Elsevier

Microvascular Research

Volume 138, November 2021, 104221
Microvascular Research

Finite element computation of magneto-hemodynamic flow and heat transfer in a bifurcated artery with saccular aneurysm using the Carreau-Yasuda biorheological model

https://doi.org/10.1016/j.mvr.2021.104221Get rights and content

Highlights

  • Unsteady 2D non-Newtonian magneto-hemodynamic heat transfer in electrically conducting blood flow in bifurcated artery with a saccular aneurysm.

  • The 6-node Taylor-Hood triangular elements have been deployed in the optimized mesh design for the robust finite element simulations.

  • The temperature decreases at the parent artery (inlet) and both the distant and prior artery with the increment in the Prandtl number.

  • A higher Reynolds number also causes a reduction in velocity as well as in pressure.

  • It has delivered a greater insight of blood flow behavior in saccular aneurysm in the distal aortic artery.

Abstract

“Existing computational fluid dynamics studies of blood flows have demonstrated that the lower wall stress and higher oscillatory shear index might be the cause of acceleration in atherogenesis of vascular walls in hemodynamics. To prevent the chances of aneurysm wall rupture in the saccular aneurysm at distal aortic bifurcation, clinical biomagnetic studies have shown that extra-corporeal magnetic fields can be deployed to regulate the blood flow. Motivated by these developments, in the current study a finite element computational fluid dynamics simulation has been conducted of unsteady two-dimensional non-Newtonian magneto-hemodynamic heat transfer in electrically conducting blood flow in a bifurcated artery featuring a saccular aneurysm. The fluid flow is assumed to be pulsatile, non-Newtonian and incompressible. The Carreau-Yasuda model is adopted for blood to mimic non-Newtonian characteristics. The transformed equations with appropriate boundary conditions are solved numerically by employing the finite element method with the variational approach in the FreeFEM++ code. Hydrodynamic and thermal characteristics are elucidated in detail for the effects of key non-dimensional parameters i.e. Reynolds number (Re = 14, 21, 100, 200), Prandtl number (Pr = 14, 21) and magnetic body force parameter (Hartmann number) (M = 0.6, 1.2, 1.5) at the aneurysm and throughout the arterial domain. The influence of vessel geometry on blood flow characteristics i.e. velocity, pressure and temperature fields are also visualized through instantaneous contour patterns. It is found that an increase in the magnetic parameter reduces the pressure but increases the skin-friction coefficient in the domain. The temperature decreases at the parent artery (inlet) and both the distant and prior artery with the increment in the Prandtl number. A higher Reynolds number also causes a reduction in velocity as well as in pressure. The blood flow shows different characteristic contours with time variation at the aneurysm as well as in the arterial segment. The novelty of the current research is therefore to present a combined approach amalgamating the Carreau-Yasuda model, heat transfer and magnetohydrodynamics with complex geometric features in realistic arterial hemodynamics with extensive visualization and interpretation, in order to generalize and extend previous studies. In previous studies these features have been considered separately and not simultaneously as in the current study. The present simulations reveal some novel features of biomagnetic hemodynamics in bifurcated arterial transport featuring a saccular aneurysm which are envisaged to be of relevance in furnishing improved characterization of the rheological biomagnetic hemodynamics of realistic aneurysmic bifurcations in clinical assessment, diagnosis and magnetic-assisted treatment of cardiovascular disease.”

