TY - JOUR JF - Applied Mathematics Letters VL - 23 Y1 - 2010/// N1 - cited By 8 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-77955925437&doi=10.1016%2fj.aml.2010.07.005&partnerID=40&md5=98b4a782dbea7e293d88c8f94a208fd0 A1 - Nugroho, G. A1 - Ali, A.M.S. A1 - Abdul Karim, Z.A. AV - none SP - 1388 IS - 11 TI - A class of exact solutions to the three-dimensional incompressible NavierStokes equations KW - Blow-up; Closed form solutions; Co-ordinate transform; Continuity equations; Energy rates; Exact solution; Functional forms; General solutions; Incompressible Navier Stokes equations; Linear differential equation; Nonlinear ordinary differential equation; Nontrivial solution; Particular solution; Potential function KW - Boundary conditions; Dynamical systems; Ordinary differential equations; Organic polymers; Partial differential equations; Three dimensional KW - Nonlinear equations N2 - An exact solution of the three-dimensional incompressible NavierStokes equations with the continuity equation is produced in this work. The solution is proposed to be in the form V=â??Φ+â??Ã?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-Ï?(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 NavierStokes 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. © 2010 Elsevier Ltd. All rights reserved. SN - 08939659 EP - 1396 ID - scholars1145 ER -