Numerical modelling of viscous and viscoelastic fluids flow through the branching channel

summary:The aim of this paper is to describe the numerical results of numerical modelling of steady flows of laminar incompressible viscous and viscoelastic fluids. The mathematical models are Newtonian and Oldroyd-B models. Both models can be generalized by cross model in shear thinning meaning. Numerical tests are performed on three dimensional geometry, a branched channel with one entrance and two output parts. Numerical solution of the described models is based on cell-centered finite volume method using explicit Runge–Kutta time integration. Steady state solution is achieved for $t \rightarrow \infty$. In this case the artificial compressibility method can be applied

Steady and unsteady 2D numerical solution of generalized Newtonian fluids flow

summary:This article presents the numerical solution of laminar incompressible viscous flow in a branching channel for generalized Newtonian fluids. The governing system of equations is based on the system of balance laws for mass and momentum. The generalized Newtonian fluids differ through choice of a viscosity function. A power-law model with different values of power-law index is used. Numerical solution of the described models is based on cell-centered finite volume method using explicit Runge--Kutta time integration. The unsteady system of equations with steady boundary conditions is solved by finite volume method. Steady state solution is achieved for $t \rightarrow \infty$. In this case the artificial compressibility method can be applied. For the time integration an explicit multistage Runge--Kutta method of the second order of accuracy in the time is used. In the case of unsteady computation two numerical methods are considered, artificial compressibility method and dual-time stepping method. The flow is modelled in a bounded computational domain. Numerical results obtained by this method are presented and compared

Effect of non-Newtonian fluid rheology on an arterial bypass graft: A numerical investigation guided by constructal design

In post-operative scenarios of arterial graft surgeries to bypass coronary artery stenosis, fluid dynamics plays a crucial role. Problems such as intimal hyperplasia have been related to fluid dynamics and wall shear stresses near the graft junction. This study focused on the question of the use of Newtonian and non-Newtonian models to represent blood in this type of problem in order to capture important flow features, as well as an analysis of the performance of geometry from the view of Constructive Theory. The objective of this study was to investigate the effects rheology on the steady-state flow and on the performance of a system consisting of an idealized version of a partially obstructed coronary artery and bypass graft. The Constructal Design Method was employed with two degrees of freedom: the ratio be- tween bypass and artery diameters and the junction angle at the bypass inlet. The flow problem was solved numerically using the Finite Volume Method with blood modeled employing the Carreau equation for viscosity. The Computational Fluid Dynamics model associated with the Sparse Grid method generated eighteen response surfaces, each representing a severe stenosis degree of 75% for specific combinations of rheological parameters, dimensionless viscosity ratio, Carreau number and flow index at two distinct Reynolds numbers of 150 and 250. There was a considerable dependence of the pressure drop on rhe- ological parameters. For the two Reynolds numbers studied, the Newtonian case presented the lowest value of the dimensionless pressure drop, suggesting that the choice of applying Newtonian blood may underestimate the value of pressure drop in the system by about 12.4% ( Re = 150) and 7.8% ( Re = 250). Even so, results demonstrated that non-Newtonian rheological parameters did not influence either the shape of the response surfaces or the optimum bypass geometry, which consisted of a diameter ratio of 1 and junction angle of 30 °. However, the viscosity ratio and the flow index had the greatest im- pact on pressure drop, recirculation zones and wall shear stress. Rheological parameters also affected the recirculation zones downstream of stenosis, where intimal hyperplasia is more prevalent. Newto- nian and most non-Newtonian results had similar wall shear stresses, except for the non-Newtonian case with high viscosity ratio. In the view of Constructal Design, the geometry of best performance was in- dependent of the rheological model. However, rheology played an important role on pressure drop and flow dynamics, allowing the prediction of recirculation zones that were not captured by a Newtonian model

Computational modelling of blood flow through sutured and coupled microvascular anastomoses

