Investigation of mixed convection magnetized Casson nanomaterial flow with activation energy and gyrotactic microorganisms

Present article addresses mixed convection magnetohydrodynamic Casson nanomaterial flow by stretchable cylinder. The effects of thermal, solutal and motile density stratifications at the boundary of the surface are accounted. Flow governing expressions are acquired considering aspects of permeability, thermal radiation, chemical reaction, viscous dissipation and activation energy. The obtained flow model is made dimensionless through transformations and then tackled by NDsolve code in Mathematica. Physical impacts of sundry variables on nanomaterial velocity, temperature distribution, volume fraction of microorganisms and mass concentration is investigated through plots. Furthermore, quantities of engineering interest like surface drag force, heat transfer rate, density number and Sherwood number are computed and analyzed. We observed that fluid velocity diminishes for higher curvature variable, Casson fluid material variable, Hartmann number and permeability parameter. Fluid temperature has a direct relation with Eckert number, thermophoresis variable, Brownian dispersal parameter, Prandtl number and Hartmann number. Volume fraction of gyrotactic microorganisms is decreasing function of bioconvection Lewis number, stratification parameter and bioconvection Peclet number. Detailed observations are itemized at the end.


Abstract
Present article addresses mixed convection magnetohydrodynamic Casson nanomaterial flow by stretchable cylinder. The effects of thermal, solutal and motile density stratifications at the boundary of the surface are accounted. Flow governing expressions are acquired considering aspects of permeability, thermal radiation, chemical reaction, viscous dissipation and activation energy. The obtained flow model is made dimensionless through transformations and then tackled by NDsolve code in Mathematica. Physical impacts of sundry variables on nanomaterial velocity, temperature distribution, volume fraction of microorganisms and mass concentration is investigated through plots. Furthermore, quantities of engineering interest like surface drag force, heat transfer rate, density number and Sherwood number are computed and analyzed. We observed that fluid velocity diminishes for higher curvature variable, Casson fluid material variable, Hartmann number and permeability parameter. Fluid temperature has a direct relation with Eckert number, thermophoresis variable, Brownian dispersal parameter, Prandtl number and Hartmann number. Volume fraction of gyrotactic microorganisms is decreasing function of bioconvection Lewis number, stratification parameter and bioconvection Peclet number. Detailed observations are itemized at the end.

Introduction
Nanofluids are engineered by colloidal suspension of nanometer-sized (1-100 nm) solid particles in carrier liquids. Solid particles of metals, carbides, oxides, or carbon nanotubes are suspended in conventional base liquids such as water, oils and ethylene glycol. Choi [1] was the first who introduced the notion of nanofluid and proved that thermal conductivity of carrier liquids can significantly improve by addition of man-sized particles.
In recent years nanoliquids got attention of researchers and scientists due to their practical usages in various fields of science and engineering such as extraction of geothermal power, nuclear reactors, industrial cooling, automotive, electronics, biomedical and energy sources. Non-Newtonian nanoliquids like paint, blood, melted butter, honey, paste, custard and various types of pints which hold relationship of nonlinear stress-strain. Well known Navior Stokes equations are no doubt insufficient to describe the characteristics of such fluids. Therefore researchers proposed various models such as Carreau fluid model, Prandtl fluid model and Maxwell fluid model to disclose the diverse features of such fluids. Casson fluid model is an important model because it exhibits non-Newtonian behavior below a critical shear stress and above that it behaves like an elastic solid. Examples of Casson fluid include sauces and juices, blood and honey. Due to its dual nature and potential usages in daily life, numerous researches investigated Casson fluid [2][3][4][5][6][7][8][9][10].
The least amount of energy required to initiate a chemical reaction is named as activation energy(AE). Svante Arrhenius in 1889 introduces the notion of AE. The AE has vital practical usages in formation of medicine, oil and mixtures. Bestman [11] explores the impact of AE on heat transmission performance with in boundary layer flow by a porous surface. Simultaneous impacts of AE and bioconvection phenomenon on radiative cross nanoliquid is analyzed by Azam et al [12]. Ullah et al [13] explored features of nanoliquid flow between extending sheets with variable viscosity, thermal conductivity and activation energy. Flow behavior of magnetized Williamson nanomaterial with AE and chemical reaction taking convective heat and mass boundary constraints is studied by Hamid et al [14]. Chu et al [15] discussed steady 2-D laminar incompressible flow of nanomaterial by stretchable surface with AE and bioconvection. Some advancements in this regard are listed in [16][17][18][19][20].
Phenomenon of bioconvection in nanofluid occurs when self-propelled gyrotactic microorganisms are added. Gyrotactic microorganisms are less dense as compared to the base liquids and move vertically upward direction. When density of microorganisms in upper surface of base liquid increases density stratification generates and microorganisms fall down. Return up swimming of these microorganisms retains status quo in process of bioconvection in the fluid. Bioconvection is useful in various fields of science like bio-microsystems, biomedicine and microbial fluid cells. Stability analysis in process of bioconvection in fluid flow that contains both solid small particles and microorganisms is reported by Kuznettsov and Avramenko [21]. Xun et al [22] explored the features of temperature dependent bioconvection in a rotating frame with variable thermal conductivity. Mahdy and Nabwey [23] examined heat transfer characteristics in bio-nanofluid in presence of stagnation point with zero mass flux boundary constraints. Impact of multi slip conditions on bioconvective flow of magnetized Carreau nanomaterial with thermal radiation is inspected by Elayarani et al [24]. Ramzan et al [25] investigated the irreversibility in bioconvective nanofluid together with CNT's past a permeable vertical cone with heat generation/absorption and Joule heating. Some more studies in this regard can be seen in [26][27][28][29][30].
The main theme of current exploration is to scrutinize the stratified mixed convection Casson fluid flow with gyrotactic microorganisms and solid nanoparticles. The effects of viscous dissipation, heat generation and permeability are accounted. Furthermore, activation energy, and chemical reaction are considered in concentration relation. Ndsolve function [25,30] in Mathematica software is executed to examine the behavior of flow.

