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

Authors: N. Thamaraikannan; S. Karthikeyan; Dinesh Kumar Chaudhary; Safak Kayikci
Journal: Complexity
 (2021)

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## Abstract

The 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. The thermal radiation term is incorporated in the heat transfer equation. The governing equations corresponding to velocity and temperature are converted from partial differential equations to a system of ordinary differential equations by employing similarity variables. The perturbation technique is applied to solve the governing flow equations. The impact of diverse parameters on flow features is graphically analyzed. The 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.


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## Body

## 1. Introduction

The foremost principle of MHD is that forces are formed by the magnetic field, which stimulates a current through a moving conducting fluid. The 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. The 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. This attempt was taken first by Choi [3] subsequently succeeded by Sandeep et al. [4] and Nisar et al. [5]. They 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.” They 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.Thermal 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]. The important visualization about the influence of magnetic field and radiation with assorted flow geometries is established in the literature [17–28].The 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.” The 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). The 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 [29–34].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. The considered problem has massive application in manufacturing and biological processes. The perturbation technique was employed to solve the resulting similarity equations. The velocity and temperature fields are illustrated for several pertinent flow parameters.


## 2. 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. The flow is driven by a pulsating pressure gradient [29].(1)1ρnf∂p∗∂x∗=−A1+εeiωt∗,where ρnf 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.” The physical flow system is noted in Figure 1.Figure 1
Physical model and coordinate system.We consider thatX∗-axis is taken along the lower plate, and the perpendicular axis is noted as Y∗-axis. The lower plate is at rest, maintaining the temperature T0. The upper plate is moving with the velocity Uw∗=A/ω1+εeiωt∗ parallel to the X∗-axis and maintains the temperature T1T0<T1. The nanofluid is injected from the lower plate into the porous channel with the velocity v0, and it is sucked out from the porous medium via the upper plate with the similar velocity. “Uniform magnetic field” (B0) is imposed along the usual direction to the flow. The periodic body acceleration is assumed as G∗=g0cosψ, where ψ is the phase difference and g0 is the amplitude of the body acceleration.The Casson nanofluid rheological equation is stated as(2)Tij=2μBnf+pyπc/2emn,πc>π,2μBnf+pyπ/2emn,πc<π,where “πc” is the critical value of “π” based on the non-Newtonian model, Tmn is the m,n-th stress tensor component, π=emn with emn being the m,n-th deformation rate, μBnf is the plastic “dynamic viscosity” of the “non-Newtonian” nanofluid, and py is the “yield stress” of the fluid.The governing equations of the flow problem are stated as follows:(3)∂u∗∂x∗=0,(4)∂u∗∂t∗+v0∂u∗∂y∗=−1ρnf∂p∗∂x∗+μnfρnf1+1β∂2u∗∂y∗2−σB02ρnfu∗−μnfΦρnfku∗+g0cosψ,(5)0=−1ρnf∂p∗∂y∗,(6)∂T∗∂t∗+v0∂T∗∂y∗=κnfρCpnf∂2T∗∂y∗2+μnfρCpnf1+1β∂u∗∂y∗2+σB02ρCpnfu∗2+4σ∗3K∗ρCpnf∂2T∗4∂y∗2+Q0ρCpnfT∗−T0.The boundary conditions for the present analysis are(7)u∗=0,T∗=T0,aty∗=0,u∗=Uw∗=Aω1+εeiωt∗,T∗=Tw,aty∗=L,where Q0 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, Cpnf is the “specific heat at constant pressure of nanofluid,” μnf is the “dynamic viscosity” of the nanofluid, ρnf is the “density” of the nanofluid, ρCpnf is the “heat capacity of the nanofluid,” νnf is the “kinematic viscosity of the Casson nanofluid,” κnf is the “thermal conductivity” of the nanofluid, K∗ is “Rosseland mean absorption coefficient,” and σ∗ is the “Stefan–Boltzmann constant” [16].(8)ρnf=1−φρf+φρn,μnf=μf1−φ2.5,ρCpnf=1−φρCpf+φρCpn,νnf=μnfρnf,κnfκf=2κf+κn−2φκf−κn2κf+κn+φκf−κn,where φ is the nanoparticle volume fraction and the subscripts nf,f, and n represent the nanofluid, base fluid, and nanoparticles correspondingly. The use of the above expression for κnf/κf is restricted to spherical nanoparticles only.We assume that the temperature variation within the fluid flow can be expanded in a Taylor’s series aboutT0. By neglecting the higher-order terms, we obtain(9)T∗4≅4T03T∗−3T04.We establish the dimensionless variables as follows:(10)x=x∗L,y=y∗L,t=t∗ω,u=u∗ωA,p=p∗ρnfAL,Uw=ωUw∗A,θ=T∗−T0Tw−T0.By applying equations (3), (5), (8), (9), and (10), in equations (4), (6), and (7),(11)1+1/β1−φ2.51−φ+φρn/ρf∂2u∂y2−Re∂u∂y−Ha21−φ+φρn/ρf+11−φ+φρn/ρf1−φ2.5Dau=H2∂u∂t+∂p∂x−Gcosψ,(12)4/3Rd+knf/kfPr∂2θ∂y2−1−φ+φρCpnρCpfRe∂θ∂y+Qθ=H21−φ+φρCpnρCpf∂θ∂t−Ec1+1/β1−φ2.5∂u∂y2+Ha2u2.The new boundary conditions are(13)u=0,θ=0,aty=0,u=Uw=1+εeit,θ=1,aty=1,where Da=k/ΦL2 is the “Darcy number of the porous media,” H=Lω/νf is the frequency parameter, Ha=B0Lσ/μf is the Hartmann number, Re=V0L/νf is the cross-flow Reynolds number, Ec=A/ω2/CpfTw−T0 is the Eckert number, Q=Q0L2/ρCpfνf is the “heat source/sink parameter,” Rd=4T03σ∗/kfK∗ is the radiation parameter, Pr=μCpf/kf is the Prandtl number, and G=L2ωg0/Aνf is the body acceleration parameter.


