Impact of Lorentz Force in Thermally Developed Pulsatile Micropolar Fluid Flow in a Constricted Channel

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Article Impact of Lorentz Force in Thermally Developed Pulsatile Micropolar Fluid Flow in a Constricted Channel

Muhammad Umar 1 , Amjad Ali 1 , Zainab Bukhari 1, Gullnaz Shahzadi 2,* and Arshad Saleem 1

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan; [email protected] (M.U.); [email protected] (A.A.); [email protected] (Z.B.); [email protected] (A.S.) 2 Department of Mechanical Engineering, École de Technologie Supérieure ÉTS, 1100 Notre-Dame W, Montreal, QC H3C 1K3, Canada * Correspondence: [email protected]

Abstract: This work aimed to analyze the heat transfer of micropolar fluid flow in a constricted channel influenced by thermal radiation and the Lorentz force. A finite difference-based flow solver, on a Cartesian grid, is used for the numerical solution after transforming the governing equations into the vorticity-stream function form. The impact of various emerging parameters on the wall shear stress, axial velocity, micro-rotation velocity and temperature profiles is discussed in this paper. The temperature profile is observed to have an inciting trend towards the thermal radiation, whereas it has a declining trend towards the Hartman and Prandtl numbers. The axial velocity profile has an inciting trend towards the Hartman number, whereas it has a declining trend towards the micropolar parameter and Reynolds number. The micro-rotation velocity escalates with the

micropolar parameter and Hartman number, whereas it de-escalates with the Reynolds number. The Nusselt number is observed to have a direct relationship with the Prandtl and Reynolds numbers.

Citation: Umar, M.; Ali, A.; Bukhari, Z.; Shahzadi, G.; Saleem, A. Keywords: micropolar fluid; constricted channel; pulsatile flow; thermal radiation; Lorentz force; Impact of Lorentz Force in Thermally heat transfer Developed Pulsatile Micropolar Fluid Flow in a Constricted Channel. Energies 2021, 14, 2173. https:// doi.org/10.3390/en14082173 1. Introduction Micropolar (MP) fluids are non-Newtonian fluids consisting of a dilute suspension Academic Editor: Guido Marseglia of an individual motion of thin, rigid cylindrical macromolecules. Incompressible MP fluids have significance in studying various phenomena, such as blood rheology in medical Received: 15 March 2021 sciences and liquid crystal, and dilute solutions of polymer in industries. The MP fluid Accepted: 6 April 2021 theory was first explained by Eringen [1]. Matiti [2] studied the free and forced convective Published: 13 April 2021 heat transfer through the horizontal parallel plate channel. They concluded that the flow velocity increases both for heating and cooling. A two-dimensional incompressible magne- Publisher’s Note: MDPI stays neutral tohydrodynamic (MHD) flow and heat transfer from electrically conducting micropolar with regard to jurisdictional claims in fluid between two parallel porous plates was presented by Ojjela and Kumar [3]. They published maps and institutional affil- deduced that the temperature profile upsurges by increasing the Prandtl number. Perdikis iations. and Raptis [4] analyzed the heat transfer in the steady flow of micropolar fluid past a plate and concluded that raising the radiation parameter causes an increase in the temperature profile. Gorla et al. [5] studied the steady heat transfer of an MP fluid over a semi-infinite flat plate. They deduced the result that thermal boundary layer thickness has an inverse Copyright: © 2021 by the authors. relation with the Prandtl number. Makinde [6] inspected the impact of nonlinear convected Licensee MDPI, Basel, Switzerland. heat on MHD boundary layer flow and melting heat transfer of an MP fluid with fluid This article is an open access article particles suspended over a stretching surface. They discussed the effects of the emerging distributed under the terms and parameters on the momentum and thermal boundary layers, and the heat transfer rate. conditions of the Creative Commons Turkyilmazoglu [7] investigated the MP fluid flow due to a permeable stretching sheet Attribution (CC BY) license (https:// with heat transfer. He concluded that the decreasing value of the MP parameter results in creativecommons.org/licenses/by/ 4.0/). an increase in the temperature profile. Ashraf and Wehgal [8] examined the incompressible

Energies 2021, 14, 2173. https://doi.org/10.3390/en14082173 https://www.mdpi.com/journal/energies Energies 2021, 14, 2173 2 of 16

laminar flow of an electrically conducting MP fluid between two porous disks. They deduced that the Nusselt number has a direct relationship with magnetic parameter, Reynolds number and Prandtl number. Rashidi [9] used the Homotopy analysis method to obtain accurate and complete analytic solutions of heat transfer of a micropolar fluid through a porous medium with radiation. The temperature profile decreases with increasing values of radiation parameter. Hydrodynamical study of an MP fluid in a permeable channel was carried out by Lu [10]. The axial velocity has a declining behaviour towards the coupling number, whereas micro-rotation velocity has an inclining trend towards the coupling number. Javadi [11] examined the heat transfer of a two-dimensional MP fluid flow in a permeable channel using the variational iteration method (VIM). They deduced that the heat transfer rate upsurges as the Reynolds number increases. Doh et al. [12] studied the effect of heat generation on the fully developed flow of an MP fluid in a porous vertical channel. The temperature profile is higher for a low Reynolds number, in case of heating the fluid, as compared to for a high Reynolds number. A finite difference-based scheme was employed to examine the convective heat transfer of an MP fluid in a horizontal wavy channel with local heating by Miroshnichenko [13]. They concluded that a low Prandtl number forms a vortex in the channel, which results in a low heat transfer rate. Mekheimer [14] investigated the effect of induced magnetic field on peristaltic transport of an incompressible conducting MP fluid in a symmetric channel. The magnetic parameter had an inverse relation with the shear stress. The study of the behaviour of human blood, as non-Newtonian fluid flow, through an artery having stenosis, plays an important role in understanding cardiovascular diseases. Young [15] studied the behaviour of flow in arteries having stenosis in detail. He discussed the flow separation region and profiles near the stenosis. Shit and Roy [16] examined the effects of magnetic field and heat on the pulsatile flow of blood taken as an MP fluid in the arteries with stenosis. Elahi et al. [17] studied the heat and mass transfer of MP fluid flow in arteries with mild stenosis having permeable walls. It was found that increasing values of coupling number and slip parameter result in a decrease in axial velocity profile. The temperature profile has a direct relation with the slip parameter, whereas it has an inverse relation with Darcy number. Haghighi and Asl [18] presented the mathematical modelling for two-dimensional pulsatile blood flow through overlapping constricted tapered vessels. They concluded that the taper angle has a direct relationship with velocity profile and volumetric flow rate, whereas it has an inverse relation with micro-rotation velocity. Ali et al. [19] investigated the impact of the magnetic field on the flow behaviour of a non-Newtonian Casson fluid in a constricted channel using Darcy’s law for the steady and pulsatile flows. The wall shear stress (WSS) had an inciting trend towards magnetic parameter, Strouhal number and Casson parameter. Further, the flow separation region could be controlled by the magnetic parameter. Bukhari et al. [20] expanded the study to analyze the heat transfer of Casson fluid in a constricted channel. The temperature profile had a declining trend towards the magnetic parameter, Casson parameter and Prandtl number, whereas it had an inciting trend towards the radiation parameter. Ali et al. [21] examined the pulsatile flow behaviour of micropolar-Casson fluid in a constricted channel. They concluded that the magnitude of micro-rotation velocity has a direct relation with the MP parameter, whereas it has an inverse relation with the Reynolds number. The objective of the present study is to analyze the heat transfer effects in the pulsatile flow of an MHD MP fluid with thermal radiation in vessels having constricted walls. The model is solved using the finite difference method in the Cartesian coordinate system. The impact of different emerging parameters, such as the magnetic field parameter, Strouhal number (i.e., the flow pulsation parameter), micropolar parameter, Radiation parameter and Prandtl number on the flow profiles are explained graphically. The effects on the Nusselt number are examined, along with the wall shear stress, velocity, and temperature profiles. The flow separation region is also discussed near the stenosis with the help of streamlines. The rest of the paper is structured as follows. Section2 describes the math- Energies 2021, 14, x FOR PEER REVIEW 3 of 17

parameter and Prandtl number on the flow profiles are explained graphically. The effects on the Nusselt number are examined, along with the wall shear stress, velocity, and

