1 Introduction

Arteries are the blood vessels that transport blood from the heart to all parts of the body (lungs, tissues, brain etc.). In many cases, some obstructions occur in the arteries during the blood flow, and one of them is the deposition of plaques on the arterial walls, narrowing the area of the flow region and leading to a disease called atherosclerosis or stenosis. Over a period, stenosis solidifies and constricts the blood vessels, limiting the oxygenated blood supply to the organs and other parts of the body which leads to severe complications, including heart attack, stroke, or even death [6].

To understand the influence of constriction in the lumen of a blood vessel, many research workers [8, 17, 31] have studied the blood flow through the constricted arteries by supposing it as a Newtonian viscous fluid. Mekheimer and Kot [18] presented a particle-fluid suspension model for blood flow through a catheterized curved artery with constriction. The effect of a catheter on entropy generation in two-phase blood flow is analytically studied by Mekheimer et al. [19]. Elnaqeeb et al. [11] examined the influence of Cu nanoparticles on the flow of blood, assuming the rheology of blood as Newtonian fluid in a catheterized arterial stenosis with a clot. But it is well recognized that as it flows through small-diameter vessels at low shear rates, blood exhibits non-Newtonian fluid behavior [9, 15, 30]. The papers [25, 28] include a brief sample of the scientific work on non-Newtonian effects on blood flow by considering blood as a Casson model fluid, in view of the non-Newtonian nature of blood. Akbar et al. [1] investigated the Reiner–Rivlin model fluid for blood flow in a tapered mild stenotic artery. In addition, Bugliarello, Sevilla [7] and Cokelet [10] have experimentally confirmed that there is a peripheral plasma layer (Newtonian viscous fluid) near the vessel wall and a central area of suspension of all erythrocytes as a non-Newtonian fluid when blood passes through the small blood vessels. Several authors [13, 20, 23] have examined the two-layered model of blood flow in a stenosed artery in which the central core region consisting of Casson fluid and the peripheral layer containing the Newtonian fluid.

In the studies as mentioned above, the blood flow through the blood vessels is considered as single or double-layered, and the vessel walls are treated as rigid. While the blood vessel walls are structurally composed primarily of three layers: tunica intima, tunica media, and tunica externa. The porous medium is a substance composed of a rigid frame of pores that are intertwined. The flow through porous media was explored by Whitaker [29] using Darcy’s law. Goharzadeh et al. [12] have experimentally examined the existence and thickness of the transition layer at a fluid porous interface. A three-layer structure in which a Newtonian fluid is overlying a transition porous layer, which overlays a layer of the porous region of Darcy, has been studied by Hill and Straughan [14], and they have discussed the instability of Poiseuille flow. Boodoo et al. [5] have proposed a two-fluid mathematical model for blood flow in a uniform tube such that the wall of the tube is assumed to be porous and consisting of a thin transition Brinkman layer followed by a Darcy region. In their model, the central core region comprises a micropolar fluid, and the peripheral plasma layer, Brinkman, and Darcy regions are filled with Newtonian fluid. The study shows that the fluid velocity in the core region and plasma layer reduces with the increase of hydraulic resistivity, and the smaller values of permeability increase the flow resistance in the porous region that causes an overall slower velocity of both the fluids in the core and plasma. Sharma and Yadav [24] have discussed a Casson–Newtonian fluid model for blood flow in a stenotic artery with the porous wall which comprises a Brinkman layer (tunica intima) surrounding a Darcy region. They concluded that as the yield stress increases, the Casson fluid velocity reduces, and the flow rate in the stenotic region has a symmetric profile for the stenosis shape parameter \(n=2\), but for other values of n, it is asymmetric.

The Kuang–Luo (K–L) model fluid, which is an improvement of the Casson model, has been proposed by Luo and Kuang [16]. It provides a better depiction of the non-Newtonian character of blood since it encompasses three parameters, such as yield stress, plasma viscosity, and other chemical variables, whereas only viscosity and yield stress are considered in the Casson model. In describing the shear-thinning behavior of blood within an extensive shear rate, this model is more effective [16]. The constitutive blood parameters were experimentally explored in various blood rheological equations by Zhang and Kuang [32]. The investigation revealed that Bi-exponent, Quemada, and K–L models are in good agreement with the hemorheological features of canine and human blood, and the K–L model can be used well in hemorheology and hemodynamics. Several authors [2, 3, 26] have been introduced mathematical models to illustrate the effect of stenosis on blood flow characteristics treating blood as K–L model fluid.

With this motivation, an attempt is made to study a two-fluid mathematical model for blood flow in a constricted artery consisting of a core region of suspension of all the erythrocytes supposed to be a K–L fluid and a peripheral layer of plasma as a Newtonian fluid. The arterial wall segment is assumed to be porous and comprising two-layers, namely Brinkman and Darcy. The layout of the paper is as follows. Section 2 deals with the formulation of the problem in which the equations of motion for all the four regions using K–L and Newtonian constitutive equations and the non-dimensionalization procedure are presented. The resulting differential equations are solved with the appropriate boundary conditions, and the analytic expressions for the physiologically significant flow quantities such as velocity profiles in all the regions, plug-core radius, flow flux, wall shear stress, and resistive impedance are obtained in Sect. 3. Some special cases, which are the limiting cases of the present model, are presented in Sect. 4. The impact of several physical parameters on the blood flow characteristics is analyzed in Sect. 5. The outcomes are summarized, and the significance of the presence of the peripheral plasma layer and the porous wall is indicated in the concluding Sect. 6.

2 Modeling and formulation

Consider one dimensional, steady, laminar, fully developed, and an axially symmetric flow of incompressible viscous fluid (blood) through the stenosed artery whose porous wall is composed of two layers, namely Brinkman and Darcy. We assume that the rheology of blood in the core region is characterized by the K–L fluid and a peripheral layer of plasma, porous wall as a Newtonian fluid. A cylindrical polar coordinate system \(({\bar{r}}, {\bar{\theta }}, {\bar{z}})\) whose origin is located on the vessel (constricted artery) axis is taken to examine the flow of blood. The geometry of the stenotic arterial segment (see Fig. 1) in dimensional form is written mathematically as

$$\begin{aligned} \frac{{\overline{R}}({\overline{z}})}{{\overline{R}}_0} = \left\{ \begin{array}{ll} 1-{\overline{\chi }}\{{\overline{L}}_0^{m-1}({\overline{z}}-{\overline{d}})-({\overline{z}}-{\overline{d}})^{m}\},\quad &{}{\overline{d}}\le {\overline{z}}<{\overline{d}}+{\overline{L}}_0;\\ \\ 1, &{} otherwise\end{array} \right. \end{aligned}$$
(1)

