In the present study, a Caputo–Fabrizio (C–F) time-fractional derivative is introduced to the governing equations to present the flow of blood and the transport of magnetic nanoparticles (MNPs) through an inclined porous artery with mild stenosis. The rheology of blood is defined by the non-Newtonian visco-elastic Jeffrey fluid. The transport of MNPs is used as a drug delivery application for cardiovascular disorder therapy. The momentum and transport equations are solved analytically by using the Laplace transform and the finite Hankel transform along with their inverses, and the solutions are presented in the form of Laplace convolutions. To display the solutions graphically, the Laplace convolutions are solved using the numerical integration technique. The study presents the impacts of different governing parameters on blood and MNP velocities, volumetric flow rate, flow resistance, and skin friction. The study demonstrates that blood and MNP velocities boost with an increase in the fractional order parameter, Darcy number, and Jeffrey fluid parameter. The volumetric flow rate decreases and flow resistance increases with enhancement in stenosis height. The non-symmetric shape of stenosis and the rheology of blood decrease skin friction, whereas enhancement in MNP concentration increases skin friction. A comparison of the present result with the previous work shows excellent agreement. The present study will be beneficial for the field of medical science to further study atherosclerosis therapy and other similar disorders.

Topics
Integral transforms, Nanoparticle, Drug delivery, Atherosclerosis, Cardiovascular system

The transport of hormones, oxygen, and nutrients to cells is carried out by the cardiovascular system. The cardiovascular system takes waste products produced by metabolic activities and delivers them to other organs for disposal. The cardiovascular system plays a crucial role in circulating blood flow in the body. However, the thickening or hardening of the arteries, known as atherosclerosis, affects the cardiovascular system. The main cause of atherosclerosis is the accumulation of fatty substances in the internal lining of the vessel, which prevents blood from flowing freely. This kind of disorder is commonly known as stenosis. According to the World Health Organization (WHO), cardiovascular diseases cause 17.6 million deaths each year, an estimated 32% of all deaths worldwide.1 The restriction of blood flow caused by stenosis affects the normal functioning of the cardiovascular system, generally leading to cardiovascular disorders. In the past few decades, practical interest in the characterization of blood flow through stenotic arteries has received the attention of many researchers. For example, Akbar et al.2 studied the flow of nanofluid through a permeable composite stenosis artery by using the homotopy perturbation technique. Xingting et al.3 demonstrated the thermal condition of blood flow through an artery with multiple stenosis and angles by employing the finite volume method. Zaman et al.4 analyzed nanoparticles transport during blood flow in the affected artery due to the combined impacts of stenosis and aneurysm. In the above studies and studies carried out by researchers in Refs. 5–9 showed the axial direction of blood flow is taken in the horizontal direction. However, in some circumstances, blood vessels can make an angle of inclination with the horizontal. Accordingly, several researchers formulated blood flow models and demonstrated the impacts caused by chemical reactions, slip velocity, and heat source,10 overlapping stenosis and dilatation,11 body acceleration and slip velocity,12 and Hall currents13 on blood flow through an inclined stenosis artery.

Recently, different procedures of therapy have been used to avoid atherosclerosis in the artery, for instance, by using percutaneous transluminal coronary angioplasty and atherectomy. However, after the application of these therapies, the emergence of restenosis in the blood vessel can occur again.14 On the other hand, traditional drug delivery systems like anti-thrombotic and anti-proliferative drugs are incapable of accumulating enough drugs at the target region and can cause side effects in non-target regions due to the nature of free drugs.15 To overcome these drawbacks, magnetic drug targeting (MDT) could be suggested. MDT with the help of MNPs to treat cardiovascular disorders under the application of an external magnetic field is an imperative area of study. During the past few decades, research interest in MDT has increased significantly because of its compatibility, noninvasive nature, and high targeting efficiency as compared to traditional drug delivery systems.16–18 Efficient targeted drug delivery plays an important role in the medication of many diseases, and among them is a cardiovascular disorder caused by defects such as stenosis and thrombosis.19 Alishiri et al.20 suggested, MDT for increasing the transport of drug carriers to the atherosclerosis disease in the artery. Nanoparticles play a vital role in delivering drugs to diseased tissue. In addition, the development of nanotechnology contributes much to the medical world by detecting and diagnosing diseases, such as cancer and atherosclerosis, at their early stages.21 

A fluid that does not obey the linear correlation between shear stress and rate of strain is known as a non-Newtonian fluid (NNF). Non-Newtonian fluids (NNFs) have become an interesting area of study because of their abundant applications in medical science, bioengineering, and industries. Some examples are oil, blood at low shear rates in small vessels, toothpaste, gels, drilling mud, and so on. Due to their complex nature, NNFs are categorized into three types of governing equations: rate, differential, and integral. Among the NNFs, the one that is described in relation to the relaxation and retardation time parameters is a Jeffrey fluid. Jeffrey fluid is a rate-type NNF that can be analyzed in terms of the impacts of the ratio between relaxation and retardation times, and the retardation time parameters.22–25 One important characteristic of the Jeffrey fluid model is that it can be reduced to other fluid models with the appropriate selection of Jeffrey fluid parameters. Having this in mind, various researchers presented the behavior of Jeffrey fluid flow of blood through stenosis arteries subject to the impacts of various physical parameters.26–29 The study of fluid flow in porous vessels has gained considerable interest from researchers due to its various practical applications. The flow in porous media has various engineering applications, for instance, blood flow in capillaries, dialysis in artificial kidneys, engine coolant systems, aircraft wings, the design of filters, and many others.30,31 Excellent findings on flow in porous media can be found in Refs. 32–35.

Fractional mathematical models containing time derivatives play an imperative role in the study of fluid flow. Recently, mathematical models of fluid flow involving time-fractional derivatives have inspired the interest of various researchers due to the impact of time memory on the outcome solution. The time-fractional derivative model contains more information than the classical derivative model of the same kind. Anwar et al.36 presented the C–F time-fractional derivative model and analyzed the influences of ramped velocity, Newtonian heating, radiation, and magnetohydrodynamics (MHD) on Jeffrey fluid flow. Imran et al.37 demonstrated the impacts of radiation and chemical reactions on MHD Jeffrey fluid flow using the C–F time-fractional derivative model without a singular kernel. Ali et al.38 demonstrated a comparative study on a generalized Jeffrey nanofluid employing C–F and Atangana–Balaenu fractional order derivative models. Other findings on fractional Jeffrey fluid models exist in Refs. 39 and 40.

