Spatio-temporal evolution of magnetohydrodynamic blood flow and heat dynamics through a porous medium in a wavy-walled artery

https://doi.org/10.1016/j.compbiomed.2021.104595Get rights and content

Highlights

  • Blood flow through an arterial segment with wall deformation is studied.

  • MHD effect on temperature regulation is investigated in the porous medium.

  • Computational model is developed to observe stream function and vortex dynamics.

  • WSS has a reducing effect on Darcy number and enhancing effect on Hartmann number.

  • Space-average Nusselt number has diminishing impact on Darcy and amplitude number.

Abstract

Background and objective

In a healthy body, the elastic wall of the arteries forms wave-like structures resulting from the continuous pumping of the heart. The systolic and diastolic phases generate a contraction and expansion pattern, which is mimicked in this study by considering a wavy-walled arterial structure. A numerical investigation of the spatio-temporal flow of blood and heat transfer through a porous medium under the action of magnetic field strength is conducted.

Method

The governing equations of the blood flow in the Darcy model are simulated by applying a vorticity-stream function formulation approach. The transformed dimensionless equations are further discretized using the finite difference method by developing the Peaceman-Rachford alternating direction implicit (P-R ADI) scheme.

Results

The computational results for the axial velocity, temperature distribution, flow visualization using the streamlines and vorticity contours, isotherms, wall shear stress and the average Nusselt number are presented graphically for different values of the physical parameters.

Conclusions

The study shows that the axial velocity increases with an increase in the Darcy number, and a similar phenomenon is observed because of an amplitude variation in the wavy wall. Both temperature and wall shear stress decreases with an increase in the Darcy number. The average Nusselt number increases with the magnetic field strength, while it has a reducing tendency due to the permeability of the porous medium.

Introduction

Investigating blood flow through a wavy permeable artery is very significant in understanding nutrition and waste exchange from the artery to or from surrounding organs through the arterial walls. When the left ventricle of the heart pumps oxygenated blood into the aorta, with each rhythmic pump, blood in the arteries is pushed under high pressure and velocity, away from the heart. This causes systolic and diastolic pressures of blood which creates the wavy walled pattern of the artery. Systolic blood pressure measures the amount of pressure that the blood exerts during heartbeat in which the optimum systolic blood pressure is 120 mm Hg. By contrast, diastolic blood pressure measures the arterial pressure between heartbeats with an optimal diastolic blood pressure of 80 mm Hg. These rhythmic phases produce the pulsatile pressure gradient in the arteries generating contraction and expansion [1].

Studies of viscous flow in wavy channels have contributed to major findings in the fields of engineering and biology. For example, Uchida and Aoki [2] investigated viscous flow in a wavy motioned wall with recurring contraction and expansion. Sankar and Sinha [3] developed the solution to the impulsive motion of a viscous fluid within a wavy wall using the perturbation technique. They found that at a low Reynolds number, the amplitude of the waviness of the wall becomes less important as the liquid is dragged along the wall. In contrast, at high Reynolds number flow, the effects of viscosity are highlighted in the narrow layer closer to the wall. Recently, Shit and Majee [4] examined blood flow and heat transfer characteristics in a wavy walled arterial segment that was subjected to a magnetic field. They used computational schemes to simulate the complex wavy structure of the arteries and noticed that the increased velocity of blood flow was proportional to the increase in the amplitude of the wavy wall. Rasoulzadech and Panfilov [5] studied the viscous inertial flow in a wavy channel with permeable walls and obtained the asymptotic solution using the perturbation method considering the channels aspect ratio as the perturbation parameter. The flow characteristics in an asymmetric wavy wall channel for the steady case have been investigated experimentally by Nishimura et al. [6]. Their study showed that the flow varies from laminar to turbulent, forming circulatory vortices at the diverging cross-section of the wavy channel. Mustapha et al. [7] numerically investigated magnetohydrodynamic blood flow in the presence of an irregular stenosed artery. Their study mainly focused on surface irregularities and vortex dynamics.

