By Eds. Theodore Y. Wu & John W. Hutchinson
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Extra info for Advances in Applied Mechanics, Vol. 25
He computes the corrections of order E and E’ for circular, elliptic, and rectangular cross sections. Timoshenko and Goodier (1951) present a concise and accessible account of the analysis for a circular section. The only nonzero components of stress are the shearing stresses T~~ and reyacting over the cross section. 26) rex = ~- where G is the modulus of rigidity. Substituting these into the two relevant t X I F FIG. 10. Loosely coiled ring sector of circular cross section. 27) 0. Here c i s a constant that is at the end of the calculation evaluated in terms of the applied torque FL by integrating the shear stress over the cross section.
Appl. Math. , Univ. British Columbia, Vancouver. Todd, L. (1979). Steady, laminar flow through curved pipes of small, constant cross-section. Tech. Rept. No. 79-1, Dept. , Sudbury, Canada. Todd, L. (1980). Steady, laminar flow through non-uniform, thin pipes. Tech. Rept. No. 80-1, Dept. , Sudbury, Canada. Van Dyke, M. (1970). Entry flow in a channel. J. Fluid Mech. 44, 813-823. Van Dyke, M. (1975). “Perturbation Methods in Fluid Mechanics,” Parabolic, Stanford, California. Van Dyke, M. (1978). Extended Stokes series: Laminar flow through a loosely coiled pipe.
The cross section of the pipe is then not quite circular in a plane normal to its own helical axis, which is what is usually understood as a coiled circular pipe, but the difference affects only third-order terms. As in the preceding example, we introduce a slightly nonorthogonal coordinate system x, y, z that spirals with the pipe, as shown in Fig. 9b. It is related to the Cartesian system 2, j , i by x = 2, + y = jj cos( E Z ) ,? sin( E X ) - a, z = 2 cos( &a)- j j sin( & a ) . 19) a. Potential Flow Again we consider first the simpler problem of potential flow.
Advances in Applied Mechanics, Vol. 25 by Eds. Theodore Y. Wu & John W. Hutchinson