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

: 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.


Introduction
Micropolar (MP) fluids are non-Newtonian fluids consisting of a dilute suspension of an individual motion of thin, rigid cylindrical macromolecules. Incompressible MP fluids have significance in studying various phenomena, such as blood rheology in medical sciences and liquid crystal, and dilute solutions of polymer in industries. The MP fluid theory was first explained by Eringen [1]. Matiti [2] studied the free and forced convective heat transfer through the horizontal parallel plate channel. They concluded that the flow velocity increases both for heating and cooling. A two-dimensional incompressible magnetohydrodynamic (MHD) flow and heat transfer from electrically conducting micropolar fluid between two parallel porous plates was presented by Ojjela and Kumar [3]. They deduced that the temperature profile upsurges by increasing the Prandtl number. Perdikis 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 relation with the Prandtl number. Makinde [6] inspected the impact of nonlinear convected heat on MHD boundary layer flow and melting heat transfer of an MP fluid with fluid particles suspended over a stretching surface. They discussed the effects of the emerging parameters on the momentum and thermal boundary layers, and the heat transfer rate. Turkyilmazoglu [7] investigated the MP fluid flow due to a permeable stretching sheet with heat transfer. He concluded that the decreasing value of the MP parameter results in an increase in the temperature profile. Ashraf and Wehgal [8] examined the incompressible 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. Section 2 describes the mathematical model and formulation. The results and discussions are presented in Section 3. Section 4 summarizes the conclusion.

Governing Equations
The pulsatile flow of an MP fluid in a channel with a pair of constrictions on the walls subject to Newtonian heating is considered (see Figure 1). A uniform magnetic field is applied perpendicular to the flow direction. The length of the constriction bump is 2 , i.e., from − to . The maximum channel width is set to be . The induced electric field's effect is considered negligible as the magnetic Reynolds number is very small. The constrictions on the lower and upper walls of the channel are defined, in non-dimensional form, as: The governing equations of the flow problem under consideration are given by: The constrictions on the lower and upper walls of the channel are defined, in nondimensional form, as: The governing equations of the flow problem under consideration are given by: Here, the velocity components alongx-axis andŷ-axis are represented byû andv, respectively. ρ,p, ν, andN represent the density, pressure, kinematic viscosity, and angular velocity, respectively. B ≡ (0, B 0 , 0), J ≡ J x , J y , J z , and σ represent the magnetic field with the uniform strength B 0 across 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 viscosity and q = − 4σ 3k ∂T 4 ∂ŷ represents the radiative heat flux. If E ≡ E x , E y , E z 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, E z ). In addition, using Ohm's law: Maxwell's equation ∇ × E = 0 for steady flow implies that E z = a, where a is a constant number. We can assume a to be zero. Then, Equation (7) gives J z = σûB 0 . Therefore, applying J × B = −σûB 2 0 , Equation (2) becomes: For Equation (5), expanding T 4 about T ∞ (free stream temperature) and ignoring higher-order terms, we get: The dimensionless form of Equations (3), (4), (6), (8), and (9) are obtained using the following: 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.

Vorticity-Stream Function Formulation
The vorticity (ω) and stream (ψ) functions are given by: Some manipulations with Equations (11) and (12) produce: The vorticity transport equation obtained using the quantities (16), is given by: And the Poisson equation for stream function ψ is:

Boundary Conditions
Equation (2) assumes the following form the steady-state flow: Substituting the terms from Equation (10) in Equation (22) and performing some manipulations gives: where C = (1 + K). The solution of Equation (23) is given by: The inlet condition for the axial velocity with M = 0 is given by: Equations (24) and (25) correspond to the steady velocity profile. A sinusoidal timedependent expression is considered for the flow with pulsation: At the walls, no-slip conditions are considered: u = 0, and v = 0. The boundary conditions for N on the walls are: S = 0, S = 1 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.

