A literature survey (Berger, Talbot & Yao 1983) indicates that laminar viscous flow in curved pipes has been extensively investigated. Most of the existing analytical results deal with the case of circular cross-section. The important studies dealing with elliptical cross-sections are mainly due to Thomas & Walters (1965) and Srivastava (1980). The analysis of Thomas & Walters is based on Dean's (1927, 1928) approach in which the simplified forms of the momentum and continuity equations have been used. The analysis of Srivastava is essentially a seminumerical approach, in which no explicit expressions have been presented.In this paper, using elliptic coordinates and following the unsimplified formulation of Topakoglu (1967), the flow in a curved pipe of elliptical cross-section is analysed. Two different geometries have been considered: (i) with the major axis of the ellipse placed in the direction of the radius of curvature; and (ii) with the minor axis of the ellipse placed in the direction of the radius of curvature. For both cases explicit expressions for the first term of the expansion of the secondary-flow stream function as a function of the ellipticity ratio of the elliptic section have been obtained. After selecting a typical numerical value for the ellipticity ratio, the secondary-flow streamlines are plotted. The results are compared with that of Thomas & Walters. The remaining terms of the expansion of the flow field are not included, but they will be analysed in a future paper.

In this paper, the analysis on secondary flow in curved elliptic pipes of Topakoglu & Ebadian (1985) has been extended up to a point where the rate-of-flow expression is obtained for any value of flatness ratio of the elliptic cross-section. The analysis is based on the double expansion method of Topakoglu (1967). Therefore, no approximation is involved in any step other than the natural limitation of the finite number of calculated terms of the expansions. The obtained results are systematically plotted against the curvature of centreline of the curved pipe for different values of Reynolds number.

Heat transfer for hydrodynamically and thermally fully developed laminar flow in a curved pipe is solved by a method of series expansion. The wall temperature around the periphery of any cross section, the mean heat flux along the pipe, and the internal heat generation are assumed to be uniform. The first few terms of the series expansion are determined and analytical expressions for the temperature distribution in the fluid and for the Nusselt number are presented; the results are applicable for small curvatures. A correlation with experiment is included.

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