Model of LDL-C concentration of blood flow through a vertical porous microchannel with multiple stenoses: computational simulation

Abstract

The accumulations of lipid low-density lipoprotein cholesterol (LDL-C) and other chemicals on the artery wall are known as atherosclerosis. Atherosclerosis can constrict the arteries and obstruct the blood flow. Our goal is to debate the model of unsteady pulsatile Cross fluid (blood model) flows through a vertical porous microchannel with multiple stenoses under the influence of thermal radiation and Joule heating. The equations of momentum, energy, and LDL-C concentration have been simplified with the help of mild stenosis approximation. Then they have been solved numerically by using the finite difference method. It is noticed that the blood velocity increases and reaches a steady state in the case of non-pulsating flow, while the blood velocity fluctuates in the case of pulsating flow. Moreover, the value of LDL-C concentration in the case of a chemical reaction is lower than in the absence of a chemical reaction.

Keywords:

• Cross fluid
• low-density lipoprotein cholesterol (LDL-C)
• multiple stenoses
• finite difference method

Nomenclature

$A_o$	=	Steady component of pressure gradient [m]
$A_1$	=	Fluctuating component of pressure gradient [m]
$B_r$	=	Brickmann number
$B_o$	=	Magnetic field [kg s${}^{-2}$ A${}^{-1}$]
$C^{\prime }$	=	Concentration of the fluid [kg m${}^{-2}$ s${}^{-2}$]
$c_p$	=	Specific heat at constant pressure [m${}^2$ s${}^{-2}$ K${}^{-1}$]
D	=	Mass diffusion coefficient [m${}^2$ s${}^{-1}$]
$D_a$	=	Darcy number
e	=	Amplitude fluctuation parameter
$f_p$	=	Pulse frequency [Hz]
$G_c$	=	Solutal Grashof parameter
$G_r$	=	Grashof parameter
g	=	Acceleration of gravity [m s${}^{-2}$]
$H_a$	=	Hartmann number
K	=	Thermal conductivity of the fluid[kg m s${}^{-3}$ K${}^{-1}$]
$K_o$	=	Chemical term [s${}^{-1}$]
$K_T$	=	Thermal-Diffusion ratio [kg m${}^{-2}$ s${}^{-2}$]
$k_1$	=	Permeability [m${}^2$]
$k_2$	=	Rosseland absorption coefficient [m${}^{-1}$]
L	=	Length of the arterial stenosis [m]
$l_o$	=	Inlet segment [m]
$l_{o1}$	=	Length of first stenoses [m]
$l_{o2}$	=	Distance between stenoses [m]
$l_{o3}$	=	Length of second stenoses [m]
n	=	power-law index number
$p^{\prime }$	=	Pressure of the fluid [kg m${}^{-1}$ s${}^{-2}$]
$P_r$	=	Prandtl number
$Q_o$	=	Heat source [kg m${}^{-1}$ s${}^{-3}$ K${}^{-1}$]
$R_a$	=	Radiation parameter
$R_e$	=	Reynolds number
$S^{\prime }$	=	Stress component [kg m${}^{-1}$ s${}^{-2}$]
$S_c$	=	Schmidt number
$S_r$	=	Soret number
$T^{\prime }$	=	Temperature of the fluid [K]
$t^{\prime }$	=	Time [s]
$T_a$	=	Mean fluid temperature [K]
$T_m$	=	Temperature of the medium [K]
$T_o{T}_1$	=	Walls temperature [K]
$u^{\prime }$	=	Axial component of velocity [m s${}^{-1}$]
$v^{\prime }$	=	Normal component of velocity [m s${}^{-1}$]
$W_e$	=	Weissenberg number
$x^{\prime }$	=	Axial distance [m]
$y^{\prime }$	=	Normal distance [m]
α	=	Dimensionless chemical reaction parameter
β	=	Dimensionless heat source
${\text{β}}_C$	=	Concentration expansion coefficient[kg${}^{-1}$ m${}^2$ s${}^2$]
${\text{β}}_T$	=	Thermal expansion coefficient [K${}^{-1}$]
${\gamma }^2$	=	Womersley frequency parameter
${\delta }_1$	=	Critical height of first stenosis [m]
${\delta }_2$	=	Critical height of second stenosis [m]
θ	=	Dimensionless temperature
${\mu }_o$	=	Viscosity of the fluid [kg m${}^{-1}$ s${}^{-1}$]
${\mu }_{\mathrm{\infty }}$	=	Infinite shear rate viscosity [kg m${}^{-1}$ s${}^{-1}$]
ρ	=	Density of the fluid [kg m${}^{-3}$]
σ	=	Electrical conductivity of the fluid [kg${}^{-1}$ m${}^{-3}$ s${}^3$ A${}^2$]
${\sigma }_o$	=	Stefan–Boltzmann constant [kg s${}^{-3}$ K${}^{-4}$]
ϕ	=	Dimensionless concentration
Γ	=	Cross time constant

1. Introduction

Stenosis of the arteries occurs due to the accumulation of fatty substances called (plaques or atheromas) in the linings of the blood vessels. Sometimes, an artery can become very tight to receive adequate oxygenated blood to the tissues and organs of the body. When this fat's sedimentation becomes so large or expanded, it may explode, and over time, harm may partially or entirely occlude the artery. This process is known as atherosclerosis and leads to coronary heart disease (CHD). Atherosclerosis is a prolonged result of the buildup of fatty material that deposits start when the lining of blood vessels becomes damaged, causing cholesterol carried on lipoproteins such as LDL (low-density lipoprotein). HDL (high-density lipoprotein) can remove cholesterol from these deposits. Diminishing LDL cholesterol, raising HDL cholesterol, and lessening other risk factors can help slow down atherosclerosis [1]. Recently, Hepatitis B and Coronavirus (COVID-19) have been considered the most common factors which increase the risk factors for atherosclerosis [2–4]. Studying blood flow in a narrowed artery helps understand circulatory disorders. The stenosed artery depends on the stenosis shape, length, and height. Hence the mathematical study of such a situation is significant [5,6].

The Pulsatile blood flow through stenosed artery has been of great interest to researchers because of its relevance to medical and biomedical sciences. The study of blood flow through the narrowed artery and the body acceleration is essential in many cardiovascular and tumour growth diseases. For the blood to reach the body's various muscles must be pumped from the heart through the arteries, which have a pulsating nature due to the pressure gradient. Hence the accumulated deposition of cholesterol, plaques and abnormal tissue growth resulting from the thickening of the artery lumen on the artery walls will lead to diseases such as stroke due to hypertension, hypotension, heart attacks, possible tumour and cancer [7].