The aforementioned literature surveys show the intensive analysis of Jeffrey fluid flow using the C–F fractional derivative. Jeffrey nanofluid flow has also been discussed by a few researchers. However, the analysis of magnetic nanoparticle transport for drug delivery applications through inclined porous arteries with mild stenosis by treating blood as a non-Newtonian Jeffery fluid model theoretically in the C–F fractional sense has not been explored yet. As a result, the aim of the present analysis is to fill this gap. The novelty of the present problem involves the following:

  • The non-Newtonian visco-elastic Jeffrey fluid represents the rheology of blood.

  • The C–F time-fractional derivative model is used to impose the memory effect.

  • The blood flow and transport of magnetic nanoparticles take place through an inclined artery with stenosis.

  • In the presence of a magnetic field, magnetic nanoparticles are used in drug delivery applications.

First, the geometry of mild stenosis along with the governing equations of blood flow and transport of MNPs are presented. These equations are transformed into dimensionless forms using dimensionless parameters. The dimensionless governing equations are converted into a C–F time-fractional derivative model.41 Second, the analytic solutions for the velocities of blood and MNPs are obtained using Laplace transform (LT) and finite Hankel transform (FHT) of order zero along with their inverses, and the solutions are written in the form of Laplace convolutions. Finally, using the MATLAB software, the numerical results are plotted for illustration. The study presents the effects of various parameters such as fractional parameter, Darcy number, Jeffrey fluid parameter, stenosis height, inclination angle, Hartmann number, shape parameter, and MNPs concentration parameter in the stenosis region. The outcomes of the present study will give insight into understanding the transport of MNPs during magnetic drug targeting and the dynamics of blood flow due to the presence of atherosclerosis.

In the present study, unsteady incompressible non-Newtonian Jeffrey fluid flow of blood along with MNPs through an inclined artery with mild stenosis is considered (Fig. 1). It is assumed that the flow field is unidirectional and that the MNPs are distributed uniformly throughout the blood. Here, the vertical axis is the radial axis, and the artery which makes an angle β with the horizontal is taken along the z - axis while the uniform magnetic field is applied in the vertical direction. At time t ¯ = 0⁠, there is no motion in the system. Maxwell's equations describe the magnetic field strength, while Newton's second law of motion demonstrates the transport of MNPs. The momentum equation describes the flow of blood along the flow axis.

FIG. 1.
Geometrical model of the problem.
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Geometrical model of the problem.

FIG. 1.
Geometrical model of the problem.
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Geometrical model of the problem.

Close modal
The electromagnetic force is governed by Maxwell's relations42,43
∇ × B → = μ 0 J → , ∇ ⋅ B → = 0 , ∇ × E → = − ∂ B → ∂ t ,
(1)
where μ 0⁠, E →⁠, and J → are the magnetic permeability, electric field intensity, and the current density, respectively. B → = B → 0 + b → 0 is the magnetic flux intensity, B 0 is the applied magnetic field strength in the radial direction, and b → 0 is the induced magnetic field. J → can be given by
J → = σ ( E → + V → × B → ) .
(2)
The induced magnetic field b → 0 is negligible due to a very small Reynolds number. Moreover, it is considered that there is no applied voltage and polarization in the system; as a result, the electric field E → is disregarded. Having these assumptions, we define the electromagnetic force F → m by
F → m = J → × B → = σ ( V → × B → ) × B → = − σ B 0 2 u ( r , t ) k → ,
(3)
where σ⁠, V → = u ( r , t ) k →⁠, and k → are the electrical conductivity, velocity field, the standard unit vector in the z - direction, respectively.
The equation that describes radially non-symmetric and axially symmetric stenosis is44,45
R ¯ ( z ¯ ) = { R 0 + R 0 Υ ¯ [ ( z ¯ − d ¯ ) w − L ¯ 0 ( w − 1 ) ( z ¯ − d ¯ ) ] , d ¯ < z ¯ < d ¯ + L ¯ 0 , R 0 , elsewhere ,
(4)
where Υ ¯ = δ ¯ w ( w / ( w − 1 ) ) R 0 L ¯ 0 w ( w − 1 )⁠, R ¯ ( z ¯ ) is the radius of the stenosis, and R 0 is the radius of the vessel, d ¯ is the location of the stenosis and L ¯ 0 is its length, w ≥ 2 is the stenosis shape parameter (where w = 2 denotes the symmetric stenosis), and δ ¯ is the maximum height of the stenosis at Z ¯ = d ¯ + L ¯ 0 w ( 1 / ( w − 1 ) ) with δ ¯ < R 0⁠.
The unsteady Jeffrey fluid flow of blood in cylindrical polar coordinates ( r ¯ , θ ¯ , z ¯ ) is governed by the momentum equation46–48 
∂ ∂ t ¯ ( u ¯ ( r ¯ , t ¯ ) ) − 1 ρ ∂ p ¯ ∂ z ¯ = 1 ρ 1 r ¯ ∂ ∂ r ¯ ( r ¯ S ¯ r ¯ z ¯ ) + K S N ρ ( v ¯ ( r ¯ , t ¯ ) − u ¯ ( r ¯ , t ¯ ) ) − μ k p ρ u ¯ ( r ¯ , t ¯ ) + Ψ ¯ ( t ¯ ) − σ B 0 2 sin β ρ u ¯ ( r ¯ , t ¯ ) + g ¯ sin β ,
(5)
where the extra tensor of the Jeffrey fluid S ¯ r ¯ z ¯ is of the form
S ¯ r ¯ z ¯ = μ ( 1 1 + λ 1 ) [ 1 + λ 2 ( u ¯ r ¯ ( r ¯ , t ¯ ) ∂ ∂ r ¯ + u ¯ ( r ¯ , t ¯ ) ∂ ∂ z ¯ ) ] ( ∂ ∂ z ¯ ( u ¯ r ¯ ( r ¯ , t ¯ ) ) + ∂ ∂ r ¯ ( u ¯ ( r ¯ , t ¯ ) ) ) .
(6)

The velocities u ¯ r ¯ ( r ¯ , t ¯ )⁠, u ¯ ( r ¯ , t ¯ )⁠, and v ¯ ( r ¯ , t ¯ ) are the radial and axial velocities of blood and velocity of MNPs, respectively. The constants ρ⁠, μ⁠, K S⁠, and k p are the density of blood, dynamic viscosity of blood, Stoke's constant, and the permeability of the porous medium, respectively. N represents the number of MNPs per unit volume, K S N ρ ( v ¯ ( r ¯ , t ¯ ) − u ¯ ( r ¯ , t ¯ ) ) describes the force between MNPs and blood that happens due to their relative motion, g ¯ is the acceleration due to gravity, λ 1 is the ratio between relaxation and retardation times, and λ 2 is the retardation time.