Transformation of Coordinates
The following relations are considered for the transformation of coordinates For computation purpose, we mapped the constriction to a straight channel, which results in mapping the domain [y 1 , y 2 ] to [0, 1]. Equations (18), (19), (14) and (20), applying Equation (28), become, The velocity components u and v become: The boundary conditions at the walls, in the (ξ, η) coordinate system, for the stream, vorticity, micro-rotation velocity and temperature functions are: where the value of determines the nature of the flow (0 for the steady case and 1 for the pulsatile case).
The boundary conditions for θ at the walls are as follows: 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:

Results and Discussion
Equations (29)-(32) are numerically solved using the alternating direction implicit (ADI) method [21] over a uniform structured grid spanned in ξ and η directions as follows: ξ ∈ [−10, 10] and η ∈ [0, 1]. For the current study, a Cartesian grid of 400 × 50 is used. The height and length of constriction on both walls are taken as h = 0.35 and x 0 = 2, respectively. The value of Re = 700 is fixed for all of the results obtained in this work unless stated otherwise. Equation (32) is solved for ψ = ψ(ξ, η) and then Equations (29)-(31) are solved for ω = ω(ξ, η), N = N(ξ, η), and θ = θ(ξ, η). The profiles of different parameters on the wall shear stress (WSS), axial velocity (u), micro-rotation velocity (N), temperature (θ) and Nusselt number (Nu) were computed. The profiles are shown at four different time levels defined as follows: t = 0.0 stands for the start of pulsation motion; t = 0.25 stands for the maximum flow rate; t = 0.50 stands for minimum flow rate; and t = 0.75 stands for the instantaneous zero flow rate. The expression of vorticity (ω) is used to calculate the WSS since both are orthogonal to each other [22]. To save the wall-clock time, the solution can be obtained on high-performance parallel computers through parallelization of the code [23].
The u profile and N profile are shown in Figure 2 at the constriction's throat (i.e., x = 0) with fixed values of M = 5, St = 0.02, K = 0.4, Pr = 0.6 and Rd = 0.7. These profiles form regular fluctuating patterns due to the flow pulsation and exhibit higher escalations for larger distances from the walls. For validation, a strong agreement was established between the current work and Bandyopadhyay and Layek [24] for different values of M with N = K = 0, as shown in Figure 3. . Figure 13b presents the effect of on profile when 0.5, i.e., weak concentration. Similar behaviour is seen in this case, as well. Figure 14a,b present the effect of and on the Nusselt number ( ). The has an inciting trend towards the and . Figure 15a presents the streamlines wit the variation of . It can be perceived that the streamlines are smooth near the con striction, and a decrease in the flow separation region can be observed with an increa ing value of . Figure 15b,c present the velocity and temperature contours, respectivel The ascending value of the magnetic parameter causes the increase in the size of tem perature contours. Figure 16a presents the streamlines for different values of . Th streamlines are smooth, and there is no flow separation region. Figure 16b,c present th velocity and temperature contours, respectively. Ascending value of the Strouhal num ber causes an increase in the size of velocity contours. Figure 17a presents the streamlines for variations of . It is perceived that th streamlines are smooth near the constriction. Figure 17b,c present the velocity and tem perature contours, respectively. The temperature contour density has an inciting tren towards . Figure 18a presents the streamlines for different values of . The stream lines are smooth near the constriction. Figure 18b,c present the velocity and temperatur contours, respectively. The temperature contour density has a declining trend toward .  The WSS on the upper wall of the channel for M = 0, 5, 10 and 15 is presented in Figure 4 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. Figure 5 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. Figure 6 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.                 Figure 10 presents the θ profile for Pr = 0.5, 1.0, 1.5 and 2.0 at x = −2, x = 0 and x = 2. The θ profile has a declining trend towards Pr. However, slight fluctuation near the lower wall can be seen. The Prandtl number is used to control the relative thickening of the momentum and thermal boundary layers. For small values of Pr, the heat transfer 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 and u profile with the variation of Rd. Figure 12 depicts the impact of different values of K on N profile. The effect of K on N profile when S = 0, i.e., micro-elements are close near the walls and 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 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 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 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.

Conclusions
The current work focuses on the heat transfer analysis of a micropolar fluid flow in a constricted channel under the impact of thermal radiation and the Lorentz force. The vorticity-stream function was used to transform the governing equations and then 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 different flow controlling parameters, including the Hartman number ( ), Strouhal number ( ), Prandtl number ( ), micropolar parameter ( ) and thermal radiation ( ) param-

Conclusions
The current work focuses on the heat transfer analysis of a micropolar fluid flow in a constricted channel under the impact of thermal radiation and the Lorentz force. The vorticity-stream function was used to transform the governing equations and then 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 different flow controlling parameters, including the Hartman number ( ), Strouhal number ( ), Prandtl number ( ), micropolar parameter ( ) and thermal radiation ( ) param-

Conclusions
The current work focuses on the heat transfer analysis of a micropolar fluid flow in a constricted channel under the impact of thermal radiation and the Lorentz force. The vorticity-stream function is used to transform the governing equations and then