Aerospace Technic and Technology

The subject of this study is the flow development region of laminar incompressible fluid flow in the boundary layer. This flow is an example where a direct application of the Navier-Stokes equations of gradient-free laminar incompressible fluid flow, in which the molecular viscosity is assumed to be a constant value independent of spatial coordinates, leads to a redefinition of the mathematical model. It is about the fluid boundary layer in the region of flow establishment in the motion problem of a semi-infinite plane, where the pressure gradient is zero. There is a situation when the number of equations is equal to three (two equations of momentum conservation and the equation of continuity), and the number of unknowns is equal to two - the number of the speed component. As a logical solution to the obtained inconsistency, it is proposed, as was already done for the problem of stationary motion of a plane and the problem of acceleration of a plane, to depart from the false statement about the constancy of molecular viscosity in the gradient-free boundary layer of an incompressible flow and consider molecular viscosity as a function of spatial coordinates. The need to consider the variable nature of molecular viscosity led to the discovery of another flaw in the Navier-Stokes theory. This non-trivial flaw was discovered during the application of the original numerical analytical method for solving the flow development region problem. The Navier-Stokes equations are supplemented by boundary conditions. The most important condition is the condition of fluid non-slipping on the surface of a solid body, which, by the way, does not follow any physical law. As a result, on the surface of a half-plane (or a moving body), the component of the velocity, which coincides with the direction of motion, has a constant value equal to the velocity of the body.  It immediately follows from the continuity equation that the normal derivative of the normal component of the velocity must be equal to zero along the surface of the plane (body), since the longitudinal derivative of the velocity becomes zero. However, it is quite obvious that the velocity component normal to the surface of the plane (body) changes across the boundary layer in the region of current development, which indicates the presence of a normal gradient (both components) of the velocity. The conflict or contradiction is overcome by moving away from the generally accepted condition of non-slipping to the condition of partial non-slipping, or essentially the presence of sliding. Even with the sudden braking of any vehicle, the complete stop does not occur instantly, but after some finite time and distance, so in the case of the motion of a body in a stationary fluid, there is not an instant sticking, but a gradual one - from complete sliding, when a particle of liquid has just met a moving plane (body), to complete non-slipping at the end (and further) of the flow development region. Research methods. This work uses purely theoretical methods based on the use of calculus of variations, laws of physics, and ideas from everyday life. Conclusions. An improved model of a viscous Newtonian fluid in the area of flow development in the boundary layer was derived. On the basis of assumptions about the variable nature of the molecular viscosity, which already has two components, and the departure from the non-slipping condition, analytical solutions for both components of the velocity and both components of the molecular viscosity were obtained. A comparison of the obtained results with the results of other studies is presented.

airplane; helicopter; area of flow development; boundary layer; partial slip

Pavlyuchenko A. M., & Shyiko O. M. Kompleksnyy metod rozrakhunku oporu tertya i teploobminu na poverkhni lʹotnykh osesymetrychnykh obʺyektiv pry polʹoti po trayektoriyi z nayavnistyu v prystinnomu prykordonnomu shari neizotermichnosti, styslyvosti, laminarno-turbulentnoho perekhodu ta relaminarizatsiyi [Complex method for calculating the friction resistance and thermal refraction on the surface of flight axisymmetric objects on flight by trajectory with availability in the wall boundary layer non-isothermal, compressive, laminar-turbulent transition and relainarization]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2018, no. 1, pp. 4-28. DOI: 10.32620/aktt.2018.1.01. (In Ukrainian).

Komarov, B., Zinchenko, D., & Andrieiev, O. Vplyv formy kryla na kharakterystyky pry vykorystanni intehrovanoyi rotornoyi sylovoyi ustanovky litaka [Influence of wing shape on characteristics of fan-wing aircraft power plant]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2023, no. 2, pp. 17-26. DOI: 10.32620/aktt.2023.2.02. (In Ukrainian).

Stokes, G. G. On the theories of the internal friction of fluids in motion, and the equilibrium and motion of elastic solids. Trans. Cambridge Philos. Soc., 1845, vol. 8, pp. 287-305.

