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On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within
In this chapter, we analyze the steady-state microscale fluid--structure
interaction (FSI) between a generalized Newtonian fluid and a hyperelastic
tube. Physiological flows, especially in hemodynamics, serve as primary
examples of such FSI phenomena. The small scale of the physical system renders
the flow field, under the power-law rheological model, amenable to a
closed-form solution using the lubrication approximation. On the other hand,
negligible shear stresses on the walls of a long vessel allow the structure to
be treated as a pressure vessel. The constitutive equation for the microtube is
prescribed via the strain energy functional for an incompressible, isotropic
Mooney--Rivlin material. We employ both the thin- and thick-walled formulations
of the pressure vessel theory, and derive the static relation between the
pressure load and the deformation of the structure. We harness the latter to
determine the flow rate--pressure drop relationship for non-Newtonian flow in
thin- and thick-walled soft hyperelastic microtubes. Through illustrative
examples, we discuss how a hyperelastic tube supports the same pressure load as
a linearly elastic tube with smaller deformation, thus requiring a higher
pressure drop across itself to maintain a fixed flow rate.Comment: 19 pages, 3 figures, Springer book class; v2: minor revisions, final
form of invited contribution to the Springer volume entitled "Dynamical
Processes in Generalized Continua and Structures" (in honour of Academician
D.I. Indeitsev), eds. H. Altenbach, A. Belyaev, V. A. Eremeyev, A. Krivtsov
and A. V. Porubo
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Mechanics of blood flow in capillaries
This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.Blood is a concentrated suspension of red blood cells (RBCs). Motion and deformation of RBCs can be analyzed based on knowledge of their mechanical characteristics. Models for single-file motion of RBCs in capillaries yield predictions of apparent viscosity in good agreement with experimental results for diameters up to about 8 μm. In living microvessels, flow resistance is also strongly influenced by the
presence of a ~ 1-micron layer of macromolecules bound to the inner lining of vessel walls, the endothelial surface layer. Two-dimensional simulations, in which each RBC is represented as a set of interconnected
viscoelastic elements, predict that off-center RBCs take asymmetric shapes and drift toward the center-line. Predicted trajectories agree closely with observations in microvessels of the rat mesentery. Realistic simulation of multiple interacting RBCs in microvessels remains as a major challenge for future work.This work was supported by NIH Grant HL034555
Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement
We present a new model and a novel loosely coupled partitioned numerical
scheme modeling fluid-structure interaction (FSI) in blood flow allowing
non-zero longitudinal displacement. Arterial walls are modeled by a {linearly
viscoelastic, cylindrical Koiter shell model capturing both radial and
longitudinal displacement}. Fluid flow is modeled by the Navier-Stokes
equations for an incompressible, viscous fluid. The two are fully coupled via
kinematic and dynamic coupling conditions. Our numerical scheme is based on a
new modified Lie operator splitting that decouples the fluid and structure
sub-problems in a way that leads to a loosely coupled scheme which is
{unconditionally} stable. This was achieved by a clever use of the kinematic
coupling condition at the fluid and structure sub-problems, leading to an
implicit coupling between the fluid and structure velocities. The proposed
scheme is a modification of the recently introduced "kinematically coupled
scheme" for which the newly proposed modified Lie splitting significantly
increases the accuracy. The performance and accuracy of the scheme were studied
on a couple of instructive examples including a comparison with a monolithic
scheme. It was shown that the accuracy of our scheme was comparable to that of
the monolithic scheme, while our scheme retains all the main advantages of
partitioned schemes, such as modularity, simple implementation, and low
computational costs
Mathematical analysis of blood flow model through channels with flexible walls
A simplified mathematical model of blood flow through flexible arteries is developed and analyzed. The resulting system of non-linear, non-homogeneous PDE\u27s is analyzed numerically using the Richtmyer Lax-Wendroff method. Numerical and theoretical results show excellent agreement suggesting that in physiologically relevant situations shocks only develop outside the domain of interest. These results suggest that when the model assumptions are satisfied the model provides sufficient regularity to yield a physically reasonable representation of flow through a flexible artery. We conclude with a discussion of future directions for this model
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