The Cross fluid model was introduced by Cross (1965) [8]. The Cross fluid model is a subclass of generalized Newtonian fluids that can suitable rheological model for predicting shear-thinning effects for both low and high shear rates [9]. The cross form is applied in synthesizing many polymeric solutions such as blood, 35$\mathrm{\%}$ aqueous solution of xanthan gum, and 40$\mathrm{\%}$ aqueous solution of polyacrylamide [10,11]. Cross fluid flows have viscoelastic behaviours and are granular containing incoherent sedimentary motions or glassy particles such as non-Newtonian fluids that are frequently and widely used in industrial, chemical and technological applications (e.g. melting polymers, biological solutions, paints, and transport and emulsification processes in microtubules, etc.) [12–15].

A magnetic field strongly influences biological fluid dynamics, with implications for bioengineering and medical technology. It is, therefore, possible that the flow of blood can get affected by the magnetic fields. Applications of magnetic fields emerging in health-related areas are demonstrated in different positions, including treating diseases. Magnetotherapy is a substitutional medicinal treatment in which magnets are used to relieve soreness and multi-health apprehensions and is applied to anyone with coronary artery disease [16].

All human body tissues have pores, and the plaque area contains a sizeable sebaceous nucleus and soft fibrous covering. If the plaque ruptures, the lining of atherosclerosis is intensely outlet to white cells and platelets. The porosity of the endothelial artery wall is thought to increase with the sedimentation of cholesterol due to the corrupted or inflamed artery wall [17].

Heat and mass transfer have many intricate operations, such as thermal conduction in tissues, metabolic thermogenesis, electromagnetic radiation emanated by mobile, and heat convection due to blood flow via tissue pores. Heat and mass transfer are essential because oxygen and nutrients are prevalent from blood vessels to adjacent tissues [18]. Radiation is vital in high-temperature applications such as boilers, furnaces, IC engines, and insulators in science and engineering. Moreover, the survey of thermal radiation has implementations in atmospheric fields for weather forecast and remote sensing as well as in medicine for diagnosing tumours in biological systems, treating cancers by nanoparticles, etc. [19].

Studying Magnetohydrodynamics (MHD), viscous dissipation, and joule heating effects are essential and significant in geophysical flow and nuclear engineering [20]. Low blood pressure leads to a reduction in blood flow rate. It is also observed that the blood flow is affected by a magnetic field because the red blood cells are an essential biomagnetic material. MHD is handy for targeting various magnetic drugs for cancer and is also beneficial in multiple engineering issues such as cooling liquid metals for nuclear reactors, plasma confinement, and electromagnetic and continuous casting process of metals [21]. The scrutinization of heat and mass transfer by chemical interactions is paramount in the chemical and metallurgical industries. Moreover, the influences of heat radiation on forced and free convection flow are principal in the purport of astronomical phenomena and high-temperature operations [22].

Efficient energy utilization and the eclectic exhaustion of resources have incentivized researchers to optimize the efficiency of manufacturing operations. Valuation of entropy generation and non-Newtonian fluids are essential and efficient techniques for investigating optimum heat transfer [23]. Entropy study is instrumental in vital thermodynamics to grasp the flow features in narrow areas such as diseased arteries. Breaking down nutrients into components ameliorate cells, tissues, and muscles in the human body and reduces entropy. The human body diverts heat to the environment due to changes in body temperature and ambient medium. Nutrients consume and emit heat into the emptiness by disposal of residue in the form of carbon dioxide, water, urine and faeces. Thus, the total entropy of the human body goes up. The rise of body entropy affects the vital fluids in the human body, such as blood flow, semen transfer, urine fluidity, and swallowing food via the oesophagus [24]. The study of entropy measures the uninitialized energy in any restricted thermodynamic system. Entropy is a gauge of the system's disturbance and is a feature of the system's situation which immediately changes with any invert variation in heat [25].

In this paper, we will shed light on the influence of the accumulation of lipid's low-density lipoprotein cholesterol (LDL-C) on the pulsatile flow of unsteady magnetohydrodynamic Cross fluid as a blood model through a vertical porous microchannel with multiple stenoses. The equations of momentum, energy, and LDL-C concentration have been simplified with the help of mild stenosis approximation. Then they have been solved numerically by using the finite difference method. Graphical results and discussions are explained, and some conclusions are included.

2. Problem formulation

Let us consider unsteady electrically conducted blood flow has low-density lipoprotein cholesterol (LDL-C) through a vertical porous artery with mild double stenoses. The effects of the buoyancy and Joule heating are taken into account. The non-Newtonian Cross fluid model was chosen to represent the blood. Let $x^{\prime }$-axis is taken in the direction of the main flow of the fluid, while $y^{\prime }$-axis is in the perpendicular direction of the fluid flow. An external magnetic field $B_o$ in $y^{\prime }$-direction is applied. The arterial wall $y^{\prime }=H^{\prime }\left(x^{\prime }\right)$ is kept at a temperature $T_1$ and concentration $C_1$, while the other wall $y^{\prime }=-H^{\prime }\left(x^{\prime }\right)$ is kept at a temperature $T_0$ and concentration $C_0$. The geometrical shape of the constricted artery is shown in Figure 1 [26]: (1) $\begin{array}{r}{H}^{\prime }\left(x\right)=\{\begin{array}{ll}{h}_o& \mathrm{if}0\le x^{\prime }\le l_o\\ \begin{array}{c}{h}_o+\frac{4{\delta }_1}{l_{o1}^2}\left[\right(x^{\prime }-l_o{)}^2\\ -l_{o1}(x^{\prime }-l_o)]\end{array}& \mathrm{if}{l}_o\le x^{\prime }\le l_o+l_{o1}\\ h_o& \begin{array}{c}\mathrm{if}{l}_o+l_{o1}\le x^{\prime }\le l_o\\ +l_{o1}+l_{o2}\end{array}\\ \begin{array}{c}{h}_o+\frac{4{\delta }_2}{l_{o3}^2}\left[\right(x^{\prime }-d{)}^2\\ -l_{o3}(x^{\prime }-d)]\end{array}& \begin{array}{c}\mathrm{if}{l}_o+l_{o1}+l_{o2}\le x^{\prime }\\ \le l_o+l_{o1}+l_{o2}+l_{o3}\end{array}\\ h_o& \mathrm{if}{l}_o+l_{o1}+l_{o2}+l_{o3}\\ & \le x^{\prime }\le L\end{array}\end{array}$(1) where $({\delta }_1,{\delta }_2)$ represent the critical altitudes of the first and second narrow respectively, $h_o$ is the artery radius in non-stenotic areas, $l_o$ is the site of multiple stenoses, $l_{o2}$ is the distance between stenoses, $(l_{o1},l_{o3})$ are the longitude of multiple stenoses, L is arterial segment length, and $d=l_o+l_{o1}+l_{o2}$. At any value of x', the pressure gradient can be expressed as: (2) $-\frac{\mathrm{d}{p}^{\prime }}{\mathrm{d}{x}^{\prime }}=A_0+A_1\cos \left({\omega }_P{t}^{\prime }\right),$(2) where $\left(A_0\mathrm{\&}{A}_1\right)$ are the steady and fluctuating elements of pressure gradient respectively, ${\omega }_p=2\pi f_p$, $f_p$ is the pulse frequency. Both $A_0,A_1$ are functions of x. The following governing equations might be used to list the problem's mathematical description:

Figure 1. Geometrical configuration.