Lukianov, P. V., & Song, L. Optimal character and different nature of flows in laminar boundary layers of incompressible fluid flow. Problems of friction and wear, 2022, no. 4(97), pp. 52-60. DOI: 10.18372/0370-2197.4(97)16959.

Navier, C. L. M. H. Memoire sur les lois du movement des fluids. Memoires de l’ Academie Royale des Science de l’ Institut de France, 1823, vol. 6, pp. 389-440. (In French).

Lauga, E., & Squires, T. M. Brownian motion near a partial-slip boundary: A local probe of no-slip condition. Physics of Fluids, 2005, vol. 17, iss. 10, article no. 103102. DOI: 10.1063/1.2083748.

Priezjev, N. V. Molecular diffusion and slip boundary conditions at smooth surfaces with periodic and random nanoscale textures. The Journal of Chemical Physics, 2011, vol. 135, iss. 20, article no. 204704. DOI: 10.1063/1.3663384.

Guo, L., Chen, S., & Robbins, M. O. Effective slip boundary conditions for sinusoidally corrugated surfaces. Phys. Rev. Fluids, 2016, vol. 1, iss. 7, article no. 074102. DOI: 10.1103/PhysRevFluids.1.074102.

Lukianov, P. V., & Song, L. Unsteady incompressible laminar boundary layer: time and space variable molecular viscosity. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2023, no. 3, pp. 50-60. DOI: 10.32620/aktt.2023.3.06.

Kulkarni, Y., Fullana, T., & Zaleski, S. Stream function solutions for some contact line boundary conditions: Navier slip, super slip and generalized Navier boundary conditions. Proc. R. Soc. A, 2023, vol. 479, iss. 2278, article no. 20230141. DOI: 10.1098/rspa.2023.0141.

Lauga, E., & Stone, H. A. Effective slip in pressure-driven Stokes flow. Journal of Fluid Mechanics, 2003, vol. 489, pp. 55-77. DOI: 10.1017/S0022112003004695.

Chen, J., Kim, Y. J., & Hwang, W. R. General criteria for estimation of effective slip length over corrugated surfaces. Microfluids and Nanofluids, 2020, vol. 24, article no. 68. DOI: 10.1007/s10404-020-02363-1.

Cui, H., & Ren, W. A note on the solution to the moving contact line problem with the no-slip boundary condition. Commun. Math. Sci., 2019, vol. 17, iss. 4, pp. 1167-1175. DOI: 10.4310/CMS.2019.v17.n4.a16.

Blasius, H. Grenzschichten in Flussigkeiten mit kleiner Reibung. Zeits. f. Math. u. Phys, 1908, vol. 56, pp. 1-37. (In German).

Stokes, G. G. On the effect of the internal friction of fluids on the motion of pendulums. Trans. Cambridge Philos. soc., 1851, vol. 9, pp. 1-86.

Rayleigh, Lord. On the motion of solid bodies through viscous liquids. Phil. Mag., Ser. VI, 1911, vol. 21, no. 6, pp. 697-711.

Lukianov, P. V., & Song, L. Compact analogs of the models of vortex flows generated by aircraft flight. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2023, no. 3, pp. 4-20. DOI: 10.32620/aktt.2023.5.01.

Belotserkovskiy, S. M., & Nisht, M. T. Otryvnoye i bezotryvnoye obtekaniye tonkikh kryl'yev ideal'noy zhidkost'yu [Separated and continuous flow of an ideal fluid around thin wings]. Moscow, Nauka Publ., 1978. 352 p. (In Russian).

Bulanchuk, G. G., Bulanchuk, O. N., & Dovgiy, S. A. Metod diskretnykh vikhrevykh ramok so vstavkoy promezhutochnykh tochek na vikhrevoy pelene [Discrete vortex frame method with insertion of intermediate points on the vortex sheet]. Visnyk Kharkivsʹkoho natsionalʹnoho universytetu – Bulletin of the Kharkiv National University, 2009, no. 863, pp. 37-46. (In Russian).