internal friction, boundary layer, incompressible flow, variable molecular viscosity, Low Reynolds numbers, analytical solutions, calculus of variations


The paper presents an original approach to the study of the problem of internal friction arising from the motion of a rigid body in an incompressible fluid. This approach takes into account the spatial variability of molecular viscosity in the boundary layer region, and the solution of the problem is based on the use of an extreme for the fluid flow rate functional. The spatial variability of molecular viscosity in the boundary layer, by a well-known analogy with the theory of heat conduction, is based on the absence of a spatial isotropy of the medium. It is shown that molecular viscosity depends on the nature of the flow - on how many forces act on the fluid. So, if the flow is unsteady and non-gradient or steady and gradient, then both of these flows are subject to the action of two forces. In such flows, the molecular viscosity due to the extreme of the fluid flow rate is a constant value. It has been fond that the distribution of velocity in a gradient stationary boundary layer has a parabolic distribution law, and all existing theories are described by this law quite accurately, with an error of maximum 5%. At the same time, in a laminar non-gradient boundary layer, only the force of internal friction acts on the fluid. This causes the spatial variability of molecular viscosity: shear stress can be constant not only due to the linearity of the velocity distribution, which is not observed in the boundary layer, but also due to the variability of molecular viscosity. The resulting exponential velocity distribution in a non-gradient boundary layer is in complete agreement with those in the problems solved by Stokes, and is also confirmed experimentally. The paper also points out that the exponential law is consistent with modern data obtained by direct numerical simulation (DNS) for flows with Low Reynolds numbers both single-phase and two-phase, in the presence of particles inside the fluid.

Author Biographies

 Pavlo Lukianov, National Aviation University

канд. ф.-м. наук, доцент кафедри гідрогазових систем Національного авіаційного університету, вул. Любомира Гузара, 1, м. Київ, 03058, Україна, тел.: +38 096 125 47 22

Lin SONG, National Aviation University

магістр, аспірант кафедри гідрогазових систем Національного авіаційного університету, вул. Любомира Гузара, 1, м. Київ, 03058, Україна, тел..: +38 093 351 99 93


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