Computational Insights of Bioconvective Third Grade Nanofluid Flow past a Riga Plate with Triple Stratification and Swimming Microorganisms

e goal of this study is to examine the heat-mass eects of a third grade nanouid ow through a triply stratied medium containing nanoparticles and gyrostatic microorganisms swimming in the ow.e heat and mass uxes are considered as a nonFourier model.e governingmodels are constructed as a partial dierential system. Using correct transformations, these systems are converted to an ordinary dierential model. Ordinary systems are solved using convergent series solutions. e eects of physical parameters for uid velocity, uid temperature, nanoparticle volume percentage, motile microbe density, skin friction coecients, local Nusselt number, and local Sherwood number are all illustrated in detail. When the values of the bioconvection Lewis number increase, the entropy rate also rises. e porosity parameter and modied Hartmann number show the opposite behaviour in the velocity prole.


Introduction
Researchers are interested in learning more about how to increase heat transmission because it is so important in design and business. ermal transfer of convectional liquids such as ethylene glycol, water, and oil can be used in a variety of mechanical assemblies, electrical devices, and heat dissipates. Despite this, the thermal conductivity of these base uids is weak. To counter this aw, experts from several sectors are attempting to improve the heat conductivity of newly cited uids by incorporating a unique type of nanosized particle into a new uid known as "nano uid," see Choi [1]. Nano uid ow on a at surface was examined by Khan and Pop [2]. ey see that the mass transfer gradient reduces for enhancing the thermophoresis parameter. Barnoon and Toghraie [3] analyze the impact of a non-Newtonian nano uid on aporous medium. Natural convective ow of nano uid past a heated porous plate was demonstrated by Ghalambaz et al. [4], and they concluded that the uid velocity creases when increasing the thermophoresis parameter. Aziz and Khan [5] demonstrated the characteristics of natural convective ow of nano uids over a plate. ey identi ed that heat transfer reduced by the impact of Brownian motion parameter. e nano uid ow over a thin needle was addressed by Ahmad et al. [6]. ey proved that the Brownian motion parameter leads to suppressing the nano uid concentration. Prasannakumara et al. [7] addressed the consequences of multiple slips of MHD Je ery nano uid past a surface. ey detected that the thermal boundary layer thickness thickens when enriching the thermophoresis parameter. e bioconvection phenomenon is a uid dynamic mechanism that occurs in macroscopic convective uid ow generated by a uid density gradient established by collective swimming of microorganisms. Because of their motility, these bacteria are classi ed as chemotactic, oxytactic, or gyrotactic. Near the top of the uid layer, these self-propelled motile bacteria clump together, forming a dense upper surface that is unstable or unstabilized. Bioconvection is used in a variety of industrial applications, including microbial improved oil recovery, sustainable fuel cell technologies, water treatment facilities, polymer synthesis, and so on. e 2D radiative flow of tangent hyperbolic nanofluid past a Riga plate with gyrotactic microorganisms was disclosed by Waqas et al. [8]. ey noted that the density of motile microorganisms decays when enriching the bioconvection Lewis number. Uddin et al. [9] portrayed the consequences of Stefan blowing of bioconvective flow of nanofluid past a porous medium. ey see that the density of motile microorganisms enriches when strengthening the wall suction parameter. MHD flow of cross nanofluids with gyrotactic motile microorganisms past wedge was scrutinized by Alshomrani et al. [10]. ey noted that the motile microorganisms suppress when escalating the Peclet number. Muhammad et al. [11] developed the mathematical model for the unsteady MHD flow of Carreau nanofluids with bioconvection. ey detected that the density of local motile number depresses when enhancing the Peclet parameter.
Due to its numerous industrial and engineering uses, such as cooling nuclear reactors, power generation, cooling of electronic equipment, energy production, and many others, the process of heat transfer has gotten a lot of attention from modern scholars. Fourier [12] was the first to present the heat transfer law. However, this law has the disadvantage of producing a parabolic energy equation. To address this flaw, Cattaneo [13] rewrote the Fourier equation by including the relaxation time heat flux component. In addition, Christov [14] tweaked the Cattaneo model by incorporating thermal relaxation time and used the Oldroyd upper convective model. e heat transport analysis of 2D flow cross nanofluid Cattaneo-Christov theory was investigated by Salahuddin et al. [15], and they proved that concentration relaxation leads to downfall of the nanofluid concentration. Farooq et al. [16] examined the impact of MHD flow of radiative nanofluids with Cattaneo-Christov theory. ey revealed that the fluid temperature diminishes when raising the thermal relaxation parameter. ermally radiative flow of hybrid nanoliquids with Cattaneo-Christov heat flux theory was implemented by Waqas et al. [17].
Despite the fact that nanofluids have been widely investigated, the third grade nanofluid flow over a stretching sheet with entropy optimization was examined by Loganathan et al. [18]. is study is extended with the effects of including the mixed convective flow of third grade nanofluids over a Riga plate with triple stratification and swimming microorganisms. e thermal radiative flow of third grade nanofluids containing microorganisms owing to the movement of the Riga plate is shown in this study to achieve this goal.
(i) e modified Fourier's law is used to frame energy and nanoparticle concentration equations (ii) e homotopy analysis method is used to compute the non-linear equations analytically (iii) e results of the simulations might have unique implications in the fields of thermal processes, heat transfer industry, energy systems, nuclear systems, and so on

