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SIGMA 4 (2008), 027, 19 pages arXiv:0802.3521
https://doi.org/10.3842/SIGMA.2008.027
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Applications of Group Analysis to the Three-Dimensional Equations of Fluids with Internal Inertia
Piyanuch Siriwat and Sergey V. Meleshko
School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand
Received October 31, 2007, in final form February 12, 2008; Published online February 24, 2008
Abstract
Group classification of the three-dimensional
equations describing flows of fluids with internal inertia,
where the potential function W = W(ρ,ρ·), is
presented. The given equations include such models as the
non-linear one-velocity model of a bubbly fluid with
incompressible liquid phase at small volume concentration of gas
bubbles, and the dispersive shallow water model. These models are
obtained for special types of the function W(ρ,ρ·).
Group classification separates out the function
W(ρ,ρ·) at 15 different cases. Another part of the
manuscript is devoted to one class of partially invariant
solutions. This solution is constructed on the base of all
rotations. In the gas dynamics such class of solutions is called
the Ovsyannikov vortex. Group classification of the system of
equations for invariant functions is obtained. Complete analysis
of invariant solutions for the special type of a potential
function is given.
Key words:
equivalence Lie group; admitted Lie group; optimal system of subalgebras; invariant and partially invariant solutions.
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