Volume 15, no. 4Pages 90 - 98

Numerical Method for Solving the Inverse Problem of Non-Stationary Flow of Viscoelastic Fluid in the Pipe

A.R. Aliev, Kh.M. Gamzaev, A.A. Darwish, T.A. Nofal
The process of unsteady flow of incompressible viscoelastic fluid in a cylindrical tube of constant cross-section is considered. To describe the rheological properties of a viscoelastic fluid, the Kelvin-Voigt model is used and the mathematical model of this process is presented as an integro-differential partial differential equation. Within the framework of this model, the problem is to determine the pressure drop along the length of the pipe, which ensures the passage of a given flow rate of viscoelastic fluid through the pipe. This problem belongs to the class of inverse problems related to the recovery of the right parts of integro-differential equations. By replacing variables, the integro-differential equation is transformed into a third-order partial differential equation. First, a discrete analog of the problem is constructed using finite-difference approximations. To solve the resulting difference problem, we propose a special representation that allows splitting the problems into two mutually independent second-order difference problems. As a result, an explicit formula is obtained for determining the approximate value of the pressure drop along the length of the pipeline for each discrete value of the time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
Full text
viscoelastic fluid; Kelvin - Voigt model; integro-differential equation; pressure drop along the length of the pipe; inverse problem.
1. Wilkinson W.L. Non-Newtonian Fluids: Fluid Mechanics. Mixing and Heat Transfer. New York, Pergamon Press, 1960.
2. Astarita G., Marrucci G. Principles of Non-Newtonian Fluid Mechanics. London, New York, McGraw-Hill, 1974.
3. Joseph D.D. Fluid Dynamics of Viscoelastic Liquids. New York, Springer, 1990.
4. Huilgol R. R., Phan-Thien N. Fluid Mechanics of Viscoelasticity: General Principles, Constitutive Modelling, Analytical and Numerical Techniques. Amsterdam, Elsevier, 1997.
5. Crochet M.J., Davies A.R., Walters K. Numerical Simulation of Non-Newtonian Flow. Amsterdam, Elsevier, 2012.
6. Aristov S.N., Skul'skii O.I. Exact Solution of the Problem of Flow of a Polymer Solution in a Plane Channel. Journal of Engineering Physics and Thermophysics, 2003, vol. 76, no. 3, pp. 577-585. DOI: 10.1023/A:1024768930375
7. Carrozza M.A., Hulsen M.A., H"utter M., Anderson P. D. Viscoelastic Fluid Flow Simulation Using the Contravariant Deformation Formulation. Journal of Non-Newtonian Fluid Mechanics, 2019, vol. 270, pp. 23-35. DOI: 10.1016/j.jnnfm.2019.07.001
8. Gamzaev Kh.M. Numerical Method of Pipeline Hydraulics Identification at Turbulent Flow of Viscous Liquids. Pipeline Science and Technology, 2019, vol. 3, no. 2, pp. 118-124. DOI: 10.28999/2514-541X-2019-3-2-118-124
9. Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin, De Gruyter, 2007.
10. Vabishchevich P.N., Vasil'ev V.I., Vasil'eva M.V. Computational Identification of the Right-Hand Side of a Parabolic Equation. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 6, pp. 1015-1021. DOI: 10.1134/S0965542515030185
11. Deng Z.C., Qian K., Rao X.B., Yang L., Luo G.W. An Inverse Problem of Identifying the Source Coefficient in a Degenerate Heat Equation. Inverse Problems in Science and Engineering, 2015, vol. 23, no. 3, pp. 498-517. DOI: 10.1080/17415977.2014.922079
12. Borukhov V.T., Zayats G.M. Identification of a Time-Dependent Source Term in Nonlinear Hyperbolic or Parabolic Heat Equation. International Journal of Heat and Mass Transfer, 2015, vol. 91, pp. 1106-1113. DOI: 10.1016/j.ijheatmasstransfer.2015.07.06
13. Ashyralyev A., Erdogan A.S., Demirdag O. On the Determination of the Right-Hand Side in a Parabolic Equation. Applied Numerical Mathematics, 2012, vol. 62, no. 11, pp. 1672-1683. DOI: 10.1016/j.apnum.2012.05.008
14. Gamzaev Kh. I. The Problem of Identifying the Trajectory of a Mobile Point Source in the Convective Transport Equation. Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 2, pp. 78-84. DOI: 10.14529/mmp210208