Taking into account the unidirectional flow and substitution of Eq. (6) into Eq. (5) yields
∂ ∂ t ¯ ( u ¯ ( r ¯ , t ¯ ) ) = − 1 ρ ∂ p ¯ ∂ z ¯ + ν 1 + λ 1 ( ∂ 2 ∂ r ¯ 2 + 1 r ¯ ∂ ∂ r ¯ ) u ¯ ( r ¯ , t ¯ ) + K S N ρ ( v ¯ ( r ¯ , t ¯ ) − u ¯ ( r ¯ , t ¯ ) ) − μ k p ρ u ¯ ( r ¯ , t ¯ ) + Ψ ¯ ( t ¯ ) − σ B 0 2 sin β ρ u ¯ ( r ¯ , t ¯ ) + g ¯ sin β ,
(7)
where ν is the kinematic viscosity.
The transport of MNPs is described in accordance with Newton's second law of motion, which is given by
M m ∂ ∂ t ¯ ( v ¯ ( r ¯ , t ¯ ) ) = K S ( u ¯ ( r ¯ , t ¯ ) − v ¯ ( r ¯ , t ¯ ) ) ,
(8)
where M m represents the average mass of MNPs.
The oscillating pressure gradient of blood flow in Eq. (7) has the form49–51 
− ∂ p ¯ ∂ z ¯ = q ¯ 0 + q ¯ 1 cos ( φ ¯ t ¯ ) , t ¯ > 0 ,
(9)
where q ¯ 0⁠, q ¯ 1⁠, and φ ¯ represent the steady amplitude, amplitude of the transient part, and the frequency of pulsation, respectively.
The periodic body acceleration Ψ ¯ ( t ¯ ) in Eq. (7) is given by52,53
Ψ ¯ ( t ¯ ) = ℓ ¯ 0 cos ( k ¯ t ¯ + γ 0 ) ,
(10)
where ℓ ¯ 0⁠, γ 0⁠, and k ¯ are the amplitude, the difference with the pulsatile pressure gradient, and the body acceleration frequency, respectively.
To introduce the C–F time-fractional derivative to Eqs. (7) and (8), we multiply the equations by Λ = R 0 ρ q 0 (with unit of time) which yields
Λ α D t ¯ α u ¯ ( r ¯ , t ¯ ) = Λ ρ ( q ¯ 0 + q ¯ 1 cos ( φ ¯ t ¯ ) ) + Λ ν 1 + λ 1 ( ∂ 2 ∂ r ¯ 2 + 1 r ¯ ∂ ∂ r ¯ ) u ¯ ( r ¯ , t ¯ ) + Λ K S N ρ ( v ¯ ( r ¯ , t ¯ ) − u ¯ ( r ¯ , t ¯ ) ) − Λ μ k p ρ u ¯ ( r ¯ , t ¯ ) + Λ ℓ ¯ 0 cos ( k ¯ t ¯ + γ 0 ) − Λ σ B 0 2 sin β ρ u ¯ ( r ¯ , t ¯ ) + Λ g ¯ sin β ,
(11)
M m Λ α D t ¯ α v ¯ ( r ¯ , t ¯ ) = Λ K S ( u ¯ ( r ¯ , t ¯ ) − v ¯ ( r ¯ , t ¯ ) ) ,
(12)
where the C–F time-fractional derivative model is given by41 
C − F D t α u ( r , t ) = 1 1 − α ∫ 0 t exp [ − α ( t − τ ) 1 − α ] u ′ ( r , τ ) d τ , 0 < α < 1.
(13)
The LT of Eq. (13) can have the form
L [ C − F D t α u ( r , t ) ] = s L [ u ( r , t ) ] − u ( r , 0 ) s + ( 1 − s ) α .
(14)
Equations (7) and (8) are subject to the following initial conditions (ICs) and boundary conditions (BCs):
u ¯ ( r ¯ , 0 ) = 0 , v ¯ ( r ¯ , 0 ) = 0 , for t ¯ = 0 and r ¯ ∈ [ 0 , R ¯ ] , u ¯ ( r ¯ , t ¯ ) = 0 , v ¯ ( r ¯ , t ¯ ) = 0 , for t ¯ > 0 and r ¯ = R ¯ ,
(15)
where R ¯ = R ¯ ( z ¯ ) .
For the purpose of the dimensionless study, we use the following dimensionless parameters:
r = r ¯ R 0 , t = t ¯ Λ , u = Λ u ¯ R 0 , v = Λ v ¯ R 0 , g = Λ 2 g ¯ R 0 , q 0 = Λ 2 q ¯ 0 ρ R 0 , q 1 = Λ 2 q ¯ 1 ρ R 0 , φ = Λ φ ¯ , k = Λ k ¯ , ℓ 0 = Λ 2 ℓ ¯ 0 R 0 , z = z ¯ R 0 , d = d ¯ R 0 , L 0 = L ¯ 0 R 0 , R = R ¯ R 0 , δ = δ ¯ R 0 . }
(16)
Upon using Eq. (16) into Eqs. (11) and (12) yields
D t α u ( r , t ) = q 0 + q 1 cos ( φ t ) + 1 Re ( 1 + λ 1 ) ( ∂ 2 ∂ r 2 + 1 r ∂ ∂ r ) u ( r , t ) + p c Re ( v ( r , t ) − u ( r , t ) ) − 1 Da Re u ( r , t ) + ℓ 0 cos ( k t + γ 0 ) − H a 2 u ( r , t ) + g sin β ,
(17)
p m D t α v ( r , t ) = u ( r , t ) − v ( r , t ) ,
(18)
where Re = R 0 2 Λ v is the Reynolds number, pc = K S N R 0 2 μ is the particle concentration parameter, Da = k p R 0 2 is the Darcy number, Ha = B 0 σ R 0 ρ sin β is the Hartman number, and p m = M m K S Λ is the mass parameter of particles.
The corresponding dimensionless ICs and BCs to Eqs. (17) and (18) are
u ( r R , 0 ) = 0 , v ( r R , 0 ) = 0 , for t = 0 and r R ∈ [ 0 , 1 ] , u ( 1 , t ) = 0 , v ( 1 , t ) = 0 , for t > 0 and r R = 1.
(19)
The stenosis equation in dimensionless form can be written as
R ( z ) = { 1 + Υ [ ( z − d ) w − L 0 w − 1 ( z − d ) ] , d < z < d + L 0 , 1 , elsewhere ,
(20)
with Υ = δ w ( w / ( w − 1 ) ) L 0 w ( w − 1 ) , z = d + L 0 w ( 1 / ( w − 1 ) )⁠, and δ < 1⁠.
Introducing the dimensionless radial transformation coordinate y = r R⁠, Eq. (19) takes the form
u ( y , 0 ) = 0 , v ( y , 0 ) = 0 , for t = 0 and y ∈ [ 0 , 1 ] , u ( 1 , t ) = 0 , v ( 1 , t ) = 0 , for t > 0 and y = 1.
(21)
The two integral transforms, LT and FHT, are used to find the analytic solutions of the time-fractional derivatives (17) and (18), subject to the ICs and BCs (21). The LT method is essentially used when the fluid model is described as a function of time. Applying the LT to Eqs. (17) and (18), and using ICs from Eq. (21) yields
s s + α ( 1 − s ) u ( y , s ) = ( q 0 1 s + q 1 s s 2 + φ 2 ) + 1 Re ( 1 + λ 1 ) R 2 × ( ∂ 2 ∂ y 2 + 1 y ∂ ∂ y ) u ( y , s ) + pc Re v ( y , s ) − ( pc Re + 1 Da Re + H a 2 ) u ( y , s ) + ℓ 0 ( cos γ 0 s s 2 + k 2 − sin γ 0 k s 2 + k 2 ) + 1 s g sin β ,
(22)