where \(\displaystyle {\overline{\chi }}=\dfrac{{\overline{\delta }}_s}{{\overline{R}}_0 {\overline{L}}_0^{m}}\left( \dfrac{m^{m/(m-1)}}{m-1}\right)\), \({\overline{R}}({\overline{z}})\), \({\overline{R}}_0\) are the radius of the duct with and without stenosis, respectively, \({\overline{R}}_{pl}\) is the plug core radius, \({\overline{d}}\) is the location of the constriction, \({\overline{L}}_0\) indicates its length, \(m(\ge 2)\) is the stenotic shape parameter, \({\overline{\delta }}_s\) is the maximum stenotic height occurs at \({\overline{z}}={\overline{d}}+\dfrac{{\overline{L}}_0}{m^{1/(m-1)}},\) such that \(\displaystyle \dfrac{{\overline{\delta }}_s}{{\overline{R}}_0}<<1\). We denote \({\overline{R}}_C, {\overline{R}}_P, {\overline{R}}_B\) and \({\overline{R}}_D\) be the radius of the duct in the stenotic region for core, plasma, Brinkman and Darcy regions, respectively. Let \({\overline{h}}_P, {\overline{h}}_B\) and \({\overline{h}}_D\) represent the thickness of the plasma, Brinkman and Darcy layers, respectively such that \({\overline{h}}_B={\overline{h}}_D/9\) [27].

Fig. 1
figure 1

Schematic diagram of a stenosed artery with different regions

Full size image

Let us assume \({\overline{P}}_C, {\overline{P}}_P, {\overline{P}}_B\) and \({\overline{P}}_D\) be the pressures in the four regions, respectively. As the flow is one dimensional (axial direction) and axially symmetric, the velocity vector is given by \(\overline{\mathbf{V }}=(0, 0, {\overline{w}}_i)\), where \({\overline{w}}_i\) is the axial velocity in the respective region (\(i=C, P, B,\) and D). Hence the basic momentum equations for the four regions can be written as:

Region I: Core region (K–L fluid)

$$\begin{aligned} -\dfrac{\partial {\overline{P}}_C}{\partial {\overline{z}}}+\dfrac{1}{{\overline{r}}}\, \dfrac{\partial }{\partial {\overline{r}}}\left( {\overline{r}}\, {\overline{\tau }}_{{\overline{r}}\,{\overline{z}}}\right) =0, \end{aligned}$$
(2)

where the shear stress \({\overline{\tau }}_{{\overline{r}}\,{\overline{z}}}\), for K–L fluid, is given by [16]

$$\begin{aligned} {\overline{\tau }}_{{\overline{r}}\,{\overline{z}}}&={\overline{\tau }}_0+{\overline{\sigma }}_1 \sqrt{\overline{{\dot{\gamma }}}}+{\overline{\sigma }}_2 \overline{{\dot{\gamma }}}, \ \ \ \ \ \ {\overline{\tau }}_{{\overline{r}}\, {\overline{z}}}> {\overline{\tau }}_0 \end{aligned}$$
(3)
$$\begin{aligned} \overline{{\dot{\gamma }}}&=0, {\overline{\tau }}_{{\overline{r}}\,{\overline{z}}}\le {\overline{\tau }}_0 \end{aligned}$$
(4)

Region II: Plasma region (Newtonian fluid)

$$\begin{aligned} -\dfrac{\partial {\overline{P}}_P}{\partial {\overline{z}}}+\dfrac{{\overline{\mu }}}{{\overline{r}}}\, \dfrac{\partial }{\partial {\overline{r}}}\left( {\overline{r}}\,\dfrac{\partial {\overline{w}}_P}{\partial {\overline{r}}} \right) =0 \end{aligned}$$
(5)

Region III: Brinkman porous region (Newtonian fluid)

$$\begin{aligned} -\dfrac{\partial {\overline{P}}_B}{\partial {\overline{z}}}+\dfrac{\overline{{\mu }}_e}{{\overline{r}}}\, \dfrac{\partial }{\partial {\overline{r}}}\left( {\overline{r}}\,\dfrac{\partial {\overline{w}}_B}{\partial {\overline{r}}} \right) -\dfrac{\overline{{\mu }}}{{\overline{k}}}{\overline{w}}_B=0 \end{aligned}$$
(6)

Region IV: Darcy porous region (Newtonian fluid)

$$\begin{aligned} -\dfrac{\partial {\overline{P}}_D}{\partial {\overline{z}}}-\dfrac{{\overline{\mu }}}{{\overline{k}}}{\overline{w}}_D=0, \end{aligned}$$
(7)

where \(\overline{{\dot{\gamma }}}\) is the shear rate, the three parameters \({\overline{\tau }}_0,\, {\overline{\sigma }}_1,\, {\overline{\sigma }}_2\) are functions of hematocrit (volume fraction of red blood cells), plasma viscosity, and other chemical variables, \({\overline{\mu }}_e\) is the effective viscosity of transition (Brinkman) layer and \({\overline{k}}\) is the permeability constant.

We introduce the following dimensionless variables:

$$\begin{aligned} z&=\dfrac{{\overline{z}}}{{\overline{R}}_0},\ \ \ r=\dfrac{{\overline{r}}}{{\overline{R}}_0},\ \ \nonumber \\ d&=\dfrac{{\overline{d}}}{{\overline{R}}_0},\ \ \ L_0=\dfrac{{\overline{L}}_0}{{\overline{R}}_0},\ \ \delta _s=\dfrac{{\overline{\delta }}_s}{{\overline{R}}_0}, \ \nonumber \\ R(z)&=\dfrac{{\overline{R}}({\overline{z}})}{{\overline{R}}_0},\nonumber \\ R_i(z)&=\dfrac{{\overline{R}}_i({\overline{z}})}{{\overline{R}}_0},\ w_i=\dfrac{{\overline{w}}_i}{{\overline{w}}_0},\ \ P_i=\dfrac{{\overline{R}}_0{\overline{P}}_i}{{\overline{w}}_0{\overline{\mu }}},\ \nonumber \\ \theta _1&=\dfrac{{\overline{\sigma }}_1}{{\overline{\mu }}}\left( \dfrac{{\overline{R}}_0}{{\overline{u}}_0}\right) ^{1/2},\ \ \theta _2=\dfrac{{\overline{\sigma }}_2}{{\overline{\mu }}},\nonumber \\ \lambda _e&=\dfrac{{\overline{\mu }}_e}{{\overline{\mu }}},\ \ Da=\dfrac{{\overline{k}}}{{\overline{R}}_0^{2}},\ \ \tau _{rz}=\dfrac{{\overline{R}}_0{\overline{\tau }}_{{\overline{r}}\, {\overline{z}}}}{{\overline{w}}_0{\overline{\mu }}},\ \nonumber \\ \tau _{0}&=\dfrac{{\overline{R}}_0{\overline{\tau }}_{0}}{{\overline{w}}_0{\overline{\mu }}},\ \ Re=\dfrac{{\overline{\rho }}{\overline{R}}_0 {\overline{w}}_0}{{\overline{\mu }}}, \end{aligned}$$
(8)

