# Physics Reports vol.273

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2) and the flux through the pipe is prescribed, in dimensionless variables, as The hydraulic approximation, with uniform axial velocity across each section, gives u = c $ ~= l/a(X)b(X). 4) 26 Milton Van Dyke The second approximation for the velocity potential was calculated by Olson (1971), and his analysis is reported by Sobey (1976), who uses it as the basis for treating inviscid flow with slight shear. 5) 0(&3). Rewritten in terms of the fractional transverse coordinates 7 = y / a ( X ) and 4‘ = z / b ( X ) ,these are = [ a2(L)’(1 ab 8 a2b a’ &-?

Laminare Stromung in Kanalen wechselnder Breite. 2. Marh. Phys. 58, 225-233. G. (1977). Numerical solution of slender channel laminar flows. Comp. Methods Appl. Mech. Eng. 11. 319-339. Chow, J. C . , and Soda, K. (1972). Laminar flow in tubes with constriction. Phys. Fluids 15, 1700- 1706. Daniels, P. , and Eagles, P. M. (1979). High Reynolds number flows in exponential tubes of slow variation. J. Fluid Mech. 90, 305-314. Dean, W. R. (1928). The stream-line motion of fluid in a curved pipe. Philos.

11 ) into these transformed conditions, that is, 6yC(K, h, p ) / ( A + 2 p ) = k p 2 ( k 2- 2)[KU-'(K, h, p ) - Uop-'] _a , ; ( ~ h, , p ) / p = - K Z ) - ' ( K , h, p ) C J K , h, p ) = 0, + fiy' = 0, + determines A and B and hence the transformed solution to the problem. The formal solutions (inversion integrals) can then be written by making use of the fact that u" = 2iu"",and 6 = 2 P . The formal solutions for the compressional strains are where Julius Miklowitz 52 The technique is basically due to Folk et al.