Energies 2021, 14, 2173 temperature profiles. The flow separation region is also discussed near the stenosis3 of 16 with the help of streamlines. The rest of the paper is structured as follows. Section 2 describes the mathematical model and formulation. The results and discussions are presented in Section 3. Section 4 summarizes the conclusion. ematical model and formulation. The results and discussions are presented in Section3. Section2. Mathematical4 summarizes Formulation the conclusion. 2.2.1. Mathematical Governing Equations Formulation 2.1. GoverningThe pulsatile Equations flow of an MP fluid in a channel with a pair of constrictions on the wallsThe subject pulsatile to Newtonian flow of an MP heating fluid inis aconsidered channel with (see a pairFigure of constrictions1). A uniform on magnetic the walls field subjectis applied to Newtonian perpendicular heating to is the considered flow direction. (see Figure The1). length A uniform of the magnetic constriction field is bump appliedis 2𝑥, perpendiculari.e., from −𝑥 to to the 𝑥 flow. The direction. maximum The lengthchannel of thewidth constriction is set to bump be 𝐿 is. The2x0, i.e.,induced fromelectric−x 0field’sto x0. Theeffect maximum is considered channel negligible width is as set the to bemagneticL. The induced Reynolds electric number field’s is very effectsmall. is considered negligible as the magnetic Reynolds number is very small.

FigureFigure 1. 1.A A schematic schematic diagram diagram of of the the problem problem geometry. geometry.

TheThe constrictions constrictions on on the the lower lower and upperand upper walls ofwalls the channelof the channel are deﬁned, are indefined, non- in dimensionalnon-dimensional form, as:form, as: ℎ ( h 𝜋𝑥 i h1 πx 1 + cos | | , |x| ≤ x0 y (x) =1 + cos2 , x0 𝑥 ≤𝑥 𝑦(𝑥) = 1 2 𝑥 0, |x| > x0 ( h i (1) 0, h2 |π𝑥x| 𝑥 1 − 2 1 + cos x , |x| ≤ x0 y2(x) = 0 ℎ 1,𝜋𝑥 |x| > x 1− 1 + cos , |𝑥| ≤𝑥 0 (1) 𝑦 (𝑥) = 2 𝑥 The governing equations of the ﬂow problem under consideration are given by: 1, |𝑥| 𝑥 + ˆ ∂uˆ + uˆ ∂uˆ + vˆ ∂uˆ = − 1 ∂pˆ + µ k ∇2uˆ + 1 (J × B) + k ∂N (2) ∂tˆ ∂xˆ ∂yˆ ρ ∂xˆ ρ ρ x ρ ∂yˆ

+ ˆ The governing∂vˆ + uˆ ∂vˆ + equationsvˆ ∂vˆ = of− 1the∂pˆ +flowµ problemk ∇2vˆ − underk ∂N consideration are given (3)by: ∂tˆ ∂xˆ ∂yˆ ρ ∂yˆ ρ ρ ∂xˆ 𝜕𝑢 𝜕𝑢 𝜕𝑢 ∂N1ˆ 𝜕𝑝̂ ∂Nˆ 𝜇+𝑘∂Nˆ k 1 ˆ ∂uˆ ∂vˆ 𝑘 𝜕𝑁γ 2 ˆ +𝑢 +𝑣 =−ˆ + uˆ +∂xˆ + vˆ ∂yˆ =∇ 𝑢−+ρj 2N(𝐉×𝐁+ ∂yˆ −) ∂x+ˆ + ρj ∇ N (4) (2) 𝜕𝑡̂ 𝜕𝑥 𝜕𝑦 ∂𝜌t 𝜕𝑥 𝜌 𝜌 𝜌 𝜕𝑦 ˆ ˆ ˆ q ∂T + uˆ ∂T + vˆ ∂T = k ∇2Tˆ − 1 ∂ (5) ∂tˆ ∂xˆ ∂yˆ ρCp ρCp ∂yˆ 𝜕𝑣 𝜕𝑣 𝜕𝑣 1 𝜕𝑝̂ 𝜇+𝑘 𝑘 𝜕𝑁 +𝑢 +𝑣 =− + ∂uˆ + ∂uˆ =∇𝑣−0 (6) (3) 𝜕𝑡̂ 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑦 ∂xˆ 𝜌∂yˆ 𝜌 𝜕𝑥 Here, the velocity components along xˆ-axis and yˆ-axis are represented by uˆ and vˆ, respectively. ρ, pˆ, ν, and Nˆ represent the density, pressure, kinematic viscosity, and angular 𝜕𝑁 𝜕𝑁 𝜕𝑁 𝑘 𝜕𝑢 𝜕𝑣 𝛾 +𝑢 +𝑣 =−velocity,2𝑁 respectively. + − B ≡+(0, B0∇, 0)𝑁,J ≡ Jx, Jy, Jz , and σ represent the magnetic field (4) 𝜕𝑡̂ 𝜕𝑥 𝜕𝑦 with𝜌𝑗 the uniform𝜕𝑦 strength𝜕𝑥 B0𝜌𝑗across the flow direction, the current density, and electric conductivity, respectively. γ = j(µ + k)/2 is the spin gradient viscosity, j and µ represent the micro inertia density and the dynamic viscosity, respectively. k denotes the vortex 4σ ∂Tˆ 4 viscosity and q = − 3k ∂yˆ represents the radiative heat flux. If E ≡ Ex, Ey, Ez is the

Energies 2021, 14, 2173 4 of 16

electric field and the direction of electric current flow is normal to the flow plane, then E ≡ (0, 0, Ez). In addition, using Ohm’s law:

Jx = 0, Jy = 0, Jz = σ(Ez + uBˆ 0) (7)

Maxwell’s equation ∇ × E = 0 for steady ﬂow implies that Ez = a, where a is a constant number. We can assume a to be zero. Then, Equation (7) gives Jz = σuBˆ 0. 2 Therefore, applying J × B = −σuBˆ 0, Equation (2) becomes:

+ −σuBˆ 2 ∂uˆ + uˆ ∂uˆ + vˆ ∂uˆ = − 1 ∂pˆ + µ k ∇2uˆ + 0 + k ∂Nˆ (8) ∂tˆ ∂xˆ ∂yˆ ρ ∂xˆ ρ ρ ρ ∂yˆ

4 For Equation (5), expanding Te about T∞ (free stream temperature) and ignoring higher-order terms, we get: ˆ 4 ∼ 3 ˆ 4 T = 4T∞T − 3T∞

2 q = − 4σ T3 ∂Tˆ ∂q = − 16σ T3 ∂ Tˆ Then 3k 4 ∞ ∂yˆ and ∂yˆ 3k ∞ ∂yˆ2 . Equation (5) becomes:

