Activation energy impact on unsteady Bio-convection nanomaterial flow over porous surface

1.   Introduction

Nanofluid is a revolutionary heat transfer medium that consists of nanoscale particles distributed in base liquids in a homogenous and stable manner. Nanofluid is the diluted expansion of nanoparticles in liquid. The thermal (conductivity, performance) of conventional materials is greatly improved by these distributed nanoscale particles composed of metal oxide or metal. Recent developments in nanofluid theory demonstrate unequivocally that nanoscale liquids efficiently improve the thermal properties of traditional heat carrier materials. Nanofluids are utilised for liquid cooling computer processors' excellent thermal conductivity. The stability and dispersion of nanoparticles in the system determine the efficiency of nanofluid. The thermophysical characteristics that predict heat transfer behaviour are critical for energy-saving and industrial applications. Nanofluid is also used to clean up pollutants for medicinal purposes, cooling vehicle engines, and cooling effective heat equipment. Medical engineering uses of nanofluids are getting a lot of interest. Some research topics include pharmacodynamics ^[1] and biofuel cells with plant inspiration ^[2]. In particular, applications for bio-inspired proton exchange membrane (PEM) fuel cells are growing ^[3]. Various studies have been conducted on nanofluids from different perspectives. In 1995, Choi and Eastman ^[4] gave the concept of nanofluids. Low thermal conductivity is a stumbling block in the progression of energy-efficient heat transport fluids that are needed in many industrial applications. Choi used nanofluid to improve material thermal conductivity. Wen and Ding ^[5] formulated the nanofluid by soaking titanium dioxide nanoparticles in filtered water, which is highly stable, and investigating their heat transfer actions under natural convection conditions. Buongiorno ^[6] proposed two essential features to increase heat transfer in nanomaterials: Brownian motion and thermophoretic diffusion. Murshed et al. ^[7] made a composite experimental and theoretical analysis of nanofluid's sensitive thermal conductivity and viscosity. They observed that nanofluid's thickness and thermal conductivity increase the thickness of the nanoparticle volume fraction. Ganguly et al. ^[8] used aluminium oxide nanoparticles in their research and found that electrical conductivity enhanced with increased volume fraction and temperature. Kuznetsov and Nield ^[9] utilised the Buongiorno model to investigate the convective flow along a vertical plate. Mustafa et al. ^[10] discussed the nanofluid's flow approaching a stagnation point towards a surface that has been stretched. Natural convective heat transfer of Ethylene Glycol nanofluids across a thin platinum wire was scrutinized by Asadzadeh et al. ^[11]. They discovered that the nanomaterials added to ethylene glycol boosted heat transmission up to 0.02 percent volume fraction. Hayat et al. ^[12] analysed thixotropic nanofluid in the direction of an inflexible stretching surface. Farooq et al. ^[13] intrigued viscoelastic nanofluid's over a stretched surface having non-linear radiative impacts. Bhatti et al. ^[14] worked on Jeffrey nanofluid. They explored the effect of clotting and a changing magnetic field on non-Newtonian fluid peristaltically generated motion, considering gyrotactic microbes across an annulus. Khan et al. ^[15] discussed the consequence of non-linear radiative MH flow of a Cross nanofluid in the flow towards a stretched surface. Nadeem et al. ^[16] investigated the Hybrid nanofluid's via a circular cylinder while taking thermal slip into account. Bhatti et al. ^[17] introduced the hybrid nanofluid flow. Bhatti et al. ^[18] disclosed the stagnation point nanofluid flow and magnetohydrodynamic effect over the stretchable sheeet. Rashidi et al. ^[19] analysed the statistical analysis of thermal conductivity nanofluid flow. Many researchers are intrested in the investigation about MHD nanofluid flow with heat tranfar and viscous desipation effect ^{[20,21,22,23,24]} against stretchable surface. Upreti et al. ^[25] explored the Sisko nanofluid flow with viscous dissipation behavior on stretchable surfaces. Joshi et al. ^[26] deliberated the impact of thermal radiative MHD flow and suction/blowing velocity on the hybrid-based nanofluid on a stretching sheet. Upreti et al. ^[27] scrutinized 2-dimensional nanofluid flow with single walled and multi walled carbon nanotubes versus a horizontal porous plate. Joshi et al. ^[28] examined the magnetically hybrid-based 3D nanofluid flow under the mixed convection phenomena against a porous bidirectional stretching sheet due to the physical properties of a higher‐order binary chemical reaction, interior heat generation. Rawat et al. ^[29] observed the heated stratified Cattaneo-Christov heat flux nanofluidic flow through cone. Rawat et al. ^[30] explored a hybrid-type MHD nanofluid with boundary layer flow versus a horizontal vertically plate has been established.

