Abstract: In this work an unsteady Poiseuille-type problem for the Vinogradov--Pokrovskii model was considered, and in the leading-order approximation in~\(\beta\) a non-autonomous equation for the velocity was derived, whose stiffness coefficient relaxes through the internal normal stress. In the case of a spatially homogeneous initial normal stress~\((\alpha_{22})_0\equiv a_0=\mathrm{const}\ge0\), an \emph{explicit} modal solution of the unsteady problem was obtained, whose modal coefficients are expressed through Bessel functions. Second, the explicit solution reveals a modal structure specific to this model. After a change of the unknown, each mode satisfies a Bessel equation whose order depends on the mode number, \(\nu_j^2=1-4\Wi^2\varkappa^2(j\pi)^2\), with the critical number~\(j^*=(2\pi\Wi\varkappa)^{-1}\) separating the modal space into three regimes: real order (\(j<j^*\)), zero order (\(j=j^*\)), and purely imaginary order (\(j>j^*\)). Since~\(j^*=(2\pi\Wi\varkappa)^{-1}\), increasing the Weissenberg number lowers the critical threshold. Consequently, more modes in any finite Fourier truncation enter the purely imaginary-order regime and exhibit exponentially damped oscillations rather than non-oscillatory asymptotic decay.

Cite: Guan X. , Mishchenko E.V.
An explicit Bessel-free solution and asymptotic stability for a reduced charged polymerflow equation
In compilation Computational Models and Technologies (CMT2026). 2026. – C.174-182.

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