An analytical solution of the Navier–Stokes equation for internal flows

Abstract

This paper derives a solution to the Navier–Stokes equation by considering vorticity generated at system boundaries. The result is an explicit expression for the velocity. The Navier–Stokes equation is reformulated as a divergence and integrated, giving a tensor equation that splits into a symmetric and a skew-symmetric part. One equation gives an algebraic system of quadratic equations involving velocity components. A system of nonlinear partial differential equations is reduced to algebra. The velocity is then explicitly calculated and shown to depend on boundary conditions only. This removes the need to solve the Navier–Stokes equation by a 3D numerical computation, replacing it by computation of 2D surface integrals over the boundary.

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

“…Instead of using the coordinate relation (2b) and the potential function (3) the following expression can be used: Φ = P (ξ , y, z) R(y)S(z), ξ = kx − ς (t) or Φ = P (x, ξ , z) R (ξ ) S(z), ξ = ky − ς (t) . (23) In particular, it is reasonable to suggest that other classes of nontrivial exact solutions may still be developed from (8) or from the original Navier-Stokes equations by more complex procedures to find more about the properties of the exact solutions [26,27].…”

Analytical Study of Non-Newtonian Fluid and Navier-Stokes Equations

“…Scientists turn to the numerical solutions according to the difficulty of the nonlinear terms in a described system of fluid flow [4]. Some scientists [5][6] turn to describe the physical problems in terms of nonlinear partial differential equations for special cases of fluid and flow properties.…”