p m s s + α ( 1 − s ) v ( y , s ) = u ( y , s ) − v ( y , s ) ,
(23)
with
u ( 1 , s ) = 0 , v ( 1 , s ) = 0.
(24)
Solving for v ( y , s ) from Eq. (23) yields
v ( y , s ) = s + α ( 1 − s ) p m s + s + α ( 1 − s ) u ( y , s ) .
(25)
Upon substitution of Eq. (25) into Eq. (22), we get
s s + α ( 1 − s ) u ( y , s ) = ( q 0 1 s + q 1 s s 2 + φ 2 ) + 1 Re ( 1 + λ 1 ) R 2 ( ∂ 2 ∂ y 2 + 1 y ∂ ∂ y ) u ( y , t ) + ( pc Re ( s + α ( 1 − s ) p m s + s + α ( 1 − s ) ) − ( pc Re + 1 Da Re + H a 2 ) ) u ( y , s ) + ℓ 0 ( cos γ 0 s s 2 + k 2 − sin γ 0 k s 2 + k 2 ) + 1 s g sin β .
(26)
Transformation of Eq. (26) using FHT of order zero and application of BCs from Eq. (24) gives
( s s + α ( 1 − s ) − pc Re ( s + α ( 1 − s ) p m s + s + α ( 1 − s ) ) + r n 2 Re ( 1 + λ 1 ) R 2 + pc Re + 1 Da Re + H a 2 ) u H ( y n , s ) = ( ( q 0 1 s + q 1 s s 2 + φ 2 ) + ℓ 0 ( cos ⁡ γ 0 s s 2 + k 2 − sin ⁡ γ 0 k s 2 + k 2 ) + 1 s g sin ⁡ β ) J 1 ( y n ) y n ,
(27)
where u H ( y n , s ) = ∫ 0 1 y u ( y , s ) J 0 ( y y n ) d y is the HT of u ( y , s )⁠, with y n ( n = 1 , 2 , 3 , … ) are the positive roots of J 0 ( . )⁠. The functions J 0 ( . ) and J 1 ( . ) are the zeroth and first order Bessel functions of the first kind.
Rearranging the coefficients of u H ( y n , s ) in Eq. (27) yields
u H ( y n , s ) = a 4 n s 2 + a 5 n s + a 6 n a 8 n s 2 + a 9 n s + a 10 n × ( ( q 0 1 s + q 1 s s 2 + φ 2 ) + ℓ 0 ( cos γ 0 s s 2 + k 2 − sin γ 0 k s 2 + k 2 ) + 1 s g sin β ) J 1 ( y n ) y n .
(28)
Applying the partial fraction decomposition of a 4 n s 2 + a 5 n s + a 6 n a 8 n s 2 + a 9 n s + a 10 n in Eq. (28) and simplifying the equation yields
u H ( y n , s ) = ( a 13 n s − a 11 n + a 14 n s − a 12 n ) ( ( q 0 + g sin β ) 1 s + q 1 s s 2 + φ 2 + ℓ 0 cos γ 0 s s 2 + k 2 − ℓ 0 sin γ 0 k s 2 + k 2 ) J 1 ( y n ) y n .
(29)
Expanding the coefficients of Eq. (29) yields
u H ( y n , s ) = { ( q 0 + g sin β ) [ a 13 n s − 1 s − a 11 n + a 14 n s − 1 s − a 12 n ] + q 1 s s 2 + φ 2 [ a 13 n 1 s − a 11 n + a 14 n 1 s − a 12 n ] + ℓ 0 cos γ s s 2 + k 2 [ a 13 n 1 s − a 11 n + a 14 n 1 s − a 12 n ] − ℓ 0 sin γ k s 2 + k 2 [ a 13 n 1 s − a 11 n + a 14 n 1 s − a 12 n ] } J 1 ( y n ) y n ,
(30)
where the corresponding parameters used in Eqs. (28) and (29) are given by
a 1 n = Re ( p m − α + 1 ) − pc ( 1 − α ) 2 , a 2 n = α ( Re − 2 pc ( 1 − α ) ) , a 3 n = − pc α 2 , a 4 n = Re ( p m ( 1 − α ) + ( 1 − α ) 2 ) , a 5 n = Re α ( p m + 2 ( 1 − α ) ) , a 6 n = Re α 2 , a 7 n = r n 2 Da + ( 1 + λ 1 ) [ R 2 Da ( 1 + Pc + Re H a 2 ) ] Re Da R 2 ( 1 + λ 1 ) , a 8 n = a 1 n + a 4 n a 7 n , a 9 n = a 2 n + a 5 n a 7 n , a 10 n = a 3 n + a 6 n a 7 n , a 11 n = ( − a 9 n + a 9 n 2 − 4 a 8 n a 10 n 2 a 8 n ) , a 12 n = ( − a 9 n − a 9 n 2 − 4 a 8 n a 10 n 2 a 8 n ) , a 13 n = a 4 n a 11 n 2 − a 5 n a 11 n + a 6 n a 11 n − a 12 n , a 14 n = a 4 n a 12 n 2 − a 5 n a 12 n + a 6 n a 12 n − a 11 n .
(31)
To apply the inverse LT to Eq. (30), the following special functions are used:54 
L − 1 [ 1 s ψ + ς ] = F ψ ( − ς , t ) = ∑ n = 0 ∞ ( − ς ) n t ( n + 1 ) ψ − 1 Γ ( ( n + 1 ) ψ ) , ψ > 0 ,
(32)
L − 1 [ s − ε s ψ + ς ] = ℜ ψ , ε ( − ς , t ) = ∑ n = 0 ∞ ( − ς ) n t ( n + 1 ) ψ − 1 − ε Γ ( ( n + 1 ) ψ − ε ) , Re ( ψ − ε ) > 0 ,
(33)
where F ψ ( − ς , t ) and ℜ ψ , ε ( − ς , t ) are the Robotnov–Hartley and Lorenzo–Hartley functions, respectively.
Upon the application of inverse LT to Eq. (30) with the help of Eqs. (32) and (33) yields
u H ( y n , t ) = [ G 11 n ( t ) + G 22 n ( t ) + G 33 n ( t ) + G 44 n ( t ) ] J 1 ( y n ) y n , 0 < α < 1 ,
(34)
where
G 11 n ( t ) = ( q 0 + g sin β ) ( a 13 n R 1 , 1 ( a 11 n , t ) + a 14 n R 1 , 1 ( a 12 n , t ) ) , G 22 n ( t ) = q 1 ( cos ( φ t ) ∗ [ a 13 n F 1 ( a 11 n , t ) + a 14 n F 1 ( a 12 n , t ) ] ) , G 33 n ( t ) = ℓ 0 cos γ ( cos ( k t ) ∗ [ a 13 n F 1 ( a 11 n , t ) + a 14 n F 1 ( a 12 n , t ) ] ) , G 44 n ( t ) = − ℓ 0 sin ⁡ γ ( sin ( k t ) ∗ [ a 13 n F 1 ( a 11 n , t ) + a 14 n F 1 ( a 12 n , t ) ] ) .
(35)
Applying the inverse HT to Eq. (34) yields the velocity of blood, which has the form
u ( y , t ) = 2 ∑ n = 1 ∞ J 0 ( y y n ) y n J 1 ( y n ) [ G 11 n ( t ) + G 22 n ( t ) + G 33 n ( t ) + G 44 n ( t ) ] .
(36)
The MNPs velocity can be determined from Eq. (25) by using inverse LT,
L − 1 [ v ( y , s ) ] = L − 1 [ α + s ( 1 − α ) α + s ( p m + 1 − α ) u ( y , s ) ] .
(37)
Simplification of Eq. (37) yields
v ( y , t ) = a 15 n ( 1 − a 16 n ) [ F 1 ( − a 15 n , t ) ∗ u ( y , t ) ] , 0 < α < 1.
(38)
The constants used in Eq. (38) are
a 15 n = α p m + ( 1 − α ) , a 16 n = 1 − α p m + ( 1 − α ) .
(39)

