Analytical Investigation of Magnetohydrodynamic Non-Newtonian Type Casson Nanofluid Flow past a Porous Channel with Periodic Body Acceleration

e consequence of periodic body acceleration and thermal radiation in the pulsating flow of MHD Casson nanofluid through a porous channel is addressed. A flow of the nanofluid injected through the lower plate is considered while sucked out through the upper plate with a similar velocity. e thermal radiation term is incorporated in the heat transfer equation. e governing equations corresponding to velocity and temperature are converted from partial differential equations to a system of ordinary differential equations by employing similarity variables. e perturbation technique is applied to solve the governing flow equations. e impact of diverse parameters on flow features is graphically analyzed. e result reveals that adding the nanoparticle has enhanced the velocity profile of the base fluid. Moreover, an increase in the periodic body acceleration results in enlarging velocity and temperature.


Introduction
e foremost principle of MHD is that forces are formed by the magnetic field, which stimulates a current through a moving conducting fluid. e importance of MHD in various areas such as astrophysics, biomedical research, and geophysics motivates us to investigate MHD flow. Concerning the shear strain-stress relationship, Casson fluid is a non-Newtonian fluid, which becomes firm when the yield stress is more significant than shear stress. However, it begins to deform when the yielded stress becomes less significant than shear stress. e MHD flow of Casson fluid in a porous channel is a field of dynamic research due to its applicability in industry and medical technology such as paper production, condensation, and blood flow in the human body. Due to its applications in heat storage beds and fossil fuels, heat and mass transfer through a porous medium have been developed into an exciting topic for the most recent decades. Tamoor et al. [1] focused on "MHD Casson fluid flow above a stretching cylinder." Reddy [2] examined the "MHD Casson fluid flow over an exponentially inclined permeable stretching surface with the influence of thermal radiation and chemical reaction on the flow of the MHD Casson fluid flow over an exponentially inclined permeable stretching surface." Fluids with suspended nanoparticles have an extensive range of applications in the heat transfer process because an increment in thermal conductivity in fluids was noticed while mixing nanoparticles.
is attempt was taken first by Choi [3] subsequently succeeded by Sandeep et al. [4] and Nisar et al. [5]. ey suggested that "the nanoparticle-containing gold can be used to treat and trounce cancer." Next, Abolbashari et al. [6] analyzed "the mass and heat transfer characteristics of Casson nanofluid over a stretching surface." Finally, Tassaddiq et al. [7] analyzed "the Newtonian heating effects for the generalized Casson fluid MHD flow with porous effects." ey found that velocity augments with escalating porosity and the Casson parameter control the fluid flow. Menni et al. [8,9], Khan et al. [10], and Maouedj et al. [11] simulated the applicability of nanofluid in the solar channels. ermal radiation has a crucial role in many manufacturing processes. Irfan et al. [12] and Vellanki [13] investigated the radiation effect on Casson nanofluid and bio-nanofluid flow through a porous medium. MHD flow across porous channel between two parallel plates has vast application such as petrochemical engineering, polymer technology, food production technology, and biomedical research. Falede and Adensanya [14] explored "the oscillatory flow of MHD fluid through a porous channel." "MHD flow through a vertical channel with porous medium" was investigated by Dwivedi et al. [15]. "MHD nanofluid flow and heat transfer in rotating horizontal annulus were numerically simulated with thermal radiation" by Peng et al. [16].
e important visualization about the influence of magnetic field and radiation with assorted flow geometries is established in the literature [17][18][19][20][21][22][23][24][25][26][27][28]. e importance of the pulsating flow in a porous channel is well known, and it has many biological and industrial applications. Srinivas et al. [29] considered the MHD pulsating Casson fluid flow with chemical reaction and magnetic field through a porous channel. Kumar et al. [30] considered the "Joule heating and thermal radiation effects on the pulsatile MHD flow of Casson nanofluid through a vertical porous space." e study related to periodic body acceleration has enormous applications, particularly analyzing blood flow in the cardiovascular system. In many situations, such as driving in a vehicle, running, and jogging, the human body accelerates periodically (vibration). e sudden velocity change in the human body disturbs the blood flow, which causes most health problems. Vishali and Sarojamma [31] investigated the "flow of a fluid through a catheterized artery with the effect of a periodic body acceleration" in which they concluded that "the body acceleration enhanced velocity." Sulochana [32] discussed "the influence of body acceleration on an unsteady pulsatile flow of MHD couple stress fluid." Over the years, nonlinear differential equations have been solved by approximate analytical methods such as the homotopy analysis method, weighted residual method, differential transformation methods, perturbation method, and so on. However, the non-perturbation approximate analytical methods have operational limitations that fussily tapered their performance domain, and they may sometimes lead to erroneous results while routinely implemented. Moreover, the conversion of the nonlinear equation and the evolution of corresponding recurrence equations using differential transformation methods are complex in some nonlinear systems. Consequently, the demand for reasonably simple, flexible, standard, and high precise total approximate analytical solutions is well acknowledged. One of such techniques applied for the quest is the perturbation method, which can solve nonlinear problems with high precision, while the perturbation method gives enhanced results for small perturbation parameters, further having a useful mathematical formulation with better accurateness, even for comparatively huge values of the perturbation parameter. Due to its unfussiness and high accuracy, the perturbation method was applied for solving various heat transfer and fluid mechanics problems [29][30][31][32][33][34].
To this extent, our literature review authenticates that there are no existing papers for MHD pulsating Casson nanofluid flow past a porous channel with periodic body acceleration effect. An examination was made for "pulsating Casson nanofluid flow through a porous channel with radiation" [35]. However, similar research for Casson nanofluid with periodic body acceleration effects was not investigated yet. Hence, we prolonged our effort to pulsative Casson nanofluid that passes through a porous channel between two parallel plates with periodic body acceleration and magnetic field impacts.
e considered problem has massive application in manufacturing and biological processes. e perturbation technique was employed to solve the resulting similarity equations. e velocity and temperature fields are illustrated for several pertinent flow parameters.