2 2 3 2 ∂Tˆ + uˆ ∂Tˆ + vˆ ∂Tˆ = k ∂ Tˆ + ∂ Tˆ + 16σT∞ ∂ Tˆ (9) ∂tˆ ∂xˆ ∂yˆ ρCp ∂xˆ2 ∂yˆ2 3kρCp ∂yˆ2

The dimensionless form of Equations (3), (4), (6), (8), and (9) are obtained using the following: µC x = xˆ y = yˆ u = uˆ v = vˆ t = tˆ p = pˆ St = L Pr = p L , L , U , U , T , ρU2 , UT , k , ˆ q ˆ 3 (10) N = NL , K = k , Re = UL , M = B L σ , θ = T−T2 , Rd = 16σT∞ U µ ν 0 ρν T1−T2 3kρCp Here T, M, Re, St, Pr, and Rd represent the ﬂow pulsation period, Hartman number, Reynolds number, Strouhal number, Prandtl number, and radiation parameter, respectively. N denotes the micro-rotation velocity, and K is the MP parameter. Equations (8), (3), (4), (9), and (6) after transformation becomes:

∂u ∂u ∂u ∂p 1+K 2 M2 K ∂N St ∂t + u ∂x + v ∂y = − ∂x + Re ∇ u − Re u + Re ∂y (11)

∂v ∂v ∂v ∂p 1+K 2 K ∂N St ∂t + u ∂x + v ∂y = − ∂y + Re ∇ v − Re ∂x (12)

∂N ∂N ∂N −K ∂u ∂v 1 K 2 St ∂t + u ∂x + v ∂y = Re 2N + ∂y − ∂x + Re + 2Re ∇ N (13)

2 2 2 St ∂θ + u ∂θ + v ∂θ = 1 1 ∂ θ + ∂ θ + Rd ∂ θ ∂t ∂x ∂y Re Pr ∂x2 ∂y2 ∂y2 (14) ∂u ∂u ∂x + ∂y = 0 (15)

2.2. Vorticity-Stream Function Formulation The vorticity (ω) and stream (ψ) functions are given by:

∂ψ ∂ψ ∂v ∂u u = ∂y , v = − ∂x , ω = ∂x − ∂y (16)

Some manipulations with Equations (11) and (12) produce:

∂ ∂v ∂u ∂ ∂v ∂u ∂ ∂v ∂u St ∂t ∂x − ∂y + u ∂x ∂x − ∂y + v ∂y ∂x − ∂y h i (17) = 1+K ∂2 ∂v − ∂u + ∂2 ∂v − ∂u − K ∂2 N − ∂2 N Re ∂x2 ∂x ∂y ∂y2 ∂x ∂y Re ∂x2 ∂y2

The vorticity transport equation obtained using the quantities (16), is given by:

h 2 2 i 2 2 2 2 St ∂ω + ∂ψ ∂ω − ∂ψ ∂ω = 1+K ∂ ω + ∂ ω + M ∂ ψ − K ∂ N − ∂ N (18) ∂t ∂y ∂x ∂x ∂y Re ∂x2 ∂y2 Re ∂y2 Re ∂x2 ∂y2 Energies 2021, 14, 2173 5 of 16

Again, using the quantities (16), Equation (13) becomes:

2 2 2 2 St ∂N + ∂ψ ∂N − ∂ψ ∂N = −K N + ∂ ψ + ∂ ψ + 1 + K ∂ N − ∂ N (19) ∂t ∂y ∂x ∂x ∂y Re 2 ∂x2 ∂y2 Re 2Re ∂x2 ∂y2

And the Poisson equation for stream function ψ is:

2 2 ∂ ψ + ∂ ψ = − (20) ∂x2 ∂y2 ω

2.3. Boundary Conditions Equation (2) assumes the following form the steady-state ﬂow:

+ 2 − 1 ∂pˆ + µ k ∂ uˆ + 1 (J × B) = ρ ∂xˆ ρ ∂yˆ2 ρ 0 (21)

Using J × B = −σ(Ez + uBˆ 0)B0 in Equation (21) gives:

+ 2 µ k ∂ uˆ − uBˆ 2 = ∂pˆ + E B ρ ∂yˆ2 σ 0 ∂xˆ σ z 0 (22)

Substituting the terms from Equation (10) in Equation (22) and performing some manipulations gives: d2u − 2 = L2 ∂pˆ + C dy2 M u ρνU ∂xˆ σEzB0 (23)

where C = (1 + K). The solution of Equation (23) is given by:

" h i # cosh( M ) cosh √M −cosh √M (y− 1 ) 1 2 2 C C 2 u(y) = , v = 0, M 6= 0 (24) 8 sinh2( M ) cosh √M 4 2 C

The inlet condition for the axial velocity with M = 0 is given by:

1 2 u(y) = C y − y , v = 0, M = 0, (25)

Equations (24) and (25) correspond to the steady velocity proﬁle. A sinusoidal time- dependent expression is considered for the ﬂow with pulsation:

u(y, t) = u(y)[1 + e sin(2πt)], v = 0 (26)

At the walls, no-slip conditions are considered: u = 0, and v = 0. The boundary conditions for N on the walls are: h i N = −S ∂u with 0 ≤ S ≤ 1 (27) ∂y y=0,1

1 S = 0, S = 2 , and S = 1 represent the strong, weak and turbulent ﬂow of concentrated particles near the walls in the ﬂuid. The inlet condition for micro-rotation velocity is considered zero.

2.4. Transformation of Coordinates The following relations are considered for the transformation of coordinates

− ( ) ξ = x, η = y y1 x (28) y2(x)−y1(x) Energies 2021, 14, 2173 6 of 16

For computation purpose, we mapped the constriction to a straight channel, which results in mapping the domain [y1, y2] to [0, 1]. Equations (18), (19), (14) and (20), applying Equation (28), become,

∂ω ∂ω ∂ω ∂ω St ∂t + u ∂ξ − Q ∂η + vD ∂η h 2 2 2 i 2 2 2 = 1+K ∂ ω − (P − QR) ∂ω − Q ∂ ω + Q2 + D2 ∂ ω + D M ∂ ψ (29) Re ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2 Re ∂η2 h 2 2 2 i − K ∂ N − (P − QR) ∂N − Q ∂ N + Q2 + D2 ∂ N Re ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2

∂N ∂N ∂N ∂N St ∂t + u ∂ξ − Q ∂η + vD ∂η h 2 2 2 i = −K N + ∂ ψ − (P − QR) ∂ψ − Q ∂ ψ + Q2 + D2 ∂ ψ (30) Re 2 ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2 h 2 2 2 i + 1 + K ∂ N − (P − QR) ∂N − Q ∂ N + Q2 + D2 ∂ N Re 2Re ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2

∂θ ∂θ ∂θ ∂θ St ∂t + u ∂ξ − Q ∂η + vD ∂η h 2 2 2 i (31) = 1 ∂ θ − (P − QR) ∂θ − Q ∂ θ + Q2 + D2 + RdD2 ∂ θ RePr ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2

2 2 2 ∂ ψ − (P − QR) ∂ψ − Q ∂ ψ + Q2 + D2 ∂ ψ = − (32) ∂ξ2 2 ∂η 2 ∂ξ∂η ∂η2 ω where