The symbol * in Eqs. (35) and (38) represents the convolution product, where ( h ∗ g ) ( t ) = ∫ 0 t h ( τ ) g ( t − τ ) d τ shows the convolution of two functions h and g.

The wall shear stress of the flow field can be calculated from
τ w = − μ Λ ( 1 1 + λ 1 ) ( ∂ u ∂ r | r = R ) .
(40)
The skin friction of the flow field in dimensionless form is
C f ( z , t ) = τ w ρ u 0 2 = 2 Re ( 1 + λ 1 ) R ∑ n = 1 ∞ [ G 11 n ( t ) + G 22 n ( t ) + G 33 n ( t ) + G 44 n ( t ) ] ,
(41)
where u 0 = R 0 / Λ is the characteristic velocity.
The pressure drop between z = 0 and z = L through the stenosis channel can be written as
Δ p = ∫ 0 L ( − ∂ p ∂ z ) d z .
(42)
The flow rate Q and the impedance λ (flow resistance) can, respectively, be written as follows:
Q ( z , t ) = 2 π ∫ 0 R r u ( r , t ) d y = 2 π R 2 ∫ 0 1 y u ( y , t ) d y = 4 π R 2 ∑ n = 1 ∞ 1 y n 2 [ G 11 n ( t ) + G 22 n ( t ) + G 33 n ( t ) + G 44 n ( t ) ] ,
(43)
λ ( z , t ) = Δ p Q ( z , t ) = 1 Q ( z , t ) [ ∫ 0 d ( − ∂ p ∂ z ) R = 1 d z + ∫ d d + L 0 ( − ∂ p ∂ z ) d z + ∫ d + L 0 L ( − ∂ p ∂ z ) R = 1 d z ] .
(44)
Equation (44) simplifies to
λ ( z , t ) = Δ p Q ( z , t ) = 1 Q ( z , t ) [ ( L − L 0 ) ( q 0 + q 1 cos ( φ t ) ) R = 1 + ∫ d d + L 0 ( q 0 + q 1 cos ( φ t ) ) d z ] .
(45)

In the present study, the non-Newtonian Jeffrey fluid flow of blood with the transport of MNPs is analyzed to explore meaningful interpretations. The governing equations are solved by using integral transforms. Equations (36), (38), (41), (43), and (45) represent the velocity of blood u ( y , t )⁠, velocity of MNPs v ( y , t )⁠, skin friction C f ( z , t )⁠, volumetric flow rate Q ( z , t )⁠, and flow resistance λ ( z , t )⁠, respectively. Corresponding to these analytic solutions, we present the graphs for u ( y , t )⁠, v ( y , t )⁠, C f ( z , t )⁠, Q ( z , t )⁠, and λ ( z , t ) by applying the use of MATLAB simulations, and the results are demonstrated in Figs. 3–13. These graphs demonstrate the impacts of fractional parameter α⁠, Darcy number Da⁠, Jeffrey fluid parameter λ 1⁠, stenosis height δ⁠, inclination angle β⁠, Hartmann number Ha⁠, shape parameter w, and MNPs concentration parameter pc in the stenosis region. A set of values for the parameters for the present study is considered from the existing literature, and it is shown in Table I. The validation of the present result is in excellent agreement with the Newtonian fluid (works of Shah et al.57) for λ 1 = β = ℓ 0 = 0⁠, Da → ∞⁠, q 0 = 0.5⁠, q 1 = 0.6⁠, p m = 0.8⁠, p c = 0.5⁠, Re = 5⁠, φ = π / 4⁠, H a = 2⁠, z = 1 and the result is shown in Fig. 2.

