Numerical Methods for Grade-Two Fluid Models: Finite-Element Discretizations and Algorithms

BackgroundNewtonian fluid model has been commonly applied in simulating cerebral blood flow in intracranial atherosclerotic stenosis (ICAS) cases using computational fluid dynamics (CFD) modeling, while blood is a shear-thinning non-Newtonian fluid. We aimed to investigate the differences of cerebral hemodynamic metrics quantified in CFD models built with Newtonian and non-Newtonian fluid assumptions, in patients with ICAS.MethodsWe built a virtual artery model with an eccentric 75% stenosis and performed static CFD simulation. We also constructed CFD models in three patients with ICAS of different severities in the luminal stenosis. We performed static simulations on these models with Newtonian and two non-Newtonian (Casson and Carreau-Yasuda) fluid models. We also performed transient simulations on another patient-specific model. We measured translesional pressure ratio (PR) and wall shear stress (WSS) values in all CFD models, to reflect the changes in pressure and WSS across a stenotic lesion. In all the simulations, we compared the PR and WSS values in CFD models derived with Newtonian, Casson, and Carreau-Yasuda fluid assumptions.ResultsIn all the static and transient simulations, the Newtonian/non-Newtonian difference on PR value was negligible. As to WSS, in static models (virtual and patient-specific), the rheological difference was not obvious in areas with high WSS, but observable in low WSS areas. In the transient model, the rheological difference of WSS areas with low WSS was enhanced, especially during diastolic period.ConclusionNewtonian fluid model could be applicable for PR calculation, but caution needs to be taken when using the Newtonian assumption in simulating WSS especially in severe ICAS cases.

Cardiovascular disease (CVD) and especially atherosclerosis are chronic inflammatory diseases which cause the atherosclerotic plaque growth in the arterial vessels and the blood flow reduction. Stents have revolutionized the treatment of this disease to a great extent by restoring the blood flow in the vessel. The present study investigates the performance of the blood flow after stent implantation in patient-specific coronary artery and demonstrates the effect of using Newtonian vs. non-Newtonian blood fluid models in the distribution of endothelial shear stress. In particular, the Navier-Stokes and continuity equations were employed, and three non-Newtonian fluid models were investigated (Carreau, Carreau-Yasuda and the Casson model). Computational finite elements models were used for the simulation of blood flow. The comparison of the results demonstrates that the Newtonian fluid model underestimates the calculation of Endothelial Shear Stress, while the three non-Newtonian fluids present similar distribution of shear stress. Keywords: Blood flow dynamics, stented artery, non-Newtonian fluid. Clinical Relevance- This work demonstrates that when blood flow modeling is performed at stented arteries and predictive models are developed, the non-Newtonian nature of blood must be considered.

The efficiency of industrial heat equipment can be increased using baffles. The shape of baffles is one of the effective parameters. In this work, the effect of shapes of asymmetric baffles on the thermal performance has been investigated. Four different shapes as rectangular diagonal, trapezoidal, triangular and semi-ellipsoid, as well as, vertical rectangle (as the base model) were used. Also, four non-Newtonian fluids were used as the working fluid. The governing equation, which models the physical phenomenon, was solved with the finite volume method. The results showed that better thermal performance could be observed with semi-ellipsoid baffle for all four non-Newtonian fluids. However, for different models of non-Newtonian fluids, the average of increasing of thermal performance with different percent was achieved. By comparing different models of non-Newtonian fluids, shear-thinning model shows better thermal performance than other models.

This study has been managed for the investigation of entropy generation of inclined magnetic field (MG) on the Jeffery nanofluid flow on a stretching surface containing viscous dissipation. Heat generation or absorption effects are likewise considered on the magnetohydromagnetic flow problem and electric field is considered negligible. The boundary layer approach is incorporated for simplification of the proposed governing equations in which the target of analysis is focused near the surface of the fluidic problem. The concept of dimensionless parameters are used for simplification of the proposed system which overcomes the complexity of the problem. The relaxation and retardation times are also considered for the non-Newtonian Jeffrey fluid model for better analysis of the entropy generation of inclined MG on the Jeffery nanofluid flow on a stretching surface containing viscous dissipation. The strength of analytical homotopy analysis approach is employed for finding the solutions of the proposed fluidic system in terms of energy, momentum and concentration which is effective in the spatial domain. Graphical explanation for flow parameters have been incorporated. The tabular description is given for the convergence analysis and comparison of velocity gradient at the sheet surface f″ (0) for analytical solution (HAM) computed in this manuscript along with the numerical solution. The aim of second law analysis can be achieved by increasing the magnitude of the finite different temperature parameter. The current study is also described for Newtonian fluid as a special case of our study. Stream lines patterns are also provided for both Newtonian and non-Newtonian fluid models.

