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

Authors: N. Thamaraikannan; S. Karthikeyan; Dinesh Kumar Chaudhary
Journal: Journal of Mathematics
 (2021)

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## Abstract

The effect of various parameters in the unsteady pulsating flow of radiative hydromagnetic Casson nanofluid through a porous channel is investigated. The governing equations were nondimensionalized by applying suitable transformations. The perturbation technique was employed to solve the resulting similarity equations. The velocity and temperature fields are illustrated for several pertinent flow parameters. The fluid velocity has been enhancing for higher values of the frequency parameter, Casson fluid parameter, nanoparticle volume fraction, and Darcy number. The reverse impact is observed for larger values of the Hartmann number. The 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.


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## Body

## 1. Introduction

Nanofluid consists of the uniform composition of solid particles whose dimension is in the range of 1 to 100 nm. The 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. This 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. They suggested that nanoparticle-containing gold can treat cancer as these materials have a lofty atomic quantity that produces the temperature guide to the handling of malignant tumours. The 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. Their 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. The 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 [5–8].The Casson fluid was first introduced in 1959 by Casson. He used pigment oil of printing ink to scrutinize the flow behaviour. The “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. The 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.” The 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. They 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. The magnetohydrodynamic boundary layer flow of the Casson fluid over an exponentially permeable shrinking sheet was investigated by Nadeem et al. [13].Thermal 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. They 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. The important visualization about thermal radiation with various flow geometries is found in the literature [20–25].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. Therefore, we expanded our effort to the pulsative Casson nanofluid that passes through a porous channel with radiation and magnetic-field impacts. The considered problem has enormous applications in many industrial as well as biological processes such as crude oil purification and blood flow through contracting muscles. The 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. The solution for some fluid flow problems can be found in [26–29].


## 2. 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. The flow is driven by an unsteady pressure gradient [18].(1)1ρnf∂p∗∂x∗=−A1+εeiωt∗,where ρnf 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 Figure1, the X∗-axis is taken along the lower wall, and the perpendicular axis is noted as the Y∗-axis. The lower wall maintains the temperature T0, and the upper wall maintains the temperature T1T0<T1. The nanofluid is injected from the lower wall into the porous channel with the velocity v0, and it is sucked out from the porous medium via the upper wall with a similar velocity. Uniform magnetic field (B0) is imposed along the usual direction to the flow. The physical flow system is noted in Figure 1.Figure 1
Physical model and the coordinate system.The Casson nanofluid’s rheological equation is stated ast(2)Tij=2μBnf+pyπc/2emn,πc>π,2μBnf+pyπc/2emn,πc<π,where “πc” is the critical value of “π” based on the non-Newtonian model. Tmn is the m,nth stress tensor component, and π=emn, with emn being the m,nth deformation rate. μBnf is the plastic “dynamic viscosity” of the “non-Newtonian” nanofluid. py is the “yield stress” of the fluid.The problem’s governing equation is as follows:(3)∂u∗∂t∗+v0∂u∗∂y∗=−1ρnf∂p∗∂x∗+μnfρnf1+1β∂2u∗∂y∗2−σB02ρnfu∗−μnfΦρnfku∗,0=−1ρnf∂p∗∂y∗,∂T∗∂t∗+v0∂T∗∂y∗=−κnfρCpnf∂2T∗∂y∗2+μnfρCpnf1+1β∂u∗∂y∗2+σB02ρCpnfu∗2−1ρCpnf∂qr∂y∗+Q0ρCpnfT∗−T0.The boundary conditions for the present analysis are [18](4)u∗=0,T∗=T0,aty∗=0,u∗=0,T∗=T1,aty∗=L,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, μnf is the dynamic viscosity of the nanofluid, ρnf is the density of the nanofluid, νnf is the kinematic viscosity of the Casson nanofluid, Cpnf is the specific heat at a constant pressure of the nanofluid, and κnf is the thermal conductivity of the nanofluid, which is given by Oztop and Abu-Nada as follows [14]:(5)ρnf=1−φρf+φρn,μnf=μf1−φ2.5,ρCpnf=1−φρCpf+φρCpn,νnf=μnfρnf,κnfκf=2κf+κn−2φκf−κn2κf+κn+φκf−κn.Here,φ is the nanoparticle volume fraction, ρCpnf is the heat capacity of the nanofluid, and ρCpf and ρCpn are the heat capacity of the base fluid and the nanoparticles, respectively. ρf and ρn are the densities of the fluid and the solid fractions, respectively. μf, νf, and Cpf are the dynamic viscosity, kinematic viscosity, and specific heat at a constant pressure of the base fluid, and κf and κn are the thermal conductivity of the base fluid and the solid fractions, respectively. The use of the abovementioned expression for κnf/κf is restricted to spherical nanoparticles only.We establish the dimensionless variables as follows:(6)x=x∗L,y=y∗L,t=t∗ω,u=u∗ωA,p=p∗ρAL,θ=T∗−T0T1−T0.By using the abovementioned dimensionless variables, we get(7)1+1/β1−φ2.51−φ+φρn/ρf∂2u∂y2−Re1−φ2.5∂u∂y−11−φ+φρn/ρfM2+11−φ2.5Dau−H2∂u∂t+∂p∂x=0,(8)knf/kf+4/3RdPr∂2θ∂y2−Re1−φ+φρCpn/ρCpf1−φ2.5∂θ∂y+Qθ=−1+1/β1−φ2.5Ec∂u∂y2−EcM2u2+H21−φ+φρCpnρCpf∂θ∂t.The new boundary conditions are(9)u=0,θ=0,aty=0,u=0,θ=1,aty=1,where Da=k/ΦL2 is the Darcy number of the porous media, H=Lω/νf is the frequency parameter, M=B0Lσ/μf is the Hartmann number, Re=V0L/νf is the cross-flow Reynolds number, Ec=A/W2/CpfT1−T0 is the Eckert number, Q=Q0L2/ρCpfνf is the heat source/sink parameter, Rd=4T03σ∗/kfK∗ is the radiation parameter, and Pr=μfCpf/kf is the Prandtl number.