Formulation
Here, we investigated flow of magnetized Casson nanofluid by stretched surface of cylinder. Flow contains gyrotactic microorganisms and solid nano-sized particles. Impacts of heat generation, viscous dissipation and mixed convection are considered. Stratification impacts at the boundary are further assimilated. Moreover mass concentration relation is modeled in view of chemical reaction with Arrhenius energy. Characteristics of the flow are inspected in ( ) r x , system of coordinates (see figure 1). The governing boundary layer model equations for bioconvective Casson nanomaterial are Figure 1. Flow geometry and coordinate system.

Velocity profile
This subsection is arranged to study the behavior of velocity field( ( )) . It is Figure 10. Sketches of ( ) q h for d .
h Figure 11. Sketches of ( ) q h for S . 1 observed here that velocity profile decays for greater values of ( ) b . In fact plastic dynamic viscosity of liquid increases via higher Casson fluid parameter values. Due to which internal resistance in the fluid raises, therefore velocity reduces. Figure 3 deliberates the behavior of ( ) Figure 4 is delineated to investigate the behavior of ( ) , increments in ( ) Gc decreases velocity graphs. Figure 5 depicts the impact of ( ) Gt on velocity profile. Increasing values of ( ) Gt enhances the velocity profile. Figure 6 shows that velocity profile decays in view of higher ( ) Ha estimation. Since ( ) Ha is linked with Lorentz force which is basically a resistive force and opposes the fluid flow. Therefore ( ( )) h ¢ f and layer thickness decays for higher ( ) Ha . Physical effect of permeability variable is captured in figure 7. It is clear from figure 7 that ( ( )) h ¢ f has inverse relation with permeability of the surface. In fact for higher ( ) K 1 resistance between fluid and surface increases and consequently fluid velocity diminishes. Figure 8 demonstrates  the characteristics of ( ) Rb on ( ( )) h ¢ f , it noticed here that velocity profile retards when ( ) Rb takes higher approximations. Improvement in ( ) q h is observed for higher ( ) Ec (see figure 9). Physically, enthalpy difference reduces and kinetic energy of nanomaterial improves for rising ( ) Ec , therefore ( ) q h decays. Behavior of ( ) q h for variable values of heat generation is studied in figure 10, here we examined that temperature curves enhances for higher ( ) d . h Since for higher ( ) d h more energy is supplied to the fluid, due to which ( ) q h enhances. Figure 11 shows that temperature field diminished in view rising ( ) S . 1 In fact difference between surface and ambient fluid regularly reduces, so ( ( )) q h decreases. Figure 12 characterizes the performance of thermal radiation parameter on ( ( )) q h .  As expected ( ( )) q h and layer thickness boosts up for larger estimation of ( ) R . Temperature profile rises for higher ( ) Ha (see figure 13). Physically, amplification in ( ) Ha produces extra Lorentz force and that provides additional heat to the system. Consequently ( ( )) q h enhances. Figures 14 and 15 shows that ( ( )) q h accelerates via ( ) Nb and ( ) Nt . It is due to the fact that higher ( ) Nb enhances nanoparticles inter collision and higher ( ) Nt boosts repulsion process from hot surface to ambient, therefore ( ( )) q h and layer thickness upsurges. Impact of ( ) Pr on ( ( )) q h is captured in figure 16, here retardation in ( ( )) q h for selected values of ( ) Pr is inspected. Since higher ( ) Pr decays thermal diffusivity with in Casson nanoliquid and thus ( ( )) q h and relevant layer decays.

Concentration profile
This segment is devoted to analyze the impact of ( ) j h in figures 17-24. Figure 17 explores that ( ( )) j h declines via rising ( ) g . Since reactive Figure 17. Sketches of ( ) j h for g. Figure 16. Sketches of ( ) q h for Pr.
species rapidly spreads with in the fluid, due to which ( ( )) j h decays. Figure 18 illustrates     Figure 29 signifies the behavior of S 3 on motile density. Clearly ( ( )) c h reduces versus rising S , 3 for higher S 3 variation in microorganisms concentration between surface and ambient liquid minimizes. Consequently motile density and thickness of motile density layer decays.

Physical quantities
In this section salient features of surface drag force( ) Cf , x heat transfer rate( ) Nu , x mass transfer rate( ) Sh x and density number( ) Nn x against involved parameters is investigated through tables 2-5. Variation in ( ) Here, we examined that ( ) Cf x raises for higher estimations of ( )       Table 4. Numerical simulations for ( ) Sh x versus g, ⁎ K , E , 1 Nb, Nt, Sc, Le, S , 2 n 1 and Pr.

Data availability statement
No new data were created or analysed in this study.