On a special class of analytical solutions to the three-dimensional incompressible Navier–Stokes equations

The three-dimensional incompressible Navier–Stokes equations with the continuity equation are solved analytically in this work. The spatial and temporal coordinates are transformed into a single coordinate ξ . 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 , ξ ) R ( ξ ) . The potential function is firstly substituted into the continuity equation to produce the solution for R and the resultant expression is used sequentially in the Navier–Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in P terms. Here, more general solutions are also obtained based on the particular solutions of P . Explicit analytical solutions are found to be mathematically similar for the cases of zero and constant pressure gradient. Two examples are given to illustrate the applicability of the method. It is also concluded that the selection of variables for the potential function can be interchanged from the beginning, resulting in similar explicit solutions.