The presented research paper illustrates the development of a new methodology to solve 2-dimensional (2D) Navier-Stoke equations, which Pukhnachev proposed by introducing unknown functions in the stream and pressure functions of fluid flow. The proposed novel method is distinguishable from the common vorticity-stream given in Navier-Stokes expression because it has a stream function that corresponds to the unknown function in the elliptic expression. The equation represents a couple of scheme in algorithmic considerations because it enables two situations to be solved using one function of the subject stream without putting new conditions on the innovative function. Here, the concept of numerical algorithm is applied in a flow under cavity to represent a benchmark task to be solved. The benchmark task gives enough representation of the subject flow.