This paper addresses the fascinating long history of the classical two
-dimensional biharmonic problem for a rectangular domain. Among variou
s mathematical and engineering approaches, the method of superposition
is effective for solving mechanical problems concerning creeping flow
of viscous fluid set up in a rectangular cavity by tangential velocit
ies applied along its walls, an equilibrium of an elastic rectangle, a
nd bending of a clamped thin rectangular elastic plate by a normal loa
d. The object of this paper is bath to clarify some purely mathematica
l questions connected with the solution of the infinite systems of lin
ear algebraic equations and to provide a considerable simplification o
f the numerical algorithm. The method is illustrated by several exampl
es of steady Stokes flow in a square cavity.