The 2nd rule of thermodynamics is often utilized to measure irreversibility (entropy generation). Entropy generation identifies the extent of irreversibility that occurs during any heating process. In a thermodynamic system, the quantity of entropy produced during irreversible processes is referred to as entropy generation. The amount of appropriate work destroyed during the process is directly proportional to the irreversibility of the process. The entropy generation connects fluid mechanics, the fundamental thermodynamic concepts, and heat transfer. These entropy-generating sources are applied to the enhancement and design of actual structures and methods constrained by (finite, temporal) limitations, heat/mass transfer, and irreversibilities in fluid flow. The primary aim of researchers in most engineering and industrial implimentations is to decline entropy formation to improve efficiency and effectiveness. Firstly, Bejan ^[31] explored entropy generation in a convective heat transport mechanism. Bejan ^[32] demonstrates that the flow geometry factors directly correlate with the degree of irreversibility of a convective heat transfer mechanism. Bejan discovered that two distinct characteristics, namely liquid friction and heat transfer owing to a temperature gradient, are responsible for entropy generation in the process of fluid flow. Also, the uses of a number that minimises irreversibility on a multidimensional level are outlined. However, the entropy creation rate must be reduced for thermodynamic functioning to be feasible. Bejan ^[33] also discussed the minimisation of entropy generation for modelling and optimising real-world systems utilising integrated heat transport and thermodynamics. Tasnim et al. ^[34] investigated entropy production inside the vertical slit formed by two parallel plates immersed in a porous material and subjected to the hydromagnetic effect. Mahmud and Fraser ^[35] worked on convective heat transfer problems. Ellahi et al. ^[36] explored various convection boundary layer flow across an inverted cone by evaluating entropy generation and nanofluid's shape impacts nanosize particles. The effect of heat radiation on entropy production in two-dimensional unsteady MHD nanofluid flow through a porous material was investigated by Shit et al. ^[37]. Sheikholeslami et al. ^[38] examined turbulent heat transfer of homogenous nanofluid considering entropy generation by inserting double twisted tapes and utilising nanomaterial. Specific recent research on the influence of entropy production is described in the references ^{[39,40,41,42,43,44,45,46,47,48,49]} and some studies within.

Bioconvection is the process utilised to explain the hazy pattern of uncertainty generated by low-density microorganisms swimming in the upper section of the fluid. By incorporating microorganisms into nanofluid, the suspension's stability increases. Bioconvection is a natural phenomenon due to the random movement of microorganisms in a single-celled or cell-form colony. Upswimming microorganisms are of two categories that are ordinarily employed in bioconvection experiments-heavy alga and certain oxytactic bacteria. These bacteria are known as gyrotactic microorganisms because they have a higher density than water and seem to cluster on the top section of the fluid. In 1996, by suspending microscopic swimming microbes, Vincent and Hill ^[50] investigated the mechanism of bioconvection. Kuznetsov ^[51] suggested the evolution of micro-organisms in micro-microsystems with a significant role in mass transfer production and stockpiling. Alqarni et al. ^[52] analyzed the impact of bioconvection nanofluid on magnetized two dimensional flows via lubricated surface with the presence of melting phenomenon and gyrotactic motile micro-organisms. Alluguvelli et al. ^[53] discussed the bioconvective nanofluid through the enclosure with the effect of viscous dissipation. Researchers ^{[54,55,56,57]} are giving much attention to nanoparticle bioconversion phenomenons.