Continuity: (3) $\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{y}^{\prime }}=0,$(3) Momentum: (4) $\begin{array}{rl}& \rho (\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{t}^{\prime }}+u^{\prime }\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+v^{\prime }\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{y}^{\prime }})\\ & =-\frac{\mathrm{\partial }{p}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+\frac{\mathrm{\partial }{S}_{\mathrm{xx}}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+\frac{\mathrm{\partial }{S}_{\mathrm{xy}}^{\prime }}{\mathrm{\partial }{y}^{\prime }}-\sigma B_o^2{u}^{\prime }-\frac{{\mu }_o}{k_1}{u}^{\prime }\\ & +\mathrm{\rho g}{\text{β}}_T(T^{\prime }-T_o)+\mathrm{\rho g}{\text{β}}_C(C^{\prime }-C_o),\end{array}$(4) (5) $\begin{array}{rl}& \rho (\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{t}^{\prime }}+u^{\prime }\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+v^{\prime }\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{y}^{\prime }})\\ & =-\frac{\mathrm{\partial }{p}^{\prime }}{\mathrm{\partial }{y}^{\prime }}+\frac{\mathrm{\partial }{S}_{\mathrm{yx}}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+\frac{\mathrm{\partial }{S}_{\mathrm{yy}}^{\prime }}{\mathrm{\partial }{y}^{\prime }}-\frac{{\mu }_o}{k_1}{v}^{\prime },\end{array}$(5) Energy equation: (6) $\begin{array}{rl}& \rho c_p(\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{t}^{\prime }}+u^{\prime }\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+v^{\prime }\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{y}^{\prime }})\\ & =S_{\mathrm{xx}}^{\prime }\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+S_{\mathrm{xy}}^{\prime }\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+S_{\mathrm{yx}}^{\prime }\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{y}^{\prime }}+S_{\mathrm{yy}}^{\prime }\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\\ & +K(\frac{{\mathrm{\partial }}^2{T}^{\prime }}{\mathrm{\partial }{x}^{\prime 2}}+\frac{{\mathrm{\partial }}^2{T}^{\prime }}{\mathrm{\partial }{y}^{\prime 2}})+\frac{16{\sigma }_o{T}_a^3}{3k_2}\frac{{\mathrm{\partial }}^2{T}^{\prime }}{\mathrm{\partial }{y}^{\prime 2}}\\ & +\sigma B_o^2{u}^{\prime 2}+\frac{{\mu }_o}{k_1}(u^{\prime 2}+v^{\prime 2})+Q_o(T^{\prime }-T_o),\end{array}$(6) Concentration equation: (7) $\begin{array}{rl}& \frac{\mathrm{\partial }{C}^{\prime }}{\mathrm{\partial }{t}^{\prime }}+u^{\prime }\frac{\mathrm{\partial }{C}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+v^{\prime }\frac{\mathrm{\partial }{C}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\\ & =D(\frac{{\mathrm{\partial }}^2{C}^{\prime }}{\mathrm{\partial }{x}^{\prime 2}}+\frac{{\mathrm{\partial }}^2{C}^{\prime }}{\mathrm{\partial }{y}^{\prime 2}})+\frac{D{K}_T}{T_m}(\frac{{\mathrm{\partial }}^2{T}^{\prime }}{\mathrm{\partial }{x}^{\prime 2}}+\frac{{\mathrm{\partial }}^2{T}^{\prime }}{\mathrm{\partial }{y}^{\prime 2}})\\ & -K_o(C^{\prime }-C_o),\end{array}$(7) where ρ is the density of the fluid, $(u^{\prime },v^{\prime })$ are axial and perpendicular velocities, consecutively, p represents the fluid pressure, σ is electrical conductivity of the fluid, ${\mu }_o$ is the fluid viscosity, $k_1(>0)$ is the constant of permeability, g is the gravity, ${\text{β}}_T$ represents the volumetric expansion, ${\text{β}}_C$ is the volumetric expansion due to concentration, $T^{\prime }$ is the temperature, $c_p$ represents the specific heat at a fixed pressure, K denotes the thermal conductivity, $k_2$ is the Rosseland absorption coefficient, ${\sigma }_o$ is the Stefan–Boltzmann constant, $T_a$ is the mean fluid temperature, $Q_o$ is the constant heat source, $C^{\prime }$ is the concentration of fluid, D is the coefficients of mass diffusivity, $K_T$ is the thermal-diffusion ratio, $T_m$ is the temperature of the medium and $K_o$ is the chemical term.

The extra stress tensor for Cross fluid is mathematically formulated as [8–10] (8) ${\mathbf{\text{S}}}^{\prime }={\mu }_{\mathrm{\infty }}+({\mu }_o-{\mu }_{\mathrm{\infty }})\left[\frac{1}{1+\{\mathrm{\Gamma }{\dot{\eta }}^{\prime }{\}}^n}\right]{\mathbf{A}}_{\mathbf{1}}^{\prime },$(8) where ${\mu }_{\mathrm{\infty }}$ and ${\mu }_o$ are the infinite shear rate and zero shear rate viscosities respectively, n is the power-law index, Γ represents the Cross time constant. The shear rate ${\dot{\eta }}^{\prime }$ and the first Rivlin–Ericksen tensor ${\mathbf{\text{A}}}_1^{\prime }$ can be expressed as (9) $\begin{array}{r}{\dot{\eta }}^{\prime }=\sqrt{\frac{{\mathrm{\Pi }}^{\prime }}{2}}=\sqrt{\frac{1}{2}\mathrm{tr}({{\mathbf{A}}_{\mathbf{1}}}^{\prime }{\mathbf{)}}^{\mathbf{2}}}\mathrm{and}{\mathbf{A}}_{\mathbf{1}}^{\prime }=\mathrm{\nabla }{\mathbf{V}}^{\prime }+(\mathrm{\nabla }{\mathbf{V}}^{\prime }{)}^{\mathrm{\top }},\end{array}$(9) where ${\mathbf{V}}^{\prime }=(u^{\prime },v^{\prime },0)$ is the velocity vector and ${\mathrm{\Pi }}^{\prime }$ is the second invariant strain rate tensor.

