Abstract:
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.