A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem
Applied and Computational Mathematics
Volume 7, Issue 1, February 2018, Pages: 11-18
Received: Dec. 4, 2017;
Accepted: Dec. 15, 2017;
Published: Jan. 12, 2018
Views 1374 Downloads 88
Yanhua Cao, School of Sciences, East China Jiaotong University, Nanchang, China
Zhendong Luo, School of Mathematics and Physics, North China Electric Power University, Beijing, China
In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.
A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem, Applied and Computational Mathematics.
Vol. 7, No. 1,
2018, pp. 11-18.
A. J. Zhao and D. P. Du, “Local well-posedness of lower regularity solutions for the incompressible viscoelastic fluid system,” Science China Math., 2010, 53(6), 1520–1530.
R. B. Bird, R. C. Armstrong and O. Hassager, “Dynamics of Polymeric Liquids,” Volume 1, Fluid Mechanics, Weiley Intersience, New York, 1987.
P. G. de Gennes and J. Prost, “The Physics of Liquid Crystals,” Oxford University Press, New York, 1993.
R. G. Larson, “The Structure and Rheology of complex fluids,” Oxford University Press, New York, 1995.
T. C. Sideris and B. Thomases, “Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit,” Comm. Pure Appl. Math., 2005, 58(6), 750–788.
Z. Lei, C. Liu and Y. Zhou, “Golbal solutions for the incompressible viscoelastic fluids,” Arch. Rational Mech. Anal., 2008, 188(3), 371–398.
I. E. Barton and R. Kirby, “Finite difference scheme for the solution of fluid flow problems on non-staggered grids,” International Journal for Numerical Methods in Fluids, 2000, 33(7), 939–959.
M. M. Khader and A. M. Megahed, “Numerical simulation using the finite difference method for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation,” Journal of King Saud University-Engineering Sciences, 2013, 25, 29–34.
S. K. Pandit, J. C. Kalita and D. C. Dalal, “A transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular,” Journal of Computational Physics, 2007, 225(1), 1100–1124.
I. Soukup, “Numerical method based on DGM for solving the system of equations describing motion of viscoelastic fluid with memory,” In: Numerical Mathematics and Advanced Applications ENUMATH 2015, B. Karasözen et al. (eds.). Lecture Notes in Computational Science and Engineering, Springer International Publishing Switzerland, 2016, 112, 205–213.
P. A. B. de Sampaio, “A stabilized finite element method for incompressible flow and heat transfer: A natural derivation based on the use of local time-steps,” Computer Methods in Applied Mechanics and Engineering, 2006, 195(44-47), 6177–6190.
P. A. B. de Sampaio, “A finite element formulation for transient incompressible viscous flows stabilized by local time-steps,” Computer Methods in Applied Mechanics and Engineering, 2005, 194(18-20), 2095–2108.
C. O. Faria, and J. Karam-Filho, “A regularized-stabilized mixed finite element formulation for viscoplasticity of Bingham type,” Computers & Mathematics with Applications, 2013, 66(6), 975-995.
H. Xia and Z. D. Luo, “Stabilized finite volume element method for the 2D nonlinear incompressible viscoelastic flow equation,” Boundary Value Problems, 2017, 2017(130), 1–17.
T. Chung, “Computational Fluid Dynamics”, Cambridge University Press, Cambridge, 2002.
R. X. Liu, and C. W. Shu, “Several New Methods for Computational Fluid Mechanics (in Chinese),” Science Press, Beijing, 2003.