By Eds. Theodore Y. Wu & John W. Hutchinson
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Extra resources for Advances in Applied Mechanics, Vol. 25
This is an essential distinction because, as we have already seen for plane strips, we can often use conventional coordinates if the centerline is straight, but most otherwise use curvilinear (and in general nonorthogonal) coordinates. 1 . 38) through a straight pipe of slowly varying circular cross section. We now consider the generalization to a pipe of elliptic cross section, whose major and minor axes may vary slowly and independently. Thus it is described, in terms of the contracted axial coordinate X = E X , by y ’ / a 2 ( X )+ z’/b’(X) = 1.
Potential Flow Again we consider first the simpler problem of potential flow. 20) + (Y + a)*4,, - ( Y + a M y - 2423 = 0. 21) and we prescribe the flux through the pipe in dimensionless form as 11 4,dydz =E 11 4 x d y d z = T. 22) Slow Variations in Continuum Mechanics 31 When E is negligibly small, the solution is a uniform stream along the axis of the helix, with 4 = x = X / E . Higher approximations proceed in powers of E’, and one easily calculates the third approximation as 1 =-X E + E U Z + -81E ~ U [ Y ~ +Z z 3 - ~ U Y Z- ( 3 + 8 a 2 ) 2 ]+ .
Phys. ) 28, 1-21. Blasius, H. (1910). 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.
Advances in Applied Mechanics, Vol. 25 by Eds. Theodore Y. Wu & John W. Hutchinson