Several other researchers [8,9] have also studied blood flow in a wavy channel both experimentally and numerically. Wang and Chen [10] demonstrated heat transfer through the wavy channel by convection. This was done with the help of a simple coordinate transformation and by the transfer of the wavy channel into parallel plane channel. The governing equation for this was solved by the spline alternating direction method. The vessel wall is composed of soft tissue having complex biomechanical properties that vary periodically, depending upon the pulsatile pressure gradient. Debbich et al. [11] developed an experimental model based on the PC-MR and US-Doppler image to estimate the velocity in the carotid arterial bifurcation. Piskin and Celebi [12] numerically investigated the patient-specific model based on computed tomography data for pulsatile blood flow in a carotid artery with stenosis. A patient-specific computational model for blood flow and its thermal analysis was developed by Li et al. [13]. The MRI data has been used to create 3D geometry for studying laminar blood flow in their model. Foong et al. [14] numerically studied the blood flow and heat transfer analysis in a straight arterial segment by considering blood as the non-Newtonian Sisko fluid model. The viscosity effect of blood flow using the power-law fluid model has been studied by Sedeh et al. [15], and Sedeh and Toghraie [16] to observe coronary congenital heart disease. They have used SIMPLE scheme to study computational fluid dynamics of laminar blood flow. However, all the studies listed above were restricted in consideration of the porous medium of blood flow with electromagnetic effects.

In bio-fluid dynamics, the blood flow through the porous medium is an interesting subject. In the biological context, when there is an accumulation of fatty plaques in the lumen of an arterial segment and artery-clogging takes place by blood clots, the clogged region during blood flow is treated as a porous medium. The porous media are designated by their specific surface (s) and porosity (k), where s is the total surface area per total volume and k is the void volume per total volume. Macroscopic equations describe the study of blood flow in a porous medium by Darcy's law and mass conservation. Darcy first proposed the flow model, which specifies the solid-liquid interaction in a porous medium in 1856. When fluid passes through a porous medium, an equal and opposite solid particle's drag force is applied, similar to the force exerted by the solid particles of the media on the fluid. The flow pressure gradient balances this drag force. Most analytical studies dealt primarily with the mathematical formulation based on Darcy's law, which neglects the effects of the inertial forces or a solid boundary on fluid flow and heat transfer through porous media. An interesting fact to note is that Darcy's law is valid for flow past a low permeability porous region, where permeability is the measure of the flow conductivity in the porous medium. Vafai and Tien [17] studied the effects of the internal forces and the solid boundary of the flow past a porous medium. They have shown that due to porosity in the medium, an impermeable boundary occurred. The Casson fluid flow through a tube which has a homogenous porous medium was investigated by Dash and Mehta [18]. They analyzed the pathological situation of blood flow in the lumen of the coronary artery caused by cholesterol's fatty plaques. Khaled and Vafai [19] proposed a model of the flow and heat transfer in the biological tissues through a porous medium. Shit and Roy [20] have analyzed the peristaltic flow through a non-uniform porous channel with the influence of magnetic field and slip velocity. Chaturvedi et al. [21] examined the flow of blood and heat through a porous medium in the presence of a magnetic field. They have examined the effect of porosity on the Nusselt number and wall shear stress. Moreover, Zheng et al. [22] investigated the patterns of flow in the case of fluid injection into a restricted porous medium. They have reported theoretical and numerical studies of the flow characteristics in a situation when fluid is injected into a porous medium with another fluid having different viscosity. However, none of these studies were carried out in arteries having contraction and expansion of the arterial wall.

The importance of the magnetic field can not be neglected in many physical phenomena such as chemical industries (oxygenation, movement of nutrients of blood etc. Saqib et al. [23] considered a model of magnetohydrodynamics (MHD) flow with heat transfer through the porous medium. Shit et al. [24,25] studied the effect of magnetic field on the blood flow through the arteries with the coupling influence of heat transfer and concentration of blood. Ellahi et al. [26] studied the flow of blood of Prandtl fluid through a stenosed artery having permeable walls with magnetic fields. They have analytically computed the effects of the Darcy number and slip parameter with the perturbation method. Jaiswal and Yadav [27] investigated the micropolar effect of blood flow (considering micro-rotation of blood cells) through porous layered arteries. They summarized the fact that flow dynamics are significantly altered for varied permeabilities of Darcy and Brinkman porous layer zones in the artery. As a sequel to our previous study (Shit and Majee [4]), the present work continues to develop a numerical model with a wavy walled arterial segment. The above rigorous literature survey reveals that no such attempt has been made earlier.

