In this paper, we propose a new paradigm for solving Navier–Stokes equations. The proposed methodology is based on a streamfunction–velocity formulation of the two-dimensional steady-state Navier–Stokes equations representing incompressible fluid flows in two-dimensional domains. Similar formulations are also possible for three-dimensional fluid flows. The main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values, especially on solid boundaries, encountered when solving the streamfunction–vorticity formulations. Our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity–pressure formulations of the Navier–Stokes equations. We describe the new formulation of the Navier–Stokes equations and use this formulation to solve a couple of fluid flow problems. We use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem. It is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem. In addition, we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature. In all cases, we obtain high accuracy solutions with little additional cost.