Mathematical Formulation
Consider the influence of uniform traverse applied magnetic field on the pulsatile incompressible flow of a Casson nanofluid in a porous channel between two parallel plates, in which one plate is at rest and another plate moves in a parallel direction. e flow is driven by a pulsating pressure gradient [29].
where ρ nf is the "density of the nanofluid," p * is the "pressure,"ε ≪ 1 is suitably chosen positive quantity, A is a constant, and ω is the "frequency." e physical flow system is noted in Figure 1.
We consider that X * -axis is taken along the lower plate, and the perpendicular axis is noted as Y * -axis. e lower plate is at rest, maintaining the temperature T 0 . e upper plate is moving with the velocity U * w � (A/ω) 1 + εe iωt * parallel to the X * -axis and maintains the temperature T 1 (T 0 < T 1 ). e nanofluid is injected from the lower plate into the porous channel with the velocity v 0 , and it is sucked out from the porous medium via the upper plate with the similar velocity. "Uniform magnetic field" (B 0 ) is imposed along the usual direction to the flow. e periodic body acceleration is assumed as G * � g 0 cos ψ, where ψ is the phase difference and g 0 is the amplitude of the body acceleration.
e Casson nanofluid rheological equation is stated as 2 Complexity 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, π � e mn with e mn being the (m, n)-th deformation rate, μ B nf is the plastic "dynamic viscosity" of the "non-Newtonian" nanofluid, and p y is the "yield stress" of the fluid. e governing equations of the flow problem are stated as follows: e boundary conditions for the present analysis are where Q 0 is "heat source/sink parameter," β is the "Casson parameter," u * is the "dimensional velocity in X-direction," k and Φ are the "permeability and porosity of the porous medium," σ is the "electrical conductivity" of the fluid, C p nf is the "specific heat at constant pressure of nanofluid," μ nf is the "dynamic viscosity" of the nanofluid, ρ nf is the "density" of the nanofluid, (ρC p ) nf is the "heat capacity of the nanofluid," ] nf is the "kinematic viscosity of the Casson nanofluid," κ nf is the "thermal conductivity" of the nanofluid, K * is "Rosseland mean absorption coefficient," and σ * is the "Stefan-Boltzmann constant" [16].
where φ is the nanoparticle volume fraction and the subscripts nf, f, and n represent the nanofluid, base fluid, and nanoparticles correspondingly. e use of the above expression for (κ nf /κ f ) is restricted to spherical nanoparticles only.

Complexity
We assume that the temperature variation within the fluid flow can be expanded in a Taylor's series about T 0 . By neglecting the higher-order terms, we obtain We establish the dimensionless variables as follows: By applying equations (3), (5), (8), (9), and (10), in equations (4), (6), and (7), e new boundary conditions are where Da � k/ΦL 2 is the "Darcy number of the porous media,"

Method of Solution
e velocity u and temperature θ can be supposed to have the form: Now, substituting equations (1), (14), and (15) into equations (11)- (13) and then equating the coefficients of various powers of ε, we get e new boundary conditions are Solving the six ODEs (16)- (21) with the boundary conditions (22), we obtain 4 Complexity where m's, A's, and B's are constants given in Appendix. e dimensionless Nusselt number at the plates is given by

