Symbolic computation of flow in a rotating pipe

A perturbation solution of the fully developed flow through a pipe of circular cross‐section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ω a 2 /ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non‐linear equations for the axial velocity ω and the streamfunction ϕ of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ϕ n , n ϕ 1, of the perturbation expansion ϕ = ∑   n = 1 ∞R n ϕ n in terms of ω n ‐1 , the ( n ‐1)‐th order of the expansion ω = ∑   n = 0 ∞R n ω n , and of the lower orders ϕ 1 ,…,ϕ n − 1 . Then it permits the computation of ω n from ω 0 ,…,ω n − 1 and ϕ 1 ,…,ϕ; n . The computation starts from the Hagen–Poiseuille flow ω 0 , i.e. the perturbation is around this flow. The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co‐ordinates: in the complex co‐ordinates chosen the two‐dimensional harmonic (Laplace, Δ) and biharmonic (Δ 2 ) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ω n , ϕ n and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co‐ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ⩽ 6 of the perturbation expansion.