Problem Development
For an incompressible fluid model with body forces, the continuity and motion equations are where ρ is the "fluid density," v * is the "velocity field," b is the "body forces," J is the "electric current," and T is the "third-grade incompressible fluids Cauchy stress tensor" [19].
where μ, (H 1 , H 2 , H 3 ) and A * 1 , c i -"viscosity coefficient", "kinematics tensors" and "material modulis" d/dt is expressed as the material time derivative e relationship between the Clausius-Duhem inequality and the thermodynamically compatible fluid is described by Fosdick and Rajagopal. [20].
Pakdemirli [21] took into consideration the Boussinesq and normal boundary layer approximations. e representation of steady flow of third grade nanofluids containing motile microorganisms is assumed. e surface is linearly stretched via velocity u w � ax, in positive x direction in its own path. Moreover, the flow is considered along the sheet while v is perpendicular, and B 0 magnetic field is taken vertical to the flow direction. e wall temperature T w , wall concentration C w , and motile microorganisms' wall concentration N w are defined. Figure 1 portrays the flow geometry of the problem. e governing equations are extended from Loganathan et al. [18] as follows: With the boundary points Here, b 1 , b 2 , d 1 , d 2 , e 1 , and e 2 are the dimensional constants, and T 0 and C 0 are the "reference temperature and concentrations," respectively. u and v are the "velocity components" in x and y directions, ρ is the "fluid density," υ is the "fluid kinematics viscosity," k p is the "permeability of the porous medium," C b is the "drag coefficient," J 0 is the "current density applied to the electrodes," M 0 is the "magnetic property of the permanent magnets," a 1 is the "magnets positioned in the interval separating the electrodes," σ * is the "Stefen-Boltzmann constant," Cp is the "specific heat capacity of the fluid," and k is the "thermal conductivity." Transformations are declared as follows: e nonlinear governing equations are  e boundary conditions are specified in the following manner:

Modelling of Entropy Generation
For the third grade nanoliquid, the entropy generation rate is as follows: Equation (13) was changed by using the boundary layer approximation.

Journal of Mathematics
As a consequence, the dimensionless entropy generation number may be calculated by using the following formula: As a result, the total entropy generation number has the corresponding dimensionless form: Expression of the Bejan number is
e HAM includes the auxiliary parameters (h f , h θ , h ϕ , and h χ ), and these are the responsible for solution convergence.

Results and Discussion
is section focused on the effects of divergent physical factors on fluid velocity, fluid temperature, nanoparticle volume fraction, motile microbe density, skin friction coefficients, local Nusselt number, and local Sherwood number. Table 1 provides the validation of the present analysis with previously published results [18,22]. From this comparison, we found that the current computation is an optimum one.
In this section, we focused on the variations of fluid velocity, fluid temperature, nanoparticle volume fraction, motile microorganism density, skin friction coefficients, local Nusselt number, and local Sherwood number for divergent physical parameters. Figures 2(a)-2(d)) provide the impact of α 1 , α 2 , Ha, P m , P e , N r , R b , and S 1 on the velocity profile. It is detected that the fluid velocity enriches when escalating the quantity of α 1 , α 2 , Ha, and P e , and it downfalls when enhancing the quantity of P m , N r , R b , and S 1 . Physically, the modified Hartmann number leads to strengthening the external electric field, and this causes to increase the fluid velocity. e temperature variations of Ha, R b , S 1 , λ, Fr, and P m are presented in Figures 3(a)-3(c)). It is seen that the fluid temperature escalates when raising the quantity of R b , Fr, and P m , and the opposite behaviour was attained when varying the values of Ha, S 1 , and λ. Figures 4(a) and  describes the graphical evaluation of the microorganism density number Nn x against the variations in Rb and Ω with other parameters as taken fixed. e microorganism density number Nn x is improved with the enhancement in Rb and Ω.

Conclusions
In this article, we analyse the performance of heat-mass effects of third grade nanofluid flow through a triply stratified medium with swimming of nanoparticles, and gyrostatic microorganisms are swum into this flow. e non-Fourier heat and mass flux's theory were used to frame the energy and nanoparticle concentration equations. e reduced models were analytically solved by applying the HAM scheme. e major outcomes are summarized as follows: (i) e fluid velocity enhances when raising the modified Hartmann number, and it suppresses for a larger quantity of the thermal relaxation parameter.

Data Availability
" e raw data supporting the conclusion of this report will be made available by the corresponding author without undue reservation."