Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows
This paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundar...
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AIMS Press
2021-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/mbe.2021193?viewType=HTML |
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| author | Soonpil Kang Sharbel Nashar Elizabeth R. Livingston Arif Masud |
| author_facet | Soonpil Kang Sharbel Nashar Elizabeth R. Livingston Arif Masud |
| author_sort | Soonpil Kang |
| collection | DOAJ |
| description | This paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundary conditions are derived via locally resolving the fine-scale variational equation facilitated by the Variational Multiscale (VMS) framework. The proposed method reproduces the consistency and stabilization terms that are present in the Nitsche type approaches. In addition, for the shear-rate fluids, two more boundary terms appear. One of these terms is the viscosity-derivative term and is a function of the shear-rate, while the other term is a zeroth-order term. These terms play an important role in attaining optimal convergence rates for the velocity and pressure fields in the norms considered. A most significant contribution is the form of the stabilization tensors that are also variationally derived. Employing edge functions the edge stabilization tensor is numerically evaluated, and it adaptively adjusts itself to the magnitude of the boundary residual. The resulting formulation is variationally consistent and the weakly imposed no-slip boundary condition leads to higher accuracy of the spatial gradients for coarse boundary-layer meshes when compared with the traditional strongly imposed boundary conditions. This feature of the present approach will be of significance in imposing interfacial continuity conditions across non-matching discretizations in blood-artery interaction problems. A set of test cases is presented to investigate the mathematical attributes of the method and a patient-specific case is presented to show its clinical relevance. |
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| format | Article |
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| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-12-17T00:02:11Z |
| publishDate | 2021-05-01 |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-adf5681d562a42d08d747a3699686f312022-12-21T22:11:03ZengAIMS PressMathematical Biosciences and Engineering1551-00182021-05-011843855388610.3934/mbe.2021193Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flowsSoonpil Kang0Sharbel Nashar 1Elizabeth R. Livingston2Arif Masud3Department of Civil and Environmental Engineering, and Department of Biomedical and Translational Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USADepartment of Civil and Environmental Engineering, and Department of Biomedical and Translational Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USADepartment of Civil and Environmental Engineering, and Department of Biomedical and Translational Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USADepartment of Civil and Environmental Engineering, and Department of Biomedical and Translational Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USAThis paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundary conditions are derived via locally resolving the fine-scale variational equation facilitated by the Variational Multiscale (VMS) framework. The proposed method reproduces the consistency and stabilization terms that are present in the Nitsche type approaches. In addition, for the shear-rate fluids, two more boundary terms appear. One of these terms is the viscosity-derivative term and is a function of the shear-rate, while the other term is a zeroth-order term. These terms play an important role in attaining optimal convergence rates for the velocity and pressure fields in the norms considered. A most significant contribution is the form of the stabilization tensors that are also variationally derived. Employing edge functions the edge stabilization tensor is numerically evaluated, and it adaptively adjusts itself to the magnitude of the boundary residual. The resulting formulation is variationally consistent and the weakly imposed no-slip boundary condition leads to higher accuracy of the spatial gradients for coarse boundary-layer meshes when compared with the traditional strongly imposed boundary conditions. This feature of the present approach will be of significance in imposing interfacial continuity conditions across non-matching discretizations in blood-artery interaction problems. A set of test cases is presented to investigate the mathematical attributes of the method and a patient-specific case is presented to show its clinical relevance.http://www.aimspress.com/article/doi/10.3934/mbe.2021193?viewType=HTMLweakly imposed dirichlet boundary conditionsnon-newtonian shear-rate dependent fluidsvariational multiscale methodinterface stabilizationblood flows |
| spellingShingle | Soonpil Kang Sharbel Nashar Elizabeth R. Livingston Arif Masud Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows Mathematical Biosciences and Engineering weakly imposed dirichlet boundary conditions non-newtonian shear-rate dependent fluids variational multiscale method interface stabilization blood flows |
| title | Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows |
| title_full | Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows |
| title_fullStr | Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows |
| title_full_unstemmed | Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows |
| title_short | Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows |
| title_sort | weakly imposed boundary conditions for shear rate dependent non newtonian fluids application to cardiovascular flows |
| topic | weakly imposed dirichlet boundary conditions non-newtonian shear-rate dependent fluids variational multiscale method interface stabilization blood flows |
| url | http://www.aimspress.com/article/doi/10.3934/mbe.2021193?viewType=HTML |
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