# zbMATH — the first resource for mathematics

The Dean equations extended to a helical pipe flow. (English) Zbl 0675.76040
Summary: The Dean equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter $$\lambda$$ /$${\mathcal R}$$, where $$\lambda$$ is the ratio of the torsion $$\tau$$ to the curvature $$\kappa$$ of the pipe axis and $${\mathcal R}$$ the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit $$\lambda$$ /$${\mathcal R}\to \infty$$ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow is recovered.
Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by H. C. Kao [ibid. 184, 335-356 (1987; Zbl 0645.76045)] are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by S. Murata, Y. Miyake, T. Inaba and H. Ogawa [Bull. JSME 24, 355 ff. (1981)] is discussed.

##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text:
##### References:
  Dean, Phil. Mag. 4 pp 208– (1927)  Cuming, Aeronaut. Res. Counc. R. M. 15 pp 461– (1952)  Choi, J. Fluid Mech. 15 pp 461– (1988)  DOI: 10.1146/annurev.fl.15.010183.002333  DOI: 10.1017/S0022112081002073 · Zbl 0483.76040  DOI: 10.1017/S0022112078001032 · Zbl 0374.76028  DOI: 10.1016/0169-5983(87)90008-6  DOI: 10.1007/BF01535586 · Zbl 0377.76026  Murata, Bull JSME 24 pp 355– (1981)  Mercier, Fusion Nucleaire 3 pp 89– (1963)  DOI: 10.1016/0045-7930(86)90017-4 · Zbl 0608.76032  DOI: 10.1017/S0022112080001310  DOI: 10.1017/S002211208700291X · Zbl 0645.76045  Germano, Atti VIII Cong. Naz. AIDAA, Torino September 125 pp 1– (1985)  DOI: 10.1017/S0022112082003206 · Zbl 0533.76029  Dean, Phil. Mag. 5 pp 673– (1928)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.