Exact solutions for unsteady isobaric flows of a viscous incompressible fluid in a layer with permeable boundaries and a constant transverse velocity

K.V. Gubareva, E.Yu. Prosviryakov, A.V. Eremin

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Abstract. The study of exact solutions to the Navier–Stokes equations remains an important task in fluid mechanics, as such solutions serve as a benchmark for verifying numerical methods and reveal fundamental flow patterns. Of particular interest are three-dimensional unsteady isobaric flows with constant transverse velocity, which model filtration processes and fluid motion in layers with permeable boundaries.
Aim. To perform a complete compatibility analysis of the overdetermined system of Navier–Stokes equations for three-dimensional unsteady isobaric flows with a constant transverse velocity in a layer with permeable boundaries, reduce it to a solvable system, and construct new families of exact solutions.
Methods. Compatibility analysis of overdetermined systems of partial differential equations, the method of characteristics for linear partial differential equations, reduction to coupled parabolic equations, construction of polynomial and self-similar solutions, the Galilean transformation for reduction to the heat equation, and numerical visualization in MATLAB using the parameters of water at 20°C.
Results. Compatibility conditions are derived, leading to a linear functional relationship between the horizontal velocity components. The original overdetermined system is reduced to a pair of quasilinear parabolic equations. New classes of exact solutions have been constructed: polynomial solutions in the characteristic variable, which generalize shear flows, as well as self-imilar solutions of the moving and diffusing Gaussian pulse type that describe the evolution of localized perturbations. An analysis of the limit transition to an ideal fluid is performed.
Conclusions. For the first time, exact solutions describing three-dimensional unsteady viscous fluid flows with a constant transverse velocity have been obtained. The results contribute to the theory of layered flows and can be used to verify numerical methods in computational fluid dynamics (CFD).

Keywords: isobaric flow, constant transverse velocity, compatibility conditions, exact solutions, reduction of Navier–Stokes equations, layered flows, self-similar solutions

For citation. Gubareva K.V., Prosviryakov E.Yu., Eremin A.V. Exact solutions for unsteady isobaric flows of a viscous incompressible fluid in a layer with permeable boundaries and a constant transverse velocity. News of the Kabardino-Balkarian Scientific Center of RAS. 2026. Vol. 28. No. 4. Pp. 33–45. DOI: 10.35330/1991-6639-2026-28-4-33-45

©  Gubareva K.V., Prosviryakov E.Yu., Eremin A.V., 2026

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Information about the authors

Kristina V. Gubareva, Candidate of Technical Sciences, Associate Professor, Department of Industrial Thermal Power Engineering, Samara State Technical University;
244, Molodogvardeyskaya street, Samara, 443100, Russia;
r.kristina2017@mail.ru, ORCID: https://orcid.org/0000-0002-9845-8372, SPIN-code: 4171-9816
Evgenii Yu. Prosviryakov, Doctor of Physical and Mathematical Sciences, Professor, Department of Information Technology and Automation, Ural Federal University;
19, Mira street, Ekaterinburg, 620002, Russia;
Head of Sector, Sector of Nonlinear Vortex Hydrodynamics, Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences;
34, Komsomolskaya street, Ekaterinburg, 620049, Russia;
evgen_pros@mail.ru, ORCID: https://orcid.org/0000-0002-2349-7801, SPIN-code: 3880-5690

Anton V. Eremin, Doctor of Technical Sciences, Associate Professor, Vice-Rector for Scientific Work, Head of the Department of Industrial Thermal Power Engineering, Samara State Technical University;
244, Molodogvardeyskaya street, Samara, 443100, Russia;
a.v.eremin@list.ru, ORCID: https://orcid.org/0000-0002-2614-6329, SPIN-code: 3892-0775

Funding

The study was performed without external funding.