The research presented in this thesis uses Computational Fluid Dynamics (CFD) to model blood flow through idealised sutured and coupled microvascular Anastomoses to investigate the affect of each surgical technique on the flow within the vessel. Local flow phenomena are examined in detail around suture and coupler sites to study characteristics that could potentially initiate thrombus formation; for example, changes in velocity profile, wall shear stress or recirculating flow (vorticity). Idealised geometries of sutured and coupled blood vessels were created using CFD software with dimensions identical to microvascular suture material and coupling devices. Vessels were modelled as non‐compliant 1mm diameter ducts, and blood was simulated as a Newtonian fluid, in keeping with previous similar studies. All analyses were steady-state and performed on arteries. Comparison of the sutured and coupled techniques in the simulated microarterial anastomoses revealed a reduced boundary velocity profile; high Wall Shear Stress (WSS); high Shear StrainRate(SSR);and elevated vorticity at the suturesites. The coupled anastomosis simulation showed a small increase in maximum WSS at the anastomotic region compared to a pristine vessel. However, this was less than half that of the sutured model. The coupled vessel displayed an average WSS equal to a pristine vessel. Taken together, these observations demonstrate an increased thrombogenic profile in the sutured anastomosis when compared to a pristine, or indeed a coupled vessel. Data from the simulations on a coupled anastomosis reveal a profile that is less thrombogenic than that of the sutured anastomosis, and one that is nearly equivalent to that of a pristine vessel. Overall, it can be concluded that, within the limits of CFD simulations and the assumptions taken in this study, a sutured anastomosis is potentially more likely to generate an intravascular thrombosis than a coupled anastomosis

Quantitative methods to regulate angiogenesis: applications to cancer and cardiovascular disease

