3D mathematical investigation of MHD flow imposed with bioconvective conditions over casson nanoflui

Abstract

Keywords: 3D MHD flow; Bioconvection; Buongiorno’s Model; Casson Fluid; Chemical reaction; Lie symmetry analysis; Nanofluid; Shooting Method

Introduction

Now days, researchers are working in the field of nanofluid because of its physical characteristics and uses in a

icine, and industry. Research greatly benefits from the characteristics of nanofluids, especially their low resistivity

*Corresponding author. *E-mail address: saloni.20.jindal@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

and attractive thermophysical properties. Usman et al. [1] presented the Darcy theory to demonstrate the impact of MHD hybrid nanofluid flow over stretching sheet exponentially. Asghar et al. [2] showed the impact of slip conditions and joule heating over hybrid nanofluid induced by shrinking surface exponentially. Haque et al. [3] presented the magnetic field effect via heat transportation of nanofluid flow numerically. Numerical computation of ternary nanofluid flow induced by energy transportation over permeable sheet has been shown by Bilal et al. [4]. Algehyne et al. [5] discussed the energy transition over bioconvective nanofluid flow numerically. Takreem et al. [6] discussed the radiative ternary nanofluid flow via shape factor analysis. Rafique et al. [7] utilizing OHAM Williamson nanofluid model induced with thermal radiation and bioconvection by porous sheet. Afzal et al. [8] demonstrated the impact of 3D MHD nanofluid flow in presence of heat radiation. Atofarati et al. [9] presented a holistic analysis for heat transfer enhancement in nanofluids. Over permeable stretching surface, a numerical study has been done by Jubair et al. [10] to identify the impact of chemical reaction. A vital area of research, fluid dynamics is essential to comprehending and improving a variety of natural and industrial systems. Complex fluids, like Casson fluids, have attracted a lot of attention among the many fluids that have been examined because of their special qualities and uses. Shear-thinning behaviour and a yield stress are characteristics of Casson fluids, a kind of non-Newtonian fluid. Research has recently focused on how Casson fluids behave when exposed to external stimuli like magnetic fields and porous heat sources, particularly when they are stretched. In order to optimize engineering applications, it is necessary to comprehend the complicated flow characteristics that can result from the interplay of various external elements with Casson fluids. Researchers have focused a great deal of emphasis on the thermo-physical features of non-Newtonian fluid applications in several scientific and engineering domains. These fluids are suitable for use in the pharmaceutical, chemical, and other industries. Nonetheless, there are some difficulties in studying non-Newtonian fluids. When stretched, Casson fluids, which are renowned for their capacity to record shear-thinning behaviour, display special properties. The first was Casson movement for printing ink type pigment oil suspensions [11]. Rao et al. [12] discussed the impact of thermal radiation in addition to chemical reaction induced by stretching surface in presence of 3D MHD nanofluid flow by applying similarity solutions. Ramamoorthy and Pallavarapu [13] presented the impact of Williamson fluid flow induced by heat radiation. Influence of Viscous and Ohmic dissipation influenced by MHD Casson nanofluid flow has been presented by Vinita et al. [14] by following Runge Kutta Fehlberg algorithm. Ahmed M. Megahed [15] found the impact of heat radiation in presence of Williamson fluid. Reeshan et al. [16] discussed the optimization of hybrid nanofluid flow induced by MHD Casson fluid flow utilizing computational