It is common to assume that the infinite shear rate viscosity ${\mu }_{\mathrm{\infty }}$ is zero [9,27]. In the light of this assumption, the Cross fluid's stress tensor has the following form (10) ${\mathbf{\text{S}}}^{\prime }={\mu }_o\left[\frac{1}{1+\{\mathrm{\Gamma }{\dot{\eta }}^{\prime }{\}}^n}\right]{\mathbf{A}}_{\mathbf{1}}^{\prime }\mathbf{.}$(10) It's noteworthy to notice that the Cross fluid exhibits shear-thinning behaviour for $(0<n<1)$, whereas the standard Newtonian model may be recovered when $\mathrm{\Gamma }\to 0$. The initial and boundary conditions are: (11) $\begin{array}{r}\begin{array}{rl}& u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_o,C^{\prime }=C_o\mathrm{at}{t}^{\prime }=0,\\ & u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_o,C^{\prime }=C_0\mathrm{at}{y}^{\prime }=-H^{\prime }\left(x^{\prime }\right),\\ & u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_1{C}^{\prime }=C_1\mathrm{at}{y}^{\prime }=H^{\prime }\left(x^{\prime }\right).\end{array}\end{array}$(11) The physical quantities of significance are skin friction coefficient $C_f^{\prime }$, Nusselt number $N_u^{\prime }$ and Sherwood number $S_h^{\prime }$ which are defined as: (12) $\begin{array}{rl}{C}_f^{\prime }& =\frac{{\tau }_w^{\prime }}{\rho {\left(\frac{A_o{h}_o^2}{{\mu }_o}\right)}^2},N_u^{\prime }=\frac{h_o{q}_w^{\prime }}{K(T_1-T_o)},\\ S_h^{\prime }& =\frac{h_o{h}_w^{\prime }}{D(C_1-C_o)}\end{array}$(12) where ${\tau }_w^{\prime }$ is the wall skin friction, $q_w^{\prime }$ is the surface heat flux and $h_w^{\prime }$ the wall mass flux given by (13) $\begin{array}{rl}{\tau }_w^{\prime }& =S_{\mathrm{xy}}^{\prime },q_w^{\prime }=-K\left(\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\right),h_w^{\prime }=-D\left(\frac{\mathrm{\partial }{C}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\right)\mathrm{at}\\ y^{\prime }& =\pm H^{\prime }\left(x^{\prime }\right)\end{array}$(13) Non-dimensional variables and parameters are defined as: (14) $\begin{array}{r}\begin{array}{rl}y& =\frac{y^{\prime }}{h_o},x=\frac{x^{\prime }}{l_o},u=\left(\frac{{\mu }_o}{A_o{h}_o^2}\right)u^{\prime },\\ v& =\left(\frac{l_o{\mu }_o}{A_o{h}_o^2\delta }\right)v^{\prime },H=\frac{h^{\prime }}{h_o},t=f_p{t}^{\prime },p=\frac{p^{\prime }}{A_o{l}_o},\\ \theta & =\frac{T^{\prime }-T_o}{T_1-T_o},\varphi =\frac{C^{\prime }-C_o}{C_1-C_o},S_{\mathrm{xx}}=\frac{l_o}{A_o{h}_o^2}{S}_{\mathrm{xx}}^{\prime },\\ S_{\mathrm{xy}}& =\frac{S_{\mathrm{xy}}^{\prime }}{A_o{h}_o},S_{\mathrm{yx}}=\frac{S_{\mathrm{yx}}^{\prime }}{A_o{h}_o},S_{\mathrm{yy}}=\frac{l_o}{A_o{h}_o^2}{S}_{\mathrm{yy}}^{\prime },\\ \dot{\eta }& =\frac{{\mu }_o}{A_o{h}_o}{\dot{\eta }}^{\prime },{\gamma }^2=\frac{\rho h_o^2{f}_p}{{\mu }_o},R_e=\frac{\rho A_o{R}_o^3}{{\mu }_o^2},\\ e& =\frac{A_1}{A_o},\mathrm{Ha}=\sqrt{\frac{\sigma }{{\mu }_o}}{B}_o{h}_o,\mathrm{Da}=\frac{k_1}{R_o^2},\\ G_r& =\frac{\mathrm{\rho g}{\text{β}}_T(T_1-T_o)}{A_o},G_C=\frac{\mathrm{\rho g}{\text{β}}_C(C_1-C_o)}{A_o},\\ W_e& =\frac{\mathrm{\Gamma }{A}_o{h}_o}{{\mu }_o},P_r=\frac{c_p{\mu }_o}{K},R_a=\frac{16{\sigma }_o{T}_a^3}{3k_{2}K},\\ B_r& =\frac{A_o^2{h}_o^4}{{\mu }_o(T_1-T_o)K},\text{β}=\frac{Q_o{h}_o^2}{K},S_c=\frac{{\mu }_o}{\mathrm{\rho D}},\\ S_r& =\frac{\mathrm{\rho D}{K}_T(T_1-T_o)}{{\mu }_o{T}_m(C_1-C_o)},\alpha =\frac{\rho K_o{h}_o^2}{{\mu }_o}.\end{array}\end{array}$(14) According to the premises and simplifications mentioned in Young (1968) for mild stenosis $(\frac{\delta }{h_o}<<1)$, $(\frac{h_o}{l_o}\sim o(1\left)\right)$ [28], Equations (3(3) $\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{x}^{\prime }}+\frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{y}^{\prime }}=0,$(3) )–(11(11) $\begin{array}{r}\begin{array}{rl}& u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_o,C^{\prime }=C_o\mathrm{at}{t}^{\prime }=0,\\ & u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_o,C^{\prime }=C_0\mathrm{at}{y}^{\prime }=-H^{\prime }\left(x^{\prime }\right),\\ & u^{\prime }=0,v^{\prime }=0,T^{\prime }=T_1{C}^{\prime }=C_1\mathrm{at}{y}^{\prime }=H^{\prime }\left(x^{\prime }\right).\end{array}\end{array}$(11) ) using the above assumption in the non-dimensional form are as follows: (15) $\begin{array}{rl}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}& =0,\end{array}$(15) (16) $\begin{array}{rl}{\gamma }^2\frac{\mathrm{\partial }u}{\mathrm{\partial }t}& =1+\mathrm{ecos}\left(2\mathrm{\pi t}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left[\frac{\frac{\mathrm{\partial }u}{\mathrm{\partial }y}}{1+{\left\{W_e\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)\right\}}^n}\right]\\ & -(H{a}^2+\frac{1}{\mathrm{Da}})u+G_r\theta +G_c\varphi ,\end{array}$(16) (17) $\begin{array}{rl}{P}_r{\gamma }^2\frac{\mathrm{\partial }\theta }{\mathrm{\partial }t}& =(1+\mathrm{Ra})\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+B_r[\frac{{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2}{1+{\left\{W_e\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)\right\}}^n}\\ & +(H{a}^2+\frac{1}{\mathrm{Da}})u^2]+\text{β}\theta ,\end{array}$(17) (18) $\begin{array}{rl}{S}_c{\gamma }^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}& =\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+S_r{S}_c\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}-\alpha S_c\varphi .\end{array}$(18) The associated boundary conditions: (19) $\begin{array}{rl}& u=0,\theta =0,\varphi =0\mathrm{at}t=0,\\ & u=0,\theta =0,\varphi =0\mathrm{at}y=-H\left(x\right),\\ & u=0,\theta =1\varphi =1\mathrm{at}y=H\left(x\right),\end{array}$(19) where (20) $\begin{array}{r}H\left(x\right)=\{\begin{array}{ll}1& \mathrm{if}0\le x\le 1\\ \begin{array}{c}1+{\displaystyle \frac{4{\delta }_1^{\ast }}{L_{o1}^2}}\left[\right(x-1{)}^2\\ -L_{o1}(x-1)]\end{array}& \mathrm{if}1\le x\le 1+L_{o1}\\ 1& \begin{array}{c}\mathrm{if}1+L_{o1}\le x\le 1\\ +L_{o1}+L_{o2}\end{array}\\ \begin{array}{c}1+{\displaystyle \frac{4{\delta }_2^{\ast }}{L_{o3}^2}}\left[\right(x-d^{\ast }{)}^2\\ -l_{o3}(x-d^{\ast })]\end{array}& \mathrm{if}{d}^{\ast }\le x\le d^{\ast }+L_{o3}\\ 1& \mathrm{if}{d}^{\ast }+L_{o3}\le x\le L^{\ast }\end{array}\end{array}$(20) where ${\delta }_1^{\ast }=\frac{{\delta }_1}{h_o},{\delta }_2^{\ast }=\frac{{\delta }_2}{h_o},L_{o1}=\frac{l_{o1}}{l_o},L_{o2}=\frac{l_{o2}}{l_o},d^{\ast }=1+L_{o1}+L_{o2},L_{o3}=\frac{l_{o3}}{l_o},L_{o4}=\frac{l_{o4}}{l_o},L^{\ast }=d^{\ast }+L_{o3}+L_{o4}$.

