Invariants of gas and fluid flows and exact solutions to boundary eigenvalue problems

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The project consists of three parts. Part A will be devoted to investigation of invariants of gas and fluid flows generalizing new invariants discovered in my paper CCV#7 for the isentropic gas dynamics in mushroom clouds appearing after the atomic and thermonuclear explosions in atmosphere. I intend: (a) To find material conservation laws for the axisymmetric gas dynamics in mushroom clouds with arbitrary equation of state; (b) To obtain invariants for dynamics of ideal incompressible fluid with vanishing swirl (atomic and thermonuclear explosions under water); (c) To derive invariants of axisymmetric and helically symmetric plasma flows in magnetohydrodynamics. The work on Part A is planed to do in 2022-2025 jointly with PhD1 student and in 2025-2027 jointly with MSc2 student. Part B will be committed to the study of chaotic dynamics of viscous fluid with applications to the turbulence problem. Based on our joint with *Post-Doctoral Fellow R. Kipka paper *CCV#1 I plan: (a) To investigate numerically different regimes of chaotic dynamics in more general non-symmetric exact solutions to the Navier - Stokes equations; (b) To derive new time-dependent exact solutions to the Navier - Stokes equations and viscous magnetohydrodynamics equations with chaotic streaklines and chaotic magnetic field lines. The work on Part B is planed to do in 2023-2026 jointly with my Post-Doctoral Fellow and in 2023-2025 jointly with MSc1 student. Part C will be dedicated to the study of exact solutions to the ideal fluid dynamics, viscous magnetohydrodynamics and Navier - Stokes equations that have a common property that fluid velocity V(x) satisfies Beltrami equation curl V(x) =gV(x). This amounts to the eigenvalue-boundary-value problem for the operator curl. Based on results of my work CCV#4 I plan: (a) To represent in elementary functions all series of eigenfields of operator curl inside a ball 0 < R < R1  and a spherical shell R2 < R < R1 with the non-penetration boundary condition. In work by Cantarella et al (2000) [18] the eigenfields were described by special functions (Bessel's and Legendre's); (b) To investigate the scale-invariant equations for the eigenvalues  gk.m; (c) To derive exact solutions to the Navier-Stokes equations with the no-slip boundary condition. The work on Part C is planed to do in 2025-2027 in collaboration with PhD2 student. Results of the research will be important to all specialists on mathematical physics and on gas and fluid mechanics. The anticipated outcomes and benefits for Canada are: (a) Young specialists in the advanced areas of science will be trained and will defend their Dissertations. Each PhD student and Post-Doctoral Fellow will publish 2 papers in USA Journal Physics of Fluids and in European Journal ZNA; each MSc student will publish at least 1 paper in the same Journals. (b) Important mathematical problems will be resolved and their practical applications will be developed.