This paper presents a variational formulation for the steady two-dimensional laminar flow of an incompressible Newtonian fluid  using cylindrical coordinates. All terms of the motion equation are considered without neglecting any term. The stationary condition of the formulation results in two sets of symmetrical equations, where  the correspondence with the equations of motion and continuity is demonstrated. The formulation is applied to the entrance region of long pipes, as an example, and a trial function is proposed. The results are compared with the numerical results from the literature, showing a good agreement.

variational principle, Navier-Stokes equation, continuity equation, fluid flow.