The skin friction coefficient, the Nusselt number, and the Sherwood number in dimensionless forms are: (21) $\begin{array}{rl}{C}_f& =\frac{1}{R_e}\left[\frac{\frac{\mathrm{\partial }u}{\mathrm{\partial }y}}{1+{\left(W_e\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^n}\right],N_u=-\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\right),\\ S_h& =-\left(\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}\right)\mathrm{at}y=\pm H\left(x\right)\end{array}$(21)

3. Analysis of entropy generation

Following [23,29,30] the entropy generation can be written as: (22) $\begin{array}{rl}{S}_{\mathrm{gen}}& =\frac{K}{T_o^2}{\left(\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\right)}^2+\frac{1}{T_o^2}\left(\frac{16{\sigma }_o{T}_a^3}{3k_2}\right){\left(\frac{\mathrm{\partial }{T}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\right)}^2\\ & +\frac{1}{T_o}\left(S_{\mathrm{xy}}^{\prime }\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\right)+\frac{\sigma B_o^2}{T_o}{u}^{\prime 2}+\frac{{\mu }_o}{k_1{T}_o}{u}^{\prime 2}.\end{array}$(22) A characteristic entropy generation is given by as (23) $S_g=\frac{K(T_1-T_o{)}^2}{T_o^2{h}_o^2}.$(23) The entropy generation in the dimensionless form can be written as follows: (24) $\begin{array}{rl}{N}_s& =\frac{S_{\mathrm{gen}}}{S_g}=(1+R_a){\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\right)}^2\\ & +\frac{B_r}{\mathrm{\Omega }}[\frac{{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2}{1+{\left\{W_e\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)\right\}}^n}+(H{a}^2+\frac{1}{\mathrm{Da}})u^2],\end{array}$(24) where $\mathrm{\Omega }=\frac{T_1-T_o}{T_o}$ is the dimensionless temperature difference.

