 Previous Article
 DCDSB Home
 This Issue

Next Article
Sheared nematic liquid crystal polymer monolayers
Evaluation of interfacial fluid dynamical stresses using the immersed boundary method
1.  Perforating Research, Schlumberger, 14910 Airline Road, Rosharon, TX 77583, United States 
2.  Department of Mathematics, Tulane University, New Orleans, LA 70118, United States 
3.  Department of Biomedical Engineering, Tulane University, New Orleans, LA 70118, United States 
To identify suitable methods for evaluating stresses, we investigate three model flow problems at very low Reynolds numbers. We compare the results of the immersed boundary calculations to those achieved by the boundary element method (BEM). The stress on an immersed boundary may be calculated either by direct evaluation of the fluid stress (FS) tensor or, for the stress jump, by direct evaluation of the locally distributed boundary force (wall stress or WS). Our first model problem is Poiseuille channel flow. Using an analytical solution of the immersed boundary formulation in this simple case, we demonstrate that FS calculations should be evaluated at a distance of approximately one grid spacing inward from the immersed boundary. For a curved immersed boundary we present a procedure for selecting representative interfacial fluid stresses using the concepts from the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump in a channel and unsteady peristaltic pumping, we present an 'exclusion filtering' technique for accurately measuring stresses. Using this technique, these studies show that the immersed boundary method can provide reliable approximations to interfacial stresses.
[1] 
Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems  B, 2011, 15 (2) : 373389. doi: 10.3934/dcdsb.2011.15.373 
[2] 
HongMing Yin. A free boundary problem arising from a stressdriven diffusion in polymers. Discrete & Continuous Dynamical Systems  A, 1996, 2 (2) : 191202. doi: 10.3934/dcds.1996.2.191 
[3] 
Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluidsstructure interaction in sperm motility. Discrete & Continuous Dynamical Systems  B, 2011, 15 (2) : 343355. doi: 10.3934/dcdsb.2011.15.343 
[4] 
Daniele Boffi, Lucia Gastaldi. Discrete models for fluidstructure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems  S, 2016, 9 (1) : 89107. doi: 10.3934/dcdss.2016.9.89 
[5] 
Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractionalorder nonNewtonian fluids with shear stress on the boundary. Discrete & Continuous Dynamical Systems  S, 2020, 13 (3) : 683693. doi: 10.3934/dcdss.2020037 
[6] 
Assane Lo, Nassereddine Tatar. Exponential stabilization of a structure with interfacial slip. Discrete & Continuous Dynamical Systems  A, 2016, 36 (11) : 62856306. doi: 10.3934/dcds.2016073 
[7] 
Rebecca Vandiver. Effect of residual stress on peak cap stress in arteries. Mathematical Biosciences & Engineering, 2014, 11 (5) : 11991214. doi: 10.3934/mbe.2014.11.1199 
[8] 
Donato Patrizia, Andrey Piatnitski. On the effective interfacial resistance through rough surfaces. Communications on Pure & Applied Analysis, 2010, 9 (5) : 12951310. doi: 10.3934/cpaa.2010.9.1295 
[9] 
Donatella Donatelli, Corrado Lattanzio. On the diffusive stress relaxation for multidimensional viscoelasticity. Communications on Pure & Applied Analysis, 2009, 8 (2) : 645654. doi: 10.3934/cpaa.2009.8.645 
[10] 
Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete & Continuous Dynamical Systems  B, 2012, 17 (4) : 11851203. doi: 10.3934/dcdsb.2012.17.1185 
[11] 
Horst Heck, Gunther Uhlmann, JennNan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 6376. doi: 10.3934/ipi.2007.1.63 
[12] 
Mark Jones. The bifurcation of interfacial capillarygravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 11831204. doi: 10.3934/cpaa.2011.10.1183 
[13] 
Tuan Hiep Pham, Jérôme Laverne, JeanJacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems  S, 2016, 9 (2) : 557584. doi: 10.3934/dcdss.2016012 
[14] 
Champike Attanayake, SoHsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete & Continuous Dynamical Systems  B, 2015, 20 (2) : 323337. doi: 10.3934/dcdsb.2015.20.323 
[15] 
Carlos Gutierrez, Víctor Guíñez. Simple umbilic points on surfaces immersed in $\R^4$. Discrete & Continuous Dynamical Systems  A, 2003, 9 (4) : 877900. doi: 10.3934/dcds.2003.9.877 
[16] 
Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems  B, 2012, 17 (4) : 11751184. doi: 10.3934/dcdsb.2012.17.1175 
[17] 
Gunther Uhlmann, JennNan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems & Imaging, 2009, 3 (2) : 309317. doi: 10.3934/ipi.2009.3.309 
[18] 
V. Torri. Numerical and dynamical analysis of undulation instability under shear stress. Discrete & Continuous Dynamical Systems  B, 2005, 5 (2) : 423460. doi: 10.3934/dcdsb.2005.5.423 
[19] 
Irena Lasiecka, Justin Webster. Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Communications on Pure & Applied Analysis, 2014, 13 (5) : 19351969. doi: 10.3934/cpaa.2014.13.1935 
[20] 
Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems  B, 2017, 22 (7) : 28312856. doi: 10.3934/dcdsb.2017153 
2018 Impact Factor: 1.008
Tools
Metrics
Other articles
by authors
[Back to Top]