## 3. The Proposed Solution Method

In order to solve equations (7) and (8) with the boundary conditions (9), we apply the perturbation technique. The velocity u and temperature ϴ can be assumed to have the form(10)u=u1y+εu2yeit+ε2u3ye2it,(11)θ=θ1y+εθ2yeit+ε2θ3ye2it.Now, we substitute equations (10) and (11) into the equations (7) and (8); then, equating the coefficients of various powers of ε, we get(12)A1u1ll−Re1−φ2.5u1l−A2u1=−H2,(13)A1u2ll−Re1−φ2.5u2l−A6u2=−H2,(14)A1u3ll−Re1−φ2.5u3l−A10u3=−H2,(15)B1θ1ll−B2θ1l−Qθ1=−Ec1+1/β1−φ2.5u1l2−EcM2u12,(16)B1θ2ll−B2θ2l−B13θ2=−2Ec1+1/β1−φ2.5u1lu2l−2EcM2u1u2,(17)B1θ3ll−B2θ3l−B25θ3=−Ec1+1/β1−φ2.5u2l2−EcM2u22.The new boundary conditions are(18)u1=0,u2=0,u3=0,θ1=0,θ2=0,θ3=0,aty=0,u1=0,u2=0,u3=0,θ1=1,θ2=0,θ3=0,aty=1.Solving the six ODEs (12)–(17) using the boundary conditions (18), we obtain(19)u1=A3em1y+A4em2y+A5,u2=A8em3y+A7em4y+A9,u3=A11em5y+A12em6y+A13,θ1=B12em7y+B11em8y+B5e2m1y+B6e2m2y+B7em1y+B8em2y+B9em1+m2y+B10,θ2=B23em9y+B24em10y+B14em1+m3y+B15em1+m4y+B16em2+m3y+B17em2+m4y+B18em1y+B19em2y+B20em3y+B21em4y+B22,θ3=B43em11y+B42em12y+B26e2m3y+B27e2m4y+B28em3+m4y+B29em1+m3y+B30em1+m4y+B31em2+m3y+B32em2+m4y+B33em1+m5y+B34em1+m6y+B35em2+m5y+B36em2+m6y+B37em3y+B38em4y+B39em5y+B40em6y+B41,where m’s, As, and Bs are the constants specified in Appendix.The dimensionless Nusselt number at the walls is given by(20)Nu=−∂θ∂yy=0,1=−θ1′y+εθ2′yeit+ε2θ3′ye2ity=0,1.


