We investigate theoretically the steady incompressible viscoelastic flow in a rigid axisymmetric cylindrical pipe with varying cross-section. We use the Oldroyd-B viscoelastic constitutive equation to model the fluid viscoelasticity. First, we derive exact general formulae: for the total average pressure-drop as a function of the wall shear rate and the viscoelastic axial normal extra-stress; for the viscoelastic extra-stress tensor and the Trouton ratio as functions of the fluid velocity on the axis of symmetry; and for the viscoelastic extra-stress tensor along the wall in terms of the shear rate at the wall. Then we exploit the classic lubrication approximation, valid for small values of the square of the aspect ratio of the pipe, to simplify the original governing equations. The final equations are solved analytically using a regular perturbation scheme in terms of the Deborah number, De, up to eighth order in De. For a hyperbolically shaped pipe, we reveal that the reduced pressure-drop and the Trouton ratio can be recast in terms of a modified Deborah number, Dem, and the polymer viscosity ratio, η, only. Furthermore, we enhance the convergence and accuracy of the eighth-order solutions by deriving transformed analytical formulae using Padé diagonal approximants. The results show the decrease of the pressure drop and the enhancement of the Trouton ratio with increasing Dem and/or increasing η. Comparison of the transformed solutions with numerical simulations of the lubrication equations using pseudospectral methods shows excellent agreement between the results, even for high values of Dem and all values of η, revealing the robustness, validity and efficiency of the theoretical methods and techniques developed in this work. Last, it is shown that the exact solution for the Trouton ratio gives a well-defined and finite solution for any value of Dem and reveals the reason for the failure of the corresponding high-order perturbation series for Dem > 1/2.

We investigate aspects of the spherical squirmer model employing both large-scale numerical simulations and asymptotic methods when the squirmer is placed in weakly elastic fluids. The fluids are modelled by differential equations, including the upper-convected Maxwell (UCM)/Oldroyd-B, finite-extensibility nonlinear elastic model with Peterlin approximation (FENE-P) and Giesekus models. The squirmer model we examine is characterized by two dimensionless parameters related to the fluid velocity at the surface of the micro-swimmer: the slip parameter $\xi $ and the swirl parameter $\zeta $. We show that, for swimming in UCM/Oldroyd-B fluids, the elastic stress becomes singular at a critical Weissenberg number, Wi, that depends only on $\xi$. This singularity for the UCM/Oldroyd-B models is independent of the domain exterior to the swimmer, or any other forces considered in the momentum balance for the fluid – we believe that this is the first time such a singularity has been explicitly demonstrated. Moreover, we show that the behaviour of the solution at the poles is purely extensional in character and is the primary reason for the singularity in the Oldroyd-B model. When the Giesekus or the FENE-P models are utilized, the singularity is removed. We also investigate the mechanism behind the speed and rotation rate enhancement associated with the addition of swirl in the swimmer's gait. We demonstrate that, for all models, the speed is enhanced by swirl, but the mechanism of enhancement depends intrinsically on the rheological model employed. Special attention is paid to the propulsive role of the pressure and elucidated upon. We also study the region of convergence of the perturbation solutions in terms of Wi. When techniques that accelerate the convergence of series are applied, transformed solutions are derived that are in very good agreement with the results obtained by simulations. Finally, both the analytical and numerical results clearly indicate that the low-Wi region is more important than one would expect and demonstrate all the major phenomena observed when swimming with swirl in a viscoelastic fluid.

Microorganisms are commonly found swimming in complex biological fluids such as mucus and these fluids respond elastically to deformation. These viscoelastic fluids have been previously shown to affect the swimming kinematics of these microorganisms in non-trivial ways depending on the rheology of the fluid, the particular swimming gait and the structural properties of the immersed body. In this report we put forth a previously unmentioned mechanism by which swimming organisms can experience a speed increase in a viscoelastic fluid. Using numerical simulations and asymptotic theory we find that significant swirling flow around a microscopic swimmer couples with the elasticity of the fluid to generate a marked increase in the swimming speed. We show that the speed enhancement is related to the introduction of mixed flow behind the swimmer and the presence of hoop stresses along its body. Furthermore, this effect persists when varying the fluid rheology and when considering different swimming gaits. This, combined with the generality of the phenomenon (i.e. the coupling of vortical flow with fluid elasticity near a microscopic swimmer), leads us to believe that this method of speed enhancement could be present for a wide range of microorganisms moving through complex fluids.