Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate

The present study deals with the swimming of gyrotactic microorganisms in a nanofluid past a stretched surface. The combined effects of magnetohydrodynamics and porosity are taken into account. The mathematical modeling is based on momentum, energy, nanoparticle concentration, and microorganisms’ equation. A new computational technique, namely successive local linearization method (SLLM), is used to solve nonlinear coupled differential equations. The SLLM algorithm is smooth to establish and employ because this method is based on a simple univariate linearization of nonlinear functions. The numerical efficiency of SLLM is much powerful as it develops a series of equations which can be subsequently solved by reutilizing the data from the solution of one equation in the next one. The convergence was improved through relaxation parameters in the study. The accuracy of SLLM was assured through known methods and convergence analysis. A comparison of the proposed method with the existing literature has also been made and found an excellent agreement. It is worth mentioning that the successive local linearization method was found to be very stable and flexible for resolving the issues of nonlinear magnetic materials processing transport phenomena.


Introduction
Recent betterments in nanotechnology arose through the investigation of the physical characteristics of matter at the nanoscale level. Multiple industrial utilizations of nanofluids established their growing use in heat transfer. The use of nanofluids has been promoted in assorted imperative subfields due to their thermal transport and captivating uses. Like their peculiarly Bio-convection has many utilizations, similar to model oil, microbial enhanced oil recovery (EOR) and gas-bearing sedimentary basins. Due to this, some researchers have analyzed the mechanisms of several bio-convection obstacles providing suspended solid particles. The microbial EOR is a new technological process for gas and oil production and enhancing oil restoration. This mechanism involves the insertion of the preferred microorganisms into the containers and the residual oil left in the reservoir is reduced through in situ amplification when secondary restoration is exhausted. The self-impelled motile microorganisms enhanced the density of the base fluid in a peculiar way to produce a bio-convection kind of stream. Based on the cause of propulsion, the motile microorganisms perhaps categorized into various kinds of microorganisms, including oxytactic or chemotaxis, gyrotactic microorganisms, and negate gravitaxis. Unlike the motile microorganisms, the nanoparticles are not self-propelled, and their movement is through the thermophoresis and Brownian motion, impacting the inward nanofluid. Kuznetsov and Avramenko [45] analyzed bio-convection into a suspension of gyrotactic microorganisms through a layer of finite depth. This conception was extended by Kuznetsov and Geng [46] to numerous bio-convection problems. Lee et al. [47] experimentally interpreted the effects of convention in heated plate-fin. Khan and Makinde [48] examined nanofluid bio-convection caused by gyrotactic-microorganisms and they perceived that the microorganisms amplify the base-fluid density through floating/swimming in a specific manner. Recently, Raees et al. [49] interpreted that bio-convection into nanofluids has made enormous contributions to the Colibri micro-volumes spectrometer and benefitted the stability of nanofluids. Some other studies relating to gyrotactic microorganisms can be viewed here [50,51].
The intention of the current analysis is to examine the impact of an activation energy on magnetized fluid comprising of nanoparticles and motile gyrotactic microorganisms, flowing through a stretchable permeable sheet, by employing a successive local linearization method [52,53] not yet available in the existing literature. The current study scrutinizes the transporting phenomena into a nanofluid consisting of self-impelled motile gyrotactic microorganisms by providing a non-uniform magnetic field and convective cooling processes. The thermophoresis, Brownian-motion, and convectively cooling phenomena are also examined. Numerical results are displayed, and comparability with previous investigations is also provided for the validity of the current results.

Modeling
Let a bi-dimensional incompressible viscous, steady, and magnetized nanofluid flow comprising gyrotactic microorganisms through a stretched porous sheet filling porous space be assumed, as shown in Figure 1. Bio-convection has many utilizations, similar to model oil, microbial enhanced oil recovery (EOR) and gas-bearing sedimentary basins. Due to this, some researchers have analyzed the mechanisms of several bio-convection obstacles providing suspended solid particles. The microbial EOR is a new technological process for gas and oil production and enhancing oil restoration. This mechanism involves the insertion of the preferred microorganisms into the containers and the residual oil left in the reservoir is reduced through in situ amplification when secondary restoration is exhausted. The self-impelled motile microorganisms enhanced the density of the base fluid in a peculiar way to produce a bio-convection kind of stream. Based on the cause of propulsion, the motile microorganisms perhaps categorized into various kinds of microorganisms, including oxytactic or chemotaxis, gyrotactic microorganisms, and negate gravitaxis. Unlike the motile microorganisms, the nanoparticles are not self-propelled, and their movement is through the thermophoresis and Brownian motion, impacting the inward nanofluid. Kuznetsov and Avramenko [45] analyzed bioconvection into a suspension of gyrotactic microorganisms through a layer of finite depth. This conception was extended by Kuznetsov and Geng [46] to numerous bio-convection problems. Lee et al. [47] experimentally interpreted the effects of convention in heated plate-fin. Khan and Makinde [48] examined nanofluid bio-convection caused by gyrotactic-microorganisms and they perceived that the microorganisms amplify the base-fluid density through floating/swimming in a specific manner. Recently, Raees et al. [49] interpreted that bio-convection into nanofluids has made enormous contributions to the Colibri micro-volumes spectrometer and benefitted the stability of nanofluids. Some other studies relating to gyrotactic microorganisms can be viewed here [50,51].
The intention of the current analysis is to examine the impact of an activation energy on magnetized fluid comprising of nanoparticles and motile gyrotactic microorganisms, flowing through a stretchable permeable sheet, by employing a successive local linearization method [52,53] not yet available in the existing literature. The current study scrutinizes the transporting phenomena into a nanofluid consisting of self-impelled motile gyrotactic microorganisms by providing a nonuniform magnetic field and convective cooling processes. The thermophoresis, Brownian-motion, and convectively cooling phenomena are also examined. Numerical results are displayed, and comparability with previous investigations is also provided for the validity of the current results.