Results and Discussion
e effect of diverse material parameters on the non-dimensional heat transfer and velocity is discussed in this section. e values of the Nusselt number are compared with previously available results obtained by Srinivas et al. [29] for the various assumptions of the Hartmann number, Reynolds number, radiation parameter, and heat source parameter to assess the validity of the current results, and the outcome is shown in Table 1.
roughout the computation, the parameters are taken as t � π/4, ε � 0.01, Re � 2, ϕ � 0.1, Ha � 1, β � 5, Da � 0.2, Ec � 0.2, Q � 0.5, Rd � 3, Pr � 14, ψ � 60°, and G � 1 unless otherwise stated. Figures 2(a) and 2(b) represent the effect of nanoparticle volume fraction (φ) and Hartmann number (Ha) on the velocity distribution u(y). It is noticed that u(y) improves with an escalation in φ. Furthermore, the velocity is diminishing function of Ha since the applied magnetic field generates the Lorentz forces, which acts opposite to the flow direction. Figures 3(a) and 3(b) exhibit the effect of H and G on velocity distribution u(y). It is noticed that u(y) improves with an escalation in frequency parameter (H) and body acceleration parameter (G). Figure 4(a) elucidates the effect of Da on velocity distribution u(y). It is evident that u(y) improves with an escalation in Darcy number (Da) because Da is inversely proportional to the Darcy drag force. Figure 4(b) shows that an augment in Casson fluid parameter (β) escalates velocity distribution u(y). Figures 5(a) and 5(b) show the influence of (Ec) and (G) on the temperature distribution θ(y). e observations conclude that the temperature distribution θ(y) upsurges with a rise in Ec. In addition, it is obvious that the escalating periodic body acceleration slightly improves the temperature.   Figure 8(a), it is observed that the enhancement in Eckert number (Ec) produces an increasing tendency in unsteady temperature θ t (y). Figure 8(b) depicts that the unsteady temperature θ t (y) is oscillating with increasing Darcy number (Da). From Figure 9(a), it is observed that the increase in body acceleration parameter (G) produces an increase in unsteady temperature θ t (y). Figure 9(b) shows that the unsteady temperature θ t (y) is oscillating with increasing Hartmann number (Ha). Figure 10(a) shows that the amplification in the nanoparticle volume fraction (φ) produces an insurgence in unsteady temperature θ t (y) at the lower half of the channel; however, it begins osculating near the upper wall. From Figure 10(b), it is evident that the increase in Pr escalates the unsteady temperature.
From Figure 11(a), the influence of Q is minimal in the lower part of the channel. However, the unsteady temperature slightly decreases on the upper part of the channel for the increasing values of Q. From Figure 11(b), it is noticed Complexity 5 that the increase in radiation parameter (Rd) produces a diminishing tendency in unsteady temperature. Figure 12(a) shows the influence of the nanoparticle volume fraction (φ) on the Nusselt number (Nu) distribution. It is observed that a given rise in φ results in the decrease of Nusselt number (Nu) at the lower plate, although it increases at the upper plate. Figure 12(b) shows the influence of the body acceleration parameter (G) on the Nusselt number (Nu) distribution. It is observed that a given increase in body acceleration at the lower plate results in the decrease of Nusselt number (Nu), while it increases at the upper plate. Figure 13 shows the influence of Rd on the Nusselt number (Nu) distribution. It is observed that a given increase in the Rd results in the decline of the Nusselt number (Nu) at the upper plate, while it increases  e greatest rate of change for Nu in both the plates was seen near the null values of Rd and Re. Figure 14 illustrates the variation of the unsteady Nusselt number (Nu t ) for Re with time variation in the upper plate. It is observed that the unsteady Nusselt number upsurges with increasing Re.
us, the unsteady Nusselt number distribution is a periodic function of "t" with period 2π. Table 2 shows the variations of the Nusselt number (Nu)for the Casson nanofluid with and without body acceleration. e periodic body acceleration is taken as the body acceleration parameter G � 1. From this table,   plate. erefore, the body acceleration reduces the heat conduction in lower temperature (T 0 ), while it is enhanced at higher temperature (T w ).

Conclusion
e present study addressed the influence of periodic body acceleration and thermal radiation on the MHD flow of Casson nanofluid through a porous channel. e pulsating flow is included in the flow problem. e analytical technique (perturbation) is utilized to compute the physical system. e considered problem is essential as the study related to periodic body acceleration in the porous channel has enormous applications in industrial and biological sectors. e generalized solutions have been found for the velocity and temperature distributions. Various graphs were depicted to emphasize vivid effects of various parameters, such as Reynolds number, frequency parameter, body acceleration parameter, Casson fluid parameter, radiation parameter, nanoparticle volume fraction, and the Darcy number: (1) e fluid velocity increased for elevated values of frequency parameter, Casson fluid parameter, body acceleration parameter, nanoparticle volume fraction, and the Darcy number. However, the reverse effect was depicted for growing values of the Hartmann number. plates. However, it reduces the heat conduction in lower temperature (T 0 ), while it increases at higher temperature (T w ).