The general solution to Stokes flow in 2D Cartesian coordinates. For the 2D case the governing equation is \(abla^{4} \psi=0\). The operator \(abla\) may be applied either in Cartesian \((x, y)\) or in polar \((r, \theta)\) 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:

\[abla^{2} \omega=0\]

and

\[abla^{2} \psi=-\omega\]

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:

\[r^{n} \cos (n \theta) ; \quad r^{n} \sin (n \theta)\]

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.