ISSN:
0271-2091
Keywords:
Computer algebra
;
Pipe flow
;
Rotating pipe
;
Perturbation expansion
;
Engineering
;
Engineering General
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
Notes:
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 = 2Ωa2/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 φn, n φ 1, of the perturbation expansion φ = ∑n = 1∞ Rnφn in terms of ωn-1, the (n-1)-th order of the expansion ω = ∑n = 0∞ Rnωn, and of the lower orders φ1,…,φn - 1. Then it permits the computation of ωn from ω0,…,ωn - 1 and φ1,…,φ;n. The computation starts from the Hagen-Poiseuille flow ω0, 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 (Δ2) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ωn, φn 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.
Additional Material:
4 Tab.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/fld.1650110302
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