### SIMULATION OF 3D TRANSIENT FLOW PASSING THROUGH AN INTESTINAL ANASTOMOSIS BY LATTICE-BOLTZMANN METHOD

#### Abstract

predictable negative consequences associated with disruptions of hydrodynamic processes in the anastomosis area. These negative consequences can be partially avoided by choosing anastomosis anatomical form based on mathematical modeling. Known mathematical models are cumbersome and do not allow to obtain results in real time. The proposed approach using lattice Boltzmann method allows solving this problem.

Objective. The purpose of the work is to develop a three-dimensional mathematical model of anastomosis for research of hydrodynamic

parameters of fluids with complex structure in real time.

Method. The method of constructing and analyzing the mathematical model of anastomosis of the digestive tract based on lattice

Boltzmann method is proposed. The method differs in that it provides simultaneous analysis of hydrodynamic parameters of the liquid and

determines the nature of movement of fine-grained inclusions in the anastomosis area. The main stages of the method are the development of technology for determining the modeling area, discretization of the three-dimensional Boltzmann equation with the choice of lattice and the nature of the collision operator, taking into account the complex structure of the liquid; development of the technology of transition from the density distribution function to the distribution of pressure at the mesoscopic level, taking into account the properties of the liquid, the creation of the process of transforming the set of mesoscopic parameters into the macroscopic parameters of the liquid. Results include determining the distribution of the velocity field in the anastomosis area to modify its geometry. The study of the

influence of gravity on the nature of motion of fine-grained inclusions has been carried out. The quantitative characteristics of the delay

of particles in the area of anastomosis, depending on the dynamic viscosity of the liquid, are determined.

Conclusions. The three-dimensional mathematical model discussed in this paper is based on the application of the lattice Boltzmann

method for calculating the hydrodynamic parameters of the motion of fluid in the study area. The distinctive feature of the model is that

it accounts for the complex nature of the liquid having fine-grained inclusions. The model allows determining the behavior of these

inclusions and the field of speed with sufficient accuracy in real time.

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Nesterenko B. B. Mathematical simulation for parallel

asynchronous methods of boundary value problems of

mathematical physics / B. B. Nesterenko, M. A. Novotarskiy //

IMACS World Congress: 16th international conference, Lausanne, 21–25 August 2000: proceedings. – Ecole Polytechnique Federale de Louzanne, 2000. – P. 116–122. 2. Rosenau Ph. Extending hydrodynamics via the regularization of the Chapman-Enskog expansion / Ph. Rosenau // Physical Review A. – 1989. – Vol. 40, Issue 12. – P. 7193–7196. DOI: 10.1103/PhysRevA.40.7193.

Temam R. Navier-stokes equations: theory and numerical analysis/ R. Temam. – Amsterdam: North-Holland, 1984. – 408 p.

Mitchell A. R. The finite difference method in partial differential equations / A. R. Mitchell, D. F. Griffiths. – New York: John Wiley, 1980. – 272 p.

Girault V. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms / V. Girault, P-A. Rivart. – Berlin: Springer-Verlag, 1980. – 375 p. DOI: 10.1007/978-3-642-61623 5.

Versteeg H. K. An Introduction to Computational Fluid Dynamics: The Finite Volume Method / H. K. Versteeg, M. Malalasekera. – London: Paerson education Ltd., 1995. – 511 p.

Vilsmeier R. Finite volumes for complex applications II: problems and perspectives / R. Vilsmeier, F. Benkhaldoun, D. H nel. – Middlesex: Hermes Science Publications, 1999. – 887 p.

Chen S. Lattice Boltzmann method for fluid flows / S. Chen,

G. Doolen // Annual Review of Fluid Mechanics. – 1998. –

Vol. 30. – P. 329–364. DOI: 10.1146/annurev.fluid.30.1.329.

Jahanshaloo L. An overview of boundary implementation in

lattice Boltzmann method for computational heat and mass

transfer / L. Jahanshaloo, N. A. C. Sidik, A. Fazeli // International Communications in Heat and Mass Transfer. – 2016. – Vol. 78, Issue 11. – P. 1–12. DOI: 10.1016/

j.icheatmasstransfer.2016.08.014.

Tan J. A parallel fluid solid coupling model using LAMMPS and Palabos based on the immersed boundary method / J. Tan, T. Sinno, S. Diamond, [Electronic resource]. – https://arxiv.org/abs/1704.04551.

#### GOST Style Citations

asynchronous methods of boundary value problems of

mathematical physics / B. B. Nesterenko, M. A. Novotarskiy //

IMACS World Congress: 16th international conference, Lausanne, 21–25 August 2000: proceedings. – Ecole Polytechnique Federale de Louzanne, 2000. – P. 116–122. 2. Rosenau Ph. Extending hydrodynamics via the regularization of the Chapman-Enskog expansion / Ph. Rosenau // Physical Review A. – 1989. – Vol. 40, Issue 12. – P. 7193–7196. DOI: 10.1103/PhysRevA.40.7193.

3. Temam R. Navier-stokes equations: theory and numerical analysis/ R. Temam. – Amsterdam: North-Holland, 1984. – 408 p.

4. Mitchell A. R. The finite difference method in partial differential equations / A. R. Mitchell, D. F. Griffiths. – New York: John Wiley, 1980. – 272 p.

5. Girault V. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms / V. Girault, P-A. Rivart. – Berlin: Springer-Verlag, 1980. – 375 p. DOI: 10.1007/978-3-642-61623 5.

6. Versteeg H. K. An Introduction to Computational Fluid Dynamics: The Finite Volume Method / H. K. Versteeg, M. Malalasekera. – London: Paerson education Ltd., 1995. – 511 p.

7. Vilsmeier R. Finite volumes for complex applications II: problems and perspectives / R. Vilsmeier, F. Benkhaldoun, D. H nel. – Middlesex: Hermes Science Publications, 1999. – 887 p.

8. Chen S. Lattice Boltzmann method for fluid flows / S. Chen,

G. Doolen // Annual Review of Fluid Mechanics. – 1998. –

Vol. 30. – P. 329–364. DOI: 10.1146/annurev.fluid.30.1.329.

9. Jahanshaloo L. An overview of boundary implementation in

lattice Boltzmann method for computational heat and mass

transfer / L. Jahanshaloo, N. A. C. Sidik, A. Fazeli // International Communications in Heat and Mass Transfer. – 2016. – Vol. 78, Issue 11. – P. 1–12. DOI: 10.1016/

j.icheatmasstransfer.2016.08.014.

10. Tan J. A parallel fluid solid coupling model using LAMMPS and Palabos based on the immersed boundary method / J. Tan, T. Sinno, S. Diamond, [Electronic resource]. – https://arxiv.org/abs/1704.04551.

DOI: https://doi.org/10.15588/1607-3274-2018-1-9

Copyright (c) 2018 M. A. Novotarskyi, S. G. Stirenko, Y. G. Gordienko

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