Advanced Engineering Research (Rostov-on-Don)

“Advanced Engineering Research (Rostov-on-Don)” is a peer-reviewed scientific and practical journal. It aims to inform the readers about the latest achievements and prospects in the field of Mechanics, Mechanical Engineering, Computer Science and Computer Technology. The journal is a forum for cooperation between Russian and foreign scientists, contributes to the convergence of the Russian and world scientific and information space.

Priority is given to publications in the field of theoretical and applied mechanics, mechanical engineering and machine science, friction and wear, as well as on methods of control and diagnostics in mechanical engineering, welding production issues. Along with the discussion of global trends in these areas, attention is paid to regional research, including issues of mathematical modeling, numerical methods and software packages, software and mathematical support of computer systems, information technology challenges.

All articles are published in Russian and English and undergo a peer-review procedure.

The journal is included in the List of peer-reviewed scientific editions, in which the main scientific results of dissertations for the degrees of Candidate and Doctor of Science are published (List of the Higher Attestation Commission under the Ministry of Science and Higher Education of the Russian Federation).

The journal covers the following fields of science:

Theoretical Mechanics, Dynamics of Machines (Engineering Sciences)
Deformable Solid Mechanics (Engineering Sciences, Physical and Mathematical Sciences)
Mechanics of Liquid, Gas and Plasma (Engineering Sciences)
Mathematical Simulation, Numerical Methods and Program Systems (Engineering Sciences)
System Analysis, Information Management and Processing, Statistics (Engineering Sciences)
Automation and Control of Technological Processes and Productions (Engineering Sciences)
Software and Mathematical Support of Machines, Complexes and Computer Networks (Engineering Sciences)
Computer Modeling and Design Automation (Engineering Sciences, Physical and Mathematical Sciences)
Computer Science and Information Processes (Engineering Sciences)
Machine Science (Engineering Sciences)
Machine Friction and Wear (Engineering Sciences)
Technology and Equipment of Mechanical and Physicotechnical Processing (Engineering Sciences)
Engineering Technology (Engineering Sciences)
Welding, Allied Processes and Technologies (Engineering Sciences)
Methods and Devices for Monitoring and Diagnostics of Materials, Products, Substances and the Natural Environment (Engineering Sciences)
Hydraulic Machines, Vacuum, Compressor Equipment, Hydraulic and Pneumatic Systems (Engineering Sciences)

The editorial policy of the journal is based on the traditional ethical principles of Russian scientific periodicals, supports the Code of ethics of scientific publications formulated by the Committee on Publication Ethics (Russia, Moscow), adheres to the ethical standards of editors and publishers, enshrined in the Code of Conduct and Best Practice Guidelines for Journal Editors, Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE).

The journal is addressed to those who develop strategic directions for the development of modern science — scientists, graduate students, engineering and technical workers, research staff of institutes, practical teachers.

About the journal

In September 2020, the scientific journal “Vestnik of Don State Technical University” (ISSN 1992-5980) changed its title.

The new title of the journal is “Advanced Engineering Research (Rostov-on-Don)” (eISSN 2687-1653).

The journal “Advanced Engineering Research (Rostov-on-Don)” is registered with the Federal Service for Supervision of Communications, Information Technology and Mass Media on August 7, 2020 (Extract from the register of registered mass media ЭЛ №ФС 77-78854 – electronic edition)

All articles of the journal have DOI index registered in the CrossRef system.

Founder and publisher: Federal State Budgetary Educational Institution of Higher Education "Don State Technical University", Rostov-on-Don, Russian Federation, https://donstu.ru/

ISSN (online) 2687-1653

Year of foundation: 1999.

Frequency: 4 issues per year (March 30, June 30, September 30, December 30).

Distribution: Russian Federation.

The journal "Advanced Engineering Research (Rostov-on-Don)" accepts for publication original articles, studies, review papers, that have not been previously published.

Website: https://www.vestnik-donstu.ru/

Editor-in-Chief: Alexey N. Beskopylny, Dr. Sci. (Engineering), Professor (Rostov-on-Don, Russia).

Languages: Russian, English

Key characteristics: indexing, peer-reviewing.

Licensing history:

The journal uses International Creative Commons Attribution 4.0 (CC BY) license.