The latest investigation adds to the work of Khan et al. ^[58], who scrutinized entropy generation in mixed convection nanomaterial flow. We added bioconvection and activation energy into the nanomaterial. We then used a simple and efficient computational method shooting technique in MATLAB software to solve the modelled nonlinear ordinary differential equations. The computed results for the physical quantities of interest are presented in graphical and tabular formats.

2.   Formulation of mathematical model and flow analysis

Assume the effect of entropy generation on bioconvective nanofluid flow across a stretching porous surface in the presence of motile microorganisms. The nonlinear thermal radiation, temperature base heat sink and exponential space-based heat source are considered. The velocity slip is also taken into account. A Buongiorno model for nanofluid, which tackles Brownian and thermophoretic diffusions effect, is taken into account. The famous Darcy relation for developing porous media properties is used. The mathematical model's flow pattern is detailed in Figure 1.

The governing equations for the flow mentioned above problem are ^[58]

Here, velocity components in $(x, y)$ direction are symbolized by $(\bar u, \bar v)$. $\rho$ and $p$ denote density and pressure of the fluid. $(\mu, \nu)$ denote (absolute, kinematic) viscosity. $\sigma$ denote the respectively. ${B_0}$ is the magnetic field's strength, $L$ denotes reference length and $\lambda$ is degree of unsteadiness. ${c_p}$ dessignates the specific heat, $\bar \tau = \frac{{{{(\rho c)}_p}}}{{{{(\rho c)}_f}}}$ characterizes the ratio of the nanoparticle's heat capacity to the base fluid's heat capacity. ${Q_0}$ denotes the coefficient of heat source/sink, ${\alpha _m}$ is thermal diffusivity, $k_r^2$ be a chemical reaction rate, ${E_a}$ denotes the activation energy, ${W_c}$ is the cell swimming speed. $(\bar T, \bar C, \bar N)$ denote the temperature, concentration, and microorganisms concentration respectively. $({\bar T_\infty }, {\bar C_\infty }, {\bar N_\infty })$ denote the ambient (temperature, concentration, microorganisms concentration). ${\bar C_w}$ is the concentration of the surface. The coefficient of Brownian diffusion, thermophoresis, and molecular diffusion are symbolized by $({\bar D_B}, {\bar D_T}, {\bar D_m})$ respectively.

According to the approximation of Rosseland, the radiative heat flux is:

in which ${\sigma ^*}$ symbolises Stefan-Boltzman constant and ${k^*}$ mean absorption coefficient.

The current flow problem's boundary conditions are:

where $({\bar u_w}, {\bar u_0})$ are the velocities of (surface, reference) and ${h_f}$ be a heat transfer coefficient. $\bar V(x, t) > 0$ indicate suction velocity and $\bar V(x, t) < 0$ designate the injection velocity. Because pressure is constant along $y$ -axis Eq (3), so there is no viscous consequence at the outer layer of the boundary. Euler's form of momentum equation can be used to calculate the pressure distribution as follows:

Where ambient velocity along the $x$ -axis is denoted by ${\bar u_\infty }$. By using Eq (9) the ambient velocity has the following structure: $\bar u = {\bar u_\infty } \to 0$, and so Eq (2) becomes

To obtain the dimensionless flow pattern, the following similarity conversions are used:

