Non‐Newtonian blood flow study in a model cavopulmonary vascular system

A transient haemodynamic study in a model cavopulmonary vascular system has been carried out for a typical range of parameters using a finite element‐based Navier–Stokes solver. The focus of this study is to investigate the influence of non‐Newtonian behaviour of the blood on the haemodynamic quantities, such as wall shear stress (WSS) and flow pattern. The computational fluid dynamics (CFD) model is based on an artificial compressibility characteristic‐based split (AC‐CBS) scheme, which has been adopted to solve the Navier–Stokes equations in space–time domain. A power law model has been implemented to characterize the shear thinning nature of the blood depending on the local strain rate. Using the computational model, numerical investigations have been performed for Newtonian and non‐Newtonian flows for different frequencies and input pulse forms. The haemodynamic quantities observed in total cavopulmonary connection (TCPC) for the above conditions suggest that there are considerable differences in average (about 25–40%) and peak (about 50%) WSS distributions, when the non‐Newtonian behaviour of the blood is taken into account. The lower WSS levels observed for non‐Newtonian cases point to the higher risk of lesion formation, especially at higher pulsation frequencies. A realistic pulse form is relatively safer than a sinusoidal pulse as it has more energy distributed in the higher harmonics, which results in higher average WSS values. The present study highlights the importance of including non‐Newtonian shear thinning behaviour for modelling blood flow in the vicinity of repaired arterial connections. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical and analytical solutions of an Oldroyd 8‐constant MHD fluid with nonlinear slip conditions

The flow of an Oldroyd 8‐constant non‐Newtonian MHD fluid is investigated analytically and numerically. The governing equations for the flow field are derived for a steady one‐dimensional flow. The effect of constant applied magnetic field is included and its influence on the flow field is studied. The nonlinear governing equation along with nonlinear boundary conditions is solved analytically and the solution is obtained in an elegant way. Numerical solutions are also obtained using higher order Chebyshev spectral methods. The influence of various non‐Newtonian parameters and applied magnetic field is investigated. Results showing the effect of various physical parameters of the flow are presented and investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study

This paper presents a numerical study for the unsteady flow of a magnetohydrodynamic (MHD) Sisko fluid in annular pipe. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field. Based on the constitutive relationship of a Sisko fluid, the non‐linear equation governing the flow is first modelled and then numerically solved. The effects of the various parameters especially the power index n, the material parameter of the non‐Newtonian fluid b and the magnetic parameter B on the flow characteristics are explored numerically and presented through several graphs. Moreover, the shear‐thinning and shear‐thickening characteristics of the non‐Newtonian Sisko fluid are investigated and a comparison is also made with the Newtonian fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐Newtonian blood flow dynamics in a right internal carotid artery with a saccular aneurysm

Flow dynamics plays an important role in the pathogenesis and treatment of cerebral aneurysms. The temporal and spatial variations of wall shear stress in the aneurysm are hypothesized to be correlated with its growth and rupture. In addition, the assessment of the velocity field in the aneurysm dome and neck is important for the correct placement of endovascular coils. This work describes the flow dynamics in a patient‐specific model of carotid artery with a saccular aneurysm under Newtonian and non‐Newtonian fluid assumptions. The model was obtained from three‐dimensional rotational angiography image data and blood flow dynamics was studied under physiologically representative waveform of inflow. The three‐dimensional continuity and momentum equations for incompressible and unsteady laminar flow were solved with a commercial software using non‐structured fine grid with 283 115 tetrahedral elements. The intra‐aneurysmal flow shows complex vortex structure that change during one pulsatile cycle. The effect of the non‐Newtonian properties of blood on the wall shear stress was important only in the arterial regions with high velocity gradients, on the aneurysmal wall the predictions with the Newtonian and non‐Newtonian blood models were similar. Copyright © 2005 John Wiley & Sons, Ltd.     

Multiphase non‐Newtonian effects on pulsatile hemodynamics in a coronary artery

Hemodynamic stresses are involved in the development and progression of vascular diseases. This study investigates the influence of mechanical factors on the hemodynamics of the curved coronary artery in an attempt to identify critical factors of non‐Newtonian models. Multiphase non‐Newtonian fluid simulations of pulsatile flow were performed and compared with the standard Newtonian fluid models. Different inlet hematocrit levels were used with the simulations to analyze the relationship that hematocrit levels have with red blood cell (RBC) viscosity, shear stress, velocity, and secondary flow. Our results demonstrated that high hematocrit levels induce secondary flow on the inside curvature of the vessel. In addition, RBC viscosity and wall shear stress (WSS) vary as a function of hematocrit level. Low WSS was found to be associated with areas of high hematocrit. These results describe how RBCs interact with the curvature of artery walls. It is concluded that although all models have a good approximation in blood behavior, the multiphase non‐Newtonian viscosity model is optimal to demonstrate effects of changes in hematocrit. They provide a better stimulation of realistic blood flow analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

A 2D boundary element method for simulating the deformation of axisymmetric compound non‐Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non‐Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non‐Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non‐Newtonian material. By transforming the integral representation for the velocity to cylindrical co‐ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two‐dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non‐Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd‐B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break‐up mechanism of compound drops in relation to the specific non‐Newtonian character of the membrane. Copyright © 1999 John Wiley & Sons, Ltd.     