A comprehensive theoretical study has been carried out on the flow behavior of both single and multiple phase non-Newtonian fluids in porous media. This work is divided into three parts: (1) development of numerical and analytical solutions; (2) theoretical studies of transient flow of non-Newtonian fluids in porous media; and (3) applications of well test analysis and displacement efficiency evaluation to field problems. A fully implicit, integral finite difference model has been developed for simulation of non-Newtonian and Newtonian fluid flow through porous media. Several commonly-used rheological models of power-law and Bingham plastic non-Newtonian fluids have been incorporated in the simulator. A Buckley-Leverett type analytical solution for one-dimensional, immiscible displacement involving non-Newtonian fluids in porous media has been developed. Based on this solution, a graphic approach for evaluating non-Newtonian displacement efficiency has been developed. The Buckley-Leverett-Welge theory is extended to flow problems with non-Newtonian fluids. An integral method is also presented for the study of transient flow of Bingham fluids in porous media. In addition, two well test analysis methods have been developed for analyzing pressure transient tests of power-law and Bingham fluids, respectively. Applications are included to demonstrate this new technology. The physical mechanisms involved in immiscible displacement withmore » non-Newtonian fluids in porous media have been studied using the Buckley-Leverett type analytical solution. The results show that this kind of displacement is a complicated process and is determined by the rheological properties of the non-Newtonian fluids and the flow conditions, in addition to relative permeability data. In another study, an idealized fracture model has been used to obtain some insights into the flow of a power-law fluid in a double-porosity medium. For flow at a constant rate, non-Newtonian flow behavior in a fractured medium is characterized by two-parallel straight lines on a log-log plot of injection pressure versus time. Transient flow of a general pseudoplastic fluid has been studied numerically and it has been found that the long time pressure responses tend to be equivalent to that of a Newtonian system.« less

The modeling of the distinct non-Newtonian fluid properties is an essential prerequisite for the computational simulation of associated flow fields. In particular, some non-Newtonian fluids reveal strong diverse viscosity response behaviours to pure elongational and simple shearing flows. Therefore, it is necessary to be able to distinguish between these flow types even in complex flow configuration. Unfortunately flow types are naturally mixed and this distinction becomes quite difficult. Only in a Lagrangian framework the tracking of the Lagrangian fluid element deformation allows an accurate strain related deformation type assignment. However, most CFD approaches prefer the Eulerian framework accepting the loss of the natural flow path alignment of the moving fluid particles. Consequently shear and elongation rates are barely separable without particular assignment methods. In this work a tensor decomposition method from vortex dynamics is discussed which allows to distinguish between these flow types. In vortex dynamics the problem occurred to separate shearing from purely rotational flows because different hydro- and aerodynamic flow phenomena are caused by shear and vortex related flow types. Thereto, various methods were proposed, among others the optimal triple tensor decomposition method which is able to separate vortical from shearing flows but also, after some modification, elongational from shearing flows. This tensor decomposition is now used to calculate elongation and shearing rates as input variables into non-Newtonian fluid models for the calculation of the local elongational and shear viscosity. The application case is a cross slot channel flow often used as reference. In this numerical simulation study the impact of the elongation rate modeling on the contraction flow topology is shown and discussed. It is shown that the modeling of different viscous elongational and shear-thinning affects the resulting flow significantly  

The primary purpose of the present study is to investigate the unsteady flow and heat transfer characteristics of non-Newtonian power-law fluid over an oscillating flat plate. The power-law fluid model is employed to distinguish the non-Newtonian fluid behavior, which has wide applications in nature and technology, such as crude oil exploitation, polymer processing and some bio applications. In this work, the governing equations are set up using the non-Newtonian fluid model. The radial basis function collocation method (RBFCM) and the finite element method (FEM) are used to solve the coupled equations. The numerical results of the fluid velocity and fluid temperature fields are analyzed in detail. For authentication of the proposed schemes, the results are compared with earlier research works. To further verify the validity of the FEM and the RBFCM results, the flow behavior of crude oil was studied and the obtained results were found to be consistent with each other using both methods. Our results provide further guidance for evaluation aspects dealing with crude oil in a pipe, which are basic tools in oil and gas explorations.

