TITLE:
Stochastic Chaos of Exponential Oscillons and Pulsons
AUTHORS:
Victor A. Miroshnikov
KEYWORDS:
The Navier-Stokes Equations, Stochastic Chaos, Helmholtz Decomposition, Exact Solution, Decomposition into Invariant Structures, Experimental and Theoretical Programming, Quantization of Kinetic Energy, Random Elementary Oscillon, Random Elementary Pulson, Random Internal Elementary Oscillon, Random Diagonal Elementary Oscillon, Random External Elementary Oscillon, Random Wave Pulson, Random Internal Wave Oscillon, Random Diagonal Wave Oscillon, Random External Wave Oscillon, Random Group Pulson, Random Internal Group Oscillon, Random Diagonal Group Oscillon, Random External Group Oscillon, Random Energy Pulson, Random Internal Energy Oscillon, Random Diagonal Energy Oscillon, Random External Energy Oscillon, Random Cumulative Energy Pulson
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.13 No.4,
December
1,
2023
ABSTRACT: An exact three-dimensional solution for stochastic
chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The
Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the
Archimedean, Stokes, and Navier problems. The exact solution is obtained
with the help of the method of decomposition in invariant structures.
Differential algebra is constructed for six families of random invariant
structures: random scalar kinematic structures, time-complementary random
scalar kinematic structures, random vector kinematic structures,
time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector
dynamic structures. Tedious computations are performed using the
experimental and theoretical programming in Maple. The random scalar and vector
kinematic structures and the time-complementary random scalar and vector
kinematic structures are applied to solve the
Stokes problem. The random scalar and vector dynamic structures are
employed to expand scalar and vector variables of the Navier problem.
Potentialization of the Navier field becomes available since vortex forces,
which are expressed via the vector potentials of the Helmholtz decomposition,
counterbalance each other. On the contrary, potential forces, which are described
by the scalar potentials of the Helmholtz decomposition, superimpose to
generate the gradient of a dynamic random pressure. Various constituents of the
kinetic energy are ascribed to diverse interactions of random,
three-dimensional, nonlinear, internal waves with a two-fold topology, which
are termed random exponential oscillons and pulsons. Quantization of the
kinetic energy of stochastic chaos is developed in terms of wave structures of
random elementary oscillons, random elementary pulsons, random internal,
diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons,
random group pulsons, random internal,
diagonal, and external group oscillons, a random energy pulson, random
internal, diagonal, and external energy oscillons, and a random cumulative
energy pulson.