00 00 ηy (ξ) + (1−η)y (ξ) ηy0 (ξ) + (1−η)y0 (ξ) P = P(ξ, η) = 2 1 , Q = Q(ξ, η) = 2 1 , y2(ξ) − y1(ξ) y2(ξ) − y1(ξ) (33) y0 (ξ) − y0 (ξ) R = R(ξ) = 2 1 , D = D(ξ) = 1 y2(ξ) − y1(ξ) y2(ξ) − y1(ξ)

The velocity components u and v become:

∂ψ ∂ψ ∂ψ u = D(ξ) ∂η , v = Q(ξ, η) ∂η − ∂ξ (34)

The boundary conditions at the walls, in the (ξ, η) coordinate system, for the stream, vorticity, micro-rotation velocity and temperature functions are:

" √ # C cosh( M )tanh √M 2 2 C ψ(η, t) = 2 M [1 + e sin(2πt)], at η = 0 4M sinh ( 4 ) √ (35) cosh M h i ( 2 ) C √M ψ(η, t) = 2 M 1 − M tanh [1 + e sin(2πt)], at η = 1 4sinh ( 4 ) 2 C

where the value of e determines the nature of the ﬂow (0 for the steady case and 1 for the pulsatile case). 2 h 2 2 ∂ ψ i ω = − Q + D 2 ∂η η=0,1 2 (36) h 2 ∂ ψ i N = − wD 2 ∂η η=0,1 The boundary conditions for θ at the walls are as follows:

θ = 1, at η = 0 (37) θ = 0, at η = 1

The outlet boundary conditions for all the variables are associated with the fully developed ﬂow. The dimensionless parameter Nusselt number, pertinent to the heat transfer analysis, is given by:

Lq Nu = w = −D · θ0(0) (38) k(T1−T2) Energies 2021, 14, x FOR PEER REVIEW 9 of 17

Energies 2021, 14, 2173 7 of 16 impact of 𝐾 on 𝑁 profile when 𝑆=0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. The influence of 𝑅𝑒 on the 𝑁 profile is shown in Figure 13. Figure 13a depicts the effect of 𝑅𝑒 on 𝑁 profile when 𝑆 = 0. The 𝑁 profile has a de- 3. Results and Discussion clining trend towards 𝑅𝑒. Figure 13b presents the effect of 𝐾 on 𝑁 profile when 𝑆= Equations (29)–(32) are numerically solved using the alternating direction implicit 0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. (ADI) method [21] over a uniform structured grid spanned in ξ and η directions as follows: ξ ∈Figure[−10, 1014a,b] and presentη ∈ [0, 1the]. For effect the currentof 𝑃𝑟 study,and 𝑅𝑒 a Cartesian on the Nusselt grid of 400 number× 50 is ( used.𝑁𝑢). The 𝑁𝑢 hasThe an heightinciting and trend length towards of constriction the 𝑃𝑟 on and both 𝑅𝑒 walls. Figure are taken 15a aspresentsh = 0.35 theand streamlinesx0 = 2, with therespectively. variation Theof 𝑀 value. It ofcanRe be= 700 perceivedis fixed for that all of the the resultsstreamlines obtained are in thissmooth work unlessnear the constriction,stated otherwise. and a decrease Equation in(32 the) is flow solved separati for ψ = ψon(ξ ,regionη) and thencan Equationsbe observed(29) –with(31) arean increas- ingsolved value for of ω𝑀=. Figureω(ξ, η), 15b,cN = N present(ξ, η), and theθ =velocityθ(ξ, η). an Thed profilestemperature of different contours, parameters respectively. on the wall shear stress (WSS), axial velocity (u), micro-rotation velocity (N), temperature The ascending value of the magnetic parameter causes the increase in the size of tem- (θ) and Nusselt number (Nu) were computed. The profiles are shown at four different time peraturelevels defined contours. as follows: Figuret =16a0.0 presentsstands for the startstreamlines of pulsation for motion; differentt = 0.25valuesstands of 𝑆𝑡. The streamlinesfor the maximum are smooth, flow rate; andt = there0.50 stands is no forflow minimum separation flow rate;region. and tFigure= 0.75 stands16b,c forpresent the velocitythe instantaneous and temperature zero flow contours, rate. The expressionrespectively. of vorticity Ascending (ω) is value used to of calculate the Strouhal the num- berWSS causes since an both increase are orthogonal in the size to each of velocity other [22]. contours. To save the wall-clock time, the solution canFigure be obtained 17a presents on high-performance the streamlines parallel for computers variations through of 𝑃𝑟 parallelization. It is perceived of the that the code [23]. streamlinesThe u areprofile smooth and N nearprofile the are constriction. shown in Figure Figure2 at 17b,c the constriction’s present the throat velocity (i.e., and tem- peraturex = 0) contours,with fixed valuesrespectively. of M = 5The, St temperat= 0.02, K ure= 0.4 contour, Pr = 0.6 densityand Rd has= 0.7 an. Theseinciting trend towardsprofiles 𝑃𝑟 form. Figure regular 18a fluctuating presents patterns the streamlines due to the for flow different pulsation values and exhibit of 𝑅𝑑 higher. The stream- linesescalations are smooth for larger near distancesthe constriction. from the walls.Figure For 18b,c validation, present a the strong velocity agreement and wastemperature contours,established respectively. between the The current temperature work and Bandyopadhyaycontour density and has Layek a declining [24] for different trend towards = = 𝑅𝑑values. of M with N K 0, as shown in Figure3.

(a) (b)

Figure Figure2. (a) 𝑢 2. ( profilea) u proﬁle and and (b)( b𝑁) Nprofileproﬁle against t (for𝑡 (for four four pulsation pulsation cycles) cycles) at x = at0 for 𝑥=0 different for ηdifferent’s. 𝜂’s.

The WSS on the upper wall of the channel for M = 0, 5, 10 and 15 is presented in Figure4 at the four time instants of a pulse cycle. The WSS had an inciting trend towards M and attains its maximum value at t = 0.25. The flow decelerates during the interval of 0.25 < t ≤ 0.75. At t = 0.50, the WSS was noted to have a declining trend towards M. Figure5 presents the u profile for K = 0.1, 0.2, 0.3 and 0.4. A symmetrical behaviour can be seen at the locations away from the constriction bump, e.g., x = −5 and x = 5. The u profile has an inciting trend towards K. At t = 0.25, u attains its peak value. Figure6 presents the u and θ profiles for M = 0, 5, 10 and 15 at x = 0 and x = 2 (constriction’s lee), respectively. The u profile has an inciting trend towards M. The u profile is maximum at the location x = 0 at t = 0.25. The thermal boundary layer shows a declining trend towards M. However, the fluctuations can be seen at x = 2.

(a) (b) Figure 3. Comparison of the wall shear stress (WSS) on (a) the upper wall and (b) the lower wall of the constricted channel for different values of 𝑀 at 𝑡 = 0.25.