Here stream function $\psi$ is $\bar u = \frac{{\partial \psi }}{{\partial y}}$ and $\bar v = \frac{{\partial \psi }}{{\partial x}}$. It upholds the continuity equation Eq (1). Further Eqs (4)–(6) and Eqs (8)–(11) takes the forms:

with

In the equations above, the dimensionless variables are described as permeability parameter $K = \frac{{L\nu }}{{{{\bar u}_w}{\kappa _p}}},$ magnetic parameter $M = \frac{{\sigma LB_0^2}}{{\rho {{\bar u}_0}}},$ unsteadiness parameter $\alpha = \frac{{L\lambda {e^{ - \frac{x}{L}}}}}{{{{\bar u}_0}}},$ mixed convection parameter $\lambda = \frac{{{\beta ^{**}}g*L\left({1 - {{\overline C }_\infty }} \right)\left({{{\bar T}_w} - {{\bar T}_\infty }} \right)}}{{\bar u_w^2}}\,$, $Nt = \frac{{\bar \tau {{\bar D}_T}}}{{\nu {{\bar T}_\infty }}}({\bar T_w} - {\bar T_\infty })$ thermophoresis parameter, the bouncy ratio parameter explain with $Nr = \frac{{\left({{\rho _p} - {\rho _f}} \right)\left({{{\bar C}_w} - {{\bar C}_\infty }} \right)}}{{{\rho _f}\left({1 - {{\overline C }_\infty }} \right)\left({{{\bar T}_w} - {{\bar T}_\infty }} \right){\beta ^{**}}}}$, the bio convection Rayleigh number for $Nc = \frac{{\gamma *\left({{\rho _m} - {\rho _f}} \right)\left({{{\bar N}_w} - {{\bar N}_\infty }} \right)}}{{{\rho _f}\left({1 - {{\overline C }_\infty }} \right)\left({{{\bar T}_w} - {{\bar T}_\infty }} \right){\beta ^{**}}}}$, $Pr = \frac{{{{(\rho c)}_p}\nu }}{{{\alpha _m}}}$ Prandtl number, $Nb = \frac{{\bar \tau {{\bar D}_B}}}{\nu }({\bar C_w} - {\bar C_\infty })$ Brownian motion variable, $Rd = \frac{{16{\sigma ^*}\bar T_\infty ^3}}{{3k{k^*}}}$ radiation variable, Brinkman number $Br = \frac{{\mu {c_p}}}{k}\frac{{\bar u_w^2}}{{{c_p}({{\bar T}_w} - {{\bar T}_\infty })}} = Pr.Ec,$ Eckert number is $Ec = \frac{{\bar u_w^2}}{{{c_p}({{\bar T}_w} - {{\bar T}_\infty })}},$ ${\theta _w} = \frac{{{{\bar T}_w}}}{{{{\bar T}_\infty }}}$ is dimensionless temperature ratio parameter, $\delta = \frac{{L{Q_0}}}{{{{\bar u}_w}\rho {c_p}}}$ heat source/sink parameter, $Le = \frac{{{\alpha _m}}}{{{{\bar D}_B}}}$ Lewis number, $Lb = \frac{\nu }{{{{\bar D}_m}}}$ indicates bioconvection Schmidt number, $ϵ\mathrm{ = }\frac{\mathrm{(}{\overline{T}}_{w}-{\overline{T}}_{\infty }\mathrm{)}}{{\overline{T}}_{\infty }}$ is temperature ratio variable, $E = \frac{{{E_a}}}{{{\kappa _p}{{\bar T}_\infty }}}$ is the dimensionless activation energy, $Pe = \frac{{{b_m}{W_c}}}{{{{\bar D}_m}}}$ is Peclet number, $\Omega = \frac{{{{\bar N}_\infty }}}{{{{\bar N}_w} - {{\bar N}_\infty }}}$ be the microorganisms difference parameter, $\gamma = {N_1}\nu \sqrt {\frac{{{{\bar u}_0}}}{{2\nu L(1 - \lambda t)}}}$ and $Bi = \frac{h}{k}L$ is Biot number.

The drag surface force $({C_f}),$ heat transport rate $N{u_x},$ mass transport rate $S{h_x},$ and local density number of microorganisms $S{n_x}$ are given as:

here ${\tau _w}$ is wall shear stress, ${q_w}$ examine a heat flux, ${j_m}$ denotes mass flux rate, and ${j_n}$ be a motile transportation flux and are given as:

using Eq (21) in Eq (17) we have

where $R{e_x} = (\frac{{L{{\bar u}_w}}}{\nu })$ is the local Reynolds number.