FIG. 2.
Comparison of velocity profile with the works of Shah et al.57
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Comparison of velocity profile with the works of Shah et al.57 

FIG. 2.
Comparison of velocity profile with the works of Shah et al.57
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Comparison of velocity profile with the works of Shah et al.57 

Close modal
FIG. 3.
Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different Da and α when λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3 and g = 9.81.
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Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different Da and α when λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3 and g = 9.81⁠.

FIG. 3.
Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different Da and α when λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3 and g = 9.81.
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Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different Da and α when λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3 and g = 9.81⁠.

Close modal
FIG. 4.
Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different λ 1 and α when q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different λ 1 and α when q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

FIG. 4.
Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different λ 1 and α when q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profiles of the velocities (a) u ( y , t ) and (b) v ( y , t ) for different λ 1 and α when q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

Close modal
FIG. 5.
Profile of volumetric flow rate Q for different δ with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c ) α = 0.8, (d) α = 1.0.
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Profile of volumetric flow rate Q for different δ with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c ) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 5.
Profile of volumetric flow rate Q for different δ with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c ) α = 0.8, (d) α = 1.0.
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Profile of volumetric flow rate Q for different δ with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c ) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
FIG. 6.
Profile of volumetric flow rate Q for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of volumetric flow rate Q for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

FIG. 6.
Profile of volumetric flow rate Q for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of volumetric flow rate Q for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

Close modal
FIG. 7.
Profile of flow resistance λ for different Ha with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of flow resistance λ for different Ha with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 7.
Profile of flow resistance λ for different Ha with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of flow resistance λ for different Ha with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
FIG. 8.
Profile of flow resistance λ for different δ with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of flow resistance λ for different δ with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 8.
Profile of flow resistance λ for different δ with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of flow resistance λ for different δ with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
FIG. 9.
Profile of flow resistance λ for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of flow resistance λ for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

FIG. 9.
Profile of flow resistance λ for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of flow resistance λ for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

Close modal
FIG. 10.
Profile of skin friction C f for different w with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different w with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 10.
Profile of skin friction C f for different w with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different w with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
FIG. 11.
Profile of skin friction C f for different λ 1 with q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different λ 1 with q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 11.
Profile of skin friction C f for different λ 1 with q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different λ 1 with q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
FIG. 12.
Profile of skin friction C f for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of skin friction C f for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

FIG. 12.
Profile of skin friction C f for different β when α = 0.6, λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81.
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Profile of skin friction C f for different β when α = 0.6⁠, λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81⁠.

Close modal
FIG. 13.
Profile of skin friction C f for different pc with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different pc with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

FIG. 13.
Profile of skin friction C f for different pc with λ 1 = 0.6, q 0 = 0.2, q 1 = 0.3, ℓ 0 = 0.8, β = π / 6, L = 5, d = 1, L 0 = 2, δ = 0.08, t = 0.3, and g = 9.81 when (a) α = 0.4, (b) α = 0.6, (c) α = 0.8, (d) α = 1.0.
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Profile of skin friction C f for different pc with λ 1 = 0.6⁠, q 0 = 0.2⁠, q 1 = 0.3⁠, ℓ 0 = 0.8⁠, β = π / 6⁠, L = 5⁠, d = 1⁠, L 0 = 2⁠, δ = 0.08⁠, t = 0.3⁠, and g = 9.81 when (a) α = 0.4⁠, (b) α = 0.6⁠, (c) α = 0.8⁠, (d) α = 1.0⁠.

Close modal
TABLE I.

Physical parameter values with references.

Physical parametersValuesReferences
pm 0.8 Maiti et al.43  
γ0  π / 3 Maiti et al.43  
pc 0.5 Maiti et al.43  
φ  π / 4 Ali et al.49  
Re 3 Ali et al.49  
k  π / 4 Bansi et al.55  
Da 0.8 Yadeta and Shaw56  
w 2 Ponalagusamy47  
Physical parametersValuesReferences
pm 0.8 Maiti et al.43  
γ0  π / 3 Maiti et al.43  
pc 0.5 Maiti et al.43  
φ  π / 4 Ali et al.49  
Re 3 Ali et al.49  
k  π / 4 Bansi et al.55  
Da 0.8 Yadeta and Shaw56  
w 2 Ponalagusamy47  
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A time-variant C–F fractional derivative concept, which gives the memory impact, is utilized in the governing equations of blood flow and transport of nanoparticles. The final solutions of the flow pattern with the corresponding physical quantities are displayed along with the fractional parameter. The present analysis considers a wide range of the values of parameter α with ( α = 0.4 , 0.6 , 0.8 , 1.0 )⁠. The lower value of α (i.e., α = 0.4⁠) supports the lower memory effect, while the larger value of α ( α = 0.8 ) supports the higher memory impact. The general time derivative is represented by α = 1.0⁠. In all the results displayed graphically, the effect of the variation of fractional parameters is incorporated, which reflects a tremendous contribution to the flow field and physical quantities. Figures 3 and 4 show the impact of the fractional parameters on the velocities of fluid and MNPs along with different values of Darcy number Da and Jeffrey fluid parameter λ 1⁠. From the figures, it is shown that the fractional parameters support the flow of blood and the transport of MNPs. Hence, the velocities of blood and MNPs increase with an increase in α⁠. It is seen that the larger value of memory effect ( α = 0.8 ) accelerates the flow of blood and the transport of MNPs, and the phenomenon continues for the general time derivative with α = 1.0⁠. On the other hand, a smaller value of α slows the velocities of both blood and MNPs, and the velocities are quickly dropped. This phenomenon is supported by the behavior of fractional parameters. Hence, a large value of α gives a quick response to the previous memories of blood flow behavior and the transport of MNPs, whereas the lower value of α takes some intermediate time to respond. Since then, it is shown that α plays an imperative role in regulating both the velocities. It is important to mention that although the velocity profile for MNPs shows the same trend as the velocity of blood, there is a significant difference in the magnitude of the velocities. The transport velocity of MNPs is comparatively lower when compared to the velocity of blood. This happens due to the additional impact of retarding and drag forces on the MNPs.49 

