Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

2009 ◽  
Vol 10 (6) ◽  
pp. 3457-3465 ◽  
Author(s):  
R. Naz ◽  
F.M. Mahomed ◽  
D.P. Mason
Keyword(s):  
Conservation Laws ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Free Jets ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Partial Lagrangian

scholarly journals First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and -Symmetries

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Gülden Gün ◽  
Teoman Özer
Keyword(s):  
Governing Equation ◽  
First Integrals ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Minimum Drag ◽  
Symmetry Properties ◽  
Group Invariant Solutions ◽  
Work First ◽  
Partial Lagrangian ◽  
Integrating Factors

We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.

scholarly journals Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
R. Naz
Keyword(s):  
Fluid Velocity ◽  
Invariant Solution ◽  
Jet Flows ◽  
Order Partial Differential Equation ◽  
Free Wall ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Third Order ◽  
Group Invariant ◽  
Group Invariant Solutions

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

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