A non‐linear dynamical system approach to finite amplitude Taylor‐Vortex flow of shear‐thinning fluids

The effect of shear thinning on the stability of the Taylor–Couette flow is explored for a Carreau–Bird fluid in the narrow‐gap limit. The Galerkin projection method is used to derive a low‐order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional non‐linear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the circular Couette flow, becomes lower as the shear‐thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow, which coincides with the onset of a supercritical bifurcation. Comparison with existing measurements of the effect of shear thinning on the critical Taylor and wave numbers show good agreement. The Taylor vortex cellular structure loses its stability in turn, as the Taylor number reaches a critical value. At this point, an inverse Hopf bifurcation emerges. In contrast to Newtonian flow, the bifurcation diagrams exhibit a turning point that sharpens with shear‐thinning effect. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of a flow around an impulsively started radially deforming circular cylinder

The development of a two‐dimensional viscous incompressible flow generated by a deformable circular cylinder impulsively started into rectilinear motion is studied numerically for the Reynolds numbers equal to 550 and 3000. The vorticity transport equation is solved by a second‐order finite difference method in both directions of the domains. The Poisson equation for the streamfunction is solved by a Fourier–Galerkin method in the direction of the flow that is assumed to remain symmetrical and a second‐order finite difference for the radial direction. The advance in time is achieved by a second‐order Adams–Bashforth scheme. The computed results are compared qualitatively with experimental and numerical results done before in the particular non‐deformable case. The comparison is found to be satisfactory. The influence of the deformation of the cylinder on the flow structure and the drag coefficient is then analyzed. Copyright © 1999 John Wiley & Sons, Ltd.     

A high‐order upwind control‐volume method based on integrated RBFs for fluid‐flow problems

This paper is concerned with the development of a high‐order upwind conservative discretization method for the simulation of flows of a Newtonian fluid in two dimensions. The fluid‐flow domain is discretized using a Cartesian grid from which non‐overlapping rectangular control volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle‐point rule. One‐dimensional integrated radial basis function schemes using the multiquadric basis function are employed to represent the variations of the field variables along the grid lines. The convection term is effectively treated using an upwind scheme with the deferred‐correction strategy. Several highly non‐linear test problems governed by the Burgers and the Navier–Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well. Copyright © 2010 John Wiley & Sons, Ltd.     

Mathematical modelling of two‐phase non‐Newtonian flow in a helical pipe

Governing equations for a two‐phase 3D helical pipe flow of a non‐Newtonian fluid with large particles are derived in an orthogonal helical coordinate system. The Lagrangian approach is utilized to model solid particle trajectories. The interaction between solid particles and the fluid that carries them is accounted for by a source term in the momentum equation for the fluid. The force‐coupling method (FCM), developed by M.R. Maxey and his group, is adopted; in this method the momentum source term is no longer a Dirac delta function but is spread on a numerical mesh by using a finite‐sized envelop with a spherical Gaussian distribution. The influence of inter‐particle and particle–wall collisions is also taken into account. Copyright © 2005 John Wiley & Sons, Ltd.     

A spectral/hp least‐squares finite element analysis of the Carreau–Yasuda fluids

A least‐squares finite element model with spectral/hp approximations was developed for steady, two‐dimensional flows of non‐Newtonian fluids obeying the Carreau–Yasuda constitutive model. The finite element model consists of velocity, pressure, and stress fields as independent variables (hence, called a mixed model). Least‐squares models offer an alternative variational setting to the conventional weak‐form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces used for the velocity, pressure, and stress fields are necessary when the polynomial order (p) used is sufficiently high (say, p > 3, as determined numerically). Also, the use of the spectral/hp elements in conjunction with the least‐squares formulation with high p alleviates various forms of locking, which often appear in low‐order least‐squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate, benchmark problems of Kovasznay flow, backward‐facing step flow, and lid‐driven square cavity flow are used. Then the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics is studied parametrically. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical analysis of three-dimensional Newtonian extrudate swell

The present paper considers the problem of predicting extrudate shapes from asymmetrical dies for Newtonian fluids. The flow is fully three-dimensional and an exploration of finite elements is made with a view to finding accurate, stable and economical schemes. A number of elements are compared and we conclude that some of the Fortin elements are most useful on the grounds of computational overhead and solution accuracy. These are used to investigate some symmetrical (square dies) and asymmetrical (unequal lip) planar and general L-shaped die flows. Finally, we show that in an unconstrained extrudate the final shape must be such that particles describe a helix in space; special cases include circular flow and rectilinear flow.     

