Significance of MHD Radiative Non-Newtonian Nanofluid Flow towards a Porous Channel: A Framework of the Casson Fluid Model

e effect of various parameters in the unsteady pulsating flow of radiative hydromagnetic Casson nanofluid through a porous channel is investigated. e governing equations were nondimensionalized by applying suitable transformations. e perturbation technique was employed to solve the resulting similarity equations. e velocity and temperature fields are illustrated for several pertinent flow parameters. e fluid velocity has been enhancing for higher values of the frequency parameter, Casson fluid parameter, nanoparticle volume fraction, and Darcy number. e reverse impact is observed for larger values of the Hartmann number. e result reveals that adding the nanoparticles has enhanced the heat transfer of the base fluid as the nanoparticles increase the heat conductivity. Furthermore, it is noticed that the temperature profile decreases rapidly for higher values of the cross Reynolds number and the radiation parameter. Finally, an excellent agreement between the current results and previous results is obtained by comparing with the available limiting results in the literature.


Introduction
Nanofluid consists of the uniform composition of solid particles whose dimension is in the range of 1 to 100 nm. e heat transfer characteristics of nanofluid will be controlled by low thermal conductivity. Due to this constraint, the heat transfer ability of conventional heat transfer fluids such as water, lubricant oil, and ethylene glycol has been highly limited since they provided low thermal conductivity. Generally, metals exhibit higher thermal conductivity than typical heat transfer fluids. Mixing low-dimension metallic particles (nanoparticles) with fluids will help increase the thermal conductivity.
is process is identified as an innovative technique for enhancing the heat transfer coefficient. Choi [1] attempted this first and succeeded. Nisar et al. [2] investigated the steady flow with heat diffusion of blood which transmits the micropolar nanoliquid with gold particles through a curved shrinking/stretched surface with the impact of radiation. ey suggested that nanoparticlecontaining gold can treat cancer as these materials have a lofty atomic quantity that produces the temperature guide to the handling of malignant tumours. e heat transfer behaviors in the presence of the magnetic field were investigated using the finite-element method by Sheikholeslami et al. [3]. Dogonchi et al. [4] examined the magnetohydrodynamic (MHD) flow of nanofluid and heat transmission between two surfaces by taking into account the Joule heat effects. eir study has identified an increase in the concentration, Nusselt number, and temperature profiles with an increase in the magnitude of the Schmidt number. Magnetohydrodynamics (MHD) is a branch of dynamics which deals with fluids during electrical conduction, such as electrolytes, plasmas, and liquid metals. e primary principle of MHD is to produce forces in the fluid when the magnetic field induces a current through a moving conducting fluid. Magnetohydrodynamics has been found in diverse engineering applications. Some other cutting-edge research reports about the nanofluid features and applications are given in [5][6][7][8].
e Casson fluid was first introduced in 1959 by Casson. He used pigment oil of printing ink to scrutinize the flow behaviour. e "Casson fluid" is noted as the most popular non-Newtonian fluid. It behaves like a Newtonian fluid above a critical stress value, and at low shear strain, it behaves like an elastic solid. e Casson fluid model can be chosen for analyzing the rheological behaviour of the ingredients of fluids such as soup, honey, coolant oil, and human blood. Abolbashari et al. [9] analyzed the flow, mass, and heat transfer characteristics of the Casson nanofluid over a stretching surface analytically. Sheikh et al. [10] investigated "the MHD free convection flow of the Casson fluid in a channel," and they found that "the larger values of the Casson parameter elevate the velocity of the Casson fluid." e rate of heat transfer of the steady MHD stagnation point flow of the Casson fluid on the shrinking/ stretching surface with the effect of thermal radiation and viscous dissipation was investigated by Lund et al. [11]. Tassaddiq et al. [12] analyzed the Newtonian heating effects for the generalized Casson fluid MHD flow with porous effects. ey found that velocity grows with an increase in the porosity parameter, and the fluid flow is controllable for all possible values of the Casson parameter. e magnetohydrodynamic boundary layer flow of the Casson fluid over an exponentially permeable shrinking sheet was investigated by Nadeem et al. [13]. ermal radiation plays a vital role in many industrial processes. Kumar et al. [14] applied the Buongiorno model to investigate the pulsatile flow of the MHD Casson nanofluid through a vertical porous channel with thermal radiation. MHD flow through a porous channel between two parallel plates has enormous applications such as in petrochemical industries, power generators, polymer technology, and food industries. Falede et al. [15] investigated "MHD oscillatory flow through a porous channel saturated with porous media." "MHD flow through a vertical channel with a porous medium" was analyzed by Dwivedi et al. [16]. Loganathan et al. [17] investigated the entropy features of a MHD third-grade nanoliquid induced by a linearly stretching plate. ey found that the Bejan number rate is enhanced due to thermal radiation effects. Srinivas et al. [18] studied the effect of the applied magnetic field, chemical reaction, and thermal radiation on the pulsating flow of the Casson fluid in a porous channel. Anwar et al. [19] explained the transient free convection phenomenon in a vertical porous channel subject to nonlinear thermal radiation. e important visualization about thermal radiation with various flow geometries is found in the literature [20][21][22][23][24][25].
To this extent, our literature review confirms that there are not existing papers for hydromagnetic pulsating Casson nanofluid flow through a porous channel with radiation effects. An examination was made for pulsating Casson fluid flow through a porous channel with radiation and chemical reaction by Srinivas et al. [18]. However, similar research for the Casson nanofluid with brief discussions is still limited. erefore, we expanded our effort to the pulsative Casson nanofluid that passes through a porous channel with radiation and magnetic-field impacts. e considered problem has enormous applications in many industrial as well as biological processes such as crude oil purification and blood flow through contracting muscles. e perturbation technique was employed to solve the resulting similarity equations. As a method for finding analytical solutions approximately, the perturbation method is an exciting tool among other analytical methods. Due to its unfussiness and high precision, the perturbation method was applied for solving many heat transfer and fluid mechanics problems. e solution for some fluid flow problems can be found in [26][27][28][29].

