Compatible pseudospectral approximations for incompressible flow in an undulating tube

The flow of an Oldroyd‐B fluid through an undulating tube is considered. The effect of elasticity and inertia on flow resistance is investigated numerically using a time‐splitting technique. This technique can be used to solve both steady and transient viscoelastic flows. A pseudospectral method based on mixed Fourier–Chebyshev expansions is used to represent the flow variables in space. An approximation space for pressure is constructed which is compatible with that for the velocity. This is achieved by removing the spurious modes using a singular value decomposition. A projection method ensures that mass is conserved identically at the collocation points. Numerical results are presented in such a way as to highlight the inertial and elastic effects. To this end two sets of results are given: the first, inertialess viscoelastic flow; the second, flow at nonzero Reynolds number holding the Weissenberg number constant. The two cases are shown to have quite opposite effects upon the flow rate and resistance.