Natural bio-convective flow of Maxwell nanofluid over an exponentially stretching surface with slip effect and convective boundary condition

The under-consideration article mainly focuses an unsteady three-dimensional Maxwell bio-convective nanomaterial liquid flow towards an exponentially expanding surface with the influence of chemical reaction slip condition. The feature of heat transport is achieving in the existenceof convective boundary condition and variable thermal conductivity. With the help of similarity variables, the flow form of equations is turned into a nonlinear form of coupled ODEs. The numerical solutions are calculated by adopting bvp4c function of MATLAB. Impact of distinct characteristics on the temperature, velocity microorganism and concentration field is graphically evaluated. Moreover, physical quantities are observed via graphs and tabulated data in details. It has been seen by the observation that the involvement of unsteadiness parameter restricts the change of laminar to turbulent flow. Further, for increasing velocity slip parameter velocity component in both directions shows lessening behavior. The Nusselt number exhibits diminishing behavior for larger values of Deborah number, and it shows the opposite behavior for larger values of convective parameter.


Scientific Reports
| (2022) 12:2220 | https://doi.org/10.1038/s41598-022-04948-y www.nature.com/scientificreports/ x, y, z Space coordinates (m) B 0 Magnetic field T, C, n Temperature, concentration, and microorganism density c p Specific heat capacity h w Heat transport coefficient A Unsteadiness parameter φ(η) Dimensionless factor for concentration W c Cell swimming speed χ(η) Dimensionless variables for microorganism density Nb Brownian motion parameter f (η), g(η) Dimensionless variables x-and y-direction k Thermal conductivity Wm −1 K −1 D m Microorganism diffusion coefficient q m , j w Heat flux and mass flux Qn To understand the rheological aspect and mechanism, many models have been established for non-Newtonian fluid in the past. The researchers have given a special attention to the nonlinear differential and rate type models. Maxwell model is the rate type model, and they discussed the characteristics of relaxation time. Due to specific application and special stress relaxation properties the non-Newtonian fluid is a talking point for the researchers. Non-Newtonian fluids are detected at chemical and nuclear industries, foodstuffs, bio engineering, polymeric liquids, and material processing. The Maxwell liquid model was proposed by Maxwell 1 to illustrate the elastic and viscous reaction of air. Zhao et al. 2 7 designated the heat and mass transport investigation of Jeffrey MHD nanofluid flow with porous medium over an extending sheet. Khan and Nadeem 8 propose a comparison of linear and exponential stretching sheets of a rotating Maxwell nanomaterials liquid flow with stratification influence. Some latest research associated to non-Newtonian liquid is given in the Refs. [9][10][11] . The transfer of heat is a natural mechanism which happens with temperature differences within the system. Recently, the heat transfer phenomenon as a wave inspired the researchers from all over the world because heat transfer prevalent biomedical and industrial application. For example, electronic device cooling, power generator, heat conduction in tissues, and nuclear reactor cooling etc. The law of energy conduction to the analysis of transport of heat is suggested by Fourier 12 . Cattaneo 13 modifying the Fourier law to avoid the heat conduction behavior by exerting time relaxation term. Magyari and Keller 14 presented the mass and heat transport analysis of flow over an exponentially extending surface. The heat transfer of unsteady 3D viscous flow of a boundary layer fluid of the series solution passes an impulsively expanding sheet is employed by Xu et al. 15  www.nature.com/scientificreports/ thin film fluid flow analytically using two different ways. Hayat et al. 18 presented the heat transport analysis of stagnation point MHD flow on a vertical sheet. Gkountas et al. 19 analyzed the heat transfer of a viscous nanofluid in the presence of various nanoparticles. The fluid dynamics by a stretching sheet are valuable in extrusion processes. The sheeting material formed in industrial production processes, and they consist of both polymer and metal sheets. The material region between the die and the collecting mechanism may logically assume that the stretching process alter with distance from the die, while cooling begins to stretch because of the solidification that ultimately happen. The current research concentrates to examine the flows by an exponentially stretching sheet. Such flow is quite widespread in applications such as paper production, crystal growing, continuous casting, glass fiber, metallurgical processes etc. The field of geophysical fluid dynamics that naturally occurring on earth is the main application of such fluid motion. The geophysical fluid dynamics contain a larger scale motion on earth, such as oceanography, meteorology, river flow, cloud's motion etc. The extensibility of the sheet is a valuable aspect of the flow which can be carried out to boost the machinelike feature of the sheet. Flow on an extending sheet first time analyzed by Crane 20 . Later on, Gupta and Gupta 21 examined the mass and heat transport of liquid flow on an extending surface. Bidin and Nazar 22 incorporated the two-dimensional viscous liquid in the regime of radiation passes an exponentially extending surface. The flow of viscous liquid passes an exponentially extending surface with MHD is premeditated by Ishak 23 . Liu et al. 24 conferred the heat transport of 3D viscous fluid flow pass an exponentially surface. Hayat et al. 25 investigated the transport of heat on stagnant point MHD flow of nanoliquid across the extending surface in the presence of nonlinear radiation. Benos et al. 26 considered the shrinking / stretching surface to elaborate the transfer of heat on flow of MHD in the existence of radiation. Some studies concerning to stretching surfaces is presented in the Refs. [27][28][29][30] .
