| Popis: |
This thesis considers steady, fully-developed flows through weakly curved pipes using the extended Stokes series method. The Stokes series for pipes of circular cross-section is expanded in powers of the Dean number, K, to 196 terms by computer. Analysis shows that the convergence is limited by an imaginary conjugate pair of square-root singularities K = ±iKc. Contrary to previous analysis of this solution, analytic continuation of the series indicates that the flux ratio in a weakly curved pipe does not vary asymptotically as K−1/10 for large K. Using generalised Padé approximants it is proposed that the singularity at iKc corresponds to a symmetry breaking bifurcation, at which three previously unreported complex branches are identified. The nature of the singularity is supported in part by numerical consideration of the governing equations for complex Dean number. It is postulated that there exists a complex solution to the governing equations for which the azimuthal velocity varies asymptotically as K−1/2, and the streamfunction as K0 near K = 0. This is supported by the results from the generalised Padé approximants. Brief consideration is given to pipes of elliptic cross-section. The Stokes series for pipes of elliptic cross-section for various aspect ratios, λ, is expanded up to the K24 term by computer. For small K, it is found that the flux ratio achieves a minimum for aspect ratio λ ≈ 1.75. This, and the behaviour of the total vorticity, is in agreement with previous studies which found that the effect of the curvature is reduced in the limit of small and large aspect ratios. The convergence of the series solution is found to be limited by an imaginary conjugate pair of square-root singularities K = ±iKc(λ), which varies with λ. |
