Hidden Invariances in Problems of Two-Dimensional and Three-Dimensional Wall Jets for Newtonian and Non-Newtonian Fluids

This work is devoted to the investigation of self-similar solutions for steady wall jets. The problem is considered in the context of two- and three-dimensional Prandtl boundary layer equations, and three-dimensional parabolized Navier--Stokes equations for Newtonian and non-Newtonian fluids. In contrast to dimensional analysis, which does not allow the determination of self-similar solutions in this case, a generating functions approach elaborated by Vinogradov [Soviet Math. Dokl., 19 (1978), pp. 144--148] enables one to derive conservation laws for the above-mentioned problems and, as a consequence, to find new self-similarities of the Navier--Stokes equations.

[1]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[2]  G. Caviglia.  Composite variational principles and the determination of conservation laws , 1988 .

[3]  M. Glauert.  The wall jet , 1956, Journal of Fluid Mechanics.

[4]  A. Tumin,et al.  Stability of a Compressible Laminar Wall-Jet With Heat Transfer , 1995 .

[5]  E. Chassignet,et al.  Mechanism of eddy separation from coastal currents , 2000 .

[6]  V. Shtern,et al.  Viscous flows with paradoxical properties , 1989 .

[7]  Stephen C. Anco,et al.  Derivation of conservation laws from nonlocal symmetries of differential equations , 1996 .

[8]  L. Sedov.  Similarity and Dimensional Methods in Mechanics , 1960 .

[9]  Alexandre M. Vinogradov,et al.  From symmetries of partial differential equations towards secondary (“quantized”) calculus , 1994 .

[10]  M. Itagaki,et al.  Analysis of Active Dissolution of Copper in Acidic Solution by EQCM/Wall Jet Split Ring Disk Electrode , 2000 .

[11]  A. Bocharov,et al.  Symmetries and conservation laws for differential equations of mathematical physics , 1999 .

[12]  H. Blasius.  Grenzschichten in Flüssigkeiten mit kleiner Reibung , 1907 .

[13]  G. Bluman,et al.  Erratum: Integrating factors and first integrals for ordinary differential equations , 1999, European Journal of Applied Mathematics.

[14]  G. Bluman,et al.  Direct Construction of Conservation Laws from Field Equations , 1997 .

[15]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[16]  DISSIPATIVE SYSTEMS, CONSERVATION LAWS AND SYMMETRIES , 1996 .

[17]  J. Murray.  The boundary layer on a flat plate in a stream with uniform shear , 1961, Journal of Fluid Mechanics.

[18]  P. Weidman,et al.  Natural convection far downstream of a heat source on a solid wall , 1997, Journal of Fluid Mechanics.

[19]  Alexandre M. Vinogradov,et al.  Symmetries and conservation laws of partial differential equations: Basic notions and results , 1989 .

[20]  W. Tollmien,et al.  Über Flüssigkeitsbewegung bei sehr kleiner Reibung , 1961 .

[21]  G. I. Barenblatt,et al.  Similarity, Self-Similarity and Intermediate Asymptotics , 1979 .

[22]  Trägårdh,et al.  An Experimental Study of the Kinetics of Particle Deposition in a Wall-Jet Cell Using Total Internal Reflection Microscopy. , 2000, Journal of colloid and interface science.

[23]  F. Hsiao,et al.  Experimental studies on flow transition of a plane wall jet , 1996, The Aeronautical Journal (1968).

[24]  John C. Tannehill,et al.  Parabolized reduced Navier-Stokes computational techniques , 1992 .

[25]  A. Tumin,et al.  Instability and Receptivity of Laminar Wall Jets , 1997 .

[26]  N. Riley,et al.  Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary , 1989 .

[27]  T. Tsujishita.  On variation bicomplexes associated to differential equations , 1982 .

[28]  H. Schlichting.  Boundary Layer Theory , 1955 .

[29]  G. Herrmann,et al.  On conservation laws for dissipative systems , 1991 .

[30]  P. Weidman.  New solutions for laminar boundary layers with cross flow , 1997 .

[31]  Jean-Raymond Abrial,et al.  On B , 1998, B.

[32]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[33]  John H. Merkin,et al.  A note on the wall-jet problem , 1986 .

[34]  R. A. Bajura,et al.  Transition in a two-dimensional plane wall jet , 1975, Journal of Fluid Mechanics.

[35]  Detection of near-wall hydrodynamic effects by electrochemiluminescence , 1999 .

[36]  J Cousteix,et al.  Three-Dimensional and Unsteady Boundary- Layer Computations , 1986 .

[37]  N. Riley,et al.  Effects of compressibility on a laminar wall jet , 1958, Journal of Fluid Mechanics.

[38]  E. Curd.  Possible applications of wall jets in controlling air contaminants. , 1981, The Annals of occupational hygiene.

[39]  M. Sun,et al.  Separation control on a thick airfoil with multiple slots blowing at small speeds , 2000 .

[40]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[41]  Chie Gau,et al.  Flow and impingement cooling heat transfer along triangular rib-roughened walls , 2000 .

[42]  V. N. Gusyatnikova,et al.  Symmetries and Conservation Laws of Navier—Stokes Equations , 1989 .

[43]  N. Goldenfeld,et al.  Intermediate asymptotics and renormalization group theory , 1989 .