Group Invariant Solutions for Radial Jet Having Finite Fluid Velocity at Orifice

The group invariant solution for Prandtl-s boundary layer equations for an incompressible fluid governing the flow in radial free, wall and liquid jets having finite fluid velocity at the orifice are investigated. For each jet a symmetry is associated with the conserved vector that was used to derive the conserved quantity for the jet elsewhere. This symmetry is then used to construct the group invariant solution for the third-order partial differential equation for the stream function. The general form of the group invariant solution for radial jet flows is derived. The general form of group invariant solution and the general form of the similarity solution which was obtained elsewhere are the same.

• I. Naeem
• R. Naz

• Two-dimensional jets
• radial jets
• group invariant solution.

[1] W. H. Schwarz, The radial free jet, Chem. Eng. Sci. 18 (1963) 779-786.
[2] H. Schlichting, Laminare Strahlausbreitung, Z. Angew. Math. Mech. 13
(1933) 260-263.
[3] W. G. Bickley, The plane jet, Phil. Mag. 23 (1937) 727-731.
[4] D. P. Mason, Group invariant solution and conservation law for free
laminar two-dimensional jet, Journal Nonlinear Math. Phys. 9 (2002) 92-
101.
[5] R. Naz, F. M. Mahomed, D. P. Mason, Conservation laws via the
partial Lagrangian and group invariant solutions for radial and twodimensional
free jets, Submitted to Journal of Nonlinear Analysis: Real
world applications.
[6] M. B. Glauert, The wall jet, J. Fluid Mech. 1 (1956) 625-643.
[7] N. Riley, Asymptotic expansions in radial jets, J.Math. and Phys. 41
(1962) 132-146.
[8] N. Riley, An Application of Mangler-s transformation, Quart. J. Mech.
Applied. Math. 14 (1961) 197-202 .
[9] N. Riley, Radial jets with swirl, Part I. Incompressible Flow, Quart. Journ.
mech. and Applied Math. 15, (1962) 435-458.
[10] E. J. Watson, The radial spread of a liquid jet over a horizontal plane,
J. Fluid Mech. 20 (1964) 481-499.
[11] R. Naz, F. M. Mahomed and D. P. Mason, Symmetry solutions of a thirdorder
ordinary differential equation which arises from Prandtl boundary
layer equations, Journal Nonlinear Math. Phys. Accepted for publication.
[12] R. Naz, D. P. Mason and F. M. Mahomed, Conservation laws and
physical conserved quantities for laminar two-dimensional and radial
jets, Accepted for publication in journal of Nonlinear Analysis: Real
World Applications.
[13] A. H. Kara and F. M. Mahomed, Relationship between symmetries and
conservation laws, Int. J. Theor. Phys. 39 (2000) 23-40.
[14] M. B. Glauert, symposium uber Grenzschichtforschung, edited by H.
Gortler (Springerverlag 1957).