In the present study, we have examined the blood flow and heat transfer through an artery with wavy walls in the presence of an externally applied magnetic field and porous medium. The vorticity and stream function formulation are employed for direct numerical simulation of the governing equations using the P-R Alternating Direction Implicit (ADI) method. The average Nusselt number, wall shear stress and flow rate with varying Hartmann number, Darcy parameter, and the amplitude of wavy walls are the important results. The quantitative and qualitative analyses were carefully made with the help of a graphical representation of our results generated from the numerical experiments. The state-of-the-art of this work is presented in table-1.

Section snippets

Mathematical formulation

In the present study, we considered a cylindrical polar coordinates system (r′, θ′, x′), in which x′−axis represents the direction of the arterial flow and all the flow characteristics are independent of azimuthal coordinate θ′. We denoted the velocity vector V in which u′ and v′ are the velocity components along with the axial and radial directions, respectively. Let us consider a uniform magnetic field of strength B0 which is applied perpendicular to the direction of the axial flow. The

Coordinate transformation

To solve the above equations numerically, the physical arterial coordinates are transformed into a rectangular computational domain byξ(x,r)=rR(x).

Using the transformation (22) into equation (18), we getu=1ξR2ψξ,andv=1ξRψxξRRxψx.

Use of Eqs. (22)–(24), the transformation of Eqs. (20), (21) reduce toωt+ωξ2R2ψxωnR3Rxψξ+1nR2ψξωx1nR2ψxωξ=1Re2ωx2ωξ2R2+3ξR2Rx2ξR2Rx2+1ξR2wξ+ξ2R2Rx2+1R22ωξ22ξRRx2ωξx+Ha2ξ2ReR3ξ2ψξ2ψξ1ReDaω,2ψx2+1ξR23ξ2Rx2ξ2R2Rx2

Numerical methodology

The rectangular computational domain is subdivided into the cartesian cells by drawing straight lines parallel to the coordinate axis. The governing equations are then discretized to convert the partial differential equations into a set of algebraic finite difference equations. The discretization process is followed as per the Peaceman-Rachford Alternating Direction Implicit (ADI) method. Eq. (26) is solved by the Successive Over Relaxation (SOR) method as it is a linear equation in ψ. The

Results and discussion

In this work, we have investigate the MHD blood flow and heat transfer through a porous medium under the effect of a magnetic field. The numerical simulation has been performed to obtain the information about flow phenomena from the stream function, vorticity, and energy equations. We have employed the standard second order ADI-Peaceman Rachford method with the help of an implicit finite difference scheme. The discretized equations are solved efficiently with the tri-diagonal matrix algorithm,

Conclusions

In the numerical investigation, the influence of magnetic field strength and permeability of the medium is analyzed on the blood flow and heat transfer past an arterial segment. The governing equations are solved by implementing the vorticity-stream function formulation. The stream function and vorticity equations are discretized using a finite difference PR-ADI scheme. The effects of the Reynolds number Re, Darcy number Da, the Hartmann number Ha and the wavy wall parameter α on the flow and

Authors contribution

S. Majee developed the numerical code and partly written the manuscript, S. Maiti did all mathematical calculation and plotted figures, G. C. Shit conceived the problem and verified the results, D.K. Maiti supervised the overall numerical computation and finalize the manuscript.

Declaration of competing interest

The authors declare that they do not have any competing interest for publication of this manuscript in this journal.

Acknowledgement

The authors are grateful to the esteemed reviewers for their constructive comments and suggestions based on which this article is improved. We are also gratefully acknowledge the SERB, DST, Government of India for financial support with grant no EEQ/2016/000050.

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