where \({\overline{\mu }}\) is the Newtonian fluid viscosity, \(\theta _1\) and \(\theta _2\) are the non-dimensional K–L fluid parameters, Da is the Darcy number, \(\lambda _e\) is the non-dimensional effective viscosity and Re is the Reynolds number. Equations (2)–(7) can be transformed to dimensionless form by employing the above non-dimensional variables:

Region I: Core region

$$\begin{aligned} -\dfrac{\partial {P}_C}{d{z}}+\dfrac{1}{{r}}\, \dfrac{\partial }{\partial r}\left( {r}\, {\tau }_{{r}{z}}\right) =0 \end{aligned}$$
(9)

where

$$\begin{aligned}&{\tau }_{{r}{z}}=\tau _0+\theta _1 \sqrt{-\dfrac{\partial w_C}{\partial r}}-\theta _2\dfrac{\partial w_C}{\partial r}, \quad \tau _{rz}> \tau _0 \end{aligned}$$
(10)
$$\begin{aligned}&-\dfrac{\partial u_C}{\partial r}=0, \tau _{rz}\le \tau _0 \end{aligned}$$
(11)

It is emphasized from the Eqs. (10) and (11) that the velocity gradients vanish in the flow region where the shear stress \((\tau _{rz})\) is less than the yield stress \((\tau _{0})\), which in turn indicates that the plug flow occurs whenever \(\tau _{rz} \le \tau _{0}\). However, normal flow occurs whenever \(\tau _{rz}>\tau _{0}\) (i.e. the shear stress is greater than the threshold stress).

Region II: Plasma region

$$\begin{aligned} -\dfrac{\partial {P}_P}{\partial {z}}+\dfrac{1}{r}\, \dfrac{\partial }{\partial {r}}\left( {r}\,\dfrac{\partial {w}_P}{\partial {r}} \right) =0 \end{aligned}$$
(12)

Region III: Brinkman porous region

$$\begin{aligned} -\dfrac{\partial {P}_B}{\partial {z}}+\dfrac{\lambda _e}{r}\, \dfrac{\partial }{\partial {r}}\left( {r}\,\dfrac{\partial {w}_B}{\partial {r}} \right) -\dfrac{1}{Da}{w}_B=0 \end{aligned}$$
(13)

Region IV: Darcy porous region

$$\begin{aligned} -\dfrac{\partial {P}_D}{\partial {z}}-\dfrac{1}{Da}{w}_D=0 \end{aligned}$$
(14)

The boundary conditions are

$$\begin{aligned}&(i)\, \tau _{rz}\,\, \text {is finite}\, \text {at}\,\, r=0\ \end{aligned}$$
(15)
$$\begin{aligned}&(ii)\, -\dfrac{\partial w_{P}}{\partial r}=\tau _0\nonumber \\&\quad +\theta _1\left( -\dfrac{\partial w_C}{\partial r}\right) ^{1/2}+\theta _2\left( -\dfrac{\partial w_C}{\partial r}\right) \,\, \text {at}\,\, r=R_C(z) \end{aligned}$$
(16)
$$\begin{aligned}&(iii)\, w_{C}=w_P\,\, \text {at}\, \, r=R_C(z) \end{aligned}$$
(17)
$$\begin{aligned}&(iv)\, w_{P}=w_B\,\, \text {at}\, \, r=R_P(z) \end{aligned}$$
(18)
$$\begin{aligned}&(v)\, \dfrac{1}{\phi }\left( \dfrac{\partial w_{B}}{\partial r}\right) -\left( \dfrac{\partial w_{P}}{\partial r}\right) =\dfrac{\beta }{\sqrt{Da}}\,w_B\,\, \text {at} \,\,\, r=R_P(z) \end{aligned}$$
(19)

where \(\phi\) is the porosity of the medium and \(\beta\) is the parameter of the stress jump [21].

$$\begin{aligned} (vi)\, \dfrac{\partial w_{B}}{\partial r}=\dfrac{\alpha }{\sqrt{Da}}\,(w_B-w_D)\,\, \text {at} \,\,\, r=R_B(z) \end{aligned}$$
(20)

where \(\alpha\) is the parameter of the Darcy slip [4].

$$\begin{aligned} (vii) \, \dfrac{\partial w_{C}}{\partial r}=0 \,\, \text {at} \,\, r=R_{pl}(z) \end{aligned}$$
(21)

The dimensionless form of the wall geometry is given by

$$\begin{aligned} R(z) = \left\{ \begin{array}{ll} 1-\chi \left[ L_0^{m-1}(z-d)-(z-d)^{m}\right] ,\quad &{}d\le z<d+L_0;\\ \\ 1, &{} otherwise\end{array} \right. \end{aligned}$$
(22)

where \(\displaystyle \chi =\dfrac{\delta _s}{L_0^{m}}\left( \dfrac{m^{m/(m-1)}}{m-1}\right)\).

