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 performedmore »
Analysis of a 3D nonlinear moving boundary problem describing fluidmeshsell interaction
Abstract. We consider a nonlinear, moving boundary, fluidstructure interaction problem between a time dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the timedependent Navier Stokes equations in a threedimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the twodimensional linearly elastic Koiter shell equations coupled to a onedimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh supported shell allows displacements in all three spatial directions. Twoway coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many ap plications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via Lie operator splitting method combined with an Arbitrary LagrangianEulerian approach, and a nontrivial extension of the AubinLionsSimon compactness result to problems on moving domains.
 Award ID(s):
 1853340
 Publication Date:
 NSFPAR ID:
 10148592
 Journal Name:
 Transactions of the American Mathematical Society
 ISSN:
 23300000
 Sponsoring Org:
 National Science Foundation
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