Modeling
Let a bi-dimensional incompressible viscous, steady, and magnetized nanofluid flow comprising gyrotactic microorganisms through a stretched porous sheet filling porous space be assumed, as shown in Figure 1.
Their respective boundary conditions can be read as is an Arrhenius function, m is the dimensionless exponent, T is the temperature, C is the concentration for nanoparticle, N is the density for motile microorganism, p the pressure, ρ f , ρ m , ρ p are the densities of nanofluid, microorganisms, D T is thermophoresis-diffusion coefficient, D M is diffusivity of microorganisms, D B is Brownian-diffusion coefficient, k is the thermal conductivity of nanofluid, σ is the electrical conductivity of nanofluid, γ is the average volume for a microorganisms, α = k/ ρc p is the thermal diffusivity, k r is chemical reaction rate, E a activation energy, C F is the Forchheimer coefficient, k 0 is the Boltzman constant, bW C are constants, τ = (ρC) p /(ρC) f is the proportion of the effected nanoparticle heat capacitance of the base-fluid, strength of magnetic field is B(x) = B 0 (x), velocity of stretched sheet is U w = ax, positive constant is a, concentration is C w, temperature of the wall is T w , motile microorganisms' densities are N ∞ and N w , ambient concentration is C ∞ and ambient temperature is T ∞ . The similarity transformation variables are defined as follows Using Equation (9) in Equations (1)-(8), we have Processes 2020, 8, 328 where The significant parameters are defined in the list of Nomenclatures. The shear stress, the local heat, the local mass, and the motile microorganisms' fluxes past the subsurface, imperative parameters, the skin-friction coefficient, the local Sherwood number, the local density number of the motile microorganisms, the local Nusselt number and the local Reynolds number, are respectively defined as

Numerical Solution
The implementation of the SLLM to the present system of differential equations needs to reduce the order of Equation (10). In view of the transformation g = h, Equations (10)-(13) can be written as By using Taylor's series expansion, the non-linear term "h 2 " can be linearized as Here, the subscript "t" stands for the previous approximated value, whereas the subscript "t + 1" stands for the current approximated value. Now, when we placed Equation (22) in Equation (18), then the non-linear system along with the corresponding boundary conditions are first decoupled by employing the Gauss-Seidel relaxation method, and then, in view of Chebyshev spectral collocation, the resulting system interims of differentiation matrix "D = 2 l D" become with their respective boundary conditions The system can be expressed in a more simplified way as where Processes 2020, 8, 328 T are vectors of sizes (N + 1) × 1, while0 is a vector of order (N + 1) × 1 and I is an identity matrix of order (N + 1) × (N + 1).
The implementation of boundary conditions on the system (23)- (27), yields the following The applicable initial guesses approximation are selected as These initial approximation assumptions satisfy the boundary conditions (28) and (29), which subsequently accomplish the approximations of g t , h t , θ t , φ t , Φ t for each t = 1, 2, . . . . . . by employing the SLLM technique.

Convergence of SLLM Technique
A significant effort was executed to obtain the convergent solutions by employing the successive over-relaxation (SOR) method for each result via this iterative scheme. If "Z" is the resolving function, then the SLLM technique at the (t + 1) iteration is Now, by revising this, the new mode of the SLLM technique is indicated as where "ω" is the convergence improving the parametric quantity whereas "B 1 " and "E 1 " represent matrices. This revised SLLM technique improves the accuracy and efficiency of numerical results.