3 Solution

Suppose the pressure gradient in all the four regions are equal.

$$\begin{aligned} -\dfrac{\partial {P}_C}{\partial {z}}=-\dfrac{\partial {P}_P}{\partial {z}}=-\dfrac{\partial {P}_B}{\partial {z}}=-\dfrac{\partial {P}_D}{\partial {z}}=P_g(z) \end{aligned}$$
(23)

Substituting Eq. (23) in Eq. (9) and integrating, we get

$$\begin{aligned} \tau _{rz}=-\dfrac{r P_g}{2}+\dfrac{A_1}{r} \end{aligned}$$
(24)

Using the boundary condition (15), Eq. (24) becomes

$$\begin{aligned} \tau _{rz}=-\dfrac{r P_g}{2} \end{aligned}$$
(25)

The solutions of Eqs. (10)–(14) can be obtained as

$$\begin{aligned} w_C&=\left( \dfrac{2\tau _0\theta _2-\theta _1^2}{2\theta _2^2}\right) r\nonumber \\&\quad -\dfrac{P_g r^2}{4\theta _2}+\dfrac{\theta _1}{6 P_g\theta _2^3}\left( \theta _1^2-4 \tau _0 \theta _2+2 P_g \theta _2 r\right) ^{3/2}+A_2 \end{aligned}$$
(26)
$$\begin{aligned} w_P&=-\dfrac{P_g r^{2}}{4}+A_3\, \ln (r)+A_4 \end{aligned}$$
(27)
$$\begin{aligned} w_B&=A_5\, I_0(\xi r)+ A_6\, K_0(\xi r)+P_g Da \end{aligned}$$
(28)
$$\begin{aligned} w_D&=P_g Da \end{aligned}$$
(29)

where \(\xi =\dfrac{1}{\sqrt{\lambda _e Da}},\ I_0 \ \text {and}\ K_0\) are modified Bessel functions of first and second kind, respectively and \(A_2, A_3, A_4, A_5\) and \(A_6\) are arbitrary constants to be determined.

The velocity in the plug core region is obtained by

$$\begin{aligned} w_{pl}&=\left( \dfrac{2\tau _0\theta _2-\theta _1^2}{2\theta _2^2}\right) R_{pl}\nonumber \\&\quad -\dfrac{P_g R_{pl}^2}{4\theta _2}+\dfrac{\theta _1}{6 P_g\theta _2^3}\left( \theta _1^2-4 \tau _0 \theta _2+2 P_g \theta _2 R_{pl}\right) ^{3/2}+A_2 \end{aligned}$$
(30)

Applying boundary condition (16), we get

$$\begin{aligned} A_3=0 \end{aligned}$$
(31)

By applying boundary conditions (17)–(20), one can obtain

$$\begin{aligned}&A_2-A_4=-\dfrac{\theta _1}{6 P_g\theta _2^3}\left( \theta _1^2-4 \tau _0\theta _2 +2 P_g\theta _2 R_{C}\right) ^{3/2} \nonumber \\&\qquad -\dfrac{1}{2\theta _2^2}\left( 2\tau _0\theta _2-\theta _1^2\right) R_{C}+\dfrac{P_g}{4\theta _2}(1-\theta _2)R_C^2 \end{aligned}$$
(32)
$$\begin{aligned}&A_4-I_0(\xi R_P)\, A_5-K_0(\xi R_P)\, A_6=\dfrac{P_g}{4} (R_P^2+4 Da) \end{aligned}$$
(33)
$$\begin{aligned}&\left[ \phi \beta I_0(\xi R_P)-\xi \sqrt{Da} I_1(\xi R_P)\right] \, A_5\nonumber \\&\qquad +\left[ \xi \sqrt{Da}K_1(\xi R_P)+\phi \beta K_0(\xi R_P)\right] \, A_6\nonumber \\&\quad =\dfrac{\phi P_0}{2} (R_P\sqrt{Da}-2\beta Da) \end{aligned}$$
(34)
$$\begin{aligned}&\left[ \alpha I_0(\xi R_B)-\xi \sqrt{Da} I_1(\xi R_B)\right] \, A_5\nonumber \\&\qquad +\left[ \alpha K_0(\xi R_B)+\xi \sqrt{Da} K_1(\xi R_B)\right] \, A_6=0 \end{aligned}$$
(35)

where \(A_2, A_4, A_5\ and \ A_6\) are arbitrary constants which can be found out by solving the Eqs. (32)–(35) using MATLAB software, and are given in “Appendix”.

Applying boundary condition (21), we can obtain the expression for plug core radius

$$\begin{aligned} R_{pl}=\dfrac{2 \tau _0}{P_g} \end{aligned}$$
(36)

The flow rate through the plug core region \(Q_{pl}\) is defined as

$$\begin{aligned} Q_{pl}=\int \limits _0^{R_{pl}} r\, u_{pl}\, dr \end{aligned}$$
(37)

which, on substituting Eq. (30), gives

$$\begin{aligned} Q_{pl}&=\dfrac{\theta _1 R_{pl}^2}{12 P_g\theta _2^3}\left( \theta _1^2-4 \tau _0\theta _2 +2 P_g R_{pl}\theta _2\right) ^{3/2}\nonumber \\&\quad +\dfrac{1}{4\theta _2^2}\left( 2\tau _0\theta _2-\theta _1^2\right) R_{pl}^3\nonumber \\&\quad -\dfrac{P_g R_{pl}^4}{8\theta _2}+\dfrac{A_2 R_{pl}^2}{2} \end{aligned}$$
(38)

Similarly, the flux through the core, plasma, Brinkman and Darcy regions may be expressed as

$$\begin{aligned} Q_{C}&=\dfrac{1}{6\theta _2^2}\left( 2\tau _0\theta _2-\theta _1^2\right) \left( R_C^3-R_{pl}^3\right) +\dfrac{P_g}{16\theta _2} \left( R_{pl}^4-R_C^4\right) \nonumber \\&\quad +\dfrac{A_2}{2}\left( R_{C}^2-R_{pl}^2\right) \nonumber \\&\quad +\dfrac{\theta _1}{210 P_g^3\theta _2^5}\left( \theta _1^2-4 \tau _0\theta _2 +2 P_g \theta _2 R_{C}\right) ^{5/2}\nonumber \\&\quad \left( -\theta _1^2+4\tau _0\theta _2+5 P_g\theta _2 R_C\right) \nonumber \\&\quad -\dfrac{\theta _1}{210 P_g^3\theta _2^5}\left( \theta _1^2-4 \tau _0\theta _2 +2 P_g\theta _2 R_{pl}\right) ^{5/2}\nonumber \\&\quad \left( -\theta _1^2+4\tau _0\theta _2+5 P_g\theta _2 R_{pl}\right) \end{aligned}$$
(39)
$$\begin{aligned} Q_{P}&=\dfrac{P_g}{16}\left( R_C^4-R_P^4\right) +\dfrac{A_4}{2}\left( R_P^2-R_C^2\right) \end{aligned}$$
(40)
$$\begin{aligned} Q_B&=\dfrac{A_5}{\xi }\left[ R_B I_1(\xi R_B)-R_P I_1(\xi R_P)\right] \nonumber \\&\quad -\dfrac{A_6}{\xi }\left[ R_B K_1(\xi R_B)-R_P K_1(\xi R_P)\right] \nonumber \\&\quad +\dfrac{P_g Da}{2}\left( R_B^2-R_P^2\right) \end{aligned}$$
(41)
$$\begin{aligned} Q_D&=\dfrac{P_g Da}{2}\left( R_D^2-R_B^2\right) \end{aligned}$$
(42)

