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Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries. (English) Zbl 1081.35073
The authors admit in their introduction various inaccuracies of their model. Blood is certainly not a Newtonian fluid, and walls of the arteries are not linear elastic membranes, and thus do not obey Navier’s elastic membrane equations, moreover wall deformations of arteries can only be described as “large”. Yet the authors still make a case for a greatly simplified study. They state the problem as that of an unsteady incompressible Newtonian fluid flow in a thin and very long elastic cylinder, with zero angular velocity along the entire length. The number $\varepsilon$ defining the ratio of radius to length, which is known to be of order 0.02--0.06 is regarded as “small”. The lateral wall is assumed to be elastic, allowing only radial deformations. The assumption of well-posedness of the three-dimensional problem, combined with well-posedness of the one-dimensional problem allows the authors to assume that their two-dimensional problem is well-posed. Clearly the domain is time-dependent, and motion of the arterial boundary which is described in Lagrangian coordinates has to be coordinated with motion of the fluid described in Eulerian coordinates. The authors do not follow the previously suggested weak formulation of such problems coupling Eulerian and Lagrangian coordinates by using directly the fixed point approach. Instead they use linearalization which does not change the values of energy, and then use fixed point arguments. The crucial assumption made by the authors includes assignment of memory in the style of {\it M. A. Biot}’s articles published in [J. Acoust. Soc. Am. 28, 168--178, 179--191 (1956)], which discussed flow through a porous medium. The existence of memory turned out to be helpful in explaining some slow-fast phenomena and the existence of some hardly explainable oscillations. Just this very brief insight into this problem illustrates the enormous difficulties faced by the authors who produced after all an algorithm, which seems to capture important aspects of two-dimensional flow phenomena including some unexpected backward flow in the boundary layer. The authors should be congratulated for pursuing such difficult topic, and making some progress at least in comprehending the difficulties of coming up with some mathematical model which is not one-dimensional. Trivial notational changes such as $v$ to denote the Poisson ratio in the discussion of shear (which is common in applied mechanics) instead of $\sigma$, would make some formulas easier to read.

35Q35PDEs in connection with fluid mechanics
76Z05Physiological flows
35Q80Applications of PDE in areas other than physics (MSC2000)
74F10Fluid-solid interactions
76D27Other free-boundary flows; Hele-Shaw flows
76M45Asymptotic methods, singular perturbations (fluid mechanics)
76M50Homogenization (fluid mechanics)
92C35Physiological flows
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