Mathematical Formulation
We consider the influence of uniform traverse applied magnetic field on the pulsatile incompressible flow of the Casson nanofluid in a porous channel. e flow is driven by an unsteady pressure gradient [18].
where ρ nf is the density of the nanofluid, p * is the pressure, ε ≪ 1 is the suitably chosen positive quantity, "A" is a constant, and ω is the frequency. As in Figure 1, the X * -axis is taken along the lower wall, and the perpendicular axis is noted as the Y * -axis. e lower wall maintains the temperature T 0 , and the upper wall maintains the temperature T 1 (T 0 < T 1 ). e nanofluid is injected from the lower wall into the porous channel with the velocity v 0 , and it is sucked out from the porous medium via the upper wall with a similar velocity. Uniform magnetic field (B 0 ) is imposed along the usual direction to the flow. e physical flow system is noted in Figure 1. e Casson nanofluid's rheological equation is stated ast where "π c " is the critical value of "π" based on the non-Newtonian model. T mn is the (m, n) th stress tensor component, and π � e mn , with e mn being the (m, n) th deformation rate. μ Bnf is the plastic "dynamic viscosity" of the "non-Newtonian" nanofluid. p y is the "yield stress" of the fluid. e problem's governing equation is as follows: e boundary conditions for the present analysis are [18] u * � 0, T * � T 0 , at y * � 0, where u * is the dimensional velocity in the X-direction, k and Φ are the permeability and porosity of the porous medium, σ is the electrical conductivity of the fluid, μ nf is the dynamic viscosity of the nanofluid, ρ nf is the density of the nanofluid, ] nf is the kinematic viscosity of the Casson nanofluid, C pnf is the specific heat at a constant pressure of the nanofluid, and κ nf is the thermal conductivity of the nanofluid, which is given by Oztop and Abu-Nada as follows [14]: Here, φ is the nanoparticle volume fraction, (ρC p ) nf is the heat capacity of the nanofluid, and (ρC p ) f and (ρC p ) n are the heat capacity of the base fluid and the nanoparticles, respectively. ρ f and ρ n are the densities of the fluid and the solid fractions, respectively. μ f , ] f , and C pf are the dynamic viscosity, kinematic viscosity, and specific heat at a constant pressure of the base fluid, and κ f and κ n are the thermal conductivity of the base fluid and the solid fractions, respectively. e use of the abovementioned expression for (κ nf /κ f ) is restricted to spherical nanoparticles only.
We establish the dimensionless variables as follows: Figure 1: Physical model and the coordinate system.