The primary purpose of the present study is to investigate the unsteady flow and heat transfer characteristics of non-Newtonian power-law fluid over an oscillating flat plate. The power-law fluid model is employed to characterize the non-Newtonian fluid behavior, which has wide applications in nature and technology, such as crude oil exploitation, polymer processing and some bio applications. In this study, the coupled governing equations are set up using the non-Newtonian fluid model. The radial basis function collocation method (RBFCM) and the finite element method (FEM) are used to solve the coupled equations. The simulation results on the fluid velocity and temperature distribution are analyzed in detail. For authentication of the proposed schemes, the numerical results are compared with earlier research works. To further verify the validity of the FEM and the RBFCM results, the flow behavior of crude oil was studied and the obtained results were found to be consistent with each other using both methods. Our results provide further guidance for evaluation aspects dealing with crude oil in a pipe, which are basic tools in oil and gas explorations.

A comprehensive theoretical study has been carried out on the flow behavior of both single and multiple phase non-Newtonian fluids in porous media. This work is divided into three parts: development of numerical and analytical solutions; theoretical studies of transient flow of non-Newtonian fluids in porous media; and applications of well test analysis and displacement efficiency evaluation to field problems. A fully implicit, integral finite difference model has been developed for simulation of non-Newtonian and Newtonian fluid flow through porous media. Several commonly-used rheological models of power-law and Bingham plastic non-Newtonian fluids have been incorporated in the simulator. A Buckley-Leverett type analytical solution for one-dimensional, immiscible displacement involving non-Newtonian fluids in porous media has been developed. An integral method is also presented for the study of transient flow of Bingham fluids in porous media. In addition, two well test analysis methods have been developed for analyzing pressure transient tests of power-law and Bingham fluids, respectively. Applications are included to demonstrate this new technology. The physical mechanisms involved in immiscible displacement with non-Newtonian fluids in porous media have been studied using the Buckley-Leverett type analytical solution. In another study, an idealized fracture model has been used to obtain some insights into the flow of a power-law fluid in a double-porosity medium. Transient flow of a general pseudoplastic fluid has been studied numerically. 125 refs., 91 figs., 12 tabs.

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Chapter 3 Similarity and Pseudosimilarity Solutions of Degenerate Boundary Layer Equations

A non-Newtonian fluid model with an objective vorticity

The characteristic scale of pore flow in tight reservoirs is generally in the range of 0.1 μm to 1 μm, which shows the obvious micro- and nanoscale effect. The traditional oil and gas seepage theory cannot accurately describe the flow law of liquid in the micro- and nanopores. The determination of seepage characteristics is crucial to the development, layout, and prediction of tight oil. Therefore, a non-Newtonian fluid model is established to discuss the flow characteristics of confined liquid in the heterogeneous pores of microtubules and reveal the nonlinear seepage law of water in micro- and nanochannels and tight reservoirs. Based on the characteristics of non-Newtonian fluid of confined fluid in micro- and nanospace, the flow model of non-Newtonian fluid under the action of shear stress was deduced. The flow velocity variation of liquid in micro- and nanochannel and dense core was analyzed, and the flow rate of water was less than that predicted by macro theory. According to the flow experiment of water in micro- and nanochannels, the flow model of power-law non-Newtonian fluid was verified. At the same time, through the flow experiment of water in the dense rock core, the non-Newtonian model was used for nonlinear fitting, and the non-Newtonian power-law parameters and average pore radius were obtained, which verified the effectiveness of the non-Newtonian flow model.

Non-Newtonian fluid model was created against the Newton’s Law of viscosity where the viscosity is no more constant and dependent on the shear rate. The existing such fluid can be found in many industrial claims especially in food manufacturing, lubrication, biomedical flows and oil and gas. Besides, the used of non-Newtonian fluid occurs in mining industry where the slurries and muds are often handled. There are many models on non-Newtonian fluid available in literature where some of them capture the specific properties. The Reiner–Philippoff (RP) fluid model is considered in this endeavour due to the capabilities of the model which can be acted in three different family of fluid which are viscous, shear thickening and the shear-thinning. Mathematical model is constructed using continuity, momentum and energy equations where in form of partial differential equations (PDEs). The complexity of the proposed model is abridged by deduced the equations into ordinary differential equations (ODEs) by adopting similarity variables before the computation is done by bvp4c function drive in MATLAB software. To ratify the validity of the proposed model as well as numerical outputs, the comparative study is performed and it found to be in very strong agreement under limiting case where the present model is condensed to be identical with the reported model previously. The consequences of pertinent parameters on fluid’s characteristics are analyzed in details through the plotted graphic visuals and tabular form.

A note on higher-order perturbative corrections to squirming speed in weakly viscoelastic fluids