Abstract
Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with time-dependent structural coefficients are developed to examine kinematic and dynamic problems for a deterministic chaos of N stochastic waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The Dirichlet problems are formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains for stochastic waves vanishing at infinity. Development of the novel method of solving partial differential equations through decomposition in invariant structures is resumed by using experimental and theoretical computation in Maple?. This computational method generalizes the analytical methods of separation of variables and undetermined coefficients. Exact solutions for the deterministic chaos of upper and lower cumulative flows are revealed by experimental computing, proved by theoretical computing, and justified by the system of Navier-Stokes PDEs. Various scenarios of a developed wave chaos are modeled by 3N parameters and 2N boundary functions, which exhibit stochastic behavior.
Highlights
The two-dimensional (2d) Navier-Stokes system of partial differential equations (PDEs) for a Newtonian fluid with a constant density ρ and a constant kinematic viscosity ν in a gravity field g is ∂v ∂t + (v ⋅∇)v = − 1 ρ ∇pt +ν∆v g, (1)How to cite this paper: Miroshnikov, V.A. (2014) Deterministic Chaos of N Stochastic Waves in Two Dimensions
To examine linear and nonlinear parts of kinematic and dynamic problems for 2d stochastic waves in the theory of Newtonian flows with harmonic velocity, the Kinematic exponential Fourier (KEF) structures, the dynamic exponential Fourier (DEF) structures, and the KEF-DEF structures with time-dependent structural coefficients are developed in the current paper
Theoretical Solutions for the Dynamic Potentials in the KEF Structures Theoretical dynamic problems in the kinematic Fourier (KF) structures for the Helmholtz and Bernoulli potentials of the cumulative flows are set by the Lamb-Helmholtz PDEs (8)-(10) in the vortical presentation with φ = 0
Summary
The two-dimensional (2d) Navier-Stokes system of partial differential equations (PDEs) for a Newtonian fluid with a constant density ρ and a constant kinematic viscosity ν in a gravity field g is. The exponential Fourier eigenfunctions obtained by the classical method of separation of variables of the 2d Laplace equation in [1] and [4] were primarily used for a linear part of the kinematic problem for free-surface waves of the theory of the ideal fluid with ν = 0 in [6] This analytical method was recently developed into the computational method of solving PDEs by decomposition into invariant structures. To examine linear and nonlinear parts of kinematic and dynamic problems for 2d stochastic waves in the theory of Newtonian flows with harmonic velocity, the KEF structures, the DEF structures, and the KEF-DEF structures with time-dependent structural coefficients are developed in the current paper.
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