3.   Entropy generation

Mathematically the generation of entropy is expressed as:

In this case, the first term on the right side of Eq (20) relates to the generation of entropy (irreversibility) via heat transfer. The second term evaluates the generation of entropy (irreversibility) due to energy dissipation in fluid friction. The third term corresponds to the entropy generation via a magnetic field and the 4^th term represents entropy generation through a porous medium. The 5^th and 6^th terms of Eq (20) depict entropy generation (irreversibility) via mass diffusion. The final term of Eq (20) displays the generation of entropy (irreversibility) via the microorganism's concentration. The dimensionless total entropy generation rate is denoted by ${N_g}$ and is given as:

The entropy generation due to heat transfer is

The entropy generation number is represented by ${N_{vjp}} = \frac{{Br}}{\varepsilon }\left({{{f''}^2}(\xi) + 2{{f'}^2}(\xi)\left({K + M} \right)} \right)$, the dimensionless entropy generation number for mass diffusion is represented by ${N_{md}} = \frac{{\Upsilon {\Omega _0}}}{\varepsilon }\phi '(\xi)\left({\frac{1}{\varepsilon }\phi '(\xi) + \theta '(\xi)} \right),$ and the entropy generation number for microorganisms concentration is represented by ${N_{moc}} = {\left({\frac{\Gamma }{\xi }} \right)^2}\chi '{(\xi)^2} + \frac{{{\Upsilon _1}{\Omega _0}}}{\xi }\left({\theta '(\xi)\chi '(\xi)} \right)$, $Br = \frac{{{{\bar u}^2}_w\mu }}{{k\left({{{\bar T}_w} - {{\bar T}_\infty }} \right)}}$ is Brinkman number, $\Omega = \frac{{{{\bar C}_w} - {{\bar C}_\infty }}}{{{{\bar C}_\infty }}}$ is dimensionless concentration ratio variable, $\varepsilon = \frac{{{{\bar T}_w} - {{\bar T}_\infty }}}{{{{\bar T}_\infty }}}$ is temperature ratio variable, and $\Upsilon = \frac{{R{{\bar D}_B}\left({{{\bar C}_\infty }} \right)}}{{{\alpha _m}}}$, ${\Upsilon _1} = \frac{{R{{\bar D}_B}\left({{{\bar N}_{_\infty }}} \right)}}{{{\alpha _m}}}$ and $\Gamma = \frac{{{{\bar N}_w} - {{\bar N}_\infty }}}{{{{\bar N}_\infty }}}$ are dimensionless diffusion number through nanoparticles concentration. The typical rate of entropy generation is characterized by ${E_{{g_0}}}$ and is defined as ${E_{{g_0}}} = \frac{{k{{\left({{{\bar T}_w} - {{\bar T}_\infty }} \right)}^2}}}{{{{\left({\frac{y}{\xi }} \right)}^2}{{\bar T}^2}_\infty }}.$

Bejan number $Be$ ranges from 0 to 1. When $Be$ fluctuates between 0 and 0.5, fluid friction irreversibility dominates. $Be = 0.5$ indicates that heat transport and liquid friction produce entropy generation rates at the same rate. $Be$ Fluctuating between 0.5 and 1 designates that heat transport irreversibility dominates. The dimensionless Bejan number is defined as:

3.1.   Computational procedure

The higher order nonlinear equations with boundary are tackled numerically by implemating the bvp4c solver in MATLAB with a shooting algorithem. For this, firstly the larger higher order equations are altered to the first order problem by applying some new variables. The convergence is rate is therefore this scheme is more powerful as comppare to other numerical schemes. This technique is more efficient compare to other numerical methods. The shooting technique is utilized to convert higher order boundary value problems (BVPs) into 1st order initial value problems (IVPs).

Consider,

and the corresponding boundary constraints are:

3.1.1.   Validation of results

Form the current results and published literature observed a good agrement between the results as shown in Table 1.

3.2.   Results and discussion

Here, the impact of distinct parameters upon the distribution of velocity $f'\left(\xi \right)$, temperature distribution $\theta \left(\xi \right),$ concentration $\phi \left(\xi \right),$ and motile microorganisms $\chi \left(\xi \right)$ are discussed. the parameters are taken in rang $0.1 \leqslant \lambda \leqslant 1.2,$ $1.0 \leqslant \alpha \leqslant 4.0,$ $0.1 \leqslant M \leqslant 1.2,$ $0.1 \leqslant Nc \leqslant 1.2,$ $0.1 \leqslant Nr \leqslant 1.2,$ $2.0 \leqslant Pr \leqslant 5.0,$ $1.5 \leqslant {\theta _w} \leqslant 1.8,$ $0.1 \leqslant {Q_E} \leqslant 1.2,$ $0.1 \leqslant {Q_T} \leqslant 3.0,$ $1.2 \leqslant Le \leqslant 2.4,$ $0.1 \leqslant Nt \leqslant 0.4,$ $0.1 \leqslant Nb \leqslant 0.4,$ $0.1 \leqslant Pe \leqslant 1.2,$ $1.2 \leqslant Lb \leqslant 2.4$ and $5.0 \leqslant Br \leqslant 20.0$.

3.2.1.   Consequence of velcoity field

The impact of the mixed convection parameter $\lambda$ upon the flow component $f'$ is shown in Figure 2. The dimensionless velocity $f'$ enhance as mixed convective number $\lambda$ rises. The behavior of unsteadiness parameter $\alpha$ upon the velocity component $f'$ is illustrated in Figure 3. The dimensionless velocity $f'$ of fluid decays with the growing unsteadiness parameter. Impact of magnetic parameter $M$ versus velocity field is demonstrated in Figure 4. The dimensionless velocity $f'$ of fluid decays with enhancing $M$. Physically by increassiing the magnetic parameter Lorentz forces are produces causes the resistaance in flow of fluid. Therefore velocity flow is declines. Due to magnetic effect momentum layer vanish. Impact of Bioconvection Rayleigh number $Nc$ on dimensionless velocity gradient $f'$ is captured in Figure 5. Bioconvection Rayleigh number includes density difference that produce a decay in the velocity field. With increase in $Nc$ decreases the velocity $f'$ of fluid. Physically buoyancy forces are develop in the occurrence of higher buoyancy ratio parameter as a result bioconvection, an enhancement bioconvection Rayleigh number reduce the fluid flow. Figure 6 depicts the effect of buoyancy ratio parameter $Nr$ upon velocity field $f'$. For greater worth of $Nr$ the velocity of the fluid decreases. Parameter of Buoyancy ratio $Nr$ involves density differences. These density differences are responsible for decrease in velocity.

3.3.   Concoquence of temperature distribution

Figure 7 showed the outcome of Prandtl number $\Pr$ against temperature field $\theta (\xi)$. There is an inverse relationship between Prandtl number $\Pr$ on thermal field $\theta (\xi)$. Larger value of $\Pr$ shows lower thermal diffusivity due to which a decay in temperature distribution $\theta$ is noted. Temperature distribution $\theta (\xi)$ and dimensionless temperature ratio ${\theta _w}$ has direct relation which is shown in Figure 8. By boosting the value of ${\theta _w}$ the temperature profile $\theta$ also increase. More details can be seen in Figures 9 and 10.

