Geodesic
Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 1, pp. 24-30 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The process of unsteady flow of viscous incompressible fluid in a pipe with a permeable wall described by a nonlinear system of partial differential equations in the velocity-pressure variables is considered. This system of equations is reduced to a single nonlinear equation of parabolic type with respect to velocity. Within the framework of the obtained model, the inverse problem is posed to determine the permeability coefficient of the pipe wall, which depends only on the time variable. In this case, an additional condition relative to the fluid pressure is set at the outlet of the pipe. A difference analogue of the coefficient inverse problem is built using finite-difference approximations. For the solution of the received difference problem, the special representation is offered allowing to split problems into two mutually independent linear difference problems of the second order on each discrete value of a time variable. As a result, an explicit formula is obtained to determine the approximate value of the wall permeability coefficient for each discrete value of the time variable. On the basis of the proposed computational algorithm, numerical experiments were carried out for model problems.
Keywords: unsteady fluid flow, pipe with permeable wall, coefficient of permeability of the pipe wall, difference problem.
Mots-clés : coefficient inverse problem 
@article{VYURM_2020_12_1_a2,
     author = {Kh. M. Gamzaev},
     title = {Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {24--30},
     year = {2020},
     volume = {12},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a2/}
}
TY  - JOUR
AU  - Kh. M. Gamzaev
TI  - Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2020
SP  - 24
EP  - 30
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a2/
LA  - ru
ID  - VYURM_2020_12_1_a2
ER  - 
%0 Journal Article
%A Kh. M. Gamzaev
%T Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2020
%P 24-30
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a2/
%G ru
%F VYURM_2020_12_1_a2
Kh. M. Gamzaev. Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 1, pp. 24-30. http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a2/

[1] E.A. Marshall, E.A. Trowbridge, “Flow of a Newtonian Fluid through a Permeable Tube: The application to the Proximal Renal Tubule”, Bulletin of Mathematical Biology, 36:5–6 (1974), 457–476 | DOI

[2] S.M. Ross, “A mathematical model of mass transport in a long permeable tube with radial convection”, Journal of Fluid Mechanics, 63:1 (1974), 157–175 | DOI

[3] C. Pozrikidis, “Stokes flow through a permeable tube”, Archive of Applied Mechanics, 80:4 (2010), 323–333 | DOI

[4] Q. Zhang, Z. Wang, “Modeling Study on Fluid Flow in Horizontal Perforated Pipes with Wall Influx”, International Journal of Fluid Mechanics Research, 41:6 (2014), 556–566 | DOI

[5] M. Elshahed, “Blood flow in capillary under starling hypothesis”, Applied Mathematics and Computation, 149:2 (2004), 431–439 | DOI

[6] P. Muthu, T. Berhane, “Mathematical model of flow in renal tubules”, International Journal of Applied Mathematics and Mechanics, 6:20 (2010), 94–107

[7] N.K. Mariamma, S.N. Majhi, “Flow of a Newtonian Fluid in a Blood Vessel with Permeable Wall — A theoretical model”, Computers Mathematics with Applications, 40:12 (2000), 1419–1432 | DOI

[8] Kabanikhin S.I., Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter, Germany, 2011, 459 pp.

[9] Samarskiy A.A., Vabishchevich P.N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, URSS, LKI Publ., M., 2014, 478 pp. (in Russ.)

[10] P.N. Vabishchevich, V.I. Vasil'ev, “Computational algorithms for solving the coefficient inverse problem for parabolic equations”, Inverse Problems in Science and Engineering, 24:1 (2016), 42–59 | DOI

[11] Kh.M. Gamzaev, “Numerical Solution of Combined Inverse Problem for Generalized Burgers Equation”, Journal of Mathematical Sciences, 221:6 (2017), 833–839 | DOI