Figure 5 depicts the influence of the fractional parameters α on volumetric flow rate Q for different values of stenosis height δ⁠. From the figure, it can be seen that the volumetric flow rate slows down for a lower value of the fractional parameter and speeds up for a higher value of the fractional parameter, which supports a long memory effect. This phenomenon happens due to the fact that the volumetric flow rate has a positive correlation with the velocity of the fluid. Figures 7 and 8 depict the influence of α on flow resistance λ⁠, for different values of Hartman number Ha and stenosis height δ⁠, respectively. From both figures, it is shown that the flow resistance decreases with an increase in the value of the fractional parameter. From Figs. 7 and 8, it is clearly indicated that the larger fractional parameter that supports the long memory effect restricts the passage of fluid through the stonesis region more than the lower fractional parameter that supports the short memory effect. Hence, the flow resistance phenomenon is inversely proportional to the volumetric flow rate, which is why the impact of fractional parameters shows such an inverse relation to flow resistance. The impacts of α along with shape parameter w, Jeffrey fluid parameter λ 1 and MNPs concentration parameter pc on skin friction C f of the inner arterial wall are demonstrated in Figs. 10, 11 and 13 respectively. From the figures, it is observed that the value of skin friction is higher for the larger fractional parameter that supports the long memory effect, whereas the lower fractional parameter that supports the short memory effect restricts skin friction. In physical terms, the skin friction is supported by the increase in the velocity field of the fluid, as observed in Fig. 3. One can clearly understand that the findings from Figs. 3–5, 7, 8, 10, 11 and 13 show the advantage of using the time-fractional derivative model in real-life applications. A proper selection of fractional order parameter α which links to the flow of blood and the transport of MNPs is a key factor that can be applicable in the medical science field for local drug targeting purposes to treat diseases such as atherosclerosis and tumor cells. Further, the application can reduce bleeding during surgery.

Figure 3 demonstrates the impact of Darcy number Da and α on the velocities of blood and MNPs. The Darcy number Da is related to the permeability nature of the vessel and it significantly affects the velocities of blood and MNPs. From the figure, it is illustrated that both velocities increase with an increase in Da⁠. Physically, this condition is associated with factors such as the permeability of the artery and the intravascular pressure, which contribute to blood flow. On the other hand, when the permeability of the medium increases, the drag force decreases, which in turn increases the velocities of blood and MNPs. From these phenomena, it can be claimed that the arterial wall with lower permeability supports the capture efficiency of the drug particles better than the artery with higher permeability. This shows the importance of the permeability of the arterial wall in regulating the velocities to trap the MNPs in the diseased region during drug targeting.

Visco-elastic fluids are non-Newtonian fluids that exhibit both viscous and elastic components in nature. The viscous components arise due to the presence of plasma, while the presence of red blood cells (RBCs) is responsible for the elastic nature.58 Further, it is clear that the higher hematocrit (percentage of RBCs in the blood) enhances the elastic nature of the blood, particularly in small blood vessels.59 In the present study, the Jeffrey fluid is used to define the rheology of blood due to its visco-elastic nature. Figure 4 demonstrates the impact of Jeffery fluid parameter λ 1 along with α on velocities of blood and MNPs. From the figure, it is seen that both the velocities of blood and MNPs increase with increases in the value of λ 1⁠. This means that a larger value of λ 1 leads to a higher relaxation time that allows more time for the fluid and MNPs to move faster. This parameter has a vital role in regulating blood flow and can be applicable during drug delivery phenomena and surgery.48 

Figures 5 and 6 display the graphs of volumetric flow rate Q with respect to the axial distance for different values of stenosis height δ (along with α), and inclination angle β⁠, respectively. The volume flow rate represents the volume of fluid that passes through a particular region of the vessel per unit of time. It is well known that the volumetric flow rate is proportional to the cross-sectional area of the vessel and the velocity of the fluid. Due to the presence of stenosis, the cross-sectional area of the blood vessel changes. Furthermore, the volumetric flow rate through a blood vessel is highly affected by the change in the radius of the vessel due to the existence of the stenosis height. From Fig. 5, one can see that Q significantly decreases with an increase in stenosis height. Hence, with an increase in δ⁠, the radius of the artery in the stenosis region decreases, and further, the cross-sectional area of the artery decreases which is responsible for the reduction of Q. In physical terms, a reduction in the volumetric flow rate of blood can cause complications in the cardiovascular system, leading to stroke, heart attack, and so forth. The phenomenon can help biomedical engineers develop suitable devices to diagnose and treat atherosclerosis during drug targeting. From Fig. 6, it is shown that the volumetric flow rate enhances with an increase in the inclination angle β from β = π 9 to β = π 3⁠. The volumetric flow rate is minimum when the blood vessel has lower inclination; however, the flow rate attains its maximum value at the maximum inclination angle. This phenomenon is related to the velocity of blood; hence, the inclination angle is positively related to the velocity of blood flow.

Figures 7 and 8 depict the influence of Hartmann number Ha and stenosis height δ along with α on the flow resistance λ⁠, respectively. Among the primary factors that describe the resistance to the flow of blood is the change in vessel diameter. For example, vessel diameter changes when the fatty substance is built up in the internal lining of the vessel, which causes a cardiovascular disorder commonly known as stenosis. The presence of the stenosis significantly enhances the flow resistance of blood.60 In both figures (Figs. 7 and 8), it is seen that the stenosis is symmetric (with shape factor w = 2⁠) and, hence, the profile of the flow resistance is symmetric along the center of stenosis ( z = 2 )⁠. Since then, we only considered the stenosis region in the figure to show a clear impact of the flow resistance.

