Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory

Abstract

A flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross-sectional area. The latter gives rise to a non-constant pressure gradient in the flow-wise direction and, hence, to a nonlinear flow rate--pressure drop relation (unlike the Hagen--Poiseuille law for a rigid tube). Many biofluids are non-Newtonian, and are well approximated by generalized Newtonian (say, power-law) rheological models. Consequently, we analyze the problem of steady low Reynolds number flow of a generalized Newtonian fluid through a slender elastic tube by coupling fluid lubrication theory to a structural problem posed in terms of Donnell shell theory. A perturbative approach (in the slenderness parameter) yields analytical solutions for both the flow and the deformation. Using matched asymptotics, we obtain a uniformly valid solution for the tube's radial displacement, which features both a boundary layer and a corner layer caused by localized bending near the clamped ends. In doing so, we obtain a ``generalized Hagen--Poiseuille law'' for soft microtubes. We benchmark the mathematical predictions against three-dimensional two-way coupled direct numerical simulations (DNS) of flow and deformation performed using the commercial computational engineering platform by ANSYS. The simulations show good agreement and establish the range of validity of the theory. Finally, we discuss the implications of the theory on the problem of the flow-induced deformation of a blood vessel, which is featured in some textbooks.

27 pages, 8 figures; v2 includes major revisions to the derivation and discussion; v3 includes further major revisions to the derivation and discussion; v4 includes minor improvements (note title change), version accepted for publication in Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)

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• Non-Newtonian fluid-structure interaction: Flow of a viscoelastic Oldroyd-B fluid in a deformable channel
• Oscillatory flows in compliant conduits at arbitrary Womersley number
• Flow rate-pressure drop relations for shear-thinning fluids in deformable configurations: theory and experiments
• Flow rate--pressure drop relations for new configurations of slender compliant tubes arising in microfluidics experiments
• An electrokinetic route to giant augmentation in load bearing capacity of compliant microfluidic channels
• Soft Hydraulics in Channels with Thick Walls: The Finite-Reynolds-Number Base State and Its Stability
• Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels
• Electroosmotic flow of viscoelastic fluids in deformable microchannels