## 4. Results and Discussion

This segment presents the Casson nanofluid pulsating flow filling through a porous channel in the appearance of thermal radiation impacts. The 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. Figures2(a)–2(e) presents the effect of the frequency parameter H, Casson fluid parameter (β),nanoparticle volume fraction (φ), Hartmann number M, and Darcy number Da on the velocity distribution uy. Throughout the computation, the parameters are taken as t=π/4,Re=1, ϕ=0.1,M=0.5, β=2,Da=0.1,ε=0.01,Ec=0.1,Q=0.5,Rd=2,Pr=21,andH=3 unless otherwise stated.Figure 2
Velocity distribution fort=π/4,ε=0.01,andRe=1. (a) Effect of H when ϕ=0.1,M=0.5,Da=0.1,andβ=2. (b) Effect of β when ϕ=0.1,M=0.5, Da=0.1,andH=3. (c) Effect of φ when M=0.5, Da=0.1,β=2,andH=3. (d) Effect of M when ϕ=0.1,Da=0.1,H=3,andβ=2. (e) Effect of Da when ϕ=0.1,M=0.5,H=3,andβ=2.
(a)(b)(c)(d)(e)Figures2(a) and 2(b) exhibit the effect of H and β on velocity distribution uy, and it is noticed that uy improves with an escalation in the frequency parameter H; furthermore, it is noted that the velocity is the increasing function of the Casson fluid parameter (β) due to the diminishing behaviour of yield stress. Figure 2(c) illustrates the effect of φ on uy. From this plot, we observed that the velocity upsurges with an escalation in the nanoparticle volume fraction (φ). Figure 2(d) exhibits the effect of M on velocity profile uy. Here, as M increases, reduction in velocity occurs because the retarding forces generated by the applied magnetic field act as resistive drag forces in the opposite direction to the flow. The influence of various values of the Darcy number Da on the velocity distribution uy is presented in Figure 2(e). It is obtained from this graph that the velocity is the enhancing function of Da. This 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 ). The 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.Figures3(a) and 3(b) display the influence of (φ) and Re on the temperature distribution θy. The observations conclude that the temperature distribution θy 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 θy. 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 θy with intensification in Rd.Figure 3
Temperature distribution forε=0.01,H=3,β=2,Ec=0.1,Da=0.1,Q=0.5,Rd=2,M=0.5,Pr=21,andt=π/4. (a) Effect of ϕ when Re=1. (b) Effect of Re when ϕ=0.1.
(a)(b)Figure 4
Temperature distribution fort=π/4,ε=0.01,H=3,β=2,Ec=0.1,Q=0.5,Da=0.1,M=0.5,Re=1,andϕ=0.1. (a) Effect of Pr when Rd=2, and (b) effect of Rd when Pr=21.
(a)(b)The effect of the radiation parameter (Rd),nanoparticle volume fraction (φ), and Hartmann number M on unsteady and steady temperature θy 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 5
Effect ofRd on (a) unsteady and (b) steady temperature distributions when t=π/4,ε=0.01,H=3,β=2,Ec=1,Q=0.5,Da=0.1,M=0.5,Re=1,Pr=21,andϕ=0.1.
(a)(b)Figure 6
Effect ofϕ on (a) unsteady and (b) steady temperature distributions when t=π/4,ε=0.01,H=3,β=2,Ec=1,Q=0.5,Da=0.1,M=0.5,Re=1,Pr=21,and Rd=2.
(a)(b)Figure 7
Effect ofM on (a) unsteady and (b) steady temperature distributions when t=π/4,ε=0.01,H=3,β=2,Ec=1,Q=0.5,Da=0.1,φ=0.1,Re=1,Pr=21,and Rd=2.
(a)(b)Figure6(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.Figure8(a) shows the influence of nanoparticle-volume fraction (φ) on the Nusselt number Nu distribution. It is observed that the given increase in nanoparticle-volume fraction (φ) results in the decrease of the Nusselt number Nu at the lower wall, while it increases at the upper wall. Figure 8(b) shows the influence of the Hartmann number M on the Nusselt number Nu distribution. From this figure, we conclude that Nu rises at the lower wall and reduces at the upper wall for increasing the values of M.Figure 8
