Determine the Hagen–Poiseuille flow through a duct of rectangular cross section (Fig. 3.5) by solving eq. (3.30) for u(y, z) as a Fourier series. This problem is analytically identical to determining the temperature distribution inside a rectangular object with internal heat generation and isothermal walls. With reference to Fig. 3.5, the problem statement is

To solve it, assume that

where u₁(y) is the Hagen–Poiseuille flow through the infinite parallel-plate channel of width 2a,

and where u₂ is the necessary correction,

 

 

Note that adding problems B and C equation by equation yields the original problem A. The advantage of decomposing the problem as A = B + C is that problem C can be solved by Fourier series expansion, whereas problem A cannot. Problem C is solvable because the equation ∇²u₂ = 0 is homogeneous and one set of boundary conditions (y) is homogeneous.