## 3. Method of Solution

The velocityu and temperature θ can be supposed to have the form:(14)u=u1y+εu2yeit+ε2u3ye2it,(15)θ=θ1y+εθ2yeit+ε2θ3y2it.Now, substituting equations (1), (14), and (15) into equations (11)–(13) and then equating the coefficients of various powers of ε, we get(16)α1u1ll−Reu1l−α2u1=−H2−Gcosψ,(17)α1u2ll−Reu2l−α3u2=−H2,(18)α1u3ll−Reu3l−α4u3=−H2,(19)β1θ1ll−β2θ1l−Qθ1=−β3u1l2−β4u12,(20)β1θ2ll−β2θ2l−β5θ2=−2β3u1lu2l−2β4u1u2,(21)β1θ3ll−β2θ3l−β6θ3=−β3u2l2−β4u22.The new boundary conditions are(22)u1=0,u2=0,u3=0,θ1=0,θ2=0,θ3=0,aty=0,u1=1,u2=1,u3=1,θ1=1,θ2=0,θ3=0,aty=1.Solving the six ODEs (16)–(21) with the boundary conditions (22), we obtain(23)u1=A1em1y+A2em2y+A3,u2=A4em3y+A5em4y+A6,u3=A7em5y+A8em6y+A9,θ1=B8em7y+B7em8y+B1e2m1y+B2e2m2y+B3em1y+B4em2y+B5em1+m2y+B6,θ2=B18em9y+B19em10y+B9em1+m3y+B10em1+m4y+B11em2+m3y+B12em2+m4y+B13em1y+B14em2y+B15em3y+B16em4y+B17,θ3=B37em11y+B36em12y+B20e2m3y+B21e2m4y+B22em3+m4y+B23em3+m1y+B24em1+m4y+B25em3+m2y+B26em2+m4y+B27em5+m1y+B28em6+m1y+B29em5+m2y+B30em6+m2y+B31em3y+B32em4y+B33em5y+B34em6y+B35,where m’s, A’s, and B’s are constants given in Appendix.The dimensionless Nusselt number at the plates is given by(24)Nu=−∂θ∂yy=0,1=−θ1′y+εθ2′yeit+ε2θ3′ye2ity=0,1.