3.4.   Concequence of volumetric concentration of nanoparticles

Notable effect of $\Pr$ on concentration $\phi (\xi)$ is captured via Figrue 11. It is assumed that an increase in $\Pr$ causes reduction in nanoparticle's volumetric concentration. The high value of Prandtl number $\Pr$ generates low thermal diffusivity which causes the reduction in solutal field. The physical feature of Lewis number $Le$ and solutal field $\phi (\xi)$ are described in Figrue 12. As the value of dimensionless Lewis number $Le$ increases, the solutal field gradually start decreasing. Figrue 13 indicate the effect of activation energy $E$ on solutal field $\phi (\xi)$. It is concluded that the larger value of activation energy $E$ corresponds to smaller rate of reaction. That's why the chemical process becomes slow down and concentration of nanoparticles increases. Figrue 14 demonstrates the behaviour of thermophoresis variable $Nt$ and concentration field $\phi (\xi)$. The concentration profile $\phi (\xi)$ enhanced with the higher amount of thermophoresis variable $Nt$. Physically, the thermophoresis phenomenon occurs as a result of nanoparticles moving from hot region to cold region which causes the resulting nanoparticle's percentage to rise. The impact of Brownian motion variable $Nb$ over solutl field $\phi (\xi)$ is illustrated in Figrue 15. The inverse relationship is observed between $Nb$ and $\phi (\xi)$. Actually, the Brownian effect push the nanoparticles in opposite direction of concentration gradient. Larger the Brownian motion variable, lower the solutal field of nanofluid.

3.5.   Consequence of microorganisms profile

Figure 16 present the consequence of Peclet number $Pe$ over the motile microorganisms concentration profile $\chi (\xi)$. A reduction in microorganisms profile is noticed by changing the value of $Pe$. Due to bioconvection effect in $Pe$, the motile microorganisms concentration decreases. The calculation of relative direction strength and hypazard swimming of microorganisms is what the Peclet number actually means. Therefore, a higher Peclet number indicates a more directional movement of the microbe, which results in a smaller motile microorganism field. The consequence of bioconvection Lewis number $Lb$ on the motile microorganisms field $\chi (\xi)$ is demonstrated via Figure 17. The higher bioconvection Lewis number $Lb$ results in decreasing microorganisms profile $\chi (\xi)$.

3.6.   Consequence of entropy generation

Figure 18 demonstrates the behaviour of entropy generation N_g via mixed convection parameter γ. There is direct relationship between mixed convection and entropy generation. The increasing value of mixed convection parameter γ boosts up entropy generation N_g. The behaviour of entropy generation N_g via unsteadiness parameter α is illustrated in Figure 19. It shows the inverse relation of α with N_g. The higher value of unsteadiness parameter reduce the generation of entropy. Entropy generation N_g via Brinkman number Br is displayed in Figure 20.

It is worth noting that N_g is proportional to Br. Br measured viscous heating affected by conductive heat transfer. Heat transport via molecular conduction outperforms heat yield via viscous impacts. As a result, the entropy of the system rises.

The skin friction coefficient is examined in Figure 21. It is analyzed that the skin friction coefficient improves with growing amounts of mixed convective number. The effects of the Sherwood Number on activation energy E and thermophoresis parameter Nt are shown in Figure 22. The effects of the Nu Nusselt Number $- \theta '(0)$ on thermophoresis parameter Nt and Rd are drawn in Figure 23. It is displayed that the Nusselt number diminishes with an increasing in Nt.

4.   Conclusions

The aim of this research is to analyze the impact of exponential based heat source in nanofluid over sheet with gyrotactic motile microorganisms. The effect of nonlinear thermal radiation with entropy of system is also investigated. The major remarks of current article are listed below:

• Greater magnetic parameter causes a reduction in flow of fluid.

• Velocity profile is boosted via larger mixed convection parameter.

• The Prandtl number Pr decreases the thermal distribution.

• The improvements in the radiation parameter, space dependent parameter, and temperature dependent parameter are responsible for the steep enhance in the temperature curve.

• Under the behavior of $\gamma$ and $Le$, the concentration function $\varphi (\eta)$ tends to decrease.

• The larger parameter $Nt$, rise in $\varphi (\eta)$ but it decline with rise in $N_b$ and S.

• The microorganisms field is reduced via larger $Pe$ and $Lb.$

Acknowledgments

This research is supported by Government College University, Faisalabad, Pakistan and Higher Education Commission Pakistan. The authors R. Jarra, H. Shanak, and J. Asad would like to thank Palestine Technical University-Kadoorie for supporting this work financially.

Conflict of interest

No conflict of interests.