intelligence approach. Numerical modeling of motile microbes induced by porous medium in presence of activation energy was presented by Waseem et al. [17]. Swain et al. [18] discussed the presence of viscous dissipation effect utilizing Homotopy perturbation method induced by stretching surface. Influence of ternary hybrid nanofluid over stretching sheet in presence of MHD flow has been presented by Swain et al. [19]. Sahoo et al. [20] showed the impact of entropy generation induced by inclined plate in presence of MHD hybrid nanofluid flow. The interaction of conducting fluids with an electromagnetic field is known as magnetohydrodynamics. Since some sheet materials lack a heat exchanger, rheostat flow kinematics is a practical application of magnetohydrodynamic in which a magnetic field is employed in traditional fluids. Additional applications include power storage, thermal insulation, and more. Given this, the magnetohydrodynamic (MHD) flow across a stretching sheet offers a wide range of industrial and manufacturing uses, such as continuous metal casting, polymer extrusion, the petroleum industry, and electrical power generators. The magnetohydrodynamic (MHD) effects that arise when Casson fluids are exposed to magnetic fields add complexity to the flow behaviour and have a major impact on fluid dynamics and heat transfer. Varatharaj et al. [21] presented the impact of heat radiation and transport phenomenon over hybrid nanofluid in presence of MHD flow. Adeyemi et al. [22] discussed the influence of thermal radiation and generation over MHD fluid flow. Akinbo et al. [23] implemented the impact of chemical reaction and heat radiation utilizing Walters’ B fluid in presence of stagnation point flow. Elhag et al. [24] demonstrated the influence of bio-convective gyrotactic microorganism induced by 3D sheet. Dang et al. [25] discussed the impact of chemical reaction over MHD nanofluid flow in presence of convective conditions by utilizing RKF technique following shooting algorithm. Sowmiya et al. [26] analysed the impact of heat flux over Casson fluid flow in presence of MHD convective conditions. Irshad et al. [27] presented the numerical study of Casson fluid flow in porous medium induced by stretching surface. Mehmood et al. [28] discussed the conical gap between cone and rotating discover blood flow in presence of Casson nanofluid computationally. Azar et al. [29] analyze the impact of Casson fluid flow in presence of MHD flow over porous medium. One useful method for examining nonlinear partial differential equations is Lie group analysis. Lie and Ackerman’s work [30] shows how this method was developed, while [31] unifies known exact integration techniques that take differential equations into account. The Lie group analysis method was used by numerous researchers to examine a variety of fluid flow problems under various conditions [32–35]. Disu and Salawu [36] presented the impact of temperature distribution influenced by fluid flow over porous medium in presence of magnetohydrodynamic flow by applying Lie symmetry analysis. Nabwey et al. [37]

discussed influence of convecting heating and Navier slip ferrofluid flow in presence of radiation effect by using Lie group analysis. Influence of tangent hyperbolic fluid flow induced by shrinking sheet has been presented by Swain et al. [38]. Impact of 3D MHD nanofluid flow induced by thermal laminated sheet has been presented by Akbar et al. [39]. Tufail et al. [40] using Lie group approach to analyze the impact of slip conditions over Maxwell fluid induced by shrinking sheet. Over straining surface, Shahbaz Ali [41] discussed the influence of slip conditions in presence of Bingham plastic flow utilizing Lie group analysis. Our problem’s primary goal is to determine the transportation behaviour of MHD nanofluids using Lie group similarity solution and RKF technique by following shooting algorithm over 3D permeable stretching surface. Metallurgical processes in the industrial sector include continuous filament drawing via static fluids, copper wire annealing and tinning, the production of plastic and rubber sheets, crystal development, continuous cooling, and fibre spinning, among many more uses. Along with the chemical reaction and gyrotactic microorganisms, the surface in this problem is permeable stretching surface, causing the viscosity to change with temperature and concentration over 3D sheet. On this system, we have additionally applied a magnetic field. Here, the viscosity exhibits a linear relationship with temperature, and the Lie group similarity technique yielded self-similar differential equations. The Lie group of transformations is used to identify all of the problem’s symmetries and then determine which of them can be employed to produce group-invariant, or more precisely, similarity solutions.

Mathematical Model

A constant bio-nanofluid stagnation point flow in three dimensions with Casson fluid flow in presence of chemical reaction, variable transport characteristics, and thermal convective boundary conditions is taken into consideration. A two component Buongiorno’s model has been followed in presence of Brownian motion and thermophoresis parameter. Because of the phenomena known as bio-convection, the imposition of microorganisms into the nanofluid is only necessary to stabilize the tiny nanoparticles. Furthermore, it is thought that the nanoparticles have no effect on the microorganisms’ ability to swim or their velocity. Additionally, it is a reasonable assumption that nanoparticle dilution (suspension) is prepared with very small amounts (i.e., 1%) in order to avoid the instability of bio-convection; otherwise, the higher concentration distribution of nanoparticles leads to the highly effective saturation (suspension) viscosity, which ultimately suppresses the given bio-convective flow. Figure 1 displays the physical model of the problem along with sheet’s velocities u and v in the x and y directions, respectively. These presumptions and the conventional boundary layer approximation result in the following governing equations.

