A class of exact solutions to the three-dimensional incompressible Navier–Stokes equations

• Published in: Applied mathematics letters 2010-11, Vol.23 (11), p.1388-1396
• Main Authors: Nugroho, Gunawan, Ali, Ahmed M.S., Abdul Karim, Zainal A.
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Continuity equation; Differential equations; Exact sciences and technology; Exact solution; Exact solutions; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Navier-Stokes equations; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Potential function; Sciences and techniques of general use; The Navier–Stokes equations; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Turbulence; Uniqueness
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2010.07.005

An exact solution of the three-dimensional incompressible Navier–Stokes equations with the continuity equation is produced in this work. The solution is proposed to be in the form V = ∇ Φ + ∇ × Φ where Φ is a potential function that is defined as Φ = P ( x , y , ξ ) R ( y ) S ( ξ ) , with the application of the coordinate transform ξ = k z − ς ( t ) . The potential function is firstly substituted into the continuity equation to produce the solution for R and S . The resultant expression is used sequentially in the Navier–Stokes equations to reduce the problem to a class of nonlinear ordinary differential equations in P terms, in which the pressure term is presented in a general functional form. General solutions are obtained based on the particular solutions of P where the equation is reduced to the form of a linear differential equation. A method for finding closed form solutions for general linear differential equations is also proposed. The uniqueness of the solution is ensured because the proposed method reduces the original problem to a linear differential equation. Moreover, the solution is regularised for blow up cases with a controllable error. Further analysis shows that the energy rate is not zero for any nontrivial solution with respect to initial and boundary conditions. The solution being nontrivial represents the qualitative nature of turbulent flows.