Stokes flow through a twisted tube

Abstract

Pressure-driven flow through a tube with helical corrugations produced either by twisting a straight tube with arbitrary cross-section, or by embossing helical corrugations on a circular tube, or by inserting a helical fin inside a circular tube, is considered. The Stokes-flow problem is formulated in non-orthogonal curvilinear helical coordinates defined with respect to the helical pitch and azimuthal wavenumber, where the latter is determined by the tube cross-section rotational symmetry. In the first part of the paper, a perturbation analysis is carried out for a circular tube with small-amplitude sinusoidal corrugations, and the solutions of the first- and second-order perturbation problems are found by analytical methods. In the second part, an asymptotic analysis is performed for large-pitched helical corrugations and tubes with arbitrary cross-section, and the solutions of the zeroth-, first- and second-order problems are computed by finite-element methods for unidirectional and two-dimensional Stokes flow over the cross-sectional plane normal to the tube axis. The results illustrate the kinematic structure of the flow and demonstrate the dependence of the flow rate on the tube geometry.