Introduction

Blood is an immensely complex aqueous ionic solution containing cellular elements and these elements include microscopic cells such as erythrocytes (red blood cells), leukocytes (white blood cells), thrombocytes, lymphocytes and lipoproteins suspended in a continuous saline plasma. It enables the sustained and efficient transportation of oxygen and CO2, nutrients, hormones and metabolic wastes, among many other functions, throughout the body to maintain cell-level metabolism. Blood circulation is also critical in maintaining the necessary regulation of the pH, osmotic pressure and temperature of the whole body and protecting it from microbial and mechanical damage (Vasu et al., 2020). The plasma generally behaves as a Newtonian fluid (Gupta et al., 2016), is composed of water (93%) and electrolytes, organic molecules, numerous proteins (3%), and waste products, whereas the whole blood (a suspension of cells and highly viscous in nature), exhibits the property of a non-Newtonian fluid, in particular in smaller vessels. At high shear rate blood usually behaves like a Newtonian fluid as observed in large arteries (Davies, 2009; Baskurt and Meiselman, 2003). Arterial blood flow is fundamental to the human circulatory system, and the presence of arterial stenosis (constriction) adversely influences the health of the cardiovascular system (Baieth, 2008). Blood flows to the body organs and body cells through a complex network of arteries, veins, and capillaries. The motion of blood is due to continuous pumping by the heart as deoxygenated blood is transported to the heart from all the body organs through veins and the heart pumps oxygenated blood to the whole body through the arteries. Over the past few decades, an impressive number of comprehensive theoretical and experimental investigations related to blood flow in arteries in the presence of a stenosis have been conducted with various methodologies (Amiri et al., 2019). Relevant examples include Criminale et al. (2018), comes with the results that the accurate identification of blood hemodynamics is an essential step in characterizing flow regimes that would govern processes in physiology and pathology. Mathur and Jain (2013), developed the mathematical model to study the blood flow behavior in stenosed artery and investigated the effects of stenosis on the blood flow analytically. Tripathi et al. (2021), investigated the pulsatile blood flow behavior in stenosed artery with the suspension of hybrid nanofluid. The simulated results of the study shows the significant effect of hybrid nanofluid on the flow rate and wall shear stress. More recently non-Newtonian hemodynamics has been addressed by a number of investigators. Reddy et al. (2014) studied blood flow by treating the blood as a polar (couple stress) fluid, showing that significant deviation in flow characteristics arise compared with the classical Newtonian model. Several investigators have also analysed theoretically and computationally the contribution of blood rheology to coronary artery disease and cerebral aneurysms. Agrawal et al. (2015) studied the shear-thinning characteristics of blood with a Carreau–Yasuda Model, for coil embolization as a mildly invasive endovascular method for treatment of a cerebral aneurysm. The study leads to the observation that the blood rheology exerts a prominent role in the performance of the coil which is aimed at reducing fluid loading of the blood vessel and delaying subsequent vessel wall deformation.

Simulation of blood flow has been widely used in recent decades for better understanding the symptomatic spectrum of various diseases in order to improve already existing treatments, or to develop new therapeutic techniques. The characteristics of blood flow in an artery can be modified significantly by arterial disease, which may include aneurysms and stenoses. The domain of hemodynamics has grown into a significant branch of modern fluid mechanics and includes an extensive number of theoretical and experimental investigations. In the initial stages of arterial disease, arteriosclerotic lesions (stenoses) are not distributed randomly within the arterial network as they usually appear around junctions of arteries, arterial curvatures and bifurcations of large and medium arteries, where the usual flow patterns are significantly altered resulting in complexities in the cardiovascular system (Chakravarty and Sen, 2008). As a result of the complex flow phenomena owing to the continuous development of stenoses, it often becomes extremely difficult to distinguish these disturbances from the normal flow characteristics in these critical regions. Hence hemodynamical studies play an essential role in elucidating blood flow around bends and bifurcations in many large arteries as well as in various arterial diseases. The various models and methodologies adopted in these studies are as diverse as the geometric parameters and hydrodynamic conditions of arterial bifurcations. These have been comprehensively reviewed in by Lou and Yang (1992) who emphasized that in low wall shear stress and recirculation regions such hemodynamic patterns play an important role in the development of atherosclerotic lesion and their subsequent progression. Furthermore, physiological risk factors such as hypertension, hyperlipidemia, high blood pressure and diabetes mellitus, are known to be major causes of atherosclerosis and aneurysms. Seo (2013) and Zhang et al. (2017) described numerical simulations of blood flow behavior in the bifurcated carotid artery. Seo (2013) discussed the wall shear stress (WSS) distributions as well as pressure profiles due to the shear thinning behavior in both the internal carotid artery and external carotid artery and showed computationally that the variation of the flow characteristics can be dependent on the arterial bifurcation geometry which exerts an important role in the development of atherosclerosis. Zhang et al. (2017) computed the wall shear stress and wall pressure gradient in the left as well as right coronary artery bifurcation identifying that the region of low wall shear stress (WSS) and magnitudes of maximum wall pressure gradient (WPG) increases with the angles of bifurcation. Further investigations also concluded that the initiation of the type of aneurysm is likely to be strongly influenced by the geometry of the arteries (Blanco et al., 2007; Dubey et al., 2020). The inner curved arterial walls and zones in the vicinity of flow separation at the bifurcation, are strongly associated with initial development of atherosclerosis and aneurysms. Therefore, the local flow patterns in curved locations and bifurcations are of significant relevance to the study of atherogenesis. In order to quantify accurately hydrodynamic characteristics in the regions of recirculation and separation at curved locations and bifurcations, numerous studies have been performed in recent years (Jou and Berger, 1998; Lee et al., 2008; Shaik et al., 2006).