Journal of Mathematics
By using the abovementioned dimensionless variables, we get e new boundary conditions are where Da � k/ΦL 2 is the Darcy number of the porous media,

The Proposed Solution Method
In order to solve equations (7) and (8) with the boundary conditions (9), we apply the perturbation technique. e velocity u and temperature W can be assumed to have the form Now, we substitute equations (10) and (11) into the equations (7) and (8); then, equating the coefficients of various powers of ε, we get e new boundary conditions are Solving the six ODEs (12)-(17) using the boundary conditions (18), we obtain 4 Journal of Mathematics where m's, As, and Bs are the constants specified in Appendix. e dimensionless Nusselt number at the walls is given by

Results and Discussion
is segment presents the Casson nanofluid pulsating flow filling through a porous channel in the appearance of thermal radiation impacts. e governing physical systems are solved with the aid of the perturbation technique. MATLAB software assists in computing the system of equations through the perturbation method.  Figure 2(e). It is obtained from this graph that the velocity is the enhancing function of Da. is is due to the Darcian drag force; it is known as the linear porous drag force and is inversely proportional to Da (see the third term in equation (7), i.e., (− u/Da) ). e Darcian drag force will enhance due to the rise in the absorbency of porous areas. Hence, there is an upsurge in velocity with higher range of Da. Figures 3(a) and 3(b) display the influence of (φ) and (Re) on the temperature distribution θ(y). e observations conclude that the temperature distribution θ(y) reduces with an upsurge in the nanoparticle volume fraction (φ) and crossflow Reynolds number (Re) . Figures 4(a) and 4(b) demonstrate the influence of the Prandtl number (Pr) and radiation parameter (Rd) on the θ(y). Figure 4(a) depicts that the temperature distribution decreases with an increase in the Prandtl number (Pr). From Figure 4(b), one can observe a rise in θ(y) with intensification in Rd. e effect of the radiation parameter (Rd),nanoparticle volume fraction (φ), and Hartmann number (M) on unsteady and steady temperature θ(y) is shown in Figures 5-7. Figure 5(a) shows that the unsteady temperature is oscillating with a decrease in the radiation parameter (Rd) and attaining maximum near the walls. On the other hand, Figure 5(b) shows that the increase in the radiation parameter produces a diminishing tendency in steady temperature. Figure 6(a) depicts the unsteady temperature growths with an increasing nanoparticle-volume fraction (φ). From Figure 6(b), it is observed that the steady temperature increases towards the upper wall with a given increase in the nanoparticle-volume fraction (φ), while the decrementing behaviour is noted in the lower wall. From Figure 7(a), we can observe that the Hartmann number (M) enhancement generated the oscillating effect in the unsteady temperature. Figure 7(b) displays the lower temperature in the lower wall, while the higher temperature is noted in the upper wall.      Table 1 shows the comparison of results of Nusselt number Nu with Srinivas et al. [18]. e matching of these results is made with the heat transfer rate of Newtonian and non-Newtonian. From this result, we can conclude that the present computation is an efficient one. Table 2 presents the variations of the Nusselt number (Nu). e nanofluid is taken as the nanoparticle volume fraction φ � 0.1. From this table, Nu enhanced at the lower wall for the higher values of M, Re, and Rd, while the inverse trend is noticed for an enormous Q value. However, this behaviour is reversed at the upper wall. It is also observed that, with increasing nanoparticle volume fraction (φ),Nu decreases at the lower wall, while it is increasing at the upper wall.    Heat source/ sink parameter Re:

Conclusions
Cross-flow Reynolds number Rd: Radiation parameter T 0 : e temperature at the lower wall T 1 : e temperature at the upper wall Φ: e porosity of the porous medium σ: Electrical conductivity of the fluid μ nf : Dynamic viscosity of the nanofluid ρ nf : e density of the nanofluid ] nf : Kinematic viscosity of the nanofluid φ: Nanoparticle volume fraction (ρC p ) nf : Heat capacity of the nanofluid ω: Frequency f: Fluid fraction n: Nanoparticle fraction nf: Nanofluid.

2A 1 ,
Pr ,    Data Availability e raw data supporting the conclusions of this article will be made available by the authors without undue reservation.

Conflicts of Interest
e authors declare that they have no conflicts of interest.