American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 181-197. doi: 10.3934/dcds.2010.28.181

A mathematical and numerical study of incompressible flows with a surfactant monolayer

Yuen-Yick Kwan 1, , Jinhae Park 2, and  Jie Shen 3,

1. 

Center for Computational Science, Tulane University, New Orleans, LA 70118, United States
2. 

Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067
3. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907

Received  March 2010 Revised  March 2010 Published  April 2010

We consider in this paper a mathematical model for the incompressible flows with a surfactant monolayer. The presence of surfactant gives rise to coupling surface terms which make the analysis and simulation challenging. We study the well-posedness of this coupled system of PDEs with physically relevant boundary conditions, as well as the stability of a numerical scheme. We also preform numerical simulations by a fast-spectral method and use it to study the effect of surfactant concentration on the motion of an incompressible fluid in a cylinder.
Keywords: Spectral-Galerkin method, Surfactant, Pressure-correction, Weak solution., Navier-Stokes, Instabilities, Existence.
Mathematics Subject Classification: Primary: 76B03, 65M70; Secondary: 76R99, 35K5.
Citation: Yuen-Yick Kwan, Jinhae Park, Jie Shen. A mathematical and numerical study of incompressible flows with a surfactant monolayer. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 181-197. doi: 10.3934/dcds.2010.28.181
[1]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[2]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[3]

Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369
[4]

Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[5]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[6]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[7]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[8]

Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053
[9]

Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315
[10]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[11]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171
[12]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[13]

Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467
[14]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[15]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087
[16]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471
[17]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191
[18]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495
[19]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[20]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences