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Article

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

1
College of Mathematics and Systems Science, Shandong University of Science & Technology, Qingdao 266590, China
2
College of Astronautics, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China
3
Department of Mathematics, University of Education Lahore, Faisalabad Campus, Faisalabad 38000, Pakistan
4
Faculty of Science, Taibah University Al-Madinah Al-Munawarah, Madinah 41411, Saudi Arabia
5
Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
6
Fulbright Fellow Department of Mechanical Engineering, University of California Riverside, Riverside, CA 92521, USA
7
Center for Modeling & Computer Simulation, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran-31261, Saudi Arabia
*
Author to whom correspondence should be addressed.
Processes 2020, 8(3), 328; https://doi.org/10.3390/pr8030328
Submission received: 16 February 2020 / Revised: 2 March 2020 / Accepted: 5 March 2020 / Published: 11 March 2020
(This article belongs to the Special Issue Rheological Study of Nanofluids)

Abstract

:
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.

1. 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 higher thermal conductivity, nanofluids have improved the stability that averts rapid establishing and choking adjacent to heat transfers across the boundaries of the materials. Exploring the stimulants in nanofluid, branches through the heat transfer enlargement into mechanisms comprising space-cooling, hybrid-powered engines, nuclear engineering, micro-manufacturing, microchips in computer processors, air-conditioners/refrigerators, fuel cells, diesel engine oil, and other higher energy equipment. Classical theory of the single-phase fluids is helpful for the nanofluids by observing the thermo-physical features of nanofluids, base fluids and their constituents. It is noteworthy that the thermal conductivity of the nanofluids is enhanced through volume fraction, particle size, temperature, and thermal conductivity. Buongiorno [1] promoted a nanofluid model to elaborate the propagation of thermal energy. Tiwari and Das [2] developed a simpler model through which thermos-physical characteristics were investigated as function of nanoparticle volume fraction. Kuznetsov and Nield [3] used a Buongiorno model to interpret the conduction of thermophoretic diffusion and Brownian motion on nanofluid flow neighboring a heating vertical surface through a pervious media and noted that both thermophoresis and Brownian-motion bring a decrement in heat transfer rate through the plate. As the standard of ultimate production relies on the heat transfer rate, as acknowledged, the nanofluids with a higher rate of thermal conductivity enhance the rate of heat transfer [4,5]. For this purpose, distinct techniques are adopted to raise the thermal conductivity of the fluids by providing the suspension of nano/micro- or large-sized particles into liquids. An inventive approach to enhance the heat transfer rate is performed by utilizing nano-scale particles into the base-fluid by Choi et al. [6]. They recorded that, by adding a tiny extent (<1%) of nanoparticles to regular heat transfer fluids, the thermal conductivity for fluids up to almost 4-times and higher was enhanced. Ellahi et al. [7] discussed the two-phase Newtonian nanofluid flow hybrid with hafnium particles under the effects of slip. Majeed et al. [8] scrutinized the stretched stretching sheet under the combine effects of suction, heat transfer and ferromagnetic viscoelastic fluid flow. Noghrehabadadi et al. [9] explored the flow and heat transfer of nanofluids past a stretched subsurface, supposing of thermal convectively boundary conditions and partial slip. Nilson and Griffiths [10] discussed the electro-osmotic flow with atomistic physics and presented a detailed analysis using density functional theory. Lee et al. [11] presented a comparative study on molecular dynamics via classical density function through a double layer nano-channels with the help of the poisson–boltzmann theory. Some important analyses on the nanofluid flow through various configurations are available in the references [12,13,14,15,16].
The magnetized stagnation flow past a stretched surface has numerous productive usages, such as glass industries, a tragedy core reducing system, and decontamination of crude oil. Theoretically, the boundary layer flow and the flow causes from a stretching plate are fairly imperative. Hiemenz [17] first introduced the stagnation point flow in a two-dimensional channel. Chiam [18] scrutinized the stretching and strain rate of the stagnation point flow of the sheet and observed that boundary layers do not occur close to the sheet. Asma et al. [19] numerically inspected the magnetized nanofluid motion over a rotating disk with activation energy and binary chemical formulation. It is important to mention here that Makinde and Animasaun [20] investigated an admirable work related to magnetized nanofluid flow alongside quartic autocatalysis chemical reaction and bio-convection, and recorded that, for a fixed numeric of magnetic parameters, the local skin friction further develops at a larger thickened parametric value, whereas the rate of local heat transfer decreases at high-temperature parametric values past an uppermost subsurface of a paraboloid of uprising. Scrutinizing the definitive utilities for nanofluid and MHD, few recent investigations can be found [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].
Activation energy is the minimum supply of energy needed to accomplish a chemical reaction. Very few researchers have studied the activation energy along with chemical reaction to date. Sajid et al. [39] scrutinized activation energy with nonlinear thermal radiation on the Maxwell Darcy-Forchheimer nanofluid flow. The impact of activation energy with radiative stagnation point flow on cross nanofluid was determined by Ijaz et al. [40]. Khan et al. [41] presented a theoretical report on tangent hyperbolic nanofluid alongside a fused electrical magnetic field, with Wu’s slip and activation energy aspects. A few other analyses of activation energy are given in [42,43,44].
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.

2. 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.
The leading equations for continuity, momentum, thermal energy, nanoparticle concentration, and microorganisms [54] are
v ˜ y ^ + u ˜ x ^ = 0 ,
u ˜ x ^ u ˜ + u ˜ y ^ v ˜ = p ˜ x ^ + ν f ( 2 u ˜ x ^ 2 + 2 u ˜ y ^ 2 ) + g ¯ β ( 1 C ˜ ) ( T ˜ T ˜ ) g ¯ ( ρ p ρ f ) ( C ˜ C ˜ ) g ¯ γ ( ρ m ρ f ) ( N ˜ N ˜ ) σ B 0 2 u ˜ ν f k u ˜ C F ρ f k u ˜ 2 ,
p ˜ y ^ = 0 ,
u ˜ T ˜ x ^ + v ˜ T ˜ y ^ = α ¯ [ 2 T ˜ x ^ 2 + 2 T ˜ y ^ 2 ] + τ ˜ [ D B C ˜ y ^ T ˜ y ^ + D T T { ( T ˜ x ^ ) 2 + ( T ˜ y ^ ) 2 } ] + μ f α ¯ k ¯ ( u ˜ y ^ ) 2 + σ α ¯ B 0 2 k ¯ u ˜ 2 ,
u ˜ C ˜ x ^ + v ˜ C ˜ y ^ = D B [ 2 C ˜ x ^ 2 + 2 C ˜ y ^ 2 ] + D T T [ 2 T ˜ x ^ 2 + 2 T ˜ y ^ 2 ] k r 2 ( C ˜ C ˜ ) ( T ˜ T ) m e ( E a k 0 T ˜ ) ,
u ˜ N ˜ x ^ + v ˜ N ˜ y ^ + b W C ( C ˜ C ˜ ) [ y ^ ( N ˜ C ˜ y ^ ) + x ^ ( N ˜ C ˜ x ^ ) ] = D M ( 2 N ˜ x ^ 2 + 2 N ˜ y ^ 2 + 2 2 N ˜ x ^ y ^ ) ,
Their respective boundary conditions can be read as
u ˜ = a x ^ , v ˜ = 0 , T ˜ = T ˜ w , C ˜ = C ˜ w , N ˜ = N ˜ w at y ^ = 0 ,
u ˜ 0 , N ˜ N ˜ , v ˜ 0 , T ˜ T ˜ , C ˜ C ˜ , as y ^ .
In Equations (1)–(8), u ˜ and v ˜ are the velocity components, ( T ˜ T ) m e ( E a k 0 T ˜ ) 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 ¯ / ( ρ c 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, b W C are constants, τ ˜ = ( ρ C ) p / ( ρ C ) 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 = a x ^ , 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
u ˜ = a x ^ g ( η ) , v ˜ = a ν g ( η ) , η = a ν y ^ , θ ( η ) = T ˜ T ˜ T ˜ w T ˜ , ϕ ( η ) = C ˜ C ˜ C ˜ w C ˜ , Φ ( η ) = N ˜ N ˜ N ˜ w N ˜ . }
Using Equation (9) in Equations (1)–(8), we have
g + g g g 2 M g β D g F r g 2 + G r R e 2 ( θ N r ϕ R b Φ ) = 0 ,
1 P r θ + θ [ g + N b ϕ ] + N t θ 2 + E c { g 2 + M g 2 } = 0 ,
ϕ + L e ϕ g + N t N b θ L e σ 1 ( 1 + δ θ ) m e ( E 1 + δ θ ) ϕ = 0 ,
Φ + L b g Φ P e ( [ Φ + Ω d ] ϕ + ϕ Φ ) = 0 .
g ( 0 ) = 0 , g ( 0 ) = 1 , θ ( 0 ) = ϕ ( 0 ) = Φ ( 0 ) = 1 ,
g ( ) = 0 , θ ( ) = ϕ ( ) = Φ ( ) = 0 ,
where
β D = ν a ρ f k ,   F r = C F x k ,   M = σ B 0 2 a ρ f ,   G r R e 2 = g ¯ β ( 1 C ˜ ) ( T ˜ T ˜ ) a U ˜ w ,   N r = ( ρ p ρ f ) ( C ˜ w C ˜ ) β ρ f ( T ˜ w T ˜ ) ( 1 C ˜ ) , P r = ν α ¯ , R b = γ ( ρ m ρ f ) ( N ˜ w N ˜ ) β ρ f ( T ˜ w T ˜ ) ( 1 C ˜ ) ,   N t = τ ˜ D T ( T ˜ w T ˜ ) ν T ˜ ,   N b = τ ˜ D B ( C ˜ w C ˜ ) ν , E c = U ˜ w 2 c p ( T ˜ w T ˜ ) ,   L e =   ν D B ,   σ 1 = k r 2 a ,   δ = ( T ˜ w T ˜ ) T ˜ ,   E = E a k 0 T ˜ L b =   ν D M , Ω d = N ˜ ( N ˜ w N ˜ ) ,   P e = b W C D M .
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
τ w = μ ( u ˜ y ^ ) y ^ = 0 , q w = κ ( T ˜ y ^ ) y ^ = 0 , q m = D B ( C ˜ y ^ ) y ^ = 0 , q n = D M ( N ˜ y ^ ) y ^ = 0 C g R e x 1 / 2 = g ( 0 ) , N u x R e x 1 / 2 = θ ( 0 ) , S h x R e x 1 / 2 = ϕ ( 0 ) , N n x R e x 1 / 2 = Φ ( 0 ) , R e x = U 0 x ν .

3. 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
h + g h h 2 F r h 2 M h β D h + G r R e 2 ( θ N r ϕ R b Φ ) = 0 ,
1 P r θ + θ [ g + N b ϕ ] + N t θ 2 + E c { h 2 + M h 2 } = 0 ,
ϕ + L e ϕ g + N t N b θ L e σ 1 ( 1 + δ θ ) m e ( E 1 + δ θ ) ϕ = 0 ,
Φ + L b g Φ P e { [ Φ + Ω d ] ϕ + ϕ Φ } = 0 .
By using Taylor’s series expansion, the non-linear term “ h 2 ” can be linearized as
h 2 t + 1 = h 2 t + 2 h t [ h t + 1 h t ] = 2 h t h t + 1 h 2 t .
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
D g t + 1 = h t ,
{ D 2 + d i a g [ d 11 ] D d i a g [ d 12 ] I d 13 I } H t + 1 = d 1 , t ,
{ 1 P r D 2 + d i a g [ d 11 ] D + N b d i a g [ ϕ t ] D + N t D 2 } θ t + 1 = d 2 , t ,
{ D 2 + d i a g [ d 32 ] D + N t N b d i a g [ θ t + 1 ] L e σ 1 d i a g [ d 33 ] e ( E 1 + δ θ ) I } ϕ t + 1 = d 3 , t ,
{ D 2 + d i a g [ d 42 ] D P e Ω d d i a g [ ϕ t + 1 ] I P e d i a g [ ϕ t + 1 ] I d i a g [ ϕ t + 1 ] D } Φ t + 1 = d 4 , t ,
with their respective boundary conditions
g t + 1 ( η N ) = 0 , h t + 1 ( η N ) = 1 = θ t + 1 ( η N ) = ϕ t + 1 ( η N ) = Φ t + 1 ( η N ) ,
h t + 1 ( η 0 ) = 0 = θ t + 1 ( η 0 ) = ϕ t + 1 ( η 0 ) = Φ t + 1 ( η 0 ) .
The system can be expressed in a more simplified way as
B 1 g t + 1 = E 1 ,
B 2 h t + 1 = E 2 ,
B 3 θ t + 1 = E 3 ,
B 4 ϕ t + 1 = E 4 ,
B 5 ϕ t + 1 = E 5 ,
where
B 1 = D , E 1 = h t ,
B 2 = D 2 + d i a g [ d 11 ] D d i a g [ d 12 ] I d 13 I , E 2 = d 1 , t ,
B 3 = 1 P r D 2 + d i a g [ d 11 ] D + N b d i a g [ ϕ t ] D + N t D 2 , E 3 = d 2 , t ,
B 4 = D 2 + d i a g [ d 32 ] D + N t N b d i a g [ θ t + 1 ] L e σ 1 d i a g [ d 33 ] e ( E 1 + δ θ ) I , E 4 = d 3 , t ,
B 5 = D 2 + d i a g [ d 42 ] D P e Ω d d i a g [ ϕ t + 1 ] I P e d i a g [ ϕ t + 1 ] I d i a g [ ϕ t + 1 ] D , E 5 = d 4 , t ,
d i a g [ d 11 ] = [ d 11 ( η 0 ) d 11 ( η N ) ] , d i a g [ d 12 ] = [ d 12 ( η 0 ) d 12 ( η N ) ] ,
d i a g [ d 1 , t ] = [ d 1 , t ( η 0 ) d 1 , t ( η N ) ] , d i a g [ d 2 , t ] = [ d 2 , t ( η 0 ) d 2 , t ( η N ) ] ,
d i a g [ d 32 ] = [ d 32 ( η 0 ) d 32 ( η N ) ] , d i a g [ d 33 ] = [ d 33 ( η 0 ) d 33 ( η N ) ] ,
d i a g [ d 42 ] = [ d 42 ( η 0 ) d 42 ( η N ) ] , d 3 , t = d 4 , t = 0 = [ 0 0 ] .
g t + 1 = [ g ( η 0 ) , g ( η 1 ) , . . . , g ( η N ) ] T , h t + 1 = [ h ( η 0 ) , h ( η 1 ) , . . . , h ( η N ) ] T ,
θ t + 1 = [ θ ( η 0 ) , θ ( η 1 ) , . . . , θ ( η N ) ] T , ϕ t + 1 = [ ϕ ( η 0 ) , ϕ ( η 1 ) , . . . , ϕ ( η N ) ] T ,
Φ t + 1 = [ Φ ( η 0 ) , Φ ( η 1 ) , . . . , Φ ( η N ) ] T are vectors of sizes ( N + 1 ) × 1 , while 0 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
B 1 = [ B 1 0 1 ¯ ] ,   g t + 1 = [ g t + 1 ( η 0 ) g t + 1 ( η 1 ) g t + 1 ( η N ) ¯ ] , E 1 = [ E 1 0 ¯ ] ,   B 2 =   [ 1 0 _ B 2 0 1 ¯ ] , h t + 1 = [ h t + 1 ( η 0 ) _ h t + 1 ( η 1 ) h t + 1 ( η N ) ¯ ] , E 2 = [ 0 _ E 2 1 ¯ ] , B 3 =   [ 1 0 _ B 3 0 1 ¯ ] ,   θ t + 1 = [ θ t + 1 ( η 0 ) _ θ t + 1 ( η 1 ) θ t + 1 ( η N ) ¯ ] ,   E 3 = [ 0 _ E 3 1 ¯ ] ,
[ 1 0 _ B 4 0 1 ¯ ] , ϕ t + 1 = [ ϕ t + 1 ( η 0 ) _ ϕ t + 1 ( η 1 ) ϕ t + 1 ( η N ) ¯ ] ,   E 4 =   [ 0 _ E 4 1 ¯ ] ,   [ 1 0 _ B 5 0 1 ¯ ] , Φ t + 1 = [ Φ t + 1 ( η 0 ) _ Φ t + 1 ( η 1 ) Φ t + 1 ( η N ) ¯ ] , E 5 =   [ 0 _ E 5 1 ¯ ] .
The applicable initial guesses approximation are selected as
g 0 ( η ) = ( 1 e η ) , h 0 ( η ) = e η , θ 0 ( η ) = ϕ 0 ( η ) = Φ 0 ( η ) = e η .
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.

4. 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
B 1 Z t + 1 = E 1 ,
Now, by revising this, the new mode of the SLLM technique is indicated as
B 1 Z t + 1 = ( 1 ω ) B 1 Z t + ω E 1 ,
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.

5. 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 N u x / R e x 1 / 2 , S h x / R e x 1 / 2 and N n x / R e x 1 / 2 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. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 have been plotted against all the leading parameters for microorganism distribution, nanoparticle concentration, temperature, and velocity distribution, respectively.
Figure 2 depicts that the velocity distribution decelerates by enhancing the permeability parameter β D , and also can be seen as a deceleration in momentum by taking increments of M , due to the existing 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 e 2 , 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 e 2 , was boosted by enlarging the values of the Richardson number.
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 Figure 6, Figure 7, Figure 8 and Figure 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 e 2 , 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 Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 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 .

6. 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.

Author Contributions

Conceptualization, R.E.; Investigation, M.M.B.; Methodology, A.S.; Validation, T.A.; Writing—review & editing, S.Z.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia (ORCP2500).

Acknowledgments

R. Ellahi gratefully thanks to King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia to honor him with the Chair Professor at KFUPM.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclatures

u ˜ , v ˜ Components of velocity
k ¯ Thermal conductivity
a , b Constants
g Dimensionless stream function
g ¯ Gravity
C g x ^ Skin friction coefficient
S h x ^ Sherwood number
N n x ^ Local-density number of motile microorganisms
N u x ^ Nusselt number
[ c p ] f Heat capacity
B 0 Magnetic field
C F Forchheimer coefficient
q w Local heat flux past the surface
T ˜ w Temperature of the wall
k 0 Boltzmann constant
T ˜ Ambient temperature
k Porosity parameter
p ˜ Pressure
( c p ) s Heat capacity of solid fraction
U ˜ w Stretching sheet velocity
x ^ , y ^ Cartesian coordinates along the surface
C ˜ The concentration for nanoparticle
P r Prandtl number
N ˜ Density for motile microorganism
D B Brownian-diffusion coefficient
D M Diffusivity of microorganisms
[ c p ] p Nanoparticles heat capacity
Re x Local Reynolds number
D T Thermophoresis diffusion coefficient
W C Heat capacitance of the nanoparticle
M Hartmann number
G r / R e 2 The local Richardson number
N r Buoyancy proportion parameter
N b Brownian motion parameter
m Dimensionless exponent
R b Bioconvection Rayleigh number
N t Thermophoresis parameter
E c Eckert number
E a Activation energy
L b Bioconvection Lewis number
k r Chemical reaction rate
L e Lewis number
P e Bio-convection Peclet number
q m Local mass flux past the surface

Greek Symbols

κ n f Thermal conductivity of nanofluid
Φ Motile microorganism profile
ϕ Concentration profile
μ n f Dynamic viscosity
θ Temperature profile
ν n f Kinematic viscosity of nanofluid
κ n f Thermal conductivity of nanofluid
ρ f Density of the fluid
σ Electrical conductivity
τ w Shear stress
ρ m , ρ p Densities of microorganisms and nanoparticles
Ω d Microorganisms concentration variance parameter
γ Average volume for a microorganism
α ¯ Thermal diffusivity
β D Permeability parameter

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Figure 1. Geometry for the flow problem.
Figure 1. Geometry for the flow problem.
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Figure 2. Variation in β D and M on velocity distribution. Solid line: M = 0 , Dotted line: M = 2.5 .
Figure 2. Variation in β D and M on velocity distribution. Solid line: M = 0 , Dotted line: M = 2.5 .
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Figure 3. Variation in F r and M on velocity distribution. Solid line: M = 0.5 , Dotted line: M = 2.5 .
Figure 3. Variation in F r and M on velocity distribution. Solid line: M = 0.5 , Dotted line: M = 2.5 .
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Figure 4. Variation in N r and G r R e 2 on velocity distribution. Solid line: G r R e 2 = 4 , Dotted line: G r R e 2 = 8 .
Figure 4. Variation in N r and G r R e 2 on velocity distribution. Solid line: G r R e 2 = 4 , Dotted line: G r R e 2 = 8 .
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Figure 5. Variation in R b and G r R e 2 on velocity distribution. Solid line: G r R e 2 = 4 Dotted line: G r R e 2 = 8 .
Figure 5. Variation in R b and G r R e 2 on velocity distribution. Solid line: G r R e 2 = 4 Dotted line: G r R e 2 = 8 .
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Figure 6. Effect of N r and G r R e 2 on temperature distribution. Solid line: G r R e 2 = 7.5 Dotted line: G r R e 2 = 14.5 .
Figure 6. Effect of N r and G r R e 2 on temperature distribution. Solid line: G r R e 2 = 7.5 Dotted line: G r R e 2 = 14.5 .
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Figure 7. Influence of the thermophoresis parametric quantity N t and Prandtl number “ P r ” on temperature distribution. Solid line: P r = 7 , Dotted line: P r = 13 .
Figure 7. Influence of the thermophoresis parametric quantity N t and Prandtl number “ P r ” on temperature distribution. Solid line: P r = 7 , Dotted line: P r = 13 .
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Figure 8. Influence of the thermophoresis parametric quantity N t and the Brownian-motion parameter “ N b ” on temperature distribution. Solid line: N b = 0.3 , Dotted line: N b = 0.6 .
Figure 8. Influence of the thermophoresis parametric quantity N t and the Brownian-motion parameter “ N b ” on temperature distribution. Solid line: N b = 0.3 , Dotted line: N b = 0.6 .
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Figure 9. Effect of the Eckert number E c and the Brownian motion parameter “ N b ” on temperature distribution. Solid line: N b = 0.3 , Dotted line: N b = 0.6 .
Figure 9. Effect of the Eckert number E c and the Brownian motion parameter “ N b ” on temperature distribution. Solid line: N b = 0.3 , Dotted line: N b = 0.6 .
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Figure 10. Influence of the bio-convection Lewis number L e and the thermophoresis parameter “ N t ” on concentration distribution. Solid line: N t = 4 , Dotted line: N t = 8 .
Figure 10. Influence of the bio-convection Lewis number L e and the thermophoresis parameter “ N t ” on concentration distribution. Solid line: N t = 4 , Dotted line: N t = 8 .
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Figure 11. Influence of the Brownian-motion parameter N b and the bio-convection Lewis number “ L e ” on concentration distribution. Solid line: L e = 1.5 , Dotted line: L e = 2.5 .
