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• W1935850636 abstract "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. The flow in radial jets is governed either by the system of two partial differential equations for the velocity components or by a single third-order partial differential equation for the stream function. The similarity solution for the third-order partial differential equation for the stream function for the radial free jet was discussed by Schwarz (1). Schlichting (2) and Bickley (3) derived the similarity solution for the third- order partial differential equation for the stream function for two-dimensional free jet. Mason in (4) constructed the group invariant solution for the same equation. In (5) authors found the group invariant solution for system of equations for the velocity components for both radial and two-dimensional free jets. Glauret (6) obtained the similarity solution for radial and two-dimensional wall jets. Riley in (7) established the similarity solution for the radial free, wall and liquid jets. In all these problems the fluid velocity at the orifice was infinite. Watson suggested the general form of similarity solution for the flows having finite velocity at the orifice. Riley (8) considered the problem of radial and two-dimensional wall jets for which the velocity remains finite at orifice and so our solution has some significance even near axis. The similarity solution was constructed for both radial and two-dimensional wall jets. The similarity solution for the radial free, wall and liquid jets with finite velocity at the orifice was studied by Riley (9). Schwarz in (1) derived the similarity solution for the stream function equation for radial free jet having finite as well as infinite velocity at orifice. Watson in (10) derived the similarity solution for system of equations for velocity components for the radial and two-dimensional liquid jets having finite velocity at the orifice. To the best of our knowledge, the group invariant solution for the third-order partial differential equation for stream function for radial free, wall and liquid jets, having finite fluid velocity at the orifice is still not attempted in the literature. It is considered in this paper. The similarity solution transforms the third-order partial differential equation to a third-order ordinary differential equa- tion. By using the certain transformations same third-order ordinary differential equation can be deduced for the radial jets whether the velocity at the orifice is finite or infinite. The third-order ordinary differential equation for radial free jet was first solved numerically by Schlichting (2) and later, Bickley (3) found the analytical solution. For the wall jet, Glauert (6) solved the third-order ordinary differential equation. Riley in (7), (9) found the solution for ordinary differential equation which appeared for radial liquid jet. The authors in (11) solved the third-order ordinary differential equations for radial free, wall and liquid jets by symmetry methods. In this paper we will derive the group invariant solution for the radial free, wall, liquid jets having finite velocity at the orifice. In (12) the conserved quantities for radial free, wall and liquid jets have been derived using the conservation laws. The symmetry associated with the conserved vector which is used to establish the conserved quantity for each jet generates the group invariant solution for the third-order partial differential equation for the stream function. This symmetry is obtained by using the approach introduced by Kara and Mahomed (13). We give explicitly the general form of group invariant solution for radial jet flows. We concluded that the group invariant solution and similarity solution are equivalent. In similarity solution method the form of stream function was assumed whereas in the group invariant method the form of stream function is derived." @default.
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• W1935850636 date "2008-07-21" @default.
• W1935850636 modified "2023-09-26" @default.
• W1935850636 title "Group invariant solutions for radial jet having finite fluid velocity at orifice" @default.
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