Energies 2021, 14, x FOR PEER REVIEW 9 of 17

impact of 𝐾 on 𝑁 profile when 𝑆=0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. The influence of 𝑅𝑒 on the 𝑁 profile is shown in Figure 13. Figure 13a depicts the effect of 𝑅𝑒 on 𝑁 profile when 𝑆 = 0. The 𝑁 profile has a declining trend towards 𝑅𝑒. Figure 13b presents the effect of 𝐾 on 𝑁 profile when 𝑆= 0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. Figure 14a,b present the effect of 𝑃𝑟 and 𝑅𝑒 on the Nusselt number (𝑁𝑢). The 𝑁𝑢 has an inciting trend towards the 𝑃𝑟 and 𝑅𝑒. Figure 15a presents the streamlines with the variation of 𝑀. It can be perceived that the streamlines are smooth near the constriction, and a decrease in the flow separation region can be observed with an increasing value of 𝑀. Figure 15b,c present the velocity and temperature contours, respectively. The ascending value of the magnetic parameter causes the increase in the size of temperature contours. Figure 16a presents the streamlines for different values of 𝑆𝑡. The streamlines are smooth, and there is no flow separation region. Figure 16b,c present the velocity and temperature contours, respectively. Ascending value of the Strouhal number causes an increase in the size of velocity contours. Figure 17a presents the streamlines for variations of 𝑃𝑟. It is perceived that the streamlines are smooth near the constriction. Figure 17b,c present the velocity and temperature contours, respectively. The temperature contour density has an inciting trend towards 𝑃𝑟. Figure 18a presents the streamlines for different values of 𝑅𝑑. The streamlines are smooth near the constriction. Figure 18b,c present the velocity and temperature contours, respectively. The temperature contour density has a declining trend towards 𝑅𝑑.

Energies 2021, 14, 2173 (a) (b) 8 of 16 Figure 2. (a) 𝑢 profile and (b) 𝑁 profile against 𝑡 (for four pulsation cycles) at 𝑥=0 for different 𝜂’s.

(a) (b) Energies 2021, 14, x FOR PEER REVIEW 10 of 17

Figure Figure3. Comparison 3. Comparison of the of the wall wall shear shear stressstress (WSS) (WSS) on (ona) the(a) upper the upper wall and wall (b) theand lower (b) wallthe oflower the constricted wall of the channel constricted channelfor for different different values values of M ofat 𝑀t = at0.25. 𝑡 = 0.25.

(a) (b)

FigureFigure 4. WSS 4. WSS distribution distribution (a) at (ta)= at0 ,(𝑡=0b) at, t(b=) at0.25 𝑡,(c =) at0.25t, =(c)0.50 at ,𝑡 and = (0.50d), atandt = (0.75d) atfor 𝑡M = 0.750, 5for, 10, 𝑀=0and,15 5, with10, and St =150.02, withK =𝑆𝑡0.4, Pr = 0.02=, 𝐾=0.40.7, and, Rd𝑃𝑟= 0.6. = ,0.7 and 𝑅𝑑 = .0.6

Energies 2021, 14, x FOR PEER REVIEW 10 of 17

(a) (b)

EnergiesFigure2021, 14 4., 2173WSS distribution (a) at 𝑡=0, (b) at 𝑡 = 0.25, (c) at 𝑡 = 0.50, and (d) at 𝑡 = 0.75 for 𝑀=0, 5, 10, and9 of 16 15 with 𝑆𝑡 = 0.02, 𝐾=0.4, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6

Energies 2021, 14, x FOR PEER REVIEW 11 of 17

(a) 𝑡=0.25 b) 𝑡=0.50( c) 𝑡( = 0.75

FigureFigure 5. 5.𝑢 uprofilesproﬁles for for 𝐾=0.1K = 0.1,, 0.2,0.2,0.3, 0.3 and, and 0.4 0.4 with withM =𝑀=55, St, =𝑆𝑡0.02, = Pr0.02,= 𝑃𝑟0.7, and = ,0.7 andRd = 𝑅𝑑0.6 at various = 0.6 at various locations locationsx. 𝑥.

(a) (b) (c) Figure 6. (a) 𝑢 profile versus 𝑦 at 𝑥=0, (b) 𝜃 profile versus 𝜂 at 𝑥=0, and (c) 𝜃 profile versus 𝜂 at 𝑥=2 for 𝑀=0, 5, 10, and 15 at 𝑡 = 0.25 with 𝐾=0.5, 𝑆𝑡 = 0.02, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6

Energies 2021, 14, x FOR PEER REVIEW 11 of 17

(a) 𝑡=0.25 b) 𝑡=0.50( c) 𝑡( = 0.75

Energies 2021, 14, 2173 10 of 16 Figure 5. 𝑢 profiles for 𝐾=0.1, 0.2, 0.3, and 0.4 with 𝑀=5, 𝑆𝑡 = 0.02, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = 0.6 at various locations 𝑥.

(a) (b) (c)

FigureFigure 6. 6.(a )(au) profile𝑢 profile versus versusy at x𝑦= at0,( 𝑥=0b) θ profile, (b) 𝜃 versus profileη atversusx = 0 ,𝜂 and at (c𝑥=0) θ profile, and versus (c) 𝜃 ηprofileat x = 2versusfor M 𝜂= at0, 5𝑥=2, 10, for and𝑀=0 15 at, 5t, =100.25, and with 15K at= 0.5,𝑡St = 0.25= 0.02,withPr 𝐾=0.5= 0.7,, and 𝑆𝑡Rd = = 0.020.6., 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6

Figure7 presents the WSS on the upper wall, u and θ profiles for St = 0.02, 0.04, 0.06 and 0.08 at x = 0. The WSS and u profile have an inciting trend towards St and attain the maximum value at t = 0.25. The θ profile has a declining trend towards St. Figure8 presents the WSS on the upper wall, u and θ profiles for K = 0.1, 0.2, 0.3 and 0.4 at x = 0. The WSS and u profile have a declining trend towards K. The WSS and u profile are maximum at t = 0.25. There is no significant effect on the θ profile with increasing values of K at x = 0. Figure9 depicts the WSS on the upper wall, u and θ profiles for Re = 500, EnergiesEnergies 20212021,, 1414,, xx FORFOR PEERPEER REVIEWREVIEW 1212 ofof 1717 700, 900 and 1100 at x = 0. The WSS and thermal boundary layer have an inciting trend towards Re, but u profile has a declining trend towards Re.

((aa)) (b) ((cc))

FigureFigureFigure 7. 7.7.( a((a)a)) WSS WSSWSS on onon the thethe upper upperupper wall, wall, (b ()bu) proﬁle𝑢 profile versus versusy at 𝑦x =atat 0𝑥=0, and,, ( candand) θ proﬁle ((cc)) 𝜃𝜃 profileprofile versus versusηversusat x = 𝜂𝜂0 atforat 𝑥=0𝑥=0St = 0.02 forfor , 𝑆𝑡0.04𝑆𝑡, = = 0.020.02, , 0.06,0.040.04, and, 0.06,0.06, 0.08 andand at t =0.080.080.25 atat with 𝑡M = 0.25= 5, withK = 0.4,𝑀=5Pr ,= 𝐾=0.40.7, and,, Rd𝑃𝑟= 0.6. = ,0.7, andand 𝑅𝑑𝑅𝑑 = = .0.6 .0.6

((aa)) (b) ((cc))

FigureFigureFigure 8. 8.8.(a )((aa WSS)) WSSWSS on theonon upperthethe upperupper wall, wall, (b) u proﬁle(b) 𝑢 profile versus yversusat x = 0𝑦, andatat 𝑥=0 (c) θ proﬁle,, andand (( versuscc)) 𝜃𝜃 profileprofileη at x =versusversus0 for K 𝜂𝜂 = atat0.1 𝑥=0𝑥=0,0.2, 0.3 for for, 𝐾=𝐾= and0.10.1,,0.2 0.40.2, at, 0.30.3t =,, andand0.25 with0.40.4 atatM 𝑡𝑡= 5, =St 0.25= with0.02, Pr𝑀=5= 0.7,, 𝑆𝑡 and Rd = 0.02=,, 0.6.𝑃𝑟 = ,0.7, andand 𝑅𝑑𝑅𝑑 = = .0.6 .0.6