Figure 1. Physical Diagram. To find similarity transformation for the current problem, the following dimensionless variables are introduced

equations (7)–(9) with the B.C’s (10) we use lie group transformations to convert the system into matching ordinary (8)

Here, ci; 1 ≤ i ≤ 9 are arbitrary real integers that are not all zero at once, and ε is the group scaling parameter. When

LIE Group Analysis

PDEs are difficult and computationally costly to solve numerically. The objective is to convert the PDEs in Equations (10)–(16) into ODEs. We accomplish this by converting PDEs to ODEs using the group approach. We apply the scaling group transformation in this work. Since it is difficult to solve the system of partial differentiation

The following characteristic equations are obtained by applying Taylor’s approach to extend in powers of ε while maintaining terms up to the order ε. (23)

(36) On solving the above equations we obtain the similarity transformation,

The following relations has been derived from group of transformation (24)

(31) Above relations give following solutions (32) Set of transformation reduces to (33) Using the Taylor’s series to expand in the power of ε and ignoring the higher power of ε results in

Numerical Solution

System of ODE’s (10)-(14) along with (15) solved numerically due to highly non-linear behavior because it is not an easy task to solve these mathematical equations containing crucial fluid parameters analytically. To solve these equations numerically, RKF-shooting algorithm is adopted by converting these equations into a set of initial value problems as shown in Figure 2. (44) (45) (46)

(56) (57) (58) (59) (60) with the initial conditions where Figure 2. Flow chart of “Shooting Technique”. (61) After being transformed into a system of first-order D.E’s, the coupled differential equations are explained as follows: (49) (50)

Results And Discussion

In present analysis, numerical scheme is validated as calculated results for -Θʹ(0) are compared with Abolbashari et al. [42], Gorla and Sidawi [43], Mankinde and Aziz [44] and Khan and Pop [45] for controlling fluid parameter Pr in the absence of magnetohydrodynamic flow (See Table 1). Additionally, bar graph has been plotted for these values as well to seen accuracy of the numerical technique (See Figure 3). It is observed that results obtained are highly accurate when compared with the existing literature outcomes. Moreover, Table 2-5 represents the resulting outcomes of (f″(0), g″(0)), -Θʹ(0), -Φʹ(0), and -χʹ(0) respectively for fixed values of controlling fluid parameters Pe = 0.1, κ = 0.5, Lb = 0.5, Bi3 = 0.1, Bi1 = 0.4, Bi2 = 0.1, K1 = 0.1, Cr = 0.7, β = 0.1, M = 1.5, Pr = 1.732, Nb = 0.1, Nt = 0.5, Le = 2.0 along with CPU time observed in seconds.

Results

Figure 3. Bar Graph showing comparison of resulting outcomes for -Θʹ(0) against three different values of Prandtl number as 0.7, 2.0 and 7.0 with those of Abolbashari et al. [42], Gorla and Sidawi [43], Mankinde and Aziz [44] and Khan and Pop [45]. [Created by authors]

Velocity Distribution Figure 4(a)-4(f) elucidates the axial fʹ(ξ) and transverse gʹ(ξ) velocity distribution against crucial fluid parameters namely magnetic parameter M, Casson fluid parameter β and permeability parameter K1 in direction of x and y-axis. Figure 4(a) and 4(b) manifests the variation in velocity distribution fʹ(ξ) and gʹ(ξ) against the parameter M in the range 1.5-3.5. These graphs show that

with augmentation in the values of M, velocity profile falls slowly with a small variation due to produced Lorentz drag force. Additionally, it has been noticed that both the magnitude as well as width of boundary layer of fʹ(ξ) and gʹ(ξ) lessens with enhancement in M as shown in Figures 4(a) and 4(b) serially. Also, Figures 4(c) and 4(d) shows the variation in velocity under the impact of Casson fluid parameter for

Table 2. Values of Cfx and Cfy for precised entries of M, β and K1 with CPU time in seconds M

Figure 4. Axial and Transverse velocity distribution against (a, b) Magnetic parameter M (1.2 ≤ M ≤ 3.5), (c, d) Casson fluid parameter β (0.1 ≤ β ≤ 0.9) and (e, f) permeability parameter K1 (0.1 ≤ K1 ≤ 0.5.

β = 0.1, 0.3, 0.5, 0.7, 0.9. . Same behavior of velocity has been noticed i.e. declines for higher β in both the graphs like magnetic parameter and permeability parameter K1. Reason behind declination in fluid velocity is the shear stress which is highly produced in non-Newtonian fluid when compared with Newtonian fluids. As Casson fluid is considered as non-Newtonian thus produces high stress and fluid resistance. Hence, this more non-Newtonian behavior helps in reduction of fluid velocity. Furthermore, the influence of permeability parameter K1 (0.1 ≤ K1 ≤ 0.5) over velocity profile has been illustrated via Figures 4(e) and 4(f) respectively. Both the velocity profiles i.e.

axial and transverse declines when K1 rises due to presence of more nanofluid far from boundary which shows the reduction in boundary layer thickness for both fʹ(ξ) and gʹ(ξ). Temperature Distribution Figure 5(a)-5(d) represent the variation in temperature against controlling fluid parameters especially Brownian motion parameter Nb in the range 0.1-5.0, thermophoresis parameter in the range 0.5-2.5, thermal Biot number in the range 0.1-2.0 and Prandtl number in the range 0.732-20. Figure 5(a) exhibits the nanoparticle

temperature profile Θ(ξ) for Nb. As Brownian motion is the random motion of poised particles which is produced by the collision of pendulous particles in the fluid that simultaneously helps to rise the boundary layer thickness. Consequently, higher values of Nb enhances nanoparticles movement that increases kinetic energy and hence temperature field rises (See Figure 5(a)) and local Nusselt number reduces (See Table 3). Moreover, larger values of Nb thickens the thermal boundary layer. Variation in the profile of temperature against thermophoresis Nt is sketched via Figure 5(b). This figure elaborates that, higher Nt (0.5 ≤ Nt ≤ 2.5) makes movement of nanoparticles from hot to cold zone due to temperature gradient and demonstrate stronger thermophoretic force that consequently mounting temperature profile. Further, Figure 5(c) manifests the impact of Biot number Bi1 over nanoparticle temperature field. With increase in Bi1, a rapid augmentation in fluid temperature is observed near the boundary. In addition, convective heating of 3D permeable sheet rises with rise in thermal Biot number. Moreover, it is observed that all plots are declines distinctly upto ξ = 2.6 in approx manner and convergence meets to ξ → ∞. Variation in temperature distribution Θ(ξ) under influence of Prandtl number Pr is illustrated via Figure 5(d) and the corresponding values of local Nusselt number for Pr at 0.732, 7, 10, 15 and 20 are shown in Table 3. Pr is characterized as the

ratio of two controlling parameters namely; momentum and thermal diffusivity. Nature of Pr is opposite to the Nb and Nt i.e. rate of thermal diffusion declines in case of higher Prandtl number due to viscous diffusion at higher level which is the main reason thinning of thermal boundary layer. Concentration Distribution Concentration profile for Brownian motion Nb varies from 0.1-5.0, thermophoresis Nt varies from0.5-2.5, concentration Biot number Bi2 varies from 1.0-5.0, Prandtl number Pr varies from 0.732-20, Lewis number Le varies from 1.0-20.0 and chemical reaction parameter Cr varies from 0.0-4.0 are depicted graphically via Figures 6(a)-6(f). Figure 6(a) examines the variation in nanofluid concentration field under the impact of Nb for five different values of Nb as 0.1, 1.0, 2.0, 3.0, and 5.0. Concentration profile for Brownian motion Nb varies from 0.1-5.0, thermophoresis Nt varies from 0.5-2.5, concentration Biot number Bi2 varies from 1.0-5.0, Prandtl number Pr varies from0.732-20, Lewis number Le varies from 1.0-20.0 and chemical reaction parameter Cr (varies from 0-4.0 are depicted graphically via Figures 6(a)-6(f). Figure 6(a) examines the variation in nanofluid concentration field under the impact of Nb for five different values of Nb as 0.1, 1.0, 2.0, 3.0, and 5.0. Augmentation in Nb will reduces the nanoparticle concentration and in consequence

Table 3. Values of Nux for precised entries of Nb, Nt, Bi1, and Pr with CPT time (in seconds) Nb

Figure 5. Temperature distribution against (a) Nb (0.1 ≤ Nb ≤ 5.0), (b) Nt (0.5 ≤ Nt ≤ 2.5), (c) Bi1 (0.1 ≤ Bi2 ≤ 2.0) and Pr (0.732 ≤ Pr ≤ 20.0).

solutal boundary layer thickness falls down. Additionally, slightly increase in Sherwood number is observed with enhancement in Nb as seen in Table 4. Also, the influence of Nt over nanoparticle concentration distribution is illustrated through Figure 6(b). This figure elaborates that higher Nt rises the mass distribution or concentration of the fluid. Physically, temperature gradient gives mass field and this Nt parameter makes temperature to be a mounting function of Nt. Thusly, an enhancement in mass and mass boundary layer thickness has been noticed with enhancement in Nt. Figure 6(c) illustrates the impact of concentration Biot number Bi2 (1.0 ≤ Bi2 ≤ 5.0) over nanoparticle concentration field. This figure shows that, concentration of nanofluid rises with rise in Bi2. Because

higher Biot number will enhances convective heating at the surface of three dimensional stretching sheet. As mass field is influences via energy field, one explained that a deeper mass dispersion is observed for higher Bi2 values and can be seen in Figure 6(c). Additionally, all plots of the figure falls distinctly upto ξ = 3 (approximately) and follows convergence criterion. Effect of Pr over nanoparticle concentration has been illustrated via Figure 6(d). As Pr is inversely proportional to thermal diffusivity which indirectly leads to enhance value of Prandtl number with lower diffusivity value and in consequence plays a major role in declination of temperature and concentration of the nanofluid. Hence, higher Pr declines concentration and simultaneously helps to increase local Sherwood number

Figure 6. Concentration distribution against (a) Nb (1.0 ≤ Nb ≤ 5.0), (b) Nt (0.5 ≤ Nt ≤ 2.5), (c) Bi2 (1.0 ≤ Bi2 ≤ 5.0) and Pr (0.732 ≤ Pr ≤ 20.0), (e) Le (1.0 ≤ Le ≤ 20 and (f) Cr (0.0 ≤ Cr ≤ 4.0).

as shown in Table 4. Lewis number is the ratio of α (thermal diffusivity) and DB (Brownian diffusion coefficient) i.e. inversely proportional to DB which in consequence shows that higher Le corresponds to lower DB. Hence, nanoparticle concentration reduces for greater Le (1.0 ≤ Le ≤ 20.0) as seen in Figure 6(e) and respective Sherwood number rises as seen in Table 4. Nanoparticle concentration is mainly influenced by chemical reaction. In present

analysis, chemical reaction of first order is discussed and is displayed via Figure 6(f). This figure elaborates that nanoparticle concentration decays with advancement in Cr and follows boundary conditions away from the surface, which in turns, becomes stable. Density of motile micro-organism distribution against controlling fluid parameters especially Peclet number Pe in the range 0.1-10.0, micro-organism concentration