Nusselt number distribution fort=π/4,ε=0.01,H=3,β=2,Ec=1,Da=0.1,Rd=2,Re=1,andPr=21,Q=0.5. (a) Effect of ϕ when M=0.5, and (b) effect of M when ϕ=0.1.
(a)(b)Figure9(a) shows the influence of the heat source/sink parameter Q on the Nusselt number Nu distribution. It is observed that the given increase in heat source at the lower wall results in the decrease of the Nusselt number Nu. In contrast, it increases at the upper wall. The increase in heat sink at the lower wall increases the Nusselt number Nu, while it decreases at the upper wall. Figure 9(b) shows the influence of radiation (Rd) parameter on the Nusselt number Nu distribution. It is observed that the given increase in radiation parameter (Rd) results in the decrease of the Nusselt number Nu at the upper wall, while it increases at the lower wall.Figure 9
Nusselt number distribution fort=π/4,ε=0.01,H=3,β=2,Ec=1,Da=0.1,M=0.5,Re=1,andPr=21. (a) Effect of Q when Rd=2andϕ=0. (b) Effect of Rd when Q=0.5andϕ=0.1.
(a)(b)Table1 shows the comparison of results of Nusselt number Nu with Srinivas et al. [18]. The 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 1
Comparison of the Nusselt number for Newtonian and non-Newtonian fluids with the work of Srinivas et al. [18] for the limiting case of ϕ=0, ε=0.01,H=3,β=2,Ec=1,Da=0.1,Pr=21,andt=π/4.
ParameterValuesNu=−∂θ/∂yy=0Nu=−∂θ/∂yy=1Newtonian [18]Newtonian (present)Non-Newtonian [18]Non-Newtonian (present)Newtonian [18]Newtonian (present)Non-Newtonian [18]Non-Newtonian (present)M0−2.7033−2.7033−2.6361−2.63613.17713.17712.74572.74572−2.2726−2.2726−2.2307−2.23073.57543.57542.72612.72614−1.5615−1.5615−1.5380−1.53802.67402.67401.81161.8116Re0−7.6829−7.6829−7.1678−7.16786.18466.18465.66955.66951−2.2726−2.2726−2.2307−2.23073.57543.57542.72612.72612−1.1585−1.1585−1.2067−1.2067−1.9577−1.9577−2.8056−2.8056Q0−2.5506−2.5506−2.4868−2.48681.67631.67631.21421.21420.5−2.6712−2.6712−2.6064−2.60643.23063.23062.75562.75561−2.8333−2.8333−2.7671−2.76715.43565.43564.94254.9425Rd0−4.2702−4.2702−4.1034−4.103419.277719.277717.418817.41881−3.3850−3.3850−3.2904−3.29046.25906.25905.49845.49842−2.8333−2.8333−2.7671−2.76713.23033.23032.75562.7556Table2 presents the variations of the Nusselt number Nu. The 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.Table 2
The Nusselt number whenε=0.01,H=3,β=2,Ec=1,Da=0.1,Pr=21,andt=π/4.
ParameterValuesNu=−∂θ/∂yy=0Nu=−∂θ/∂yy=1M0−3.31244.54732−3.02824.45154−2.42143.6967Re0−6.72375.08121−3.02824.45152−1.57211.7170Q0−3.15413.65560.5−3.29304.54661−3.46585.6458Rd0−5.562024.76311−4.10858.41832−3.29304.5466


## 5. Conclusions

The present study deals with the Casson nanofluid flow through a porous channel with radiation and MHD effects. 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 flow of nanofluids in the porous channel has enormous applications in industrial and biological research such as crude oil purification and blood flow in the heart and lungs. The necessary upshots of this research are noted as follows:(1)
The fluid velocity enhances for higher values of the frequency parameter, Casson fluid parameter, nanoparticle volume fraction, and Darcy number. On the other hand, the reverse impact is observed for larger values of the Hartmann number.(2)
Improved values of the cross Reynolds number and the radiation parameter produce a decrement in temperature.(3)
The Nusselt number declines at the lower wall and increases at the upper wall with increasing nanoparticle volume fraction.(4)
Diminishing behaviour of the Nusselt number is noted at the upper wall, and it is increasing at the lower wall with an augmentation in the Hartmann number, heat sink, and radiation parameters.(5)
The hydrodynamic case can be obtained for the present problem by takingM=0.



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