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

ANALYTICAL SOLUTION TO THE GENERALIZED NUSSELT–COUETTE–POISEUILLE PROBLEM FOR A MULTILAYER INHOMOGENEOUS SHEAR FLOW

This paper presents an investigation of an inhomogeneous shear flow of a viscous fluid in the gap between two parallel plates affected by a pressure gradient and gravity. It discusses a generalized formulation of the classical Nusselt–Couette–Poiseuille problem, supplemented by nontrivial boundary conditions including velocity derivatives at the moving boundary. This formulation allows one to model complex physical interactions at the interface and leads to the emergence of multilayer flow structures with alternating directions. The emphasis is on establishing analytical conditions under which the longitudinal velocity profile acquires multiple zeros within the fluid layer, thus corresponding to the formation of stable counter-current zones. The methodological foundation for the study combines an analytical solution to the full system of Navier–Stokes equations with subsequent analysis and parametric investigations. The possibility of the existence of stationary laminar flows with two and three internal zeros of longitudinal velocity, corresponding to two- and three-layer flow stratification, is rigorously proved for the first time. The flow regimes are systematically classified based on the introduced dimensionless parameters, the spatial evolution of the velocity field is analyzed, and the structural stability of multilayer configurations is studied. The obtained results are of significant importance for a deeper understanding of the physics of complex shear flows and open new possibilities for controlling the flow structure in problems of heat and mass transfer in thin layers, microfluidic systems, and modern technologies of applying functional coatings.

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