4. Numerical solution technique

Clearly, the problem stated above is nonlinear. It also appeared to be difficult to discover a closed-form or exact solution for the problem under consideration. As a result, utilizing the finite difference method to discover a numerical solution is the best alternative method. To proceed, the non-rectangular physical domain is turned into a rectangular uniform domain that may be used to execute the finite difference method using the lateral coordinate transformation $\eta =\frac{y}{H\left(x\right)}$. Equations (16(16) $\begin{array}{rl}{\gamma }^2\frac{\mathrm{\partial }u}{\mathrm{\partial }t}& =1+\mathrm{ecos}\left(2\mathrm{\pi t}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left[\frac{\frac{\mathrm{\partial }u}{\mathrm{\partial }y}}{1+{\left\{W_e\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)\right\}}^n}\right]\\ & -(H{a}^2+\frac{1}{\mathrm{Da}})u+G_r\theta +G_c\varphi ,\end{array}$(16) )–(19(19) $\begin{array}{rl}& u=0,\theta =0,\varphi =0\mathrm{at}t=0,\\ & u=0,\theta =0,\varphi =0\mathrm{at}y=-H\left(x\right),\\ & u=0,\theta =1\varphi =1\mathrm{at}y=H\left(x\right),\end{array}$(19) ) will be: (25) $\begin{array}{rl}{\gamma }^2\frac{\mathrm{\partial }u}{\mathrm{\partial }t}& =1+\mathrm{ecos}\left(2\mathrm{\pi t}\right)\\ & +\frac{[1+(1-n\left){\left(\frac{W_e}{H}\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)}^n\right]\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\eta }^2}}{H^2{[1+{\left(\frac{W_e}{H}\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)}^n]}^2}\\ & -(H{a}^2+\frac{1}{\mathrm{Da}})u+G_r\theta +G_c\varphi ,\end{array}$(25) (26) $\begin{array}{rl}{P}_r{\gamma }^2\frac{\mathrm{\partial }\theta }{\mathrm{\partial }t}& =\frac{(1+\mathrm{Ra})}{H^2}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\eta }^2}\\ & +B_r\{\frac{\frac{1}{H^2}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)}^2}{1+{\left(\frac{W_e}{H}\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)\right)}^n}+\left(H{a}^2+\frac{1}{\mathrm{Da}}\right)u^2\}\\ & +\text{β}\theta ,\end{array}$(26) (27) $\begin{array}{rl}{S}_c{\gamma }^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}& =\frac{1}{H^2}\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\eta }^2}+\frac{S_r{S}_c}{H^2}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\eta }^2}-\alpha S_c\varphi .\end{array}$(27) The corresponding boundary conditions: (28) $\begin{array}{rl}& u=0,\theta =0,\varphi =0\mathrm{at}t=0,\\ & u=0,\theta =0,\varphi =0\mathrm{at}\eta =-1,\\ & u=0,\theta =1\varphi =1\mathrm{at}\eta =1.\end{array}$(28) Suppose that $\eta =\mathrm{\Delta \eta }\ast s,t=\mathrm{\Delta \tau }\ast r\mathrm{where}s=0,1,2,\dots ,M_{\eta },\mathrm{and}r=0,1,2,\dots ,N_{\tau }$, where $\mathrm{\Delta \eta }$ and $\mathrm{\Delta \tau }$ are the space and time increments. We seek to find $u(t,\eta ),\theta (t,\eta ),\varphi (t,\eta )$ at the mesh point $[r,s]$ and they are denoted by $u[r,s],\theta [r,s],\varphi [r,s]$. The discretized version of the governing equations and the boundary conditions Equations (25(25) $\begin{array}{rl}{\gamma }^2\frac{\mathrm{\partial }u}{\mathrm{\partial }t}& =1+\mathrm{ecos}\left(2\mathrm{\pi t}\right)\\ & +\frac{[1+(1-n\left){\left(\frac{W_e}{H}\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)}^n\right]\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\eta }^2}}{H^2{[1+{\left(\frac{W_e}{H}\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }\right)}^n]}^2}\\ & -(H{a}^2+\frac{1}{\mathrm{Da}})u+G_r\theta +G_c\varphi ,\end{array}$(25) )–(28(28) $\begin{array}{rl}& u=0,\theta =0,\varphi =0\mathrm{at}t=0,\\ & u=0,\theta =0,\varphi =0\mathrm{at}\eta =-1,\\ & u=0,\theta =1\varphi =1\mathrm{at}\eta =1.\end{array}$(28) ) will be: (29) $\begin{array}{rl}& {\gamma }^2\left(\frac{u[r+1,s]-u[r,s]}{\mathrm{\Delta }\tau }\right)\\ & =1+\mathrm{ecos}\left(2\mathrm{\pi r}\mathrm{\Delta \tau }\right)\\ & +\frac{\begin{array}{c}[1+(1-n\left){\left(\frac{W_e}{H}\frac{\left(u\right[r,s+1]-u[r,s-1\left]\right)}{2\mathrm{\Delta }\eta }\right)}^n\right]\\ \frac{\left(u\right[r,s+1]-2u[r,s\left]\right)+u[r,s-1])}{\mathrm{\Delta }{\eta }^2}\end{array}}{H^2{[1+{\left(\frac{W_e}{H}\frac{\left(u\right[r,s+1]-u[r,s-1\left]\right)}{2\mathrm{\Delta }\eta }\right)}^n]}^2}\\ & -(H{a}^2+\frac{1}{\mathrm{Da}})u[r,s]+G_r\theta [r,s]+G_c\varphi [r,s],\end{array}$(29) (30) $\begin{array}{rl}& P_r{\gamma }^2\left(\frac{\theta [r+1,s]-\theta [r,s]}{\mathrm{\Delta }\tau }\right)\\ & =\frac{(1+\mathrm{Ra})}{H^2}\frac{\left(\theta \right[r,s+1]-2\theta [r,s\left]\right)+\theta [r,s-1])}{\mathrm{\Delta }{\eta }^2}\\ & +B_r\{\frac{\frac{1}{H^2}{\left(\frac{\left(u\right[r,s+1]-u[r,s-1\left]\right)}{2\mathrm{\Delta }\eta }\right)}^2}{1+{\left(\frac{W_e}{H}\frac{\left(u\right[r,s+1]-u[r,s-1\left]\right)}{2\mathrm{\Delta }\eta }\right)}^n}\\ & +(H{a}^2+\frac{1}{\mathrm{Da}})u[r,s{]}^2\}+\text{β}\theta [r,s],\end{array}$(30) (31) $\begin{array}{rl}& S_c{\gamma }^2\left(\frac{\varphi [r+1,s]-\varphi [r,s]}{\mathrm{\Delta }\tau }\right)\\ & =\frac{1}{H^2}\frac{\left(\varphi \right[r,s+1]-2\varphi [r,s\left]\right)+\varphi [r,s-1])}{\mathrm{\Delta }{\eta }^2}\\ & +\frac{S_r{S}_c}{H^2}\frac{\left(\theta \right[r,s+1]-2\theta [r,s\left]\right)+\theta [r,s-1])}{\mathrm{\Delta }{\eta }^2}\\ & -\alpha S_c\varphi [r,s].\end{array}$(31) The initial and boundary conditions: (32) $\begin{array}{rl}& u[0,s]=0,\theta [0,s]=0,\varphi [0,s]=0,\\ & u[r,0]=0,\theta [r,0]=0,\varphi [r,0]=0,\\ & u[r,M_{\eta }]=0,\theta [r,M_{\eta }]=1\varphi [r,M_{\eta }]=1.\end{array}$(32)

5. Graphical results and discussion

In this part, we investigate the influence of different parameters on the velocity, heat transfer, concentration, and entropy generation. The range of the physical parameters can be taken as follows [24,26,28,30]: The Hartmann number ($0\le \mathrm{Ha}\le 5$), Weissenberg number $0\le W_e<1$, Womersley frequency parameter $0<\gamma \le 1$, the Radiation parameter ($0\le \mathrm{Ra}\le 4$), the amplitude fluctuation parameter $0\le e\le 1$, the critical altitudes $0<{\delta }_1<{\delta }_2<1$, the Grashof number ($0\le \mathrm{Gr}\le 4$), the heat source parameter ($0\le \text{β}\le 2$), the chemical parameter $0\le \alpha \le 2$, the Soret number $0.1\le S_r\le 2$, and the Schmidt number $0.1\le S_c\le 2$. Table 1 shows the values of the skin friction coefficient, Nusselt number, and Sherwood numbers on the walls at fixed values of the parameters. It is noticed that the absolute value of the skin friction coefficient at the two walls decreases by increasing Ra, Ha and $W_e$. While the Nusselt number and Sherwood numbers vary with Ra and $H_a$ at the walls and no variation is noticed with $W_e$.