The role of hemodynamics in the growth of aneurysms has in particular stimulated considerable interest among researchers. Valencia et al. (2006) studied the effect of saccular (intercranial) aneurysms on blood flow in the artery with non-Newtonian and Newtonian fluid models. Kumar et al. (2004) studied the blood flow in an asymmetrically dilated fusiform artery under pulsatile inflow conditions for a full cycle of period, T. Increasingly the Carreau-Yasuda bio rheological fluid model has attracted considerable attention from mathematicians and engineers due to its broad applications in quantifying non-Newtonian behavior of real blood. The Carreau-Yasuda fluid model has been implemented in blood flow computation by Kumar et al. (2004) for a diseased artery. Ali et al. (2019) have investigated the biological interactions between Carreau fluid and micro-swimmers in undulating conduits (vessels) with a modified Taylor swimming sheet model, magnetic field and porous medium effects, motivated by microbot treatment of hemotological disorders.

“The presence of both ions in blood and iron in the haemoglobin molecule produces electrically conducting properties in blood. Streaming blood, can, therefore, be manipulated via the application of extracorporeal magnetic fields, which may be static or alternating. Arterial diseases such as arteriosclerosis and aneurysms, may, therefore, be treated via biomagnetic therapy. Magnetohydrodynamics (MHD) involves the motion of electrically conducting fluids under the influence of an applied magnetic field and arises in both Newtonian and non-Newtonian fluid flows. The emergence of new diverse technological applications of MHD, in medical engineering (magnetic blood separation, biomagnetics etc), chemical engineering, energy systems and materials processing, etc. have stimulated high interest in magnetic fluid dynamic simulations in recent years. Extracorporeal magnetic field has a significant effect in reducing the flow velocity when needed which can be critical in flow regulation to mitigate disease. Gireesha et al. (2015) studied the MHD fluid flow with the suspension of nanoparticles over a stretched sheet and investigated the influence of nanoparticle volume fraction and magnetic field on heat and mass transfer. Other studies have addressed the application of magnetic fields to manipulate nanoparticle concentration in fluid flow of relevance to nano-drug delivery (Mahanthesh et al., 2020a; Mahanthesh et al., 2020b). Mahanthesh et al. (2017), presented a detailed mathematical model for unsteady three-dimensional Eyring-Powell fluid flow under static magnetic field. They obtained extensive numerical results using a shooting technique coupled with a fourth-fifth order Runge–Kutta–Fehlberg scheme. Recently, Sreedevi et al. (Sreedevi et al., 2018; Sreedevi and Reddy, 2019), studied the effect of MHD heat and mass transfer on nanofluids containing single walled water-based carbon nanotubes (SWWNT) and multi-walled water-based carbon nanotubes in external flow from a vertical cone and internal flow between two stretchable rotating disks, respectively. They showed that with an increase in magnetic field parameter velocity is diminished and temperature is increased for both nanofluids. Reddy and Chamkha (2018) considered the three-dimensional hydromagnetic flow of alumina-water nanofluid over a stretching sheet, observing that with an increment in magnetic field heat transfer is strongly modified. MHD blood flows also feature in electromagnetic medical pumps, where for some specific cardiac operations, magnetic fields can be used to regulate flow rates. In diseased arteries, the effect of vessel tapering, in addition to, the shape of stenosis also constitutes an exciting scenario for magnetic blood flow simulation. Nadeem et al. (2014), discussed the effects of induced magnetic field on blood flow through stenosed vessels. This and other investigations have shown that the imposition of a magnetic field to streaming blood induces both electric and magnetic fields, which interact to generate a Lorentzian body force, which is resistive in nature and opposes the movement of blood (Mekheimer, 2008; Mekheimer and El Kot, 2008). Vasu et al. (2020), have more recently computed the MHD effect on blood flow through a stenosed coronary artery with extensive visualization, noting that blood velocity decreases with an increase in the magnetic field due to the Lorentz hydromagnetic drag force. Many different