difference parameter in the range0.1-5.0, bio-convective Lewis number in the range 0.1-0.9, chemical reaction in the range 0-4.0, micro-organism Biot number in the range 1-10 and Prandtl number in the range 0.732-20 is visualized in Figure 7(a)-7(f) serially. Figure 7(a) elucidates the impact of Peclet number Pe on motile density χ(ξ). This plot shows that, with increase in values of Pe (0.1 ≤ Pe ≤ 10.0), fluid particles moves freely leads to thickens micro-organism and hence reduces the motile density of micro-organisms. Moreover, graphs are distinct within 0.0 ≤ ξ ≤5.0 in approx manner and after that they meet and converges as ξ → ∞ by following boundary conditions. variation in gyrotactic micro-organism density χ(ξ) against micro-organism concentration difference parameter κ is sketched via Figure 7(b). It is observed from the figure that χ(ξ) declines for

higher values of κ because of lower micro-organism concentration difference between Nw & N∞ and their respective local wall motile micro-organism number has same nature as that of χ(ξ) i.e. declines with augmentation in κ as seen in Table 5. Figure 7(c) manifests the motile density against bio-convective Lewis number Lb. As bio-convective Lewis number is inversely proportional to the micro-organisms diffusivity DN, which in turn enhances Lb value with lower DN and hence reduces the micro-organism density of nanofluid flow as seen in Figure 7(c). However, local wall motile micro-organism number Mnx mounting as visualized in Table 5. Also, the impact of chemical reaction parameter Cr (0 ≤ Cr ≤ 4) over motile micro-organism distribution is illustrated via Figure

Table 4. Values of Shx for precised entries of b, Nt, Bi2, Pr, Le and Cr with CPU time (in seconds) Nb

7(d). Gyrotactic micro-organism distribution χ(ξ) is also influenced by chemical reaction Cr as that of nanoparticle concentration profile. Thus a decay in motile concentration is observed with enhancement in Cr. Further, Figure 7(e) represents the variation of χ(ξ) under the influence of micro-organism Biot number Bi3 and it shows that motile density rises with rise in Bi3 from 1.010.0. Consequently, intensification in local wall motile micro-organism number is noticed for higher Bi3 (See Table 5). Figure 7(f) portraits the variation in motile density of micro-organism χ(ξ) against Prandtl number. This figure elucidates that χ(ξ) drops down suddenly when Prandtl varies from 0.732-7 and then decreases slowly from 7 ≤ Pr ≤20. Overall reduction in χ(ξ) is observed for higher Prandtl value as seen in Figure 7(f)

and their respective Mnx increases as shown in Table 5. Additionally, graph meets its convergence criterion at ξ = 3 (approximately) as clearly visualized in the figure. Local Skin Friction, Nusselt Number, Sherwood Number & Wall Motile Micro-Organism Figure 8(a) and 8(b) examines the impact of f″(0) and g″(0) against controlling parameters β (0.1 ≤ β ≤ 0.5) and M (1.5 ≤ M ≤ 3.5) . It is observed from the figures that, higher β declines skin friction at the wall monotonically and in addition more magnetism helps to decrease the value of skin friction as well. This is due to the reason that larger magnetism and non-Newtonian fluid reduces the velocity profile which ultimately helps to reduce skin friction coefficient as depicted in both the graphs.

Figure 7. Motile density micro-organism distribution against (a) Pe (0.1 ≤ Pe ≤ 10.0), (b) κ (1.0 ≤ κ ≤ 5.0), (c) Lb (0.1 ≤ Lb ≤ 0.9), (d) Cr (0 ≤ Cr ≤ 4), (e) Bi3 (1 ≤ Bi3 ≤ 10) and (f) Pr (0.732 ≤ Pr ≤ 20.0).