The total volumetric flux is calculated as

$$\begin{aligned} Q=Q_{pl}+Q_C+Q_P+Q_B+Q_D \end{aligned}$$
(43)

Substituting Eqs. (38)–(42) in Eq. (43), the expression for flux can be written as

$$\begin{aligned} Q&=\dfrac{1}{12\theta _2^2}\left( 2\tau _0\theta _2-\theta _1^2\right) \left( R_{pl}^3+2 R_{C}^3\right) \nonumber \\&\quad -\dfrac{P_g}{16\theta _2} \left( R_{pl}^4+(1-\theta _2) R_C^4+\theta _2 R_P^4\right) \nonumber \\&\quad +\dfrac{P_g Da}{2}\left( R_D^2-R_P^2\right) +\dfrac{A_4}{2}\left( R_P^2-R_C^2\right) \nonumber \\&\quad +\dfrac{A_5}{\xi }\left[ R_B I_1(\xi R_B)-R_P I_1(\xi R_P)\right] \nonumber \\&\quad -\dfrac{A_6}{\xi }\left[ R_B K_1(\xi R_B)-R_P K_1(\xi R_P)\right] \nonumber \\&\quad +\dfrac{\theta _1 R_{pl}^2}{12 P_g \theta _2^3}\left( \theta _1^2-4 \tau _0 \theta _2+2 P_g\theta _2 R_{pl}\right) ^{3/2}\nonumber \\&\quad -\dfrac{\theta _1}{210 P_g^3\theta _2^5}\Big [\left( \theta _1^2-4 \tau _0 \theta _2\nonumber \right. \\&\left. \quad +2 P_g \theta _2 R_{C}\right) ^{5/2}\left( \theta _1^2-4\tau _0\theta _2-5 P_g\theta _2 R_C\right) \nonumber \\&\quad -\left( \theta _1^2-4 \tau _0 \theta _2+2 P_g\theta _2 R_{pl}\right) ^{5/2}\left( \theta _1^2-4\tau _0\theta _2-5 P_g \theta _2 R_{pl}\right) \Big ]\nonumber \\&\quad +\dfrac{A_2 R_{C}^2}{2} \end{aligned}$$
(44)

The shear stress at the stenosed arterial wall \(\tau _w\) can be defined as

$$\begin{aligned} \tau _w=\left( -\dfrac{d w_P}{dr}\right) _{r=R_P}+\left( -\dfrac{dw_B}{dr}\right) _{r=R_B}-\left( -\dfrac{dw_B}{dr}\right) _{r=R_P} \end{aligned}$$
(45)

In which on using Eqs. (27) and (28) gives

$$\begin{aligned} \tau _w&=\dfrac{P_g R_P}{2}+\xi \Big [A_5 \big (I_1(\xi R_P)-I_1(\xi R_B)\big )\nonumber \\&\quad -A_6 \big (K_1(\xi R_B)-K_1(\xi R_P)\big )\Big ] \end{aligned}$$
(46)

The flow impedance \(\lambda\) is defined as

$$\begin{aligned} \lambda =\int \limits _0^z\dfrac{P_g(z)}{Q}\ dz \end{aligned}$$
(47)

For various values of the parameters involved in this analysis, numerical values of resistance to blood flow can be computed by using the numerical integration on Eq. (47).

4 Special cases

When the blood flows through a constricted artery with porous wall, the velocity profiles for different blood flow models, namely Newtonian–Newtonian, Bingham–Newtonian and Casson–Newtonian are obtained as limiting cases as follows:

4.1 Newtonian–Newtonian fluid model

The velocity distribution of Newtonian–Newtonian fluid model can be calculated by taking the limit of the velocities \(w_C, w_P, w_B\) and \(w_D\) in Eqs. (26)–(29) as the yield stress \((\tau _0)\) and K–L fluid parameter \((\theta _1)\) tend to zero. As \(\tau _0\rightarrow 0\) and \(\theta _1 \rightarrow 0\), then the constitutive equation of core (K–L) fluid reduces to the Newtonian viscous fluid, which is given by

$$\begin{aligned} \tau _{rz}=-\theta _2\left( \dfrac{\partial w_C}{\partial r}\right) \end{aligned}$$
(48)

Therefore, the velocity profiles in each layer can be given by

$$\begin{aligned} w_C&=-\dfrac{P_g r^2}{4\theta _2}+B_2 \end{aligned}$$
(49)
$$\begin{aligned} w_P&=-\dfrac{P_g r^{2}}{4}+B_3 \ln (r)+B_4 \end{aligned}$$
(50)
$$\begin{aligned} w_B&=B_5\, I_0(\xi r)+ B_6\, K_0(\xi r)+P_g Da \end{aligned}$$
(51)
$$\begin{aligned} w_D&=P_g Da \end{aligned}$$
(52)

4.2 Bingham–Newtonian fluid model

The velocity distribution of Bingham–Newtonian fluid model can be calculated by taking the limit of the velocities \(w_C, w_P, w_B, w_D\) and \(w_{pl}\) in Eqs. (26)–(30) as the K–L fluid parameter \((\theta _1)\) tends to zero. As \(\theta _1 \rightarrow 0\), then the constitutive equation of core (K–L) fluid reduces to the Bingham plastic fluid, which is given by

$$\begin{aligned} \tau _{rz}=\tau _0+\theta _2\left( -\dfrac{\partial w_C}{\partial r}\right) \end{aligned}$$
(53)

Therefore, the velocity profiles in each layer can be given by

$$\begin{aligned} w_C&=\frac{1}{\theta _2}\left( \tau _0 r-\dfrac{P_g r^2}{4}\right) +C_2 \end{aligned}$$
(54)
$$\begin{aligned} w_P&=-\dfrac{P_g r^{2}}{4}+C_3 \ln (r)+C_4 \end{aligned}$$
(55)
$$\begin{aligned} w_B&=C_5\, I_0(\xi r)+ C_6\, K_0(\xi r)+P_g Da \end{aligned}$$
(56)
$$\begin{aligned} w_D&=P_g Da \end{aligned}$$
(57)
$$\begin{aligned} w_{pl}&=\frac{1}{\theta _2}\left( \tau _0 R_{pl}-\dfrac{P_g R_{pl}^2}{4}\right) +C_2 \end{aligned}$$
(58)

