its Convergence

We study the unsteady mechanical response of an incompressible viscoelastic polymeric fluid in a flat channel, governed by the Vinogradov–Pokrovskii rheological model. The motion arises from an electrohydrodynamic reduction of Poiseuille type, after which the mechanical subsystem decouples from the electric field; the velocity then depends on time and on the transverse coordinate only. Treating the rheological parameter as small, we reduce the governing system in the leading-order approximation to a non-autonomous second-order evolution equation whose stiffness coefficient relaxes exponentially in time, so that the nonstationarity is driven by the internal relaxation of the normal stress rather than by an external force. For spatially homogeneous initial normal stress, we diagonalize the Galerkin system in the sine basis and obtain an explicit modal representation in which each mode satisfies a Bessel equation whose order depends on the mode number. This yields a critical index that splits the modes into three regimes—real order, zero order, and purely imaginary order—a structure absent from the classical UCM and Oldroyd-B solutions. Using the explicit representation, we prove convergence of the modal series and show that the solution decays in the long-time limit, so that the rest state is asymptotically stable in the natural energy phase space. The analytical solution is confirmed numerically.