Bioconvection is a phenomenon that is used to describe the instability and unstructured pattern formed due to the microorganisms, as a result the lesser density particles are swimming to the uppermost portion of a liquid. These complex microorganisms, such as gyrotactic microorganisms like algae, tend to cluster at the upper section of the fluid layer as they swim upwards, resulting in an unstable top heavy density stratification. Moreover, microorganisms are the microscopic organisms that lived everywhere in the surrounding such as deep sea, rocks, equator, deserts etc. The area of oil recovery and geophysical fact the bio-convection has a notable role. Kuznetsov 31 manifested the oxytactic microorganisms along similarity of finite depth shallow horizontal surface. The micropolar nanofluid with bio-convection recently suggested by Xu and Pop 32 . The mass and heat transport rate of convective flow of Nano liquid on a stretching sheet with microorganism is presented by Shafiq et al. 33 . Nadeem et al. 34 highlighted the 3D bio-convection nanomaterial liquid flow through an exponentially extending surface with micropolar fluid. Rashed and Nabwey 35 scrutinized the mixed bioconvection flow of nanomaterial liquid with convective conditions over a circular cylinder. Amer et al. 36 investigated the dynamical motion of a symmetric rigid body around a principal axis containing the viscous fluid in the existence of gyrostatic moment. Some recent study about gyrotactic microorganisms is found in the Refs. [37][38][39] .
Motivation of the present work is to examine the three-dimensional bio-convective unsteady Maxwell nanomaterial liquid flow with the convective condition past an exponentially extending surface. The mass and heat transport investigation is represented with the influence of variable thermal conductivity and chemical reaction. The main finding of the current problem is to analyze the convective and concentration boundary condition together on the exponential stretching surface of a Maxwell nanofluid, which in not currently investigated in the literature yet. The transferred equations are tackled by applying bvp4c technique. Graphical outcomes of emerging characteristics are sketched and discussed. Physical behaviors of microorganisms, mass, and heat transport rate are analyzed through graphs and tabulated data.

Mathematical formulation
We studied an unsteady, incompressible, and three-dimensional flow of chemically reactive Maxwell bioconvective nanomaterials liquid towards an exponentially extending surface with z = 0. The convective and slip boundary condition also taken into account to analyze the mass and heat transport. The flow attends the region z > 0 shown in Fig. 1. Let the extending velocities of an exponentially stretching sheet is u w = aExp x+y l 1−α 0 t andv w = bExp x+y l 1−α 0 t in the direction of x and y respectively. Inside the boundary layer C , T and n denotes the nanoparticle volume concentration, temperature and microorganism density respectively. Furthermore, nanoparticle volume concentration, temperature and microorganism at the wall is defined by C w , T w and n w respectively and away from the wall they are C ∞ , T ∞ and n ∞ respectively. Using above mentioned assumption the flow model takes the following form, Here D T is the thermal diffusivity, D B is the mass diffusivity, D m microorganism diffusivity, k 0 is the chemical reaction constant, k(T) is the variable thermal conductivity, ρE + j × B is the body forces, b is the chemotaxis constant, ρ is the density of fluid, W c is the cell swimming speed, and c p is specific heat. The S is the extra stress tensor for Maxwell liquid model, which is characterized as, where A 1 is the Rivlin-Ericksen tensor A 1 = (∇V) t + ∇V , µ is the viscosity, 1 is the relaxation time, and D Dt is the material derivative. The governing equations of mass, momentum, energy, concentration, and microorganism by using the boundary layer approximation and Eqs. (1-6) takes the following form 40 , The velocity components in x, y, and z directions are u, v, and w respectively. The symbols H is the concentration slip factor, τ is the ratio between heat capacity of nanoparticles to the base fluid, and h w illustrates the heat transport coefficient. Furthermore, k(T) = k ∞ 1 + rajθ 42 is signified the variable thermal conductivity in which k ∞ indicates the thermal conductivity of the surrounding.