Figure 11. Influence of the Brownian-motion parameter N b and the bio-convection Lewis number “ L e ” on concentration distribution. Solid line: L e = 1.5 , Dotted line: L e = 2.5 .
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Figure 12. Variation in σ 1 and E on concentration distribution. Solid line: E = 0.5 , Dotted line: E = 1.0 .
Figure 12. Variation in σ 1 and E on concentration distribution. Solid line: E = 0.5 , Dotted line: E = 1.0 .
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Figure 13. Variation in δ and E on concentration distribution. Solid line: E = 0.5 , Dotted line: E = 2.0 .
Figure 13. Variation in δ and E on concentration distribution. Solid line: E = 0.5 , Dotted line: E = 2.0 .
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Figure 14. Effect of the bio-convection Lewis number L b and Peclet number P e on the density of motile microorganisms. Solid line: P e = 2.5 , Dotted line: P e = 4.5 .
Figure 14. Effect of the bio-convection Lewis number L b and Peclet number P e on the density of motile microorganisms. Solid line: P e = 2.5 , Dotted line: P e = 4.5 .
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Figure 15. Effect of the Peclet number P e and the microorganisms concentration variance parametric quantity Ω d on the density of motile microorganisms. Solid line: Ω d = 0.25 , Dotted line: Ω d = 0.5 .
Figure 15. Effect of the Peclet number P e and the microorganisms concentration variance parametric quantity Ω d on the density of motile microorganisms. Solid line: Ω d = 0.25 , Dotted line: Ω d = 0.5 .
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Table 1. Numerical convergent values of Nusselt number, Sherwood Number, and the local density number of the motile microorganisms across N , N t and N b by fixing M = 1 , β D = F r = E c = σ 1 = δ = E = 0 , N r = 0.5 , R b = 0.5 , G r R e 2 = 0.5 , P r = 10 , L e = 10 , L b = 2 , P e = 0.5 , Ω d = 1.0 .
Table 1. Numerical convergent values of Nusselt number, Sherwood Number, and the local density number of the motile microorganisms across N , N t and N b by fixing M = 1 , β D = F r = E c = σ 1 = δ = E = 0 , N r = 0.5 , R b = 0.5 , G r R e 2 = 0.5 , P r = 10 , L e = 10 , L b = 2 , P e = 0.5 , Ω d = 1.0 .
N N t N b N u x R e x 1 / 2 S h x R e x 1 / 2 N n x R e x 1 / 2
500.30.11.75736.66468.4301
600.30.11.75846.66588.4311
700.30.11.75876.66628.4314
800.30.11.75876.66628.4314
1000.30.11.75876.66628.4314
500.50.51.14307.71759.3273
600.50.51.14467.71899.3290
700.50.51.14527.71989.3294
800.50.51.14527.71989.3294
1000.50.51.14527.71989.3294
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 , M = β D = F r = G r R e 2 = N r = R b = 0 , E c = σ 1 = δ = E = 0 .
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 , M = β D = F r = G r R e 2 = N r = R b = 0 , E c = σ 1 = δ = E = 0 .
N b N t Current Outcomes for θ ( 0 ) Khan and Pop [55] Current Outcomes for ϕ ( 0 ) Khan and Pop [55]
0.10.10.9525230.95242.1294132.1294
0.20.6934110.69322.2741162.2740
0.30.5201340.52012.5286222.5286
0.30.10.2521770.25222.4101242.4100
0.20.1816370.18162.5150652.5150
0.30.1356100.13552.6089242.6088

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Bhatti, M.M.; Shahid, A.; Abbas, T.; Alamri, S.Z.; Ellahi, R. Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate. Processes 2020, 8, 328. https://0-doi-org.brum.beds.ac.uk/10.3390/pr8030328

AMA Style

Bhatti MM, Shahid A, Abbas T, Alamri SZ, Ellahi R. Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate. Processes. 2020; 8(3):328. https://0-doi-org.brum.beds.ac.uk/10.3390/pr8030328

Chicago/Turabian Style

Bhatti, Muhammad Mubashir, Anwar Shahid, Tehseen Abbas, Sultan Z Alamri, and Rahmat Ellahi. 2020. "Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate" Processes 8, no. 3: 328. https://0-doi-org.brum.beds.ac.uk/10.3390/pr8030328

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