The general solution to Stokes flow in 2D Cartesian coordinates

The general solution to Stokes flow in 2D Cartesian coordinates. For the 2D case the governing equation is ∇4ψ = 0. The operator ∇ may be applied either in Cartesian (x, y) or in polar (r, θ) coordinates. In either case it would be appropriate to seek a general form of the solution to this biharmonic operator. The problem of finding a solution to the biharmonic operator can be broken down into two sub-problems: ∇2ω = 0 and ∇2ψ = −ω The first problem is similar to the case of potential flow in the 2D case and admits a class of solutions of the following type: r n cos(nθ); r n sin(nθ) The second problem is then the solution to a Poisson equation with the non-homogeneous terms corresponding to each of the above functions. Solve these equations to derive a general solution to Stokes flow in 2D Cartesian coordinates.