To transform the flow model PDEs into non-dimensionalized form, we introduced the following non dimensional variable 43 , Here T 0 , C 0 , and n 0 all are the constants. Using Eq. (14), the equation of continuity automatically holds, and other Eqs. (8)(9)(10)(11)(12) take the following form, The dimensionless form of the boundary conditions is, Here prime stand for derivative with respect to η . The symbols A , β, Pr, Sc, , δ 1 ,raj and γ are represented the unsteadiness parameter, Deborah number, Prandtl number, Schmidt number, stretching ratio characteristic, concentration slip parameter, thermal conductivity parameter and convection characteristic respectively. The (20)   Table 1 demonstrated the comparison of a temperature gradient with the previously published result of Nadeem et al. 44 .
The system of equations of first order,
Scientific Reports | (2022) 12:2220 | https://doi.org/10.1038/s41598-022-04948-y www.nature.com/scientificreports/ parameter (γ ) . Physically, γ is the proportion of the hot to colder fluid convection resistance. As γ increases the thermal resistance fall down and hence temperature raises. The impact of unsteadiness characteristic A on the concentration and temperature distribution is deliberated in the Fig. 7. The reduction occurs in temperature and concentration plots by boosting the values of A . The Fig. 8 examined the outcome of thermophoresis parameter (Nt) and concentration slip parameter (δ 1 ) on the temperature sketch and concentration sketch respectively. It is seemed that the temperature sketch enlarged for greater values of Nt , while concentration sketch declines for the higher values of δ 1 . Physically, thermophoresis force occurs due to the temperature gradient, which leads to fast flow far off the sheet. The thermal boundary layer becomes dense by the large values of Nt . The diversion in concentration sketch against the distinct values of Brownian motion parameter (Nb) is visualized in Fig. 9. It is represented that the concentration sketch shows decreasing behavior for the greater values of Nb . Figure 9 reported the variation in Schmidt number (Sc) via concentration sketch. The sketch identified that the concentration sketch diminished by the enhancement of Sc . Physically, the ratio between viscous diffusion to molecular diffusion rate is said to be Schmidt number. The reduction is noted in Schmidt number by rising the mass diffusivity, hence the concentration sketch shrinks for Sc . The Fig. 10 found the impact of bio-convection Schmidt number (Sb) and Peclet number (Pe) via microorganism sketch. It is portrayed that the various values of Sb and Pe , the microorganism sketch shows decreasing behavior. Physically, Pe have an inverse relation with microorganism diffusivity, as microorganism diffusivity boosts then the devaluation is occurred in the microorganism density profile. The amplification in unsteadiness parameter (A) shows the diminishing behavior for microorganism sketch, which is illustrated in Fig. 11. The graphical description of Nusselt number, Sherwood number, and microorganism number for several parameters is presented in Fig. 12a-c. The Fig. 12a described that as growing the values of and γ the Nusselt number shows the increasing behavior in both cases. Figure 12b clearly shows that the Sherwood number decreases for distinct values of Nb and . Figure 12c revealed the impact of microorganism number for distinct values of Sb and . It is portrayed that the microorganism transfer rate enhances for the higher values of Sb and . The heat and mass transfer rate shows the effectiveness of heat and mass convection at the surface.

Concluding remarks
The 3D MHD flow of chemically reactive Maxwell bio-convective nanomaterial liquid embedded by an exponentially extending surface in the presence of viscous dissipation and joule heating effect. The thermal and solutal energy aspect has been addressed with the influence of Brownian motion and thermophoresis effect. The thermal convective and concentration slip boundary conditions are imposed on the boundary of the sheet. The specific observation of distinct characteristics is analyzed and summarized. The main outcomes of paper are: • By the enhancement of the magnetic characteristic and Deborah number the fluid velocity is declines due to occurrence of retardation effect. • By the increment of the unsteadiness parameter, the fluid velocity shows diminishing behavior.
• The higher values of Pr reduce the fluid temperature, while opposite trend is noted in the case of stronger γ.