## 4. Results and Discussion

The effect of diverse material parameters on the non-dimensional heat transfer and velocity is discussed in this section. The 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. Throughout 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°,andG=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 uy. It is noticed that uy 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 uy. It is noticed that uy improves with an escalation in frequency parameter H and body acceleration parameter G. Figure 4(a) elucidates the effect of Da on velocity distribution uy. It is evident that uy 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 uy.Table 1
Comparison of Nusselt number for Newtonian and non-Newtonian fluids with Srinivas et al. [29] for the limiting case of φ=0,G=0,ε=0.01,H=3,β=2,Ec=1,Da=0.1,Pr=21,andt=π/4.
ParameterValuesNu=−∂θ/∂yy=0Nu=−∂θ/∂yy=1NewtonianNon-NewtonianNewtonianNon-Newtonian[29]Present[29]Present[29]Present[29]PresentHa0−2.7033−2.7033−2.6361−2.63613.17713.17712.74572.74572−2.2726−2.2726−2.2307−2.23073.57543.57542.72612.72614−1.5615−1.5615−1.5380−1.53802.67402.67401.81161.8116Re0−7.6829−7.6829−7.1678−7.16786.18466.18465.66955.66951−2.2726−2.2726−2.2307−2.23073.57543.57542.72612.72612−1.1585−1.1585−1.2067−1.2067−1.9577−1.9577−2.8056−2.8056Q0−2.5506−2.5506−2.4868−2.48681.67631.67631.21421.21420.5−2.6712−2.6712−2.6064−2.60643.23063.23062.75562.75561−2.8333−2.8333−2.7671−2.76715.43565.43564.94254.9425Rd0−4.2702−4.2702−4.1034−4.103419.277719.277717.418817.41881−3.3850−3.3850−3.2904−3.29046.25906.25905.49845.49842−2.8333−2.8333−2.7671−2.76713.23033.23032.75562.7556Figure 2
(a) Effect ofHa on the “velocity distribution” when φ=0.1. (b) Effect of φ on the “velocity distribution” when Ha=1.
(a)(b)Figure 3
(a) Effect ofH on the “velocity distribution” when G=1. (b) Effect of G on the velocity distribution when H=5.
(a)(b)Figure 4
(a) Effect ofDa on the “velocity distribution” when β=5. (b) Effect of β on the “velocity distribution” when Da=0.2.
(a)(b)Figures5(a) and 5(b) show the influence of Ec and G on the temperature distribution θy. The observations conclude that the temperature distribution θy upsurges with a rise in Ec. In addition, it is obvious that the escalating periodic body acceleration slightly improves the temperature.Figure 5
(a) Effect ofEc on the “temperature distribution” when G=1. (b) Effect of G on the “temperature distribution” when Ec=0.2.
(a)(b)Figures6(a) and 6(b) displays persuade of Re and Pron the temperature distribution θy. From Figure 6(a), the observations show that the temperature distribution θy reduces with an upsurge in cross-flow Reynolds number. Figure 6(b) depicts that the temperature diminished for increasing values of Pr. The Prandtl number diminishes the thermal boundary layer thickness. In addition, it controls the thickness of the thermal boundary layer. Fluids with lower Prandtl number have higher thermal conductivity so that the heat can diffuse from the plate faster than higher Prandtl number fluids [29]. Figures 7(a) and 7(b) reveal the influence of Q and Rd on θy. Figure 7(a) depicts that the temperature distribution boosts up with a rise in the heat source/sink parameter Q. From Figure 7(b), one can observe a rise in θy with intensification in Rd. This is in agreement with the physical reality that the thermal boundary layer thickness enhances with growing Rd.Figure 6
(a) Effect ofRe on the “temperature distribution” when Pr=14. (b) Effect of Pr on the temperature distribution when Re=2.
(a)(b)Figure 7
(a) Effect ofQ on the “temperature distribution” when Rd=3. (b) Effect of Rd on the “temperature distribution” when Q=0.5.
(a)(b)The effect of Eckert numberEc, Darcy number Da, body acceleration parameter G, radiation parameter (Rd), the nanoparticle volume fraction (φ), Hartmann number Ha, heat source/sink parameter Q, and Prandtl number Pr on unsteady temperature θty is shown in Figures 8–11. From Figure 8(a), it is observed that the enhancement in Eckert number Ec produces an increasing tendency in unsteady temperature θty. Figure 8(b) depicts that the unsteady temperature θty 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 θty. Figure 9(b) shows that the unsteady temperature θty is oscillating with increasing Hartmann number Ha.Figure 8
(a) Effect ofEc on “unsteady temperature distribution” for Da=0.2. (b) Effect of Da on “unsteady temperature distribution” for Ec = 0.2.
(a)(b)Figure 9
