Magnetohydrodynamic Time-Dependent Bio-Nanofluid Flow in a Porous Medium with Variable Thermophysical Properties

In this work, a theoretical model with a numerical solution is brought forward for a bio-nanofluid with varying fluid features over a slippery sheet. )e partial differential equations (PDEs) involving temperature-dependent quantities have been translated into ordinary differential equations (ODEs) by using similarity variables. Numerical verifications have been done in three different methods: finite difference method, shooting method, and bvp4c. To figure out the influence of parameters on the flows, the graphs are plotted for the velocity, temperature, concentration, and microorganism curves. )e boundary layer thickness of the microorganism profile reduces with the Schmidt number and Peclet number. In addition to adding radiative heat flux, we added heat generation, rate of chemical reaction, and first-order slip. Adding these parameters brought new aspects to the underlying profiles. Moreover, the obtained data of the skin friction coefficient, the local Nusselt number, the local Sherwood number, and the local density of motile microorganisms are tabulated against various parameters for the physical parameters. From the results, it is apparent that the local Nusselt number decreases with the Brownian and thermophoretic parameters. )e data obtained for physical parameters have a close agreement with the published data. Finally, the graphs for slip conditions are significantly different when the comparison is drawn with no-slip condition.


Introduction
ermal analysis has attracted attention from the scientific community because of its role in our daily lives. e applications of heat transfer range from electrical devices and power plants to the heating and cooling devices, boiler, condenser, and evaporators within houses where efficiency of these devices plays a key role. e efficient devices not only reduce energy consumption but also give additional life to it.
Nanofluid is a mixture of a base fluid with 100 nm-size nanoparticles. Since the work by Choi and Eastman [1] on nanofluid, the research in this direction took a huge stride. e thermal conductivity is significantly higher than that of the traditional fluids as it was reported in Lee et al. [2]. ere are many applications in the field of nanofluids including lubricants, automation, electronics, and biomedicine. For the list of references which took multiple paths considering nanofluid in their study, one is referred to in [3][4][5][6][7].
Bioconvection is another phenomenon which occurs due to the density difference of the fluid. Raees et al. [8] recorded the homotopy analysis method (HAM) solution for an unsteady bioconvection flow in a channel and showed that the velocity component decreases with the increase in time. Uddin et al. [9] discussed bioconvection nanofluid over a wavy surface with slip flow in application to nano-biofuel cells. Khan and Makinde [10] explored bioconvection flow due to gyrotactic microorganisms. ey noticed that, with rising the values of the convective variable, the dimensionless temperature on the surface rises. Uddin et al. [11] investigated Stefan blowing with multiple slip effects in bioconvection. For finding similarity transformation, they used Lie group analysis. e resources for further study on this topic can be found in [12][13][14][15][16].
One of the ways through which heat transfer occurs is thermal radiation. It has diverse technological applications in combustion, furnace design, turbines, and solar collectors. e thermal radiation with variable fluid properties is reported in [17]. e author found that the skin friction coefficient increases with viscosity parameter. RamReddy and Naveen [18] reported results for activation energy and thermal radiation. Aziz et al. [19] discussed free convection flow in nanofluids with microorganisms. ey discovered that the bioconvection parameter affects heat transfer rate. Mutuku and Makinde [20] discussed hydromagnetic fluid flow in microorganisms. Sk et al. [21] presented multiple slip effects in the presence of microorganisms. Anwar et al. [22] discussed MHD flow in a porous channel with generalized conditions. For further study on this topic, one is referred to in [23][24][25].
e magnetic field has many applications including geothermal energy extractions, plasma studies, chemical engineering, and magnetic resonance imaging (MRI) equipment [26]. Ali et al. [27] discussed hybrid nanofluid with slip conditions for Jeffrey fluid. Mburu et al. [28] reported magnetic and thermal radiation effects over an inclined cylinder. Mabood et al. [29] combined electrical and magnetic flows for non-Newtonian nanofluids over a thin needle. e viscosity of the fluid generally depends on pressure and temperature. However, less effect in fluid flow is observed with pressure. erefore, viscosity is dependent on the temperature variation. Fatunmbi and Adeniyan [30] reported nonlinear thermal radiation in fluid flow with variable properties. Dandapat et al. [31] discussed thin film unsteady flow with variable fluid properties. Vajravelu et al. [32] discussed unsteady convective flow in a vertical surface with variable fluid properties. Shahsavar et al. [33] investigated the impact of variable fluid properties in hybrid nanofluid. Naganthran et al. [34] found results of the stretching and shrinking sheet with variable fluid properties. ey discussed dual solutions in this rotating disk. Salahuddin et al. [35] discussed variable fluid properties for viscoelastic fluid between two rotating plates.
Shafiq et al. [36] presented second-grade bioconvective nanofluid flow and computed the solution from the shooting method. In another study, Rasool and Shafiq [37] discussed Powell-Eyring nanofluid flow in a porous medium over a nonlinear surface. e porosity factor is enhancing the drag force. In another work by Shafiq et al. [38], the numerical solution of the bioconvective tangent hyperbolic nanofluid was found. e effect of temperature-dependent viscosity, thermal radiation, and gyrotactic phenomenon in a convection flow over a cylinder has been discussed in [39][40][41], respectively. In another work by Khan et al. [42], the bioconvection flow was discussed in a truncated cone. For critical review about nanofluid and its effects on viscosity along with thermal conductivity, the reader is referred to in [43][44][45][46]. For experimental investigation on nanofluids, the reader is referred to in [47].
Most theoretical studies mentioned above are focused on the idea of constant fluid properties in fluid flows. e viscosity of a fluid, however, relies heavily on temperature than on other factors, such as pressure. It comes out that the use of variable properties offers distinct effects on fluid flow motion.
is paper is ordered in the following way: the flow model is presented in Section 2. e numerical procedure for the solution is presented in Section 3. Results and discussion are given in Section 4. Conclusion of the paper is drawn in Section 5.