((aa)) (b) ((cc)) FigureFigure 9.9. ((aa)) WSSWSS onon thethe upperupper wall, (b) 𝑢 profile versus 𝑦 atat 𝑥=0,, andand ((cc)) 𝜃𝜃 profileprofile versusversus 𝜂𝜂 atat 𝑥=0𝑥=0 forfor 𝑅𝑒𝑅𝑒 == 500, 500, 700700,, 900900,, andand 11001100 atat 𝑡𝑡 = 0.25 with 𝑀=5, 𝐾=0.4, 𝑆𝑡 = 0.02,, 𝑃𝑟 = = ,0.7 ,0.7 andand 𝑅𝑑𝑅𝑑 = = .0.6 .0.6

((aa)) (b) ((cc)) FigureFigure 10.10. 𝜃𝜃 profileprofile versusversus 𝜂,, ((a) at 𝑥=−2, (b) at 𝑥=0,, andand ((c)) atat 𝑥=2𝑥=2,, forfor 𝑃𝑟𝑃𝑟 = = ,0.5 ,0.5 1.01.0,, 1.51.5,, andand 2.02.0 withwith 𝑀=5𝑀=5, , 𝐾=0.4𝐾=0.4,, 𝑆𝑡𝑆𝑡 = = 0.020.02,, andand 𝑅𝑑 = 0.6 at 𝑡 = 0.25.

Energies 2021, 14, x FOR PEER REVIEW 12 of 17

(a) (b) (c) Figure 7. (a) WSS on the upper wall, (b) 𝑢 profile versus 𝑦 at 𝑥=0, and (c) 𝜃 profile versus 𝜂 at 𝑥=0 for 𝑆𝑡 = 0.02, 0.04, 0.06, and 0.08 at 𝑡 = 0.25 with 𝑀=5, 𝐾=0.4, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6

Energies 2021, 14, x FOR PEER REVIEW 12 of 17

(a) (b) (c) Energies 2021, 14, 2173 11 of 16 Figure 8. (a) WSS on the upper wall, (b) 𝑢 profile versus 𝑦 at 𝑥=0, and (c) 𝜃 profile versus 𝜂 at 𝑥=0 for 𝐾= 0.1,0.2, 0.3, and 0.4 at 𝑡 = 0.25 with 𝑀=5, 𝑆𝑡 = 0.02, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6

(a) (b) (c)

FigureFigure 9. 9.( a()a WSS) WSS on on the the upper upper wall, wall, (b )(bu)proﬁle 𝑢 profile versus versusy at x 𝑦= at0, and𝑥=0 (c), andθ proﬁle (c) 𝜃 versus profileη at versusx = 0 for𝜂 atRe 𝑥=0= 500 ,for700 ,𝑅𝑒900, = 500, and700 1100, 900 at, andt = 0.251100 with at M𝑡= =5, 0.25K =with0.4, St𝑀=5= 0.02,, 𝐾=0.4Pr = 0.7,, 𝑆𝑡 and Rd = 0.02=, 𝑃𝑟0.6. = ,0.7 and 𝑅𝑑 = .0.6

Figure 10 presents the θ profile for Pr = 0.5 , 1.0, 1.5 and 2.0 at x = −2, x = 0 and (a) x = 2. The θ profile has a declining(b) trend towards Pr. However, slight(c fluctuation) near the lower wall can be seen. The Prandtl number is used to control the relative thickening Figure 8. (a) WSS on the upper wall, (b) 𝑢 profile versus 𝑦 at 𝑥=0, and (c) 𝜃 profile versus 𝜂 at 𝑥=0 for 𝐾= of the momentum and thermal boundary layers. For small values of Pr, the heat transfer 0.1,0.2, 0.3, and 0.4 at 𝑡 = 0.25 with 𝑀=5, 𝑆𝑡 = 0.02, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6 is quicker compared to velocity, which implies that the thermal boundary layer is thicker than the momentum boundary layer. Therefore, the cooling rate can be controlled by Pr. There is no effect on the WSS and u profile with the variation of Pr. Figure 11 presents the θ profile for Rd = 0.4, 0.8, 1.5 and 2.0 at x = −2, x = 0 and x = 2. The θ profile has an inciting trend towards Rd. This behaviour verifies the physical fact that the thickness of the thermal boundary layer has a direct relation with Rd. There is no effect on the WSS (a) (b) (c) and u profile with the variation of Rd. Figure 12 depicts the impact of different values of Figure 10. 𝜃 profile versusK on𝜂, (Na) atprofile. 𝑥=−2 The, (b) effect at 𝑥=0 of K, andon (Nc) profileat 𝑥=2 when, for 𝑃𝑟S = 0 = ,,0.5 i.e., 1.0, micro-elements 1.5, and 2.0 with are 𝑀=5 close, 𝐾=0.4, 𝑆𝑡 = 0.02, and 𝑅𝑑near the = 0.6 at walls 𝑡 and= 0.25.unable to rotate, is shown in Figure 12a. The N profile has an inciting trend towards K. Figure 12b presents the impact of K on N profile when S = 0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. The influence of Re on the N (a) profile is shown in Figure 13.(b Figure) 13a depicts the effect of Re on N profile(c) when S = 0. The N profile has a declining trend towards Re. Figure 13b presents the effect of K on Figure 9. (a) WSS on the upper wall, (b) 𝑢 profile versus 𝑦 at 𝑥=0, and (c) 𝜃 profile versus 𝜂 at 𝑥=0 for 𝑅𝑒 = 500, N profile when S = 0.5, i.e., weak concentration. Similar behaviour is seen in this case, 700, 900, and 1100 at 𝑡 = 0.25 with 𝑀=5, 𝐾=0.4, 𝑆𝑡 = 0.02, 𝑃𝑟 = ,0.7 and 𝑅𝑑 = .0.6 as well.

(a) (b) (c)

FigureFigure 10. 10.θ 𝜃proﬁle profile versus versusη,( a𝜂), at (ax) =at −𝑥=−22,(b) at, (xb=) at0 ,𝑥=0 and (c,) and at x (=c)2 at, for 𝑥=2Pr =, for0.5, 1.0𝑃𝑟, 1.5, = and,0.5 1.02.0, with1.5, andM = 2.05, K with= 0.4 𝑀=5, , St𝐾=0.4= 0.02,, 𝑆𝑡 and Rd = =0.02, 0.6and at 𝑅𝑑t = 0.25. = 0.6 at 𝑡 = 0.25.

EnergiesEnergies2021 2021, 14, 14, 2173, x FOR PEER REVIEW 12 of13 16 of 17 Energies 2021, 14, x FOR PEER REVIEW 13 of 17 Energies 2021, 14, x FOR PEER REVIEW 13 of 17

((aa)) ((bb)) ((cc)) (a) (b) (c) FigureFigureFigure 11. 11.11.θ proﬁle𝜃𝜃 profileprofile versus versusversusη,( a 𝜂)𝜂, at, ((aax)) =atat −𝑥=−2𝑥=−22,(b) at,, ((xbb=)) atat0 , 𝑥=0 and𝑥=0 (c,,) andand at x (=(cc)) 2 atat, for 𝑥=2𝑥=2Rd =,, forfor0.4 ,𝑅𝑑𝑅𝑑0.8, 1.5 =, = and,0.4,0.4 0.80.82.0,, 1.51.5with,, andandM = 2.02.05, Kwithwith= 0.4 𝑀=5𝑀=5, ,, Pr𝐾=0.4𝐾=0.4Figure= 0.7,, , and11. 𝑃𝑟𝑃𝑟 𝜃St profile= = = 0.02,0.7,0.7 andand versus at 𝑆𝑡𝑆𝑡t = 0.25. 𝜂 = =, (0.020.02a )at atat 𝑡𝑡𝑥=−2 = = 0.250.25,. . (b) at 𝑥=0, and (c) at 𝑥=2, for 𝑅𝑑 = ,0.4 0.8, 1.5, and 2.0 with 𝑀=5, 𝐾=0.4, 𝑃𝑟 = ,0.7 and 𝑆𝑡 = 0.02 at 𝑡 = 0.25.

