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A fractal graph model of capillary type systems
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
St Petersburg State Univ, Russia; St Petersburg State Polytech Univ, Russia; RAS, Russia.
Steklov Math Inst, Russia.
2018 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 7-8, p. 1044-1068Article in journal (Refereed) Published
##### Abstract [en]

We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.

##### Place, publisher, year, edition, pages
TAYLOR & FRANCIS LTD , 2018. Vol. 63, no 7-8, p. 1044-1068
##### Keywords [en]
Fractal graph; blood vessel; capillary system; percolation; quiet flow; ideal liquid; Reynolds equation
##### National Category
Mathematical Analysis
##### Identifiers
ISI: 000439381000010OAI: oai:DiVA.org:liu-150318DiVA, id: diva2:1239449
##### Note

Funding Agencies|Swedish Research Council (VR) [EO418401]; Russian Foundation for Basic Research [15-01-02175]; Linkoping University (Sweden); Linkoping University; RFBR [15-31-20600, 16-31-60112]

Available from: 2018-08-16 Created: 2018-08-16 Last updated: 2018-08-16

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Mathematics and Applied MathematicsFaculty of Science & Engineering
##### In the same journal
Complex Variables and Elliptic Equations
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Mathematical Analysis

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Cite
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