Flow Model
Consider the movement of a nanofluid containing gyrotactic microorganisms past a stretching sheet with variable physical properties. e magnetic field β 2 o is applied normal to the surface. Due to low magnetic Reynolds number, the induced magnetic field is assumed negligible. e stretching velocity is U w � ax(1 − A 1 t) − 1 . e governing model is [48] and the boundary condition corresponding to the considered model is taken as where all the variables are defined in the glossary. e similarity variables are defined as Inserting equation (7) into equations (1)-(6), we get Following Amirsom et al. [48], the physical quantities consisting of viscosity, thermal conductivity, nanoparticle, and microorganism diffusivities are written as Mathematical Problems in Engineering Equation (13) when used into equations (8)-(11), one can get All these parameters are grouped into e physical quantities of the interest in this study are the local skin friction coefficient C fx , the local Nusselt number Nu x , the local Sherwood number Sh x , and the local density number of motile microorganisms Nn x defined as Inserting equation (7) into equation (13) yields the following expressions: where the local Reynolds number is defined as

Shooting Method. A boundary value problem ((8)-(12))
can be solved with the shooting method. e stable iterative scheme, Newton-Raphson method, has been used in locating the roots followed by obtaining the solution from the fifth-order Runge-Kutta solver. e system of first-order ODEs is 4 Mathematical Problems in Engineering e results' verification is achieved from the bvp4c solver. For details on bvp4c, the reader is referred to in [49].

Finite Difference Method.
In this section, we present the finite difference method to solve boundary value problem (8)- (12). e spatial discretization is given by first defining f ′ � F in the momentum equation: and the boundary conditions are Mathematical Problems in Engineering 5

