We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiĭ operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.
Accepté le :
DOI : 10.1051/m2an/2017043
Mots clés : Non-Newtonian fluid, variable power-law index, synovial fluid, finite element method
@article{M2AN_2018__52_2_509_0,
author = {Ko, Seungchan and Pust\v{e}jovsk\'a, Petra and S\"uli, Endre},
title = {Finite element approximation of an incompressible chemically reacting {non-Newtonian} fluid},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {509--541},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2017043},
zbl = {1404.65270},
mrnumber = {3834434},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017043/}
}
TY - JOUR AU - Ko, Seungchan AU - Pustějovská, Petra AU - Süli, Endre TI - Finite element approximation of an incompressible chemically reacting non-Newtonian fluid JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 509 EP - 541 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017043/ DO - 10.1051/m2an/2017043 LA - en ID - M2AN_2018__52_2_509_0 ER -
%0 Journal Article %A Ko, Seungchan %A Pustějovská, Petra %A Süli, Endre %T Finite element approximation of an incompressible chemically reacting non-Newtonian fluid %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 509-541 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017043/ %R 10.1051/m2an/2017043 %G en %F M2AN_2018__52_2_509_0
Ko, Seungchan; Pustějovská, Petra; Süli, Endre. Finite element approximation of an incompressible chemically reacting non-Newtonian fluid. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 509-541. doi : 10.1051/m2an/2017043. http://archive.numdam.org/articles/10.1051/m2an/2017043/
[1] and , An approximation lemma for W1,p functions. In Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 1–5 | MR | Zbl
[2] and , Regularity results for discrete solutions of second order elliptic problems in the finite element method. Calcolo 23 (1986) 327–353 (1987) | DOI | MR | Zbl
[3] , , and , On the finite element approximation of p-Stokes systems. SIAM J. Numer. Anal. 50 (2012) 373–397 | DOI | MR | Zbl
[4] , and , Convergence analysis for a finite element approximation of a steady model for electrorheological fluids. Numer. Math. 132 (2016) 657–689 | DOI | MR | Zbl
[5] and , The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, 3rd edition (2008) | DOI | MR | Zbl
[6] and , Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics. Springer Verlag, New York (1991) | MR | Zbl
[7] , and , Mathematical results concerning unsteady flows of chemically reacting incompressible fluids. In Partial differential equations and fluid mechanics, volume 364 of London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Cambridge (2009) 26–53 | MR | Zbl
[8] and , On existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index. J. Math. Anal. Appl. 402 (2013) 157–166 | DOI | MR | Zbl
[9] and , Existence analysis for a model describing flow of an incompressible chemically reacting non-Newtonian fluid. SIAM J. Math. Anal. 46 (2014) 3223–3240 | DOI | MR | Zbl
[10] , , , and , Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L1. Numer. Math. 105 (2007) 337–374 | DOI | MR | Zbl
[11] , and , The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003) 223–238 | MR | Zbl
[12] , Neumann and mixed problems on curvilinear polyhedra. Integral Equ. Oper. Theory 15 (1992) 227–261 | DOI | MR | Zbl
[13] , , and , Lebesgue and Sobolev spaces with variable exponents, volume 2017 of Lect. Notes Math. Springer, Heidelberg (2011) | DOI | MR | Zbl
[14] , and , Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51 (2013) 984–1015 | DOI | MR | Zbl
[15] , and , On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optimiz. Calc. Var. 14 (2008) 211–232 | DOI | Numdam | MR | Zbl
[16] and , On the key estimate for variable exponent spaces. Azerb. J. Math. 3 (2013) 62–69 | MR | Zbl
[17] , and , Convergence of the finite element method for the porous media equation with variable exponent. SIAM J. Numer. Anal. 51 (2013) 3483–3504 | DOI | MR | Zbl
[18] , Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. In: Numerical Solution of Partial Differential Equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md. 1975). Edited by (1976) 207–274 | DOI | MR | Zbl
[19] , , and , On the modeling of the synovial fluid. Advances in Tribology (2010)
[20] , and , Hardy spaces with variable exponent. In Harmonic analysis and nonlinear partial differential equations. RIMS Kôkyûroku Bessatsu. Res. Inst. Math. Sci. (RIMS). Kyoto B42 (2013) 109–136 | MR | Zbl
[21] , and , Finite element approximation of an incompressible chemically reacting non-Newtonian fluid. [math.NA] (2017) | arXiv | MR
[22] and , Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index. [math.NA] (2017) | arXiv | MR
[23] , and , Rheological equations for synovial fluids. J. Biomech. Eng. 100 (1978) 169–186 | DOI
[24] , Extension of range of functions. Trudy Mat. Inst. Steklov. 40 (1934) 837–842 | JFM | MR | Zbl
[25] , Biochemical and mechanical processes in synovial fluid – modeling, analysis and computational simulations. Ph.D. Thesis. Charles University in Prague and Heidelberg University (2012). Available electronically from the URL: https://www-m2.ma.tum.de/foswiki/pub/M2/Allgemeines/PetraPustejovska/PhD˙pustejovska.pdf. | Zbl
[26] , Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 49 (2004) 565–609 | DOI | MR | Zbl
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