4.3 Casson–Newtonian fluid model

The velocity of distribution Casson–Newtonian fluid model can be calculated by taking the limit of the velocities \(w_C, w_P, w_B, w_D\) and \(w_{pl}\) in Eqs. (26)–(30) as the K–L fluid parameter \((\theta _1)\) tends to \(2\sqrt{\tau _0 \theta _2}\). If \(\theta _1=2\sqrt{\tau _0 \theta _2}\), then the constitutive equation of core (K–L) fluid reduces to the Casson fluid, which is given by

$$\begin{aligned} \sqrt{\tau _{rz}}=\sqrt{\tau _0}+\sqrt{\theta _2\left( -\dfrac{\partial w_C}{\partial r}\right) } \end{aligned}$$
(59)

Therefore, the velocity profiles in each layer can be given by

$$\begin{aligned} w_C&=-\frac{1}{\theta _2}\left( \tau _0 r+\dfrac{P_g r^2}{4}-\frac{\sqrt{8 \tau _0 P_g}}{3 \theta _2} r^{3/2}\right) +D_2 \end{aligned}$$
(60)
$$\begin{aligned} w_P&=-\dfrac{P_g r^{2}}{4}+D_3 \ln (r)+D_4 \end{aligned}$$
(61)
$$\begin{aligned} w_B&=D_5\, I_0(\xi r)+ D_6\, K_0(\xi r)+P_g Da \end{aligned}$$
(62)
$$\begin{aligned} w_D&=P_g Da \end{aligned}$$
(63)
$$\begin{aligned} w_{pl}&=-\frac{1}{\theta _2}\left( \tau _0 R_{pl}+\dfrac{P_g R_{pl}^2}{4}-\frac{\sqrt{8 \tau _0 P_g}}{3 \theta _2} R_{pl}^{3/2}\right) +D_2 \end{aligned}$$
(64)

The expressions for the arbitrary constants \(B_2\)–\(B_6\), \(C_2\)–\(C_6\) and \(D_2\)–\(D_6\) can be obtained as the limiting cases of the arbitrary constants \(A_2\)–\(A_6\) for the respective model. It is worth noting that the analytical expressions for velocity profiles [Eqs. (60)–(64)] are coincide with the expressions obtained by Sharma and Yadav [24].

5 Graphical results and discussion

The objective of the present study is to investigate the change in flow pattern due to stenosis and the effect of a porous wall on the blood flow in arterial stenosis whose porous wall is composed of two layers, namely Brinkman and Darcy. It is also intended to bring out the simultaneous effects of non-Newtonian behavior of blood, yield stress, plasma layer thickness, Darcy number, porosity, Darcy slip parameter, and stress jump parameter on physiologically pivotal flow characteristics such as velocity profile, plug core radius, flow flux, wall shear stress, and resistive impedance. For computational purpose, the range of the values of different parameters involved in the present analysis is chosen as [22, 27, 31]: length of the stenosis \(L_0=1\); location of the stenosis in the axial direction \(d=4\), \(z=4-5\); stenosis shape parameter \(m=2-5\); maximum height of the stenosis \(\delta _s=0-0.2\); porosity \(\phi =0.5\); Darcy slip parameter \(\alpha =0.01-0.1\); stress jump parameter \(\beta =0.1\); effective viscosity \(\lambda _e=1.1\); K–L fluid parameters \(\theta _1=0.2-2, \theta _2=0.8-1.2\); yield stress \(\tau _0=0-0.1\); plasma layer thickness \(h_P=0-0.1\); Darcy number \(Da=0.01-0.1\); Darcy region thickness \(h_D=0.4-1.8\).

Fig. 2
figure 2

Comparison of the axial velocity profiles in the stenotic zone for various fluid models

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Fig. 3
figure 3

Radial distribution of blood velocity in the stenotic zone with plasma layer thickness \((h_P)\) and yield stress \((\tau _0)\)

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Fig. 4
figure 4

Variation of velocity of blood with parameter constants in K–L fluid \((\theta _1\) and \(\theta _2)\) in the stenotic zone

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Fig. 5
figure 5

Distribution of blood velocity in the stenotic zone with Darcy number (Da) and slip parameter of Darcy \((\alpha )\)

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When the fluid flows in a stenosed artery with porous wall, the velocity profiles for different fluid models such as Newtonian–Newtonian \((\tau _0=0, \theta _1=0)\), Bingham–Newtonian \((\tau _0=0.05, \theta _1=0)\), Casson–Newtonian \((\tau _0=0.05, \theta _1=0.4899)\), and K.L-Newtonian \((\tau _0=0.05, \theta _1=0.2\) and \(\theta _1=0.8)\) are compared and displayed graphically in Fig. 2. It is obvious that, relative to the fluids with yield stress, the Newtonian fluid velocities are much greater. Another notable observation is that the plot of the velocity profile of Casson fluid \((\tau _0=0.05, \theta _1=\sqrt{2\tau _0\theta _2})\) [24] lies between those of the K.L fluid with \(\tau _0=0.05, \theta _1<\sqrt{2\tau _0\theta _2}\) and \(\tau _0=0.05, \theta _1>\sqrt{2\tau _0\theta _2}\). Figure 3 shows the radial variation of axial velocity with the yield stress \((\tau _0)\) and plasma layer thickness \((h_P)\). It depicts that the axial velocity declines as the yield stress increases. Additionally, it may be observed that with the increase in plasma layer thickness, the axial velocity continues to increase. The distribution of axial velocity with the radial distance (r) for different values of K–L fluid parameters \((\theta _1\) and \(\theta _2)\) is shown in Fig. 4. It may be noticed that the rise in the K–L fluid parameters contributes to a decrease in velocity and for the lower values of \(\theta _1 (\theta _1<2)\), the rate of decrease in velocity is observed to be greater.

Fig. 6
figure 6

Distribution of velocity of blood in the stenotic zone with Darcy layer thickness \((h_D)\)

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Fig. 7
figure 7

Axial distribution of plug flow radius \((R_{pl})\) with plasma layer thickness \((h_P)\) and yield stress \((\tau _0)\)

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Fig. 8
figure 8

Axial variation of plug flow radius \((R_{pl})\) with the maximal constriction height \((\delta _s)\)

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Figure 5 describes the effect of Darcy number (Da) and Darcy slip parameter \((\alpha )\) on the radial distribution of the axial velocity of the fluid. The figure shows that the velocity of the fluid (blood) in the stenotic region increases rapidly for the higher values of Da, whereas the velocity increases gradually with the increase of \(\alpha\). We infer from the result that in contrast to the Darcy number, the Darcy slip parameter is a weak parameter in the sense that it induces less variation in the magnitude of the axial velocity. Figure 6 reveals the combined effect of a thickness of Brinkman and Darcy layers on the radial variation of the axial velocity of the fluid. We note that the Brinkman layer thickness \((h_B)\) is altered by the change in the Darcy region thickness \((h_D)\) as the relationship between \(h_B\) and \(h_D\) is considered to be \(h_B=h_D/9\) [27]. It is clear from the figure that the axial velocity in the core, plasma, and Brinkman regions increases as the thickness of the Darcy region decreases, but it starts to decrease at the end of the Brinkman region.

