Numerical simulation for bioconvectional flow of burger nanofluid with effects of activation energy and exponential heat source/sink over an inclined wall under the swimming microorganisms

Nanofluids has broad applications such as emulsions, nuclear fuel slurries, molten plastics, extrusion of polymeric fluids, food stuffs, personal care products, shampoos, pharmaceutical industries, soaps, condensed milk, molten plastics. A nanofluid is a combination of a normal liquid component and tiny-solid particles, in which the nanomaterials are immersed in the liquid. The dispersion of solid particles into yet another host fluid will extremely increase the heat capacity of the nanoliquid, and an increase of heat efficiency can play a significant role in boosting the rate of heat transfer of the host liquid. The current article discloses the impact of Arrhenius activation energy in the bioconvective flow of Burger nanofluid by an inclined wall. The heat transfer mechanism of Burger nanofluid is analyzed through the nonlinear thermal radiation effect. The Brownian dispersion and thermophoresis diffusions effects are also scrutinized. A system of partial differential equations are converted into ordinary differential equation ODEs by using similarity transformation. The multi order ordinary differential equations are reduced to first order differential equations by applying well known shooting algorithm then numerical results of ordinary equations are computed with the help of bvp4c built-in function Matlab. Trends with significant parameters via the flow of fluid, thermal, and solutal fields of species and the area of microorganisms are controlled. The numerical results for the current analysis are seen in the tables. The temperature distribution increases by rising the temperature ratio parameter while diminishes for a higher magnitude of Prandtl number. Furthermore temperature-dependent heat source parameter increases the temperature of fluid. Concentration of nanoparticles is an decreasing function of Lewis number. The microorganisms profile decay by an augmentation in the approximation of both parameter Peclet number and bioconvection Lewis number.

The process of bioconvection can be described as the swimming up of microbes in materials, which are less dense than water. owing to the advanced concentration of microorganisms, above that the layer of substances happens to too thick and delicate, which allows the microorganisms to break down owing to the bioconvection flow. Microorganisms, many of which are older organisms on the globe known as human beings, are very important in many ways. It is defined as a type of growth of microorganism substances, such as bacteria or algae, due to the upswimming microorganism. Bioconvection has many uses in the world of biochemistry and bioinformatics. The Bioconvection process is used by bioengineering in diesel fuel goods, bioreactors, and fuel cell engineering. Platt [20] was the very first person to describe bioconvection phenomena. Unstable density distributions were adopted as a technique for the arrangement of suspensions of swim motile microorganisms and the term bioconvection was created. Kuznetsov [21] subsequently introduced this idea based on nanofluids, namely gyrotactic motile microorganisms, suggesting that the resultant large-scale flow of fluid produced by self-propelled motile gyrotactic microorganisms increases the mixture and prevents nanomaterials aggregation in nanofluids. Haq et al. [22] studied the flow properties of Cross Nanoparticles across expanded surfaces subject to Arrhenius activation energy and magnetization field. Ahmad et al. [23] examined a bioconvection nanofluid flow comprising gyrotactic motile microorganisms with a chemical reaction allowance through a porous medium past a stretched surface. Elanchezhian et al. [24] worked on the rate of motile gyrotactic microorganisms in the bioconvective nanofluid flow of Oldroyd-B past a stretching sheet with a mixing convective and inclination magnetization area. Bhatti et al. [25] performed a mathematical analysis on the migration of motile swimming microorganisms in non-Newtonian blood-based nanoliquids by anisotropic artery restriction. Khan et al. [26] illustrated the essential rheological characteristics of Jeffrey's gyrotactic motile microorganism-like nanofluid by rapid development. Shafiq et al. [27] assessed the rate of heat and mass transition of gyrotactic microorganisms with the second-grade nanofluid flow. Kotnurkar et al. [28] addressed the bioconvection of 3rd-grade nanoliquids flowing by copper-blood nanofluids in porous walls, consisting of motile species. Muhammad et al. [29] recognized the time-dependent motion of thermophysical magnetization Carreau nanofluids, which convey motile microorganisms via a spinning wedge through velocity slip as well as thermal radiation features. Farooq et al. [30] have introduced an entropic example of the 3-D bioconvective movement of nanoliquids across a linearly spinning plate in the absence of magnetic influences. Hosseinzadeh et al. [31] investigated the flow of motile microorganisms and nanotechnology through a 3-D stretching cylinder. Any important and most recent work of bioconvection swimming fluid microorganisms has been analyzed analytically by a variety of fascinating investigators [32][33][34][35][36][37].

Mathematical Formulation
This model appraises the two-dimensional Bioconvectional flow of Burgers nanofluid containing swimming gyrotactic microorganisms over a vertical inclined wall. In Burgers, a fluid phenomenon also examined the Brownian motion and thermophoresis diffusion. Heat and mass transfer aspects are found to be associated with the non-uniform internal heat generation/absorption process. The velocity of the wall is   s U x cx  and the magnetic field is along the transverse direction. The inclined wall is clarified in Figure. 1. Basic laws describing the conservation of mass and momentum yield. , The extra stress tensor of burger fluid model is With relative boundary conditions: Here in the above equation is Brownian motion, and   T D is thermophoresis effect.
Following suitable similarity transformations are used for normalizing the system of PDE: , .
The reduced system will: is the Prandtl number, Here Le Here Lb With dimensionless boundary constraints: Here Hence, the dimensionless form of engineering quantities is given by Re Re

Numerical Approach
The two-dimensional nanofluid movement of Burgers over the inclined wall is discussed in this section. In nature, the wide applications of a given model are mixed and strongly non-linear.

Results and Discussion
In this section, the physical behavior of various parameters (buoyancy ratio parameter, Hartman  Fig.9. The reduction in the concentration field  is scrutinized by growing the magnitude of the Marangoni number C and Marangoni ratio parameter D . Fig. 10 illustrates the impact of activation energy parameter E and concentration Biot number 2 S on the concentration of nanoparticles . It is analyzed that the concentration of species  boosted up with larger activation energy parameter E and concentration Biot number 2 S . Fig.11 is captured to scrutinize the behavior Nt and Brownian motion parameter Nb against the rescaled density of the concentration profile  . The concentration profile  upsurges for thermophoresis parameter Nt while reducing for Brownian motion parameter Nb . Physically when we increase the thermophoresis and Brownian motion, the thermal efficiency of fluid rises.
From this scenario noticed that the thermophoresis is also increased which tends to move nanoparticles from warm to cold sections. Features of concentration profile  over the Prandtl number Pr and Lewis's number Le for concentration are plotted in Fig.12. 6. Figures   Fig.2 Significance of C& D for f 

Tables
In this slice, the numerical outcomes of versus parameters via