Short Communication
Mathematical Modeling, Numerical Methods and Software Complexes

Steady-state non-uniform Poiseuille shear flows with Navier boundary condition

N. V. Burmasheva^ab, E. Yu. Prosviryakov^ab, M. Yu. Alies<sup>c

a</sup> Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg, 620002, Russian Federation
^b Institute of Engineering Science, Urals Branch, Russian Academy of Sciences, Ekaterinburg, 620049, Russian Federation
^c Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, Izhevsk, 426067, Russian Federation

Abstract: This study presents an exact solution to the Navier–Stokes equations for a steady non‑uniform Poiseuille shear flow in an infinite horizontal fluid layer. For this class of flows, the governing system reduces to a nonlinear overdetermined set of partial differential equations. A nontrivial exact solution is constructed within the Lin–Sidorov–Aristov class, wherein the velocity field is given by linear forms of two horizontal coordinates with coefficients depending on the vertical coordinate. The boundary‑value problem employs the Navier slip condition at the lower wall and a non‑uniform velocity profile at the upper boundary. The resulting polynomial solution is analyzed, revealing that counter‑flows can emerge due to the presence of stagnation points. It is shown that the Navier condition can lead to a maximum stratification of the velocity field into four distinct zones (three stagnation points). In the limiting case of perfect slip, the analysis demonstrates the possibility of two stagnation points.

Keywords: exact solution, non-uniform shear flow, Poiseuille flow, overdetermined system, Lin–Sidorov–Aristov class, Navier slip condition, perfect slip

UDC: 517.9:532.5.013.3:532.5.032

MSC: 76D05, 76D10, 35Q30

Received: May 11, 2025
Revised: September 13, 2025
Accepted: October 20, 2025
First online: December 8, 2025

DOI: 10.14498/vsgtu2214

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