Results and Discussion
An excellent agreement with published results is obtained for a comparison of the skin friction coefficient − f ″ (0) which is shown in Tables 1-3. e data in Table 4 show computational results for the local Nusselt number, the local Sherwood number, and the local density number of motile microorganisms obtained with bvp4c. e local Nusselt number Nu x is reduced against Brownian motion parameter Nb, thermophoretic parameter Nt, Eckert number Ec, heat source parameter s, and thermal conductive parameter h 4 .
With increasing values of Prandtl number Pr ∞ and radiation parameter Rd, the local Nusselt number shows an upward trend. e physical parameter, the local Sherwood number Sh x , depicts an upward trend against Brownian motion parameter Nb, thermophoretic parameter Nt, Schmidt number Sc, and chemical reaction parameter Kr. However, a decreasing trend for the local Sherwood number is observed for rising values of mass diffusivity parameter h 6 .
Finally, the values of the local density number of motile microorganisms Nn x decline with the increase of mass diffusivity parameter h 6 and microorganism diffusivity parameter h 8  e same argument holds for Kp. Figure 3 is plotted to perceive the effect of Prandtl number Pr ∞ on the temperature profile. It is noted that an enhancement in Prandtl number Pr ∞ causes reduction in the temperature distribution.
e smaller values of Pr ∞ correspond to the increase in thermal conductivities which causes reduction in a thermal boundary layer. For Prandtl number (Pr ≥ 1), the momentum diffusivity is dominant in fluid behavior. us, less thermal diffusivity contributes to lowering the thermal boundary layer thickness. Figure 4 depicts the influence of radiation parameter Rd on the temperature profile. It is seen that an increase in Rd enhances the temperature of the fluid. Larger values of radiation parameter transfer more heat to the fluid which overall increases the temperature and its profile. Figure 5 reports the influence of Eckert number Ec on the temperature profile. e higher values of Eckert number Ec cause an increase in the thermal boundary layer thickness. e Eckert number Ec enhances kinetic energy, which increases fluid's temperature. Figure 6 illustrates the impact of heat source parameter s on the temperature distribution. It is observed that temperature of the fluid increases with an increment in the heat generation parameter. e higher values of s provide more heat to the fluid resulting in the rise of the temperature of the fluid. Figure 7 examines the effect of temperature-dependent thermal conductivity parameter h 4 on temperature. It is noted that the thermal boundary layer thickness increases by increasing parameter h 4 . Figures 8 and 9 are drawn to perceive the effect of Brownian motion parameter Nb on the temperature and concentration profiles. It is revealed in the figure that, by increasing Brownian motion parameter Nb, thermal boundary layer thickness rises, while concentration boundary layer thickness declines. e Brownian parameter appears due to the presence of nanoparticles' concentration. Figures 10 and 11 convey the impacts of thermophoresis parameter Nt on temperature and concentration distributions. e temperature and concentration profile rise for rising values of Nt. e thermophoresis term appears due to the temperature gradient in particulate flows. Larger values of Nt transmit more temperature to the fluid along with the concentration profile. Figure 12 portrays the influence of chemical reaction parameter Kr on the concentration profile. e rising values of Kr suppress diffusion which lowers the concentration boundary layer. Figure 13 depicts the effects of Schmidt number Sc on the concentration distribution. e rise in Sc causes reduction in the concentration profile. e higher the Schmidt number, the lower the mass diffusivity which is the reason for reduction in the concentration boundary layer thickness. Figure 14 presents the influence of mass diffusivity parameter h 6 on the concentration profile. One can observe that rise in mass diffusivity parameter h 6 results in an increase of the concentration profile. Figure 15 describes the influence of Peclet number Pe on the density of motile microorganism profile. e incremental values of Peclet number Pe cause reduction in motile microorganisms' boundary layer thickness.
e Peclet number appears in the study of transport processes. It measures the importance of convection over diffusion. For larger values of the Peclet number, the convection is dominant and diffusion is negligible which is happening here in the motile microorganisms' boundary layer thickness. Figure 16 investigates the impact of bioconvection Schmidt number Sb on the density of motile microorganism profile. It is shown that rising values of bioconvection Schmidt number Sb lower the boundary layer thickness of 6 Mathematical Problems in Engineering the motile microorganism profile. In high values of Sb, the particles are giant which means these diffuse slowly. Figures 17 and 18 are drawn to perceive the effect of mass diffusivity parameter h 6 and microorganism diffusivity parameter h 8 . Increasing the values of mass diffusivity parameter and microorganism diffusivity parameter elevates the boundary layer thickness of the motile microorganism profile.

Conclusion
e focus of the paper involves unsteady MHD flow of bionanofluid in a permeable medium taking thermal radiation and chemical reaction into account over a stretching sheet with variable thermophysical properties. e notable findings of the problem are outlined in the following [ Applied magnetic field (Nm − 1 A − 1 ) μ: Dynamic viscosity (Pas) ρ: Fluid's viscosity (kgm − 3 ) σ: Fluid's electrical conductivity (Sm − 1 ) (S is siemens) M: Dimensionless magnetic variable Kp: Porosity variable T: Fluid's temperature (K) T w : Constant wall temperature (K) T ∞ : Free-stream temperature (K) k(T): Variable thermal conductivity (Wm − 1 K − 1 ) α: ermal diffusivity (m 2 s − 1 ) δ: Slip parameter C p : Heat capacity at constant pressure (Jkg − 1 K − 1 ) q r : Flux due to radiation (Wm − 2 ) Q: Internal heat generation/absorption C f : Skin friction coefficient Nu x : Local Nusselt parameter MHD: Magnetohydrodynamics PDEs: Partial differential equations