A pseudo‐viscoelastic approach for transient polymeric flow exiting a channel

The influence of elasticity of a fluid exiting a channel is examined on transient coating downstream. A hybrid spectral/boundary element approach is proposed to solve the problem. The flow inside the channel is assumed to be fully developed. A viscoelastic instability of one‐dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The Johnson–Segalman equation is used as the constitutive model. The velocity profile inside the channel is taken as the exit profile for the emerging free‐surface flow. The flow is assumed to be Newtonian as it emerges from the channel. An estimate of the magnitude of the rate‐of‐strain tensor components in the free‐surface region reveals that they are generally smaller than the shear rate inside the channel. The evolution of the flow front is simulated using the boundary element method. For the channel flow, the problem is reduced to a nonlinear dynamical system using the Galerkin projection method. Stability analysis indicates that the channel velocity may be linear or non‐linear depending on the range of the Weissenberg number. The evolution of the coating flow at the exit is examined for steady as well as transient (monotonic and oscillatory) channel flow. It is found that adverse flow can exist as a result of fluid elasticity, which can hinder the process of blade coating. Copyright © 2003 John Wiley & Sons, Ltd.     

Monolithic Newton‐multigrid solution techniques for incompressible nonlinear flow models

We present special Newton‐multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB‐stable Q₂P₁ element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well‐known flow around cylinder benchmark configuration for generalized Newtonian as well as non‐Newtonian flows including non‐isothermal, shear/pressure dependent and viscoelastic effects.Copyright © 2012 John Wiley & Sons, Ltd.     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems

The use of an adjoint technique for goal‐based error estimation described by Hartit et al. (Int. J. Numer. Meth. Fluids 2005; 47 :1069–1074) is extended to the numerical solution of free boundary problems that arise in elastohydrodynamic lubrication (EHL). EHL systems are highly nonlinear and consist of a thin‐film approximation of the flow of a non‐Newtonian lubricant which separates two bodies that are forced together by an applied load, coupled with a linear elastic model for the deformation of the bodies. A finite difference discretization of the line contact flow problem is presented, along with the numerical evaluation of an exact solution for the elastic deformation, and a moving grid representation of the free boundary that models cavitation at the outflow in this one‐dimensional case. The application of a goal‐based error estimate for this problem is then described. This estimate relies on the solution of an adjoint problem; its effectiveness is demonstrated for the physically important goal of the total friction through the contact. Finally, the application of this error estimate to drive local mesh refinement is demonstrated. Copyright © 2010 John Wiley & Sons, Ltd.     

Flow effects of blood constitutive equations in 3D models of vascular anomalies

The effects of different blood rheological models are investigated numerically utilizing two three‐ dimensional (3D) models of vascular anomalies, namely a stenosis and an abdominal aortic aneurysm model. The employed CFD code incorporates the SIMPLE scheme in conjunction with the finite‐volume method with collocated arrangement of variables. The approximation of the convection terms is carried out using the QUICK differencing scheme, whereas the code enables also multi‐block computations, which are useful in order to cope with the two‐block grid structure of the current computational domain. Three non‐Newtonian models are employed, namely the Casson, Power‐Law and Quemada models, which have been introduced in the past for modelling the rheological behaviour of blood and cover both the viscous as well as the two‐phase character of blood. In view of the haemodynamical mechanisms related to abnormalities in the vascular network and the role of the wall shear stress in initiating and further developing of arterial diseases, the present study focuses on the 3D flow field and in particular on the distribution as well as on both low and high values of the wall shear stress in the vicinity of the anomaly. Finally, a comparison is made between the effects of each rheological model on the aforementioned parameters. Results show marked differences between simulating blood as Newtonian and non‐Newtonian fluid and furthermore the Power‐Law model exhibits different behaviour in all cases compared to the other models whereas Quemada and Casson models exhibit similar behaviour in the case of the stenosis but different behaviour in the case of the aneurysm. Copyright © 2005 John Wiley & Sons, Ltd.     

Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T‐junction

An investigation of laminar steady and unsteady flows in a two‐dimensional T‐junction was carried out for Newtonian and a non‐Newtonian fluid analogue to blood. The flow conditions considered are of relevance to hemodynamical applications and the localization of coronary diseases, and the main objective was to quantify the accuracy of the predictions and to provide benchmark data that are missing for this prototypical geometry. Under steady flow, calculations were performed for a wide range of Reynolds numbers and extraction flow rate ratios, and accurate data for the recirculation sizes were obtained and are tabulated. The two recirculation zones increased with Reynolds number, but the behaviour was non‐monotonic with the flow rate ratio. For the pulsating flows a periodic instability was found, which manifests itself by the breakdown of the main vortex into two pieces and the subsequent advection of one of them, while the secondary vortex in the main duct was absent for a sixth of the oscillating period. Shear stress maxima were found on the walls opposite the recirculations, where the main fluid streams impinge onto the walls. For the blood analogue fluid, the recirculations were found to be 10% longer but also short lived than the corresponding Newtonian eddies, and the wall shear stresses are also significantly different especially in the branch duct. Copyright © 2007 John Wiley & Sons, Ltd.