Laminar flow in a curved pipe with varying curvature

The steady laminar motion of fluid through pipes of circular cross-section, the curvature of whose centre-line varies locally, is analysed theoretically. The flow in three kinds of pipes whose centre-lines are specified by \[ \hat{y} = a(1+\kappa^2\hat{x}^2)^{\frac{1}{2}},\quad\hat{y} = a\tan h\kappa\hat{x}\quad{\rm and}\quad\hat{y} = a\sin\kappa\hat{x} \] are treated as the examples of once-, twice- and periodically-curved pipes, respectively. The analysis is valid for any other two-dimensionally curved pipes, when centre-line curvature is small. At very small Reynolds number, the position of maximum axial velocity shifts towards the inner side of the pipe section; at large Reynolds number, on the contrary, it tends to the outer side, owing to centrifugal force. Furthermore, in the latter case, adaptation of the flow follows the change of mean-flow direction, with a phase lag.