OPTIMAL CHARACTER AND DIFFERENT NATURE OF FLOWS IN LAMINAR BOUNDARY LAYERS OF INCOMPRESSIBLE FLUID FLOW
Keywords: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.
Stokes G.G. On the theories of the internal friction of fluids in motion, and the equilibrium and motion of elastic solids/ G.G. Stokes // Trans. Cambridge Philos. soc. 1845, Vol. 8, p.287-305.
Schlichting H. Boundary Layer Theory / H. Schlichting. -- New York: McGraw-Hill, seventh ed., 1979.
Stokes G.G. On the Effect of the Internal Friction of Fluids on Motion of Pendulums / G.G. Stokes// Trans. Cambridge Philos. soc. 1851, Vol. 9, p.1-106.
Drawan S. Direct measurement of skin friction / S. Drawan // NACA Tech. Mem. 1953 Tech. Rep. 1121,
Driest van E.R. 1956. On Turbulent Flow Near a Wall / E.R. van Driest // Journal of Aeronautic Science,1956, November, p. 1007-1010, 1036.
Prandtl L. Uber Flussigkeitsbewegung bei sehr kleiner Reibung / L. Prandtl // Verhandlung d. III Untern. Math. Kongr. Heidelberg, 1904, p. 484-491.
Blasius H. Grenzschichten in Flussigkeiten mit kleiner Reibung / H. Blasius // Z. Math. Phys. 1908, v. 56, p. 1-37.
Karman von Th. Laminare and Turbulent Reibung / Th. von Karman // Zeitschrift für Angewandte Mathematik und Mechanik, 1921, Vol. 1, p. 233-252.
Pohlhausen K. Zur Waherungsweisen Integration der Laminaren Reibungsschicht / K. Pohlhausen // Zeitschrift für Angewandte Mathematik und Mechanik , 1921, Vol. 1, p . 252--
Sohrab S.H. A Modified Theory of Turbulent Flow over a Flat Plate / S .H. Sohrab // Proc. Of the 5th IASME /WSEAS Conference of Fluid Mechanics and Aerodynamics, Athens, Greece, 2007, August 25-27. p. 72-80.
Weyburne, D. A mathematical description of the fluid boundary layer /D. Weyburne // Applied Mathematics and Computation, 2006, Vol. 175, pp. 1675–1684.
Weyburne D. New thickness and shape parameters for the boundary layer velocity profile / D. Weyburne // Experimental Thermal and Fluid Science, 2014, Vol. 54, pp. 22–28.
Abdul-Ghafour Q.A. A general velocity profile foe a laminar boundary layer over flat plate with zero incidence / Q.A. Abdul-Ghafour // Journal of Engineering, 2011, Vol. 17, 11 p.
Maupertuis P.L.M. Accord de differentes loix de la nature qui avoient jusqu’ici paru inpompatibles / P.L.M. Maupertuis // Memoires de l'Academie Royale des Sciences de Paris, 1744, 15 April, p. 417-426.
Mandre S. Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid / S. Mandre // J. Fluid Mech. , 2022, vol. 939, A27.
Bretheim J.U. Standard logarithmic mean velocity distribution in a band-limited restricted model of turbulent flow in a half-channel / J.U. Bretheim, C. Meneveau, D.F. Gayme // Phys. Fluids, 2015, 27, 011702.
Mofakham A. A. Particles dispersion and deposition in inhomogeneous turbulent flows using continuous random walk models / A. A. Mofakham , G. Ahmadi // Phys. Fluids, 2019, 31, 083301