Angiogenesis, the growth of new microvasculature from pre-existing blood vessels, is essential for tumor growth and metastasis in several cancers, including breast cancer. The vascular endothelial growth factor (VEGF) is the primary signaling molecule promoting angiogenesis. As such, the VEGF signaling axis is a potential target to inhibit tumor angiogenesis. However, full tumor vascular inhibition has yet to be achieved, attributed to the complexity of the vascular environment. Conversely, the ability to induce angiogenesis to vascularize ischemic tissue would provide treatment options for vascular diseases, including peripheral artery disease and coronary artery disease. Hemodynamic forces drive vascular disease progression, and contribute to the induction and directionality of vessel growth. Thus, full vascular control can be obtained by targeting VEGF signaling to inhibit angiogenesis and by targeting hemodynamic forces to promote angiogenesis. To this end, I developed computational approaches to individually understand the effects of VEGF signaling and hemodynamic forces on angiogenesis. Firstly, VEGF signaling models enable anti-angiogenic treatment efficacy to be correlated to features of cells or the microenvironment, which can help us understand a major challenge in tumor vascular inhibition: tumor heterogeneity. Indeed, cell population heterogeneity has been identified as an important consideration in cellular response to VEGF treatment, and is also a major factor in angiogenic drug resistances. However, there are few techniques available to represent and explore how heterogeneity is linked to population response. Recent high-throughput genomic, proteomic, and cellomic approaches offer opportunities for profiling heterogeneity on several scales. We have recently examined heterogeneity in VEGFR membrane localization in endothelial cells. We and others processed the heterogeneous data through ensemble averaging and integrated the data into computational models of anti-angiogenic drug effects in breast cancer. Here we show that additional modeling insight can be gained when cellular heterogeneity is considered. We present comprehensive statistical and computational methods for analyzing cellomic data sets and integrating them into deterministic models. We present a novel method for optimizing the fit of statistical distributions to heterogeneous data sets to preserve important data and exclude outliers. We compare methods of representing heterogeneous data and show methodology can affect model predictions up to 3.9-fold. We find that VEGF levels, a target for tuning angiogenesis, are more sensitive to VEGFR1 cell surface levels than VEGFR2; updating VEGFR1 levels in the tumor model gave a 64% change in free VEGF levels in the blood compartment, whereas updating VEGFR2 levels gave a 17% change. Furthermore, we find that subpopulations of tumor cells and tumor endothelial cells (tEC) expressing high levels of VEGFR (> 35,000 VEGFR/cell) negate anti-VEGF treatments. We show that lowering the VEGFR membrane insertion rate for these subpopulations recovers the anti-angiogenic effect of anti-VEGF treatment, revealing new treatment targets for specific tumor cell subpopulations. This novel method of characterizing heterogeneous distributions shows for the first time how different representations of the same data set lead to different predictions of drug efficacy. Secondly, to understand how to better promote angiogenesis, accurate quantification of hemodynamic forces is essential. Numerical simulations allow for this quantification. However, due to the complexity of numerical simulations, blood is often assumed to be Newtonian, despite being non-Newtonian in nature. To ensure accurate representation of hemodynamic forces, we compare hemodynamics between Newtonian and non-Newtonian models of blood. We test these models in both healthy and atherosclerotic arteries. For the non-Newtonian model, we employ a shear-rate dependent fluid (SDF) constitutive model, based on the works by Yasuda et al in 1981. We first verify our stabilized finite element numerical method with the benchmark lid-driven cavity flow problem. Numerical simulations show that the Newtonian model gives similar velocity profiles in the 2-dimensional cavity given different height and width dimensions, given the same Reynolds number. Conversely, the SDF model gave dissimilar velocity profiles, differing from the Newtonian velocity profiles by up to 25% in velocity magnitudes. This difference can affect estimation in platelet distribution within blood vessels or magnetic nanoparticle delivery. Wall shear stress (WSS) is an important quantity involved in vascular remodeling through integrin and adhesion molecule mechanotransduction. The SDF model gave a 7.3-fold greater WSS than the Newtonian model at the top of the 3-dimensional cavity. The SDF model gave a 37.7-fold greater WSS than the Newtonian model at artery walls located immediately after bifurcations in the idealized femoral artery tree. The pressure drop across arteries reveals arterial sections highly resistive to flow which correlates with stenosis formation. Numerical simulations give the pressure drop across the idealized femoral artery tree with the SDF model which is approximately 2.3-fold higher than with the Newtonian model. In atherosclerotic lesion models, the SDF model gives over 1 Pa higher WSS than the Newtonian model, a difference correlated with over twice as many adherent monocytes to endothelial cells from the Newtonian model compared to the SDF model. Together, these computational approaches provide a necessary step towards obtaining full vascular control, through inhibiting or promoting angiogenesis, respectively

Geometric optimization of complex thermal-fluid dynamic system by means of constructal design

In this work the Constructal Theory is exposed in its generality, trying to approach it through examples mostly of a physical-engineering nature. Constructal Theory proposes to see living bodies as elements subject to constraints, which are built with a goal, an objective, which is to obtain maximum efficiency. Constructal Theory is characterized by Constructal Law, which states that if a system has the freedom to morph it develops over time a flow architecture that provides easier access to the currents that pass through it. The Constructal Law is as general as the First and Second Laws of Thermodynamics, but it has a very different purpose which makes it unique and complementary to those laws. While the First Law points to the conservation of energy, both the Constructal Law and the Second Law point to change, that is, to a direction in time. Contrary to the Second Law, the Constructal Law applies to systems that are out of balance, that is, to systems that evolve over time. While the second law deals with state variables, the Constructal Law combines flows and design. The thesis continues with the application of the Constructal Theory for a cardiac bypass shape optimization. Through the Constructal Theory the constraints under which the system is free to morph are defined and, through the classical engineering optimization processes (numerical simulations and optimization algorithms) the optimum conditions are defined, i.e., those conditions that guarantee the minimum resistance to the passage of the fluid. The characterization of the blood flow was an important step in the study of this system, as the heartbeat induces a pulsed regime inside the veins. Therefore, the simulations conducted in transient regime consider the deformed velocity profile according to the conditions dictated by the pressure gradient established by the heartbeat