Symmetry solutions and conservation laws for some partial differential equations in fluid mechanics - Naz, Rehana (My University, 2009)
ABSTRACT In jet problems the conserved quantity plays a central role in the solution process. The conserved quantities for laminar jets have been established either from physical arguments or by integrating Prandtl's momentum boundary layer equation across the jet and using the boundary conditions and the continuity equation. This method of deriving conserved quantities is not entirely systematic and in problems such as the wall jet requires considerable mathematical and physical insight. A systematic way to derive the conserved quantities for jet °ows using conservation laws is presented in this dissertation. Two-dimensional, ra- dial and axisymmetric °ows are considered and conserved quantities for liquid, free and wall jets for each type of °ow are derived. The jet °ows are described by Prandtl's momentum boundary layer equation and the continuity equation. The stream function transforms Prandtl's momentum boundary layer equation and the continuity equation into a single third- order partial di®erential equation for the stream function. The multiplier approach is used to derive conserved vectors for the system as well as for the third-order partial di®erential equation for the stream function for each jet °ow. The liquid jet, the free jet and the wall jet satisfy the same partial di®erential equations but the boundary conditions for each jet are di®erent. The conserved vectors depend only on the partial di®erential equations. The derivation of the conserved quantity depends on the boundary conditions as well as on the di®erential equations. The boundary condi- tions therefore determine which conserved vector is associated with which jet. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived. This approach gives a uni¯ed treatment to the derivation of conserved quantities for jet °ows and may lead to a new classi¯cation of jets through conserved vectors. The conservation laws for second order scalar partial di®erential equations and systems of partial di®erential equations which occur in °uid mechanics are constructed using di®erent approaches. The direct method, Noether's theorem, the characteristic method, the variational derivative method (mul- tiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example. The conservation laws for the non-linear di®usion equa- tion for the spreading of an axisymmetric thin liquid drop, the system of two partial di®erential equations governing °ow in the laminar two-dimensional jet and the system of two partial di®erential equations governing °ow in the laminar radial jet are discussed via these approaches. The group invariant solutions for the system of equations governing °ow in two-dimensional and radial free jets are derived. It is shown that the group invariant solution and similarity solution are the same. The similarity solution to Prandtl's boundary layer equations for two- dimensional and radial °ows with vanishing or constant mainstream velocity gives rise to a third-order ordinary di®erential equation which depends on a parameter. For speci¯c values of the parameter the symmetry solutions for the third-order ordinary di®erential equation are constructed. The invariant solutions of the third-order ordinary di®erential equation are also derived.
Keywords: conserved quantities; conservation laws; group invariant solution; symmetry solution