mathematical and computational studies have been reported on the influence of magnetization in arterial blood flow. Ponalagusamy and Selvi (2015) investigated the effect of magnetic field on the two-phase oscillatory blood flow by assuming core and plasma regions as a Newtonian fluid in the arterial stenosis, showing that an increment in magnetic field elevates flow resistance of the blood flow in a stenosed artery. Ponalagusamy and Priyadharshini (2018) extended the study in (Ponalagusamy and Selvi, 2015) to consider tapered stenotic and non-Newtonian effects in magnetized oscillatory two-phase blood flow. These studies however often neglected heat transfer effects which are also important since a key function of circulating blood is the transportation of heat. Prandtl numbers of streaming blood are known to be significantly higher than pure water and are critical to achieving thermo-regulatory functions in the cardiovascular system.”

The theoretical and numerical studies dealing with the effects of heat transfer and magnetic field on the pulsatile flow of blood in a saccular aneurysm at the distant bifurcated aorta, with blood considered as a non-Newtonian fluid, have received comparatively less attention. Most studies are either experimental or three-dimensional computational simulation of aneurysm in the cerebral region neglecting rheological, biomagnetic and thermal effects. It has been observed that the blood flow velocity as well as wall shear stress decreases by exposing biological systems to an external magnetic field which permits a powerful mechanism for flow control in saccular aortic aneurysm treatment. Motivated by extending these studies to more realistic cases, the present article describes a detailed mathematical and numerical study of the unsteady rheological magnetohydrodynamic blood flow, heat transfer in a bifurcated artery featuring a saccular aneurysm. Also known as berry or inter-cranial aneurysms, they exhibit a characteristic rounded shape and are the most frequent contributor to non-traumatic subarachnoid hemorrhages. The Carreau-Yasuda (Khan and Hashim, 2015; Ali et al., 2019) model is utilized for non-Newtonian (hemo-rheological) characteristics. A Fourier heat conduction model is deployed for thermal conduction heat transfer and unsteady nonlinear coupled convective heat transfer is considered in streaming blood flow. With appropriate boundary conditions the normalized conservation equations are solved with the finite element method using a variational approach provided in the commercial software, FreeFem++ (Vasu et al., 2020). These aspects constitute the novelties of the present work. The results elaborate on the influence of several non-dimensional parameters (Reynolds number (Re) and Prandtl number (Pr)) and magnetic body force parameter on velocity, skin friction coefficient, temperature profile and volumetric flow rate at the aneurysmic section, in addition to throughout the remainder of the bifurcated artery domain. The simulations of the present study are envisaged to be of relevance in furnishing improved characterization of the biomagnetic hemodynamics of realistic aneurysm bifurcations which will be of benefit in more detailed assessment, diagnosis and magnetic-assisted treatment of cardiovascular diseases. This article has therefore been motivated by the growing clinical applications of non-intrusive magnetic-assisted techniques in 21st century treatments. The advantage of numerical blood flow simulation is that it provides almost limitless (and relatively inexpensive) insights which can aid decision-making processes during the treatment of cardiovascular diseases. Although a conventional method for treating the aneurysm is to deploy a stent or catheter inside the artery, however, nowadays targeting the drugs at desired locations is increasingly becoming the new standard. This also triggers the process of clotting formation at the diseased part and the effects of such post-treatment processes can also be predicted by computational simulation. Detailed interpretation of the computations is also provided of direct relevance to the magnetohydrodynamic treatment of rheological blood flow in diseased arterial systems. Additionally, the numerical simulations provide a useful compliment to clinical studies and may prove beneficial in testing the hypothesis of disease formation and furthermore may be of benefit in the design of cardiovascular devices, heart valves, stents, probes etc.