Table 1. The skin friction coefficient, Nusselt number, and Sherwood numbers on the walls at fixed values of the parameters: $\text{β}=0.5,\mathrm{Da}=2,n=0.6,\gamma =0.5,e=0G_r=2,G_c=2,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=0.02,{\delta }_1=0.5,{\delta }_2=0.8,t=1$.

$W_e$	$H_a$	Ra	$C_f\left(0\right)$	$C_f\left(1\right)$	$-N_u\left(0\right)$	$-N_u\left(1\right)$	$S_h\left(0\right)$	$S_h\left(1\right)$
0	0	0	0.947449	−0.713976	0.262116	0.706766	0.977254	0.0998503
0	0	0.5	0.947364	−0.713909	0.385864	0.58733	0.729236	0.340186
0	2	0.5	0.688096	−0.56942	0.385485	0.58781	0.729982	0.339231
0.1	2	0.5	0.628471	−0.541481	0.385485	0.58781	0.729982	0.339231

5.1. Grid independence test

To validate the numerical simulations the grid independence test was performed with three different mesh elements as shown in Figure 2. The results do not change with changing the meshes. Hence, we will adopt $200\times 200$ mesh in this study.

Figure 2. Grid independence test for different mesh sizes for u, θ and ϕ at $\mathrm{Ha}=2,\mathrm{Ra}=0.5,\text{β}=0.5,\mathrm{Da}=2,W_e=0.4,n=0.6,\gamma =0.5,e=0G_r=2,G_c=2,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8,t=1$.

5.2. Velocity profile

Figure 3 illustrates the variance of velocity against time t, for different values of Weissenberg number $W_e$ and the amplitude fluctuation parameter e. According to the simulation, as time goes on, the fluid velocity progressively increases and reaches a steady state for the non-pulsating flow $(e=0)$ and fluctuates for the pulsating flow $(e=0.5)$. The blood velocity decreases as the fluid parameter $W_e$ increases. Due to the elastic force being greater than the viscous force. Furthermore, the values of axial velocity through a Newtonian fluid $(W_e=0)$ are substantially higher than that through a Cross fluid $(W_e\ne 0)$. The variation of velocity versus time for diversified values of Womersley frequency coefficient γ is shown in Figure 4 at two specific locations, the first one $(x=1.5)$ at the tipping altitude of the first narrow and the other one $(x=5)$ at the tipping altitude of the second narrow. It can be seen that the fluid passes more quickly in the first contraction than in the second contraction. Consequently, the velocity decelerates from $(x=1.5)$ to $(x=5)$ due to gravity. We can see that the blood velocity at the critical height of the first constriction $(x=1.5)$ increases sharply at the beginning of the time period, then gradually decreases with time fleeting and then becomes stable from $(t=2)$ to $(t=4)$ while the curve of fluid velocity at the tipping altitude of the second constriction $(x=5)$ decreases quite a bit in the period $(0\le t\le 0.2)$ approximately, and then it appears to be stable from $(t=0.2)$ to $(t=4)$. Figure 5(a,b) exhibits the velocity profiles versus η for different values of Grashof parameter $G_r$ and solutal Grashof parameter $G_c$. The Grashof parameter signifies the prorated effect of the thermal buoyancy force on the viscous hydrodynamic force. The flow is accelerated due to the enhancement in the buoyancy force due to the increase in the thermal Grashof parameter. It is evident that the velocity elevates with an increase in either the Grashof parameter or the Grashof solutal parameter. Furthermore, the blood velocity is higher for the free convection $(G_r\ne 0$ or $G_c\ne 0)$ than that in the absence of the free convection $(G_r=0$ or $G_c=0)$. Figure 6(a) demonstrates that the fluid velocity decreases with an increment in Hartmann number Ha. This situation is due to that the magnetic field will produce a reluctance force analogous to the drag force, which works to combat the flow and thus reduce its velocity. Figure 6(b) displays that the velocity augments with a rise in Darcy's number $D_a$. This can be explained as the permeability of the medium increases the velocity of the fluid due to the decline of resistance to flow within the blood vessels. Figure 7(a,b) indicates the effectiveness of the power-law index n and the radiation number $R_a$ on blood velocity. In Figure 7(a), an increment in the power-law index n decreases the axial fluid velocity. This circumstance is a result of an increase in the effective viscosity. Figure 7(b) shows that as the radiation number increases, lead to an increase in the axial fluid velocity. Moreover, the blood velocity is higher in the existence of radiation $(R_a\ne 0)$ than in the absence of radiation $(R_a=0)$.

Figure 3. The velocity versus time t, for different values of $W_e$ and e at $\mathrm{Ha}=2,\mathrm{Ra}=0.5,\text{β}=0.5,\mathrm{Da}=2,n=0.6,\gamma =0.5,G_r=2,G_c=2,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 4. The velocity versus time t, for different values of γ and x at $\mathrm{Ha}=2,\mathrm{Ra}=0.5,\text{β}=0.5,\mathrm{Da}=2,n=0.6,e=0,W_e=0.2,G_r=2,G_c=2,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 5. The velocity versus η, for different values of $G_r$ (Panel (a)) and $G_c$ (Panel (b)) at $\mathrm{Ha}=2,\mathrm{Ra}=0.5,\text{β}=0.5,\mathrm{Da}=2,W_e=0.4,n=0.6,\gamma =0.5,e=0,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8,t=1.5$.

Figure 6. The velocity versus t, for different values of Da (Panel (a)) and Ha (Panel (b)) at $\mathrm{Ra}=0.5,\text{β}=0.5,G_r=2,G_c=1,W_e=0.4,n=0.6,\gamma =0.5,e=0,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 7. The velocity versus η, for different values of n (Panel (a)) and Ra (Panel (b)) at $\mathrm{Da}=2,\text{β}=0.5,G_r=4,G_c=1,W_e=0.5,n=0.6,\gamma =0.5,e=0,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

5.3. Temperature

The distribution of temperature versus the time t, for diversified values of the radiation parameter $R_a$ is displayed in Figure 8. It is clear that in the case of pulsating flow $(e=2)$, the temperature sharply increases to reach its maximum values and then fluctuates or vibrates in a horizontal position along the time period. Moreover, the figure shows that the radiation number $R_a$ achieves a rise in the temperature at the beginning of the time period after this period an opposite behaviour is seen. This phenomenon can be explained by the fact that at the beginning of the period of time, radiation absorption occurs, which in turn leads to a rise in temperature, but over time, an equilibrium occurs in the temperature, and therefore, the effect of the radiation coefficient is reflected. The effects of the heat source parameter and Womersley frequency parameter on the temperature distribution in the case of non-pulsating flow $(e=0)$ are illustrated in Figure 9(a,b) respectively. We can conclude that the heat augments when the heat source augments. While an increase in the Womersley frequency parameter γ delays the temperature from going to its steady state.