((aa)) ((bb)) (a) (b) Figure 12. 𝑁 𝜂 𝑆=0 𝑆=0.5 𝐾=0.1 0.2 0.3 0.4 𝑆𝑡 = 0.02 𝑀=5 𝑃𝑟 = Figure 12. 𝑁 profileprofile versusversus 𝜂 ((a=a)) 𝑆=0 andand =((bb)) 𝑆=0.5= forfor 𝐾=0.1,, 0.2,, 0.3,, andand =0.4 withwith =𝑆𝑡 = =0.02,, 𝑀=5,, 𝑃𝑟 = FigureFigure 12. 12.N proﬁle𝑁 profile versus versusη (a )𝜂S (a)0 𝑆=0and ( band) S (b0.5) 𝑆=0.5for K for0.1, 𝐾=0.10.2, 0.3,, and 0.20.4, 0.3with, andSt 0.40.02 with, M 𝑆𝑡5, Pr = 0.02, 0.7𝑀=5, and, 𝑃𝑟 = 0.70.7,, andand 𝑅𝑑𝑅𝑑 = = 0.6.0.6. Rd0.7=, 0.6.and 𝑅𝑑 = 0.6.

(a) (b) ((aa)) (b) FigureFigure 13.13. 𝑁𝑁 profileprofile versusversus 𝜂𝜂 ((aa)) 𝑆=0𝑆=0 andand ((bb)) 𝑆=0.5𝑆=0.5 forfor 𝑅𝑒𝑅𝑒 == 500, 500, 700700,, 900900,, andand 11001100 withwith 𝑆𝑡𝑆𝑡 = = 0.020.02,, 𝐾=𝐾= FigureFigure 13. 13.N proﬁle𝑁 profile versus versusη (a) S𝜂= (a0) and𝑆=0 (b ) andS = 0.5(b)for 𝑆=0.5Re = 500for, 700𝑅𝑒, 900 ,= and 500, 7001100, 900with, Stand= 0.021100, K with= 0.4 𝑆𝑡, Pr = =0.7 0.02, 𝐾= 0.40.4,, 𝑃𝑟𝑃𝑟 ==, 0.7, 0.7 andand 𝑅𝑑𝑅𝑑 = = 0.6.0.6. and0.4Rd, 𝑃𝑟= 0.6. =, 0.7 and 𝑅𝑑 = 0.6.

Energies 2021, 14, 2173 13 of 16

Figure 14a,b present the effect of Pr and Re on the Nusselt number (Nu). The Nu has an inciting trend towards the Pr and Re. Figure 15a presents the streamlines with the variation of M. It can be perceived that the streamlines are smooth near the constriction, and a decrease in the ﬂow separation region can be observed with an increasing value of M. Figure 15b,c present the velocity and temperature contours, respectively. The ascending value of the magnetic parameter causes the increase in the size of temperature contours. Figure 16a presents the streamlines for different values of St. The streamlines are smooth, Energies 2021, 14, x FOR PEER REVIEWand there is no ﬂow separation region. Figure 16b,c present the velocity and temperature14 of 17

contours, respectively. Ascending value of the Strouhal number causes an increase in the Energies 2021, 14, x FOR PEER REVIEW 14 of 17 size of velocity contours.

(a) (b) Figure 14. Influence(a )of (a) 𝑃𝑟 and (b) 𝑅𝑒 on the Nusselt number with fixed 𝑀=5, (𝑆𝑡b) = 0.02, 𝐾=0.4, and 𝑅𝑑 = 0.6. Figure 14. Inﬂuence of (a) Pr and (b) Re on the Nusselt number with ﬁxed M = 5, St = 0.02, K = 0.4, and Rd = 0.6. Figure 14. Influence of (a) 𝑃𝑟 and (b) 𝑅𝑒 on the Nusselt number with fixed 𝑀=5, 𝑆𝑡 = 0.02, 𝐾=0.4, and 𝑅𝑑 = 0.6.

(a) (b) (c) (a) (b) (c)

Figure 15. (a) Streamlines, (b) vorticity, and (c) temperature contours for M at t = 0.25 with K = 0.4, St = 0.02, Pr = 0.7, Figure 15. (a) Streamlines, (b) vorticity, and (c) temperature contours for 𝑀 at 𝑡 = 0.25 with 𝐾=0.4, 𝑆𝑡 = 0.02, 𝑃𝑟 = Figureand 15.Rd (=a) 0.6.Streamlines, (b) vorticity, and (c) temperature contours for 𝑀 at 𝑡 = 0.25 with 𝐾=0.4, 𝑆𝑡 = 0.02, 𝑃𝑟 = 0.7, and 𝑅𝑑 = . 0.6 0.7, and 𝑅𝑑 = . 0.6 Figure 17a presents the streamlines for variations of Pr. It is perceived that the streamlines are smooth near the constriction. Figure 17b,c present the velocity and temperature contours, respectively. The temperature contour density has an inciting trend towards Pr. Figure 18a presents the streamlines for different values of Rd. The streamlines are smooth near the constriction. Figure 18b,c present the velocity and temperature contours, respectively. The temperature contour density has a declining trend towards Rd.

(a)( a) (b) (b) (c) (c) FigureFigure 16. 16. (a) ( Streamlines,a) Streamlines, (b) (vorticity,b) vorticity, and and (c) temperature(c) temperature contours contours for 𝑆𝑡for at 𝑆𝑡 𝑡 at =𝑡 0.25 =with 0.25 with𝑀=5 𝑀=5, 𝐾=0.4, 𝐾=0.4, 𝑃𝑟, 𝑃𝑟 = ,0.7 = ,0.7 andand 𝑅𝑑 𝑅𝑑 = . 0.6 = . 0.6

Energies 2021, 14, x FOR PEER REVIEW 14 of 17

(a) (b) Figure 14. Influence of (a) 𝑃𝑟 and (b) 𝑅𝑒 on the Nusselt number with fixed 𝑀=5, 𝑆𝑡 = 0.02, 𝐾=0.4, and 𝑅𝑑 = 0.6.