Figure 7 depicts the graph of the flow resistance against the axial distance for different values of Ha. When the fluid is under the impact of an applied magnetic field, the Hartman number Ha is considerably increases when the intensity of the Lorentz force increases. The increase in Ha tends to slow down the velocity of flow and enhance the flow resistance. Therefore, an essential, regulated magnetic field is required for therapeutic applications to increase the capture efficiency of MNPs close to the atherosclerosis region.50 

Figure 8 shows the effect of stenosis height δ on the flow resistance λ⁠. The flow resistance of a fluid is inversely proportional to the flow rate of the fluid as shown in Eq. (45). From the figure, it is shown that the flow resistance λ significantly increases with enhancement in stenosis height. Hence, the flow resistance is considerably larger at the peak of the stenosis where the height of the stenosis is maximum. Therefore, it is important to analyze and characterize the impact of δ on the flow resistance λ that will help the medical field to carry out further research to therapy atherosclerosis.

Figure 9 displays the graph of flow resistance with respect to axial distance for different values of inclination angle β⁠. From the figure, it is seen that the flow resistance significantly decreases with enhancement in inclination angle. Hence, an opposite trend of inclination angle on the volumetric flow rate is observed.

Basically, skin friction is a physical quantity that appears due to the viscous nature of the fluid and measures the resistance of the fluid at the surface. It arises due to the wall shear stress formed by the fluid flow tangential to the inner wall of the artery. For laminar fluid flow and without any deformation of the artery, the skin friction is uniform. However, due to variations in wall thickness or the presence of stenosis, skin friction may significantly affect the arterial wall. The generation of maximum shearing at the inner wall of the artery may lead to an aneurysm and damage the arterial wall.48 Therefore, it is beneficial to understand, analyze, and characterize the nature of skin friction. Figure 10 depicts the graph of skin friction C f vs axial distance for different values of stenosis shape parameter w along with α. It is important to note that stenosis shows a symmetric shape with w = 2 and the non-symmetric nature is to appear for the other values of w. A symmetric nature of the stenosis shows a higher skin friction, whereas the skin friction reduces with enhancement in other values of the shape parameter w of the stenosis. Hence, the variation of the stenosis shape significantly changes the value of skin friction. Moreover, the profile of the skin friction resembles the shape of the stenosis, and it is larger at the peak of the stenosis. On the other hand, we may conclude that the blood flow restriction is more because of the symmetric nature of the stenosis than the non-symmetric shapes.

Figure 11 depicts the graph of skin friction of the arterial wall with respect to axial distance for different values of λ 1⁠. From the figure, it is demonstrated that the skin friction C f decreases with enhancement in Jeffrey fluid parameter. This happens due to the fact that the Jeffrey fluid parameter is inversely proportional to skin friction. This phenomenon shows that blood rheology significantly influences the skin friction of the arterial wall of the stenosis region. Figure 12 shows the influence of inclination angle β on the skin friction of the arterial wall. It is demonstrated that the skin friction of the arterial wall boosts with enhancement in inclination angle.

Figure 13 demonstrates the graph of skin friction C f with respect to axial distance for different values of concentration of MNPs pc along with α⁠. Nanoparticles have various applications in medicine and the bioengineering fields. Some of the applications include tumor destruction by means of heating (hyperthermia treatment),61 separation and purification of biological molecules and cells,62 and drug and gene delivery.63,64 During the drug delivery phenomenon, the dose of drug concentration accumulated in the disease region plays a crucial role in recovering the malignant tissue. In this direction, the controlled-release drug delivery system has an advantage over the conventional drug delivery system. Because in conventional drug delivery systems, the drugs are distributed randomly, are less absorbed, and damage unaffected areas.65 In our work, we would like to know the effect of MNPs concentration on the skin friction of the arterial wall of the stenosis region. From Fig. 13, it is shown that the skin friction significantly increases with a boost in MNPs concentration because, during the application of a magnetic field, the particles concentration enhances the viscosity of the suspension.48 The increase in viscose force dominates the velocity of magnetic particles, which decreases due to this force and results in the enhancement of skin friction. Hence, this phenomenon tells us the advantage of MNPs concentration in regulating the flow velocity during drug transport through the artery and hence its capture efficiency in the stenosis region. In addition, it is worth mentioning that the percentage of increase of skin friction at the midpoint of the stenosis is 0.66%, 0.59%, 0.42%, and 0.13% for the fractional parameters (memory effects) α = 0.4⁠, α = 0.6⁠, α = 0.8 and integer order α = 1.0⁠, respectively. The percentage increase in skin friction for the short memory effect is significantly higher than the long memory effect. On the other hand, it is important to note that for a particular value of the MNPs concentration parameter, the skin friction outside the stenosis region in the artery remains constant without variation.

The findings of the present study are concerned with the flow of blood and the transport of MNPs through an inclined porous artery with mild stenosis under the application of a magnetic field. The Jeffrey fluid represents the rheology of blood flow through the artery. The MNPs are used as drug delivery applications for atherosclerosis therapy. The governing equations are written in terms of a C–F fractional model and solved using integral transforms. The presence of the stenosis significantly impacts the flow of blood, which may be a cause for diseases like heart attacks and strokes. From the present study, the important findings are as follows:

  • Increasing the memory effect increases the velocity of blood, the velocity of MNPs, volumetric flow rate, and skin friction.

  • The velocity of blood is significantly higher than the velocity of MNPs.

  • The volumetric flow rate decreases with elevation in stenosis height. This phenomenon can cause complications in the cardiovascular system, like stroke and heart attack.

  • The flow resistance decreases with the enhancement of the memory effect.

  • The velocity of blood and velocity of MNPs boost with enhancement in the Darcy number and Jeffrey fluid parameter, which shows an important application to the therapy of atherosclerosis.

  • The flow resistance increases with an increase in stenosis height and Hartman number.

  • The skin friction increases with an increase in MNPs concentration; however, it decreases with an enhancement in stenosis shape and Jeffrey fluid parameters.

The present study is beneficial to the fields of medical science and biomedical engineering in carrying out further research to treat atherosclerosis through nanoparticle-based magnetic drug targeting.

Habtamu Bayissa Yadeta gratefully acknowledges the funding received from the Research Initiation Grant (Project No. S00437) at Botswana International University of Science and Technology (BIUST).

The authors have no conflicts to disclose.

Habtamu Bayissa Yadeta: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Sachin Shaw: Conceptualization (equal); Project administration (lead); Supervision (lead); Validation (lead); Writing – review & editing (lead).

The data that support the findings of the study are available within the article.

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2023
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