Figure 8. The temperature versus t, for different values of Ra at $\text{β}=0.5,G_r=2,G_c=1,W_e=0.4,n=0.6,\gamma =0.5,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 9. The temperature versus t, for different values of γ (Panel (a)) and β (Panel (b))at $e=0,\mathrm{Ra}=0.5,G_r=2,G_c=1,W_e=0.5,n=0.6,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

5.4. Concentration

Figures 10, 11, and 12 depict the variance in the concentration profiles for many values of the Schmidt number $S_c$, the Soret number $S_r$, the chemical parameter α, and the Womersley frequency parameter γ. In Figure 10, we monitor that the concentration profile increases with a raise in Schmidt number $S_c$, and the Soret number $S_r$ at the beginning of the period of time then the opposite happens after that period. Figure 11 captures the effect of the chemical parameter α on the concentration. The concentration decreases when the chemical parameter α increases. The values of concentration are lower in the existence of chemical reaction $(\alpha \ne 0)$ than in the absence of chemical reaction $(\alpha =0)$. If the chemical reaction of the fluid molecules increases, the temperature increases but the concentration decrease. Because the fluid temperature and concentration depend on the speed of a the chemical reaction. Figure 12 shows that an increase in the Womersley frequency parameter γ postpones the concentration to reach its steady state.

Figure 10. The concentration versus t, for different values of $S_r$, and $S_c$ at $e=0,\mathrm{Ra}=0.5,G_r=2,G_c=1,W_e=0.5,n=0.6,\gamma =0.5,\alpha =0.2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 11. The concentration versus η, for different values of α at $e=0,\mathrm{Ra}=0.5,G_r=2,G_c=1,W_e=0.5,n=0.5,\gamma =0.5,S_r=2,S_c=3,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 12. The concentration versus t, for different values of γ at $e=0,\mathrm{Ra}=0.5,G_r=2,G_c=1,W_e=0.5,n=0.5,\alpha =2,S_r=0.1,S_c=2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

5.5. Entropy generation

The effect of the Weissenberg number $W_e$ on the entropy generation is drawn in Figure 13. The result shows that the increase in the Weissenberg number $W_e$ causes an augmentation in the entropy generation parameter. Furthermore, the values of entropy generation through a Newtonian fluid $(W_e=0)$ are substantially lower than that through Cross fluid $(W_e\ne 0)$. Figure 14 demonstrates the behaviour of entropy generation for the Grashof number $G_r$. The figure depicts that the entropy generation sharply increases at the beginning of the time period and decreases again, and goes to its steady state for non-pulsating flow $(e=0)$ and fluctuates for pulsating flow $(e=2)$. The result shows that an increment in the Grashof parameter $G_r$ raises the entropy generation. Moreover, the entropy generation is sizable in the existence of free convection $(G_r\ne 0)$ than without convection $(G_r=0)$. Figure 15(a,b) depicts the behaviour of the entropy generation against time (t) for different values of the radiation parameter $R_a$ and the Hartmann number Ha. Figure 15(a) shows that the entropy generation is a decreasing function with $R_a$ also the values of the entropy generation are lower for radiative heat flux $(R_a\ne 0)$ than that for non-radiative heat flux $(R_a=0)$. Figure 15(b) displays that the entropy generation increases by increasing the Hartmann number Ha. The figure also shows that entropy generation achieves a steady state after a short period of time, and this is consistent with the second law of thermodynamics.

Figure 13. The entropy generation versus η, for different values of $W_e$ at $e=0,\mathrm{Ra}=0.5,\text{β}=0.5,G_r=2,G_c=2,n=0.5,\alpha =2,S_r=2,S_c=1,\gamma =1,\mathrm{Da}=2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8,t=2$.

Figure 14. The entropy generation versus t, for different values of $G_r$ and e at $\mathrm{Ra}=0.5,\text{β}=0.5,G_c=2,n=0.6,\alpha =0.2,S_r=2,S_c=1,\mathrm{Ha}=2,\mathrm{Da}=2,\gamma =1,\mathrm{Da}=2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

Figure 15. The entropy generation versus t, for different values of Ra (Panel (a)) and Ha (Panel (b)) at $\text{β}=0.5,G_r=3,G_c=2,n=0.6,\alpha =0.2,S_r=2,S_c=1,\mathrm{Da}=2,\gamma =0.3,\mathrm{Da}=2,P_r=20,B_r=2,{\delta }_1=0.5,{\delta }_2=0.8$.

5.6. Validation of the numerical procedure

Due to the lack of experimental data on the problem investigated in this study, we cannot validate the predictions against the experiment. But, to validate our results we resolved the problem numerically by the function NDSolve Mathematica software in the special case ($(W_e=0)$) and the results are shown in Figure 16. The results show a good agreement between the two solutions for velocity, temperature, and concentration.

Figure 16. A comparison between the numerical solutions by NDSolve Mathematica and finite difference method for u, θ and ϕ at $\mathrm{Ha}=2,\mathrm{Ra}=0.5,\text{β}=0.5,\mathrm{Da}=2,W_e=0.,n=0.6,\gamma =0.5,e=0G_r=2,G_c=2,S_r=2,S_c=1,\alpha =0.2,P_r=20,B_r=0.002,{\delta }_1=0.5,{\delta }_2=0.8$.

6. Conclusions

In this article, entropy generation of the pulsatile flow of unsteady magnetohydrodynamic Cross fluid (a blood model) through a vertical porous stenosed microchannel under the influence of thermal radiation and Joule heating has been investigated. The resulting nonlinear coupled partial differential equations have been simplified with the help of mild stenosis approximation. Moreover, we used the finite difference method to solve these equations. The expressions of the blood velocity, the temperature, the concentration, and the entropy generation have been calculated numerically. The significant outcomes for the present analysis are:

• At the beginning of the time interval, the blood velocity in the first stenosed region has a strong impact.

• As time goes on, the blood velocity increases and reaches a steady state for non-pulsating flow, and fluctuates for pulsating flow.

• Fluid velocity through a Newtonian fluid is substantially higher than that through a Cross fluid.

• The medium's permeability raises the blood velocity because of the discounted resistance to flow within the blood vessels.

• At the beginning of the time interval, the temperature is elevated by increasing the radiation parameter until a critical time the reverse effect is noticed due to the heat absorption in the hemodynamic.

• The value of LDL concentration in the existence of a chemical reaction is lower than in the absence of a chemical reaction.

• The values of entropy generation through a Newtonian fluid are substantially lower than that through a Cross fluid.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number G.R.P.72/43.