(a) (b) (c)

Energies 2021, 14, 2173 14 of 16 Figure 15. (a) Streamlines, (b) vorticity, and (c) temperature contours for 𝑀 at 𝑡 = 0.25 with 𝐾=0.4, 𝑆𝑡 = 0.02, 𝑃𝑟 = 0.7, and 𝑅𝑑 = . 0.6

(a) (b) (c)

FigureFigure 16. 16.(a) (Streamlines,a) Streamlines, (b) (b vorticity,) vorticity, and and (c ()c )temperature temperature contours contours for St𝑆𝑡at att =𝑡0.25 = 0.25with withM =𝑀=55, K =, 𝐾=0.40.4, Pr =, 𝑃𝑟0.7, and = ,0.7 Energiesand 2021 𝑅𝑑, 14, x FOR= . 0.6 PEER REVIEW 15 of 17 Energies 2021Rd , =14, 0.6.x FOR PEER REVIEW 15 of 17

(a) (b) (c) (a) (b) (c) FigureFigure 17. 17.(a)( aStreamlines,) Streamlines, (b (b)) vorticity, vorticity, andand ( c()c temperature) temperature contours contours for Prforat 𝑃𝑟t = at0.25 𝑡with = 0.25St with= 0.02 𝑆𝑡, M = = 50.02, ,K 𝑀=5= 0.5,, and𝐾= Figure 17. (a) Streamlines, (b) vorticity, and (c) temperature contours for 𝑃𝑟 at 𝑡 = 0.25 with 𝑆𝑡 = 0.02, 𝑀=5, 𝐾= 0.5,Rd and= 𝑅𝑑0.6. = . 0.6 0.5, and 𝑅𝑑 = . 0.6

(a) (b) (c) (a) (b) (c) Figure 18. (a) Streamlines, (b) vorticity, and (c) temperature contours for 𝑅𝑑 at 𝑡 = 0.25 with 𝑆𝑡 = 0.02, 𝑀=5, 𝐾= 0.4Figure,Figure and 18. 𝑃𝑟 18.(a)( aStreamlines,) = Streamlines,.0.7 (b (b)) vorticity, vorticity, andand ( c()c temperature) temperature contours contours for Rdforat 𝑅𝑑t = at0.25 𝑡with = 0.25St with= 0.02 𝑆𝑡, M = = 50.02, ,K 𝑀=5= 0.4,, and𝐾= 0.4,Pr and= 0.7.𝑃𝑟 = .0.7 4. Conclusions 4. 4.Conclusions Conclusions The current work focuses on the heat transfer analysis of a micropolar fluid flow in TheThe current current work work focuses focuses on on the the heat heat transfer transfer analysis analysis of of a amicropolar micropolar fluid ﬂuid flow ﬂow in a constricted channel under the impact of thermal radiation and the Lorentz force. The a constrictedin a constricted channel channel under under the impact the impact of thermal of thermal radiation radiation and andthe theLorentz Lorentz force. force. The vorticity-stream function was used to transform the governing equations and then vorticity-streamThe vorticity-stream function function was isused used to totr transformansform the the governing governing equationsequations andand thenthen solved numerically by a finite difference based flow solver. Unlike the other studies in solved numerically by a finite difference based flow solver. Unlike the other studies in the literature, the solutions were computed on the Cartesian grid. The influence of dif- the literature, the solutions were computed on the Cartesian grid. The influence of different flow controlling parameters, including the Hartman number (𝑀), Strouhal number ferent flow controlling parameters, including the Hartman number (𝑀), Strouhal number (𝑆𝑡), Prandtl number (𝑃𝑟), micropolar parameter (𝐾) and thermal radiation (𝑅𝑑) param- (𝑆𝑡), Prandtl number (𝑃𝑟), micropolar parameter (𝐾) and thermal radiation (𝑅𝑑) parameters, on the WSS, axial velocity (𝑢), micro-rotation velocity (𝑁) and temperature (𝜃) pro- eters, on the WSS, axial velocity (𝑢), micro-rotation velocity (𝑁) and temperature (𝜃) profiles were examined graphically. The streamlines, vorticity and temperature contours are files were examined graphically. The streamlines, vorticity and temperature contours are also depicted. The key findings of the current work are as follows: also depicted. The key findings of the current work are as follows: 1. The thermal boundary layer is observed to have an inverse relationship with the 1. The thermal boundary layer is observed to have an inverse relationship with the Hartman number and Prandtl number during the whole of the pulsation cycle. Hartman number and Prandtl number during the whole of the pulsation cycle. 2. The WSS has an inciting trend towards 𝑀. At 𝑡 = 0.25, the WSS attains its peak 2. The WSS has an inciting trend towards 𝑀. At 𝑡 = 0.25, the WSS attains its peak value for a given value of 𝑀. Similar behaviour is seen on varying 𝑆𝑡 and 𝑅𝑒. The value for a given value of 𝑀. Similar behaviour is seen on varying 𝑆𝑡 and 𝑅𝑒. The WSS has a declining trend towards 𝐾. However, there is no effect of 𝑅𝑑 and 𝑃𝑟 WSS has a declining trend towards 𝐾. However, there is no effect of 𝑅𝑑 and 𝑃𝑟 on the WSS. on the WSS. 3. The 𝑢 profile has an inciting trend towards 𝑀 and 𝑆𝑡, but it has a declining trend 3. The 𝑢 profile has an inciting trend towards 𝑀 and 𝑆𝑡, but it has a declining trend towards 𝐾. There is no effect of 𝑅𝑑 and 𝑃𝑟 on the 𝑢 profile. towards 𝐾. There is no effect of 𝑅𝑑 and 𝑃𝑟 on the 𝑢 profile. 4. The 𝑁 profile has an inciting trend towards 𝐾 and 𝑀, whereas a declining trend 4. The 𝑁 profile has an inciting trend towards 𝐾 and 𝑀, whereas a declining trend towards 𝑅𝑒 is observed. No significant impact of other parameters on the 𝑁 pro- towards 𝑅𝑒 is observed. No significant impact of other parameters on the 𝑁 profile is observed. file is observed.

Energies 2021, 14, 2173 15 of 16

solved numerically by a finite difference based flow solver. Unlike the other studies in the literature, the solutions are computed on the Cartesian grid. The influence of different flow controlling parameters, including the Hartman number (M), Strouhal number (St), Prandtl number (Pr), micropolar parameter (K) and thermal radiation (Rd) parameters, on the WSS, axial velocity (u), micro-rotation velocity (N) and temperature (θ) profiles are examined graphically. The streamlines, vorticity and temperature contours are also depicted. The key findings of the current work are as follows: 1. The thermal boundary layer is observed to have an inverse relationship with the Hartman number and Prandtl number during the whole of the pulsation cycle. 2. The WSS has an inciting trend towards M. At t = 0.25, the WSS attains its peak value for a given value of M. Similar behaviour is seen on varying St and Re. The WSS has a declining trend towards K. However, there is no effect of Rd and Pr on the WSS. 3. The u profile has an inciting trend towards M and St, but it has a declining trend towards K. There is no effect of Rd and Pr on the u profile. 4. The N profile has an inciting trend towards K and M, whereas a declining trend towards Re is observed. No significant impact of other parameters on the N profile is observed. 5. The θ profile has a declining trend towards M, St, Re and Pr. In contrast, it has an inciting trend towards K and Rd. Since a declining trend in the θ profile towards Pr is observed, it can be deduced that a thinner thermal boundary layer is caused due to a larger value of Pr. 6. The dimensionless parameter Nu has an inciting trend towards Pr and Re, whereas it has a declining trend towards Rd. The finding of such a study might be helpful in the process of designing biomedical equipment for the treatment of cardiovascular diseases.

Author Contributions: Conceptualization, M.U. and A.A.; methodology, M.U., Z.B. and A.S.; soft- ware, M.U. and A.S.; validation, G.S.; formal analysis, M.U., A.A. and Z.B.; investigation, M.U., A.A. and Z.B.; writing—original draft preparation, M.U. and A.A.; writing—review and editing, A.A. and G.S.; visualization, M.U., Z.B. and A.S.; supervision, A.A. and G.S.; project administration, A.A. and G.S.; All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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