The combined influence of Brownian motion Nb (1 ≤ Nb ≤ 5.0) and thermal Biot number Bi 1 (0.5 ≤ Bi 1 ≤ 0.6 over -Θʹ(0) is elucidates via Figure 8(c). Nb and Bi1 shows reverse impact on -Θʹ(0) i.e. higher enhances -Θʹ(0) due to convective heating whereas higher Nb reduces -Θʹ(0) due to fast collision of nanoparticles. Also, the impact of Prandtl number Pr (1 ≤ Pr ≤5) and concentration Biot number Bi 2 (1 ≤ Bi 2 ≤ 5) on -Φʹ(0) is sketched via Figure 8(d). Both the controlling fluid parameters Pr and Bi 2 enhances value of -Φʹ(0) as

clearly visualized in Figure 8(d). Combined impact of Peclet number Pe (1 ≤ Pe ≤ 5.0) and κ (0.1 ≤ κ ≤ 0.5) over -χʹ(0) is shown by Figure 8(e) and it is observed that -χʹ(0) decreases with increase in κ while rises with rise in Pe. Figure 8(f ) manifests the variation in -χʹ(0) against bio-convection Lewis number Lb (0.1 ≤ Lb ≤ 0.9) and chemical reaction Cr (0 ≤ Cr ≤ 1). Both the parameters enhance wall motile micro-organism number with their higher values.

Figure 8. Graph for (a) f″(0) against M & β, (b) g″(0) against M & β, (c) -Θʹ(0) against Nb & Bi1, (d) -Φʹ(0) against Bi2 & Pr, (e) -χʹ(0) against Pe & κ and (f) -χʹ(0) against Lb & Cr.

Conclusion

This investigation deals with 3D flow consisting of nanoparticles along with gyrotactic microorganisms and bioconvective chemical reaction. Final outcomes of current research are summarized as: • Horizontal velocity fʹ(ξ) as well as transversal velocity gʹ(ξ) declines with inclination in the values of M, β and K1.

Augmentation in temperature profile is noticed for greater thermal Biot number Bi1 whereas higher Prandtl number Pr helps to falls temperature Φ(ξ).

Concentration profile Φ(ξ) declines with higher Cr, Le, Pr, Nb. On the other hand, higher Bi2 and Nt amplifies concentration field.

Higher values of crucial fluid parameter Pe and κ de-escalates the motile concentration of micro-organisms. • Skin friction coefficients f″(0) & g″(0) drop-off under the combined influence of controlling fluid parameters magnetic parameter M and Casson parameter β. Local wall motile micro-organism number accelerates with Lb and lessens with κ. • Motile density Biot number falls down by 41.57% with augmentation in value of gyrotactic microorganism Biot number ranging from 1.0-10.0. Current research can be extended for various non-Newtonian fluids by utilizing porous medium, Stefan blowing impact, entropy generation, viscous dissipation, etc. •

Non-dimensional concentration Non-dimensional temperature Ratio of heat capacities Thermal diffusivity

Subscripts ∞ Ambient condition p Particle f Fluid w Wall Superscript ʹ Prime denotes derivative w.r.t to ξ

Nomenclature

x, y Cw C L Le c β B0 Bi3 Nux qm Nt M u C∞ qw Tw Bi2 DT uw T Nb DB v Shx Pr T∞ Lb Bi1 K1 Mnx Pe

Cartesian coordinates Nanoparticle volume fraction (Kgm-3) Concentration (Kgm-3) Characteristic length Lewis number Stretching parameter Casson fluid parameter Magnetic field intensity (T) Motile microorganism Biot number Nusselt number Mass flux Thermophoresis parameter Magnetic field parameter (Tesla) Horizontal velocity Ambient nanoparticle volume fraction (Kgm-3) Heat flux (W-2m) Temperature at the sheet (K) Concentration Biot number Thermophoresis diffusion coefficient (m2s-1) Stretching velocity Temperature (K) Brownian motion parameter (m2s-1) Brownian diffusion coefficient Vertical velocity (ms-1) Sherwood number Prandtl number Ambient temperature (K) Bioconvective Lewis number Thermal Biot number Permeability parameter Local wall motile microorganism number Peclet number

Greek symbols ν kinematic viscosity (m2s-1) β Casson fluid parameter σ Velocity slip parameter ξ Similarity variable χ Non-dimensional motile concentration Density of base fluid (Kgm-3) ρf

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Statement On The Use Of Artificial Intelligence

Artificial intelligence was not used in the preparation of the article.

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