We develop an algorithm for solving the general grade-two model of non-Newtonian fluids which for the first time includes inflow boundary conditions. The algorithm also allows for both of the rheological parameters to be chosen independently. The proposed algorithm couples a Stokes equation for the velocity with a transport equation for an auxiliary vector-valued function. We prove that this model is well posed using the algorithm that we show converges geometrically in suitable Sobolev spaces for sufficiently small data. We demonstrate computationally that this algorithm can be successfully discretized and that it can converge to solutions for the model parameters of order one. We include in the appendix a description of appropriate boundary conditions for the auxiliary variable in standard geometries.
Keywords: Non-Newtonian flow, grade-two fluid flow, inflow boundary conditions, convergence analysis
@article{M2AN_2022__56_3_1007_0,
author = {Pollock, Sara and Scott, L. Ridgway},
title = {An algorithm for the grade-two rheological model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1007--1025},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {3},
doi = {10.1051/m2an/2022024},
mrnumber = {4411482},
zbl = {1490.76013},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022024/}
}
TY - JOUR AU - Pollock, Sara AU - Scott, L. Ridgway TI - An algorithm for the grade-two rheological model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1007 EP - 1025 VL - 56 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022024/ DO - 10.1051/m2an/2022024 LA - en ID - M2AN_2022__56_3_1007_0 ER -
%0 Journal Article %A Pollock, Sara %A Scott, L. Ridgway %T An algorithm for the grade-two rheological model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1007-1025 %V 56 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022024/ %R 10.1051/m2an/2022024 %G en %F M2AN_2022__56_3_1007_0
Pollock, Sara; Scott, L. Ridgway. An algorithm for the grade-two rheological model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 3, pp. 1007-1025. doi: 10.1051/m2an/2022024
[1] , and , Analysis and finite element simulations of a second-order fluid model in a bounded domain. Numer. Methods Part. Differ. Equ. Int. J. 23 (2007) 1468–1500. | MR | Zbl
[2] , and , An Introduction to Rheology. Vol. 3. Elsevier (1989). | Zbl
[3] , Solutions globales variationnelles et classiques des fluides de grade deux. C. R. Acad. Sci. Ser. I Math. 327 (1998) 953–958. | MR | Zbl
[4] , Stationary problem of second-grade fluids in three dimensions: existence, uniqueness and regularity. Math. Methods Appl. Sci. 22 (1999) 655–687. | MR | Zbl
[5] , Problem of second grade fluids in convex polyhedrons. SIAM J. Math. Anal. 44 (2012) 2018–2038. | MR | Zbl
[6] , Steady transport equation in the case where the normal component of the velocity does not vanish on the boundary. SIAM J. Math. Anal. 44 (2012) 993–1018. | MR | Zbl
[7] , Solutions in of the steady transport equation in a bounded polygon with a full non-homogeneous velocity. J. Math. Pures App. 107 (2017) 697–736. | MR | Zbl
[8] , Fully nonhomogeneous problem of two-dimensional second grade fluids. Math. Methods Appl. Sci. 41 (2018) 6772–6792. | MR | Zbl
[9] and , The Mathematical Theory of Finite Element Methods. 3rd edition. Springer-Verlag (2008). | Zbl | MR
[10] , and , Mechanics and Mathematics of Fluids of the Differential Type. In Vol. 35 of Advances in Mechanics and Mathematics. Springer (2016). | MR
[11] and , Stress-deformation relations for isotropic materials. Arch. Ration. Mech. Anal. 4 (1955) 323–425. | MR | Zbl
[12] , Non-Newtonian effects of multigrade oils on journal bearing performance. Tribol. Trans. 33 (1990) 384–394.
[13] and , Finite Element Methods for Navier-Stokes Equations. Springer Verlag, Berlin (1986). | MR | Zbl
[14] and , Finite element discretizations of a two-dimensional grade-two fluid model. ESAIM: M2AN 35 (2001) 1007–1053. | MR | Zbl | Numdam
[15] and , Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition. J. Math. Pures Appl. 78 (1999) 981–1011. | MR | Zbl
[16] and , Wellposedness of some Oldroyd models that lack explicit dissipation. Research Report UC/CS TR-2017-04, Dept. Comp. Sci., Univ. Chicago (2017).
[17] and , Rheology of aqueous solutions of food additives: effect of concentration, temperature and blending. J. Food Eng. 56 (2003) 387–392.
[18] and , Fluid Mechanics. Pergamon Press (1959). | MR
[19] , Rheology of Industrial Polysaccharides: Theory and Applications. Springer Science & Business Media (2012).
[20] , Low-shear-rate rheometry and polymer quality control. Chem. Eng. Commun. 32 (1985) 1–60.
[21] , and , The hole–pressure problem. IMA J. Appl. Math. 46 (1991) 39–66. | MR | Zbl
[22] and , Towards a unified finite element method for the Stokes equations. SIAM J. Sci. Comput. 40 (2018) A130–A141. | MR | Zbl
[23] , , and , Hyperbolic contraction measuring systems for extensional flow. Mech. Time-Depend. Mater. 21 (2017) 455–479.
[24] and , Transport equations with inflow boundary conditions. Submitted (2022). | MR
[25] , Théorie des Distributions. Hermann, Paris (1966). | MR | Zbl
[26] , C1 piecewise polynomials satisfying boundary conditions. Research Report UC/CS TR-2019-18, Dept. Comp. Sci., Univ. Chicago (2019).
[27] , The non-Newtonian characteristics of lubricating oils. ASLE Trans. 1 (1958) 68–81.
[28] , , , and , Modeling and simulation of mucus flow in human bronchial epithelial cell cultures – Part I: idealized axisymmetric swirling flow. PLoS Comput. Biol. 12 (2016) 1–28.
Cité par Sources :
