The method of lines is applied to simulate the motion of viscous incompressible fluid in a driven cavity and around an aerofoil at a large angle of attack. The numerical algorithms are based on the solution of an initial-boundary value problem for the full incompressible Navier-Stokes equations in the form of the fourth order equation for the stream function. Calculations for the cavity flow configuration have been obtained for Reynolds numbers ranging from 5000 to 30 000 on the non-uniform grids 100 x 100 and a uniform grid 150 x 150. Computations for the flow around the aerofoil have been performed at Reynolds numbers Re <= 1000.