Fig. 9
figure 9

Axial distribution of plug flow radius \((R_{pl})\) with the K–L fluid parameter \((\theta _1)\)

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Fig. 10
figure 10

Axial distribution of plug flow radius \((R_{pl})\) with Darcy number (Da) and slip parameter of Darcy \((\alpha )\)

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Fig. 11
figure 11

Axial variation of plug flow radius \((R_{pl})\) with the thickness of Darcy region \((h_D)\)

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Fig. 12
figure 12

Axial variation of plug flow radius \((R_{pl})\) for different values of shape parameter of constriction (m)

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Fig. 13
figure 13

Axial distribution of wall shear stress \((\tau _w)\) with plasma layer thickness \((h_P)\) and yield stress \((\tau _0)\)

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It is noticed that the axial velocity is very much decreased as yield stress increases, and therefore the plug flow becomes prominent in the flow of blood. The effects of various parameters \((\tau _0, h_P, \theta _1, Da, \alpha , h_D, \delta _s,\) and m) on the plug flow radius \((R_{pl})\) are shown in Figs. 7, 8, 9, 10, 11 and 12. It is illustrative that as z increases from 4 to 4.5, the plug flow radius decreases, and as z increases from 4.5 to 5, it increases. It can be observed from Figs. 7 and 8 that the plug core radius \((R_{pl})\) increases when yield stress increases and the same behavior is noted as plasma layer thickness \((h_P)\) increases, whereas its variation with \(\delta _s\) is always having an opposite nature when other parameters are held fixed. Further, one notices from Fig. 8 that the plug core radius is found to be constant along the axial direction (z) in the absence of stenosis \((\delta _s=0)\).

The effect of parameter constant in K–L model fluid \((\theta _1)\) on plug core radius along the axial direction is shown in Fig. 9. It may be pointed out that the plug core radius increases as \(\theta _1\) decreases. Figure 10 sketches the variation of plug core radius along the axial direction with Darcy number (Da) and Darcy slip parameter \((\alpha )\). It is clear from the results that the plug core radius enhances with the increase in the values of Da and \(\alpha\). Figure 11 shows that the joint impact of the thickness of the Darcy region \((h_D)\) and Brinkman layer \((h_B)\) on the plug core radius. It may be noted that as the value of \(h_D\) raises, the plug core radius upgrades. Figure 12 reveals that the plug core radius escalates in the upstream region of the constriction and reaches its minimum value at some stage, and further, it decreases in the downstream region with the advancement in the value of stenotic shape parameter (m). Also, it is seen that the point at which the plug core radius takes the least value is shifting towards downstream along the axial direction as the value of n increases, which forms the new evidence added to the literature.

Fig. 14
figure 14

Axial variation of wall shear stress \((\tau _w)\) with parameter constant in K–L fluid \((\theta _1)\)

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Fig. 15
figure 15

Axial variation of wall shear stress \((\tau _w)\) in the stenotic zone for different values of Darcy number (Da) and slip parameter of Darcy \((\alpha )\)

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Fig. 16
figure 16

Wall shear stress \((\tau _w)\) distribution in the stenotic zone for different values of the Darcy region thickness \((h_D)\)

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Fig. 17
figure 17

Axial variation of wall shear stress \((\tau _w)\) in the stenotic region for different values of shape parameter of constriction (m)

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Wall shear stress or skin friction signifies a tangential force applied to the wall of the blood vessel by the flowing blood which is produced because of the traction between the wall and the fluid (blood) along the wall. The axial variation of wall shear stress \((\tau _w)\) for different values of \(\tau _0\), \(h_P\), \(\theta _1\), Da, \(\alpha\) and m in the stenotic region has been studied and illustrated in Figs. 13, 14, 15, 16, and 17. Figure 13 shows the variation of the wall shear stress with z-axis for different values of the yield stress \((\tau _0)\) and thickness of the plasma layer \((h_P)\). It is observed that the shear stress at the arterial wall decreases as the plasma layer thickness increases while it increases with the enhancement in yield stress for a fixed value of \(h_P\). It is noticed that the rate of increase or decrease of wall shear stress with respect to the tube axis is found to be higher in the case of yield stress in comparison with the plasma layer thickness. Figure 14 is drawn to analyze the influence of parameter constant in K–L fluid \((\theta _1)\) on the axial variation of skin friction. It is established that the skin friction in axial direction increases considerably with the increase of parameter constant in K–L fluid \((\theta _1)\). One can see that as z moves from 4 to 4.5, the skin friction along the axial direction upsurges, and then it declines symmetrically as z moves further from 4.5 to 5.

Fig. 18
figure 18

Axial variation of flow resistance \((\lambda )\) with plasma layer thickness \((h_P)\) for different values of yield stress \((\tau _0)\) in the stenotic zone

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Fig. 19
figure 19

Distribution of flow impedance \((\lambda )\) in the stenotic zone for different values of parameter constant in K–L fluid \((\theta _1)\)

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In Fig. 15, how the combined role of Darcy number (Da) and Darcy slip parameter \((\alpha )\) in changing the pattern of skin friction \((\tau _w)\) has been displayed. It is seen that the skin friction decreases rapidly with the increase of Darcy number (Da), but it decays marginally as the Darcy slip parameter \((\alpha )\) increases for a fixed value of Da. Figure 16 is drawn to explore the combined effect of the thickness of Brinkman layer \((h_B)\) and Darcy region \((h_D)\) on the skin friction. When the value of \(h_D\le 1.0\), the wall shear stress increases, and it starts decreasing for \(h_D>1.0\). Therefore, the understanding of the joint role of \(h_B\) and \(h_D\) may be helpful in normalizing the blood flow. The drive of Fig. 17 is to illustrate the effect of stenosis shape parameter (m) on the axial distribution of skin friction. It is important to note that in the upstream of the stenotic zone, skin friction decreases and then increases in the downstream of the region as the m value increases. A rise in the magnitude of m tends to change the position where the wall shear stress achieves its maximum towards the downstream.

