By Eds. Theodore Y. Wu & John W. Hutchinson
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Additional info for Advances in Applied Mechanics, Vol. 25
Ed. (1963). ” Oxford Univ. Press, London. Sobey, I. J. (1976). Inviscid secondary motions in a tube of slowly varying ellipticity. J. Fluid Mech. 13, 621-639. Sobieczky, H. (1977). Kompressible Stromung in einer ebenen Schicht variabler Dicke. Z. Angew. Mafh. Mech. 57, T 207-T 209. , and Goodier, J. N. (1951). ” McGraw-Hill, New York. Todd, L. (1977). Some comments on steady, laminar flow through twisted pipes. J. Eng. Math. 11, 29-48. Todd, L. (1978). Steady, laminar flow through non-uniform, curved pipes of small cross-section.
6 b . Whereas we previously assumed that the curvature K of the centerline was small as well as slowly varying by setting K = E g ( E s ) , we now assume it to be slowly varying but not necessarily small, with K = G(Es). 3) Then Laplace's equation ( 2 . 55), the only slightly more complicated form a -(1+ an With terms in E* a+ Gn)-+ an a 1 a4 = 0. 59) shows that this is the exact solution for a circular annulus that at each station fits the local curvature of the strip, as indicated in Fig. 13.
The resulting correction of order E’ is found in terms of elementary functions-powers of (1 + G n ) and its logarithm (and this would appear to be true of all subsequent approximations). However, the results is too complicated to be given here. 3. 60). ) a. 60) shows that the first approximation satisfies a -(1+ an a i a alC, Gn)--(1 + Gn)- = 0. an dn 1 + Gn an Milton Van Dyke 42 Four quadratures yield + = A( 1 + G n ) 2In( 1 + G n ) + B(1 + G n ) 2+ C In( 1 + G n ) + D. 62) for laminar flow in an annulus except that now the coefficients A, B, C, D, which are determined by imposing the boundary conditions at the walls, depend on the axial coordinate through the slowly varying curvature function G (E S ) .
Advances in Applied Mechanics, Vol. 25 by Eds. Theodore Y. Wu & John W. Hutchinson