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

Abstract

a b s t r a c tAn 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 = kz − 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.

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“…There are many phenomena in physics which are described by Navier-Stokes equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The mathematical modeling of the weather, ocean currents, water flow in a pipes, channels and air flow around wings are described by Navier-Stokes equations.…”

“…The mathematical modeling of the weather, ocean currents, water flow in a pipes, channels and air flow around wings are described by Navier-Stokes equations. Many problems have been formulated in nonlinear partial differential equations, which face some difficulties in the way of analytical solutions [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Scientists turn to the numerical solutions according to the difficulty of the nonlinear terms in a described system of fluid flow [5].…”

“…A finite-difference method for solving the timedependent Navier-Stokes equations for an incompressible fluid is introduced by Alexandre Chorin [6]. An exact solution of the three-dimensional incompressible Navier-Stokes equations with the continuity equation is produced by Gunawan Nugroho [7]. Mats et al [8] derived a solution to the Navier-Stokes equation by considering vorticity generated at system boundaries.…”

“…The transformation of the Navier-Stokes equations to the Schrödinger equation performed by application of the Riccati equation [9]. A particular class of solutions of nonlinear differential equations can be obtained by several procedures [7]. The linear partial differential equations (PDEns) are solved by the similarity parameters method [13].…”

“…In the previous efforts, scientists studied many problems of peristaltic fluid flow in several shapes under suitable boundary conditions. Most of the previous problems are described by the nonlinear Navier-Stokes equations, which are approximately solved for long wavelength = 0 and low Reynolds number [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In the present example the proposed problem is solved analytically.…”