For more insight into the physical characteristics of the shear stress on the wall geometry of the artery, the response of shear stress in separate regions and total wall shear stress for the two cases such as \(\delta _s=0.0\) (uniform artery) and \(\delta _s=0.1\) (stenosed artery) is depicted in Table 1. From this table, we can see that the magnitude of shear stress is positive at the end of the plasma layer \(\big (\left( \tau _w\right) _P>0\big )\), whereas it is negative in the Brinkman region \(\big (\left( \tau _w\right) _B<0\big )\). It is due to the fact that the axial velocity profiles continue to decrease till the end of the plasma layer, and increase gently in the Brinkman region (see Figs. 2, 3, 4, 5 and 6). However, the magnitude of total shear stress \((\tau _w)\) exerted by the flowing blood on the porous wall of the artery is positive, and is observed to be higher in the case of stenosed arteries as compared to the uniform vessels.

Table 1 Shear stress on the wall geometry in separate regions for different values of stenotic height
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Fig. 20
figure 20

Axial variation of resistive impedance \((\lambda )\) in the stenotic zone with the Darcy region thickness \((h_D)\)

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Fig. 21
figure 21

Variation of flow impedance \((\lambda )\) in the stenotic zone with the Darcy number (Da) and slip parameter of Darcy \((\alpha )\)

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Fig. 22
figure 22

Axial variation of resistive impedance \((\lambda )\) with the maximal stenotic height \((\delta _s)\)

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The axial variation of resistive impedance \((\lambda )\) experienced by the flow of blood is shown in Figs. 18, 19, 20, 21 and 22 for various values of the critical parameters involved in the study. The joint effect of yield stress \((\tau _0)\) and plasma layer thickness \((h_P)\) on flow impedance has been studied from Fig. 18. The resistance to flow enhances as the yield stress \((\tau _0)\) increases for a fixed value of \(h_P\), whereas its variation with plasma layer thickness \((h_P)\) is of inverse nature. Figure 19 displays the axial distribution of the resistive force (flow impedance) for different values of parameter constant in K–L fluid \((\theta _1)\). It is seen that the flow resistance is increased as the value of parameter constant in K–L fluid increases. Figure 20 exhibits the combined effect of the thickness of the Brinkman layer \((h_B)\) and Darcy region \((h_D)\) on the resistive impedance. As the value of \(h_D\le 0.8\), the flow resistance increases while it has an opposite nature when \(h_D>0.8\). Thus, the collective relevant role of \(h_B\) and \(h_D\) could be provided suitable eminence in updating the blood flow model.

Figure 21 is drawn to see the influence of Darcy number (Da) and parameter of Darcy slip \((\alpha )\) on the axial distribution of resistive impedance. One can easy to observe that the impedance to flow decays rapidly as the Darcy number (Da) enhances, but it decreases marginally with the rise of the parameter of Darcy slip \((\alpha )\) for a fixed value of Da. Figure 22 depicts how the flow resistance is affected by the severity of the constriction. The maximal constriction height has been modified by the variation of parameter \(\delta _s\). It is found that for \(\delta _s = 0\), i.e., when there is blood flow through the artery in the absence of stenosis (uniform artery), the curve representing the variation of the flow impedance becomes perfectly linear. This linearity of the results is gradually lost with enhancing the maximal constriction height \((\delta _s)\), and the results are observed to grow nonlinear. Further, one can see that the flow impedance increases gradually throughout the constricted region as the maximum height of the constriction \((\delta _s)\) increases. These interpretations are self-explanatory in the sense that when the maximal stenotic height \((\delta _s)\) is drastically changed, say for \(\delta _s = 0.2\), the resistive force exerted by the streaming blood appears, as predicted, to be a maximum. Hence, the height of the constriction plays a vital role in the flow impedance encountered to characterize the flow behavior of blood in a stenosed artery.

Fig. 23
figure 23

Streamlines for different values of Darcy region thickness: a \(h_D=0.6\), b \(h_D=1.2\)

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Fig. 24
figure 24

Streamlines for different values of stenosis shape parameter: a \(m=2\), b \(m=5\)

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To get more insight into the flow behavior, streamlines are plotted for the whole region in Figs. 23 and 24. The impact of the thickness of the porous wall on the streamline pattern is shown in Fig. 23. As the Darcy region thickness increases from \(h_D=0.6\) to \(h_D=1.2\), the number of bolus increases but the size of the trapping bolus diminishes, whereas the inverse behavior of the streamline pattern is noticed in the Brinkman region. Figure 24 elucidates that the blood flow pattern (streamlines) in the stenotic region when the stenosis shape changes from symmetric \((m=2)\) to asymmetric \((m=5)\). It is observed from this pattern that the trapping bolus is shifting towards the downstream along the z-axis as the value of m increases.

6 Conclusion

The theoretical study of a two-fluid model of blood flow in a stenosed artery with porous wall treating blood in the core region as non-Newtonian fluid and peripheral plasma, Brinkman and Darcy regions filled with Newtonian fluid has been done in the present work. Due to the presence of the Brinkman and Darcy layers in the porous wall, the study brings out several curious fluid mechanical phenomena. It is found that blood velocity in the stenotic region attenuates with the increasing values of the K–L fluid parameters. Further, the velocity in both plug flow and non-plug flow regions increases with the increase of plasma layer thickness, but it slows down by the yield stress.

The plug core radius rises in the upstream region of the constriction, and then it dwindles in the downstream region with the enhancement in the value of the shape parameter of constriction (m). Further, the point at which the plug core radius takes a minimum value is shifting towards downstream along the axial direction as the value of m enhances. The flow impedance and skin friction exerted by the streaming blood are found to reduce with the increasing values of Darcy number and Darcy slip parameter, but they are enhanced with the K–L fluid parameter. The precious outcome of the present study is that the joint effect of thickness of the Brinkman layer and the Darcy region on the skin friction and flow impedance. It is found that for lower values of \(h_D\) (Darcy region thickness), the skin friction and resistive impedance are enhanced with the rise in \(h_D\), while these are found to be opposite nature for higher values of \(h_D\), which forms the new information, at least to the authors’ knowledge, added to the literature. Hence, looking at the significance of the hemodynamic factors in the understanding of blood flow in stenosed arteries, we can conclude that the information about the effects of the rheology of K–L fluid, the thickness of the plasma layer and porous wall on the flow characteristics and the precious way of selecting the appropriate dimensional values of these parameters mentioned above can be utilized to normalize the blood flow in the abnormal arteries and lead to the development of new diagnostic tools.