Discussion
This section is dedicated to the numerical results, their validation, and the discussion. To inspect the presence of all the leading parameters numerically, the computational software MATLAB was used for the numerical simulations. Table 1 shows the computed convergent outcomes of Nu x /Re 1/2 x , Sh x /Re 1/2 x and Nn x /Re 1/2 x across the number of collocation points N, N t , and N b by fixing other parameters, whereas Table 2 depicts the comparability of −θ (0), −φ (0) across N t and N b with the preceding investigations by fixing the other parameters of the governing equations. Figures 2-15 have been plotted against all the leading parameters for microorganism distribution, nanoparticle concentration, temperature, and velocity distribution, respectively.   Table 2. Comparison of the current outcomes for −θ (0) and −φ (0) with the previous investigations across N t and N b by taking P r = L e = 10,  body-force brought through the magnetic field, well-known as the Lorentz force, causing a decrement in the velocity overshooting and momentum boundary-layer thickness. In Figure 3, it was reported that the velocity distribution decelerates for both parameters by enhancing the numeric value of these parameters, i.e., the Forchheimer parameter F r and M. In Figure 4, it is recorded that, by taking the increment in N r , the velocity distribution decreases as a result of an increment in the negate buoyancy generated through the existence of nanoparticles, while the Richardson number G r /R 2 e , was boosted by enhancing the values of the Richardson number. Figure 5 portrays that, through an increment in R b , the velocity distribution decreases because the power of convection due to the bio-convection worked against the convection of buoyancy force, whereas the Richardson number G r /R 2 e , was boosted by enlarging the values of the Richardson number.  Figure 15, and the density of motile microorganisms is decreased by enhancing both the parameters, i.e., the Peclet number and the microorganism concentrations' varying parametric quantity .    Figure 15, and the density of motile microorganisms is decreased by enhancing both the parameters, i.e., the Peclet number and the microorganism concentrations' varying parametric quantity .         The impact of the buoyancy proportion parameter N r , Prandtl number P r , Hartmann number M, the Brownian-motion parameter N b , the thermophoresis parameter N t , local Eckert number E c , for various numeric values are shown in Figures 6-9. From Figure 6, can be seen that by taking the increment in N r , the temperature distribution decreases as a result of an increment in the negate buoyancy generated through the existence of nanoparticles, while the Richardson number G r /R 2 e , it is boosted by enhancing the values of the Richardson number. Figure 7 shows that, by taking an increment in Prandtl number P r , the temperature distribution slows, although enhancing the thermophoresis parameter N t accelerates the temperature distribution. Figure 8 shows the effect of thermophoresis parameter N t and the Brownian-motion parameter N b of the temperature distribution, and also noticed that the temperature distribution boosts both parameters by enhancing the numeric value of these parameters. The influence of Eckert number E c and the Brownian-motion parameter N b of temperature distribution is shown in Figure 9, and the temperature distribution is boosted for both parameters by enhancing the numeric value of these parameters. Further heating is due to the interaction of the fluid and nanoparticles because of the Brownian-motion, thermophoresis, and viscous dissipation impact. Therefore, the thickness of the thermal boundary layer becomes higher across the larger numeric of N t , N b and E c and temperature overshoots into the neighborhood of the stretched permeable sheet. The impact of the bio-convection Lewis number L e , the Brownian-motion parameter N b , the thermophoresis parameter N t , the chemical reaction constant σ 1 , relative temperature parameter δ, the parameter for activation energy E, the bio-convection L b , Peclet number P e and the microorganisms' concentration difference parameter Ω d for concentration distribution and the density of motile microorganisms are shown, respectively, through Figures 10-15. Figure 10 shows the effect of bio-convection Lewis number L e and the thermophoresis parameter N t of the concentration distribution and also shows that the concentration distribution decelerates by enhancing the numeric value of Lewis number L e , because the convection of nanoparticles enhances if we add larger values to Lewis number L e and are incremented through increases in thermophoresis parameter N t . Therefore, we suggested that the nanoparticle's boundary layer thickens with N t . From Figure 11, it observed that, by enlarging the Brownian-motion parameter N b and the bio-convection Lewis number L e , the concentration profile slows for both parameters. Figure 12 portrays the influence of the chemical reaction constant σ 1 and the parameter for activation energy E, and shows that φ is decelerated with enlarging values of σ 1 , while it is incremented with larger values of E. Figure 13 depicts the impact of the relative temperature parameter δ and the parameter for activation energy E, and shows that φ earns the largest values for δ = −0.5, −0.1, −1.5, −2.0 and enhances with increments in E. The graphical behavior of various values of the bio-convection L b and Peclet number P e in Figure 14 show that a decrement in the density of motile microorganisms quickly occurs by enhancing the bio-convection L b and Peclet number P e . That is, the density of motile microorganisms decreases strongly, and by enhancing the bio-convection Lewis number L b and Peclet number P e the decrement in microorganisms' diffusion can be calculated, hence the density and boundary layer thickness downturn for motile microorganisms with rising values of L b and P e . The power of the Peclet number P e and the microorganism concentrations' varying parametric quantity Ω d are shown in Figure 15, and the density of motile microorganisms is decreased by enhancing both the parameters, i.e., the Peclet number P e and the microorganism concentrations' varying parametric quantity Ω d .

Conclusions
The notable results of the current investigation are: i. The successive local linearization method is found to be very stable and flexible for resolving nonlinear magnetic materials' processing transport phenomena problems; ii. The numerical efficiency of SLLM is powerful, because it develops in a series of equations which are solved by reutilizing the data from the solution of one equation in the next equation; iii. Due to its accuracy, efficiency, and smoothness, it is visualized that the proposed SLLM technique could be employed as a feasible technique for solving certain classes of boundary layer fluid flow problems; iv. Furthermore, in the present investigation, we have ignored the behavior of non-Newtonian nanofluid models and double-diffusive convection flows, which can be considered in the upcoming articles.