Current issue

Vol 26, No 1 (2026)

MECHANICS

Analytical Solution of the Navier–Stokes Equations for Describing Inhomogeneous Couette Flow with a Quadratic Velocity Profile in a Layer with Permeable Boundaries

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

An exact analytical solution to the Navier–Stokes equations is derived for the first time, describing Couette flow between permeable plates subject to a quadratic velocity boundary condition. Parametric analysis demonstrates that the linear inhomogeneity coefficient specifies the asymmetry observed in both the velocity and vorticity fields. The quadratic inhomogeneity of the boundary condition controls the nonlinearity of the transverse flow distribution across the channel. Viscosity determines the thickness of the shear layer, governing the transition from a nearly linear velocity profile to pronounced localization of shear near the walls. Numerical simulations demonstrate a reversal in flow direction upon changing the sign of the parameters, accompanied by a twofold variation in velocity gradients. These findings are relevant for flow manipulation in microfluidics, membrane systems, and tribological applications.

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Abstract

Introduction. Flow control in microfluidic systems, membrane technologies, and porous bearings requires an understanding of the synergy between boundary permeability, their spatial inhomogeneity, and the viscosity of the working fluid. Each of these factors is actively studied separately. However, a comprehensive analytical description of their combined effect on the flow is needed. No such publications exist. The presented article fills this gap. Research objectives are as follows: to obtain an analytical solution for the velocity field in Couette flow with permeable boundaries and a nonlinear boundary condition; to study the formation of hydrodynamics under the influence of permeability (α), dynamic viscosity (μ), linear (A) and quadratic (B) inhomogeneity of the boundary condition.

Materials and Methods. The analytical solution is based on the stationary Navier–Stokes equations for an incompressible Newtonian fluid, with a quadratic expansion of velocity along the transverse coordinate. The axial, linear, and quadratic modes of the velocity profile were investigated using numerical modeling in MATLAB. For a stationary, laminar, isothermal flow of a Newtonian viscous incompressible fluid, the distance between permeable plates was h = 1 m. The lower plate was stationary, while the upper plate moved with a velocity of W = 0.3 m/s. The liquid filtration rate was V_w = 0.001 m/s, and μ = 0.01 Pa·s for A = ±0.03 s^–1 and B = ±0.005 m^–1·s^–1. Water, motor oil, and crude oil were studied at temperatures of 20 °C, 40 °C, or 60 °C. For this case, h = 0.02 m, W = 0.05 m/s, A = 0.1 s^–1, B = 0.02 m^–1·s^–1, V_w = 0.0005 m/s. Depending on the fluid and temperature, μ ranged from 0.05 to 9.15·10^–3 Pa·s.

Results. Asymmetry of the flow, deviation from the channel axis, and variability of the vorticity amplitude ω_y were visualized. Zero filtration velocity was observed at the lower plate in the plane z = 0 and increased with this parameter, reaching a maximum at z = h (distance between the plates). For water, the streamlines exhibited minimal deviation from the horizontal, while for oil at 20 °C, they curved near the upper wall. Two-dimensional vorticity fields for water, oil, and petroleum at various temperatures were compared. Weak ω_y and reduced viscosity resulted in negative values ω_y for water and petroleum. For oil, the situation was reversed: positive values corresponded to elevated ω_y.

Discussion. The calculation results allow us to conclude:

− changing the sign of A inverts the directions of the maxima for velocity and vorticity;

− the sign of B determines the curvature of the isolines;

− the thickness of the layer with the maximum velocity gradient changes by two orders of magnitude when transitioning from water to oil.

The identified patterns are explained by the physical meaning of the parameters: A defines the macroscopic flow asymmetry, B governs the distribution of the transverse flow, and viscosity, through α, controls the depth of boundary perturbations.

Conclusion. For the first time, an exact analytical solution to the stationary Navier–Stokes equations was obtained for generalized Couette flow of a Newtonian fluid between permeable plates with a quadratic velocity profile at the boundary. A parametric analysis has shown that coefficient A determines the asymmetry of the velocity and vorticity fields, while B determines their nonlinearity. Viscosity controls the thickness of the shear layer: for high-viscosity media, the velocity drop is localized near the walls, while for low-viscosity media, the profile is linear. The results provide a foundation for applications in microfluidics, membrane technologies, and tribology. Future prospects are associated with accounting for non-Newtonian fluid properties, unsteady regimes, and flow stability.

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ISSN 2687-1653 (Online)

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