(a) Effect ofG on unsteady “temperature distribution” when Ha=1. (b) Effect of Ha on “unsteady temperature distribution” for G=1.
(a)(b)Figure 10
(a) Effect ofφ on “unsteady temperature distribution” when Pr=14. (b) Effect of Pr on “unsteady temperature distribution” for φ=0.1.
(a)(b)Figure 11
(a) Effect of Q on “unsteady temperature distribution” whenRd=3. (b) Effect of Rd on “unsteady temperature distribution” for Q = 0.5.
(a)(b)Figure10(a) shows that the amplification in the nanoparticle volume fraction (φ) produces an insurgence in unsteady temperature θty 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 Figure11(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 ofQ. From Figure 11(b), it is noticed that the increase in radiation parameter Rd produces a diminishing tendency in unsteady temperature.Figure12(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 12
(a) Effect ofφ on “Nusselt number distribution” when G=1,Ec=1,Ha=0.5,Rd=2,andPr=21. (b) Effect of G on “Nusselt number distribution” for φ=0.1, Ec = 1, Ha = 0.5, Rd = 2, and Pr = 21.
(a)(b)Figure13 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 at the lower plate. The greatest rate of change for Nu in both the plates was seen near the null values of Rd and Re.Figure 13
Effect of Rd and Re on “the Nusselt number distribution” whenEc=1,Ha=0.5,and Pr=21.Figure14 illustrates the variation of the unsteady Nusselt number (Nut) for Re with time variation in the upper plate. It is observed that the unsteady Nusselt number upsurges with increasing Re. Thus, the unsteady Nusselt number distribution is a periodic function of “t” with period 2π.Figure 14
Effect of Re over time variation on the “unsteady Nusselt number distribution” forEc=1,Rd=2,Ha=0.5,and Pr=21.Table2 shows the variations of the Nusselt number Nufor the Casson nanofluid with and without body acceleration. The periodic body acceleration is taken as the body acceleration parameter G=1. From this table, the Nusselt number Nu enhances at the lower plate for both the cases with the higher values of Re and Rd, while the inverse trend is noticed for larger value of Q. However, this behaviour is reversed at the upper plate for Re,Rd,andQ, while it increases for enlarging values of Ha at both the plates. It is also observed that with increasing periodic body acceleration G, the Nu decreases at the lower plate, while it increases at the upper plate. Therefore, the body acceleration reduces the heat conduction in lower temperature (T0), while it is enhanced at higher temperature (Tw).Table 2
Comparison of Nusselt number for channel with and without periodic body acceleration, whenε=0.01,H=3,β=5,Ec=1,Da=0.2,Pr=21,andt=π/4.
ParameterValuesNu=−∂θ/∂yy=0Nu=−∂θ/∂yy=1Without body acceleration (G = 0)With body acceleration (G = 1)Without body acceleration (G = 0)With body acceleration (G = 1)Ha0−5.0478−5.57196.28288.17451−4.7406−5.23448.798710.72172−3.9957−4.415815.004316.9792Re0−28.7997−31.24949.608610.78702−4.9676−5.48396.94078.83764−1.4012−1.5520−8.8930−7.7016Q0−4.8737−5.38054.07575.73050.5−4.9676−5.48396.94078.83761−5.0705−5.597110.494912.6975Rd1−5.3459−5.906613.202516.38282−4.9676−5.48396.94078.83763−4.6292−5.10534.07325.3677


## 5. Conclusion

The present study addressed the influence of periodic body acceleration and thermal radiation on the MHD flow of Casson nanofluid through a porous channel. The pulsating flow is included in the flow problem. The analytical technique (perturbation) is utilized to compute the physical system. The considered problem is essential as the study related to periodic body acceleration in the porous channel has enormous applications in industrial and biological sectors. The 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)
The 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.(2)
Diminishing behaviour of fluid flow is observed for diverse values of cross Reynolds number and the Prandtl number. Prandtl number has an essential role in the heat transfer process.(3)
The Nusselt number declines at the lower plate and increases at the upper plate with the rise in the nanoparticle volume fraction, Darcy number, and body acceleration parameter.(4)
The unsteady Nusselt number distribution is a periodic function of “t” with period 2π with its lowest values near 2nπ and higher values near (2n+1) π for all integer values of n.(5)
The body acceleration enhances the velocity at both plates. However, it reduces the heat conduction in lower temperature (T0), while it increases at higher temperature (Tw).



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