Section snippets

Non-Newtonian thermo-magnetic blood flow model

An unsteady two-dimensional mathematical model for blood flow and coupled heat transfer in a bifurcated artery is considered wherein blood flow is modelled as non-homogeneous fluid. Blood rheology is simulated with the Carreau-Yasuda fluid model. Thermophysical properties are assumed constant. For the simulation, the pulsatile nature of blood has also been incorporated. The velocity is taken as zero at the walls of the vessel which is modelled as a bifurcated system with a singular saccular

Finite element simulation with FreeFEM++

The non-dimensional magnetic bio-rheological blood flow boundary value problem defined by Eqs. (31), (32), (33), (34) with boundary conditions (35) is formidable owing to strong nonlinearity, the coupling of many different variables, inclusion of two space variables and time. A robust computational scheme is, therefore, essential to obtain fast and rapidly convergent solutions. Finite element has been used vastly over different physical problems. The finite element method involves dividing the

Mesh independence analysis

By conducting several different finite element mesh (grid) distribution tests, it may be established whether the calculated numerical results are grid-independent or not. The numerical values for skin-friction coefficients, at the aneurysm, for various designs comprising unstructured fixed mesh elements with vertices and triangular elements, are provided in Fig. 3. Twelve different mesh distributions have been tested to ensure the simulated numerical results are mesh independent. Therefore, the

FREEFEM++ results and discussion

In this section, the quantitative effect of selected parameters i.e., magnetohydrodynamic body force parameter (M), Reynolds number (Re), and Prandtl number (Pr), on the velocity, temperature, pressure and skin-friction coefficient distributions with the variation of time are examined in detail. The results are all visualized via tables, contour plots, and graphs. In the computations, the default values of various parameters are as documented in Table 2.

Fig. 4(a)–(d) depicts the non-dimensional

Conclusions

In this study, motivated by providing a deeper insight into diseased cardiovascular flow dynamics, a finite element simulation of two dimensional magnetohydrodynamic heat conducting blood flow with coupled convective heat transfer through a bifurcated artery with a saccular (intercranial) aneurysm has been presented. The model generalizes previous studies by amalgamating the Carreau-Yasuda model, heat transfer and magnetohydrodynamics with complex geometric features in realistic arterial

CRediT authorship contribution statement

Ankita Dubey: Methodology, Software, Formal analysis, Investigation, Writing - Original Draft. B. Vasu: Conceptualization, Supervision, Writing - Review & Editing, Funding acquisition. O. Anwar Bég: Writing - Review & Editing, Visualization. Rama S R Gorla: Resources, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are grateful to the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India for undertaking the research work under the research project File Number: ECR/2017/001053 dated 12/03/2018.

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