Higher order asymptotic and numerical analyses of low Reynolds number flow in slowly varying tubes

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Abstract
The flow of a Newtonian fluid through long, slender, axisymmetric tubes with spatially varying diameters is investigated in this study. Employing a higher order asymptotic analysis, the velocity and pressure fields are systematically derived for steady, incompressible flow through an arbitrary shaped tube, with the analysis restricted to cases in which the physical Reynolds number is less than unity. Numerical simulations are performed to validate the asymptotic solutions and ensure their accuracy. Based on the asymptotic analysis, tube geometries are classified into two categories: unconditioned tubes, where the wall surface is not sufficiently flat at the ends, and conditioned tubes, where the wall surface is sufficiently flat at both inlet and outlet. Deviations between asymptotic and numerical results arise at the tube inlet for unconditioned geometries due to mismatched boundary conditions stemming from the absence of conditioning. These deviations are shown to vanish when a non-parabolic velocity profile is imposed as the inlet condition in numerical simulations. Additionally, it is shown that asymptotic solutions remain valid over broader parameter ranges for flows within unconditioned tubes, while their applicability is more restricted for flows within conditioned tubes. These results are extended to the case of fluid transport through periodically constricted wavy-walled tubes, commonly encountered in biological systems. Finally, numerical results for a specific tube geometry are compared with experimental data on incipient laminar flow separation, offering further insight into flow behavior under adverse pressure gradients in constricted geometries.

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