Exploring the nanomechanical concepts of development through recent updates in magnetically guided system

This article outlines an analytical analysis of unsteady mixed bioconvection buoyancy-driven nanofluid thermodynamics and gyrotactic microorganisms motion in the stagnation domain of the impulsively rotating sphere with convective boundary conditions. To make the equations physically realistic, zero mass transfer boundary conditions have been used. The Brownian motion and thermophoresis effects are incorporated in the nanofluid model. Magnetic dipole effect has been implemented. A system of partial differential equations is used to represent thermodynamics and gyrotactic microorganisms motion, which is then transformed into dimensionless ordinary differential equations. The solution methodology is involved by homotopy analysis method. The results obtained are based on the effect of dimensionless parameters on the velocity, temperature, nanoparticles concentration and density of the motile microorganisms profiles. The primary velocity increases as the mixed convection and viscoelastic parameters are increased while it decreases as the buoyancy ratio, ferro-hydrodynamic interaction and rotation parameters are increased. The secondary velocity decreases as viscoelastic parameter increases while it increases as the rotation parameter increases. Temperature is reduced as the Prandtl number and thermophoresis parameter are increased. The nanoparticles concentration is increased as the Brownian motion parameter increases. The motile density of gyrotactic microorganisms increases as the bioconvection Rayleigh number, rotation parameter and thermal Biot number are increased.


Problem formulation
The two-dimensional time-state, laminar incompressible, mixed convection boundary layer flow of viscous electrically conducting second grade nanofluid with swimming gyrotactic microorganisms in the stagnation domain of the rotating sphere and thermal convective boundary condition is scrutinized. Recognizing the Buongiorno nanofluid model, comprising of Brownian motion and thermophoresis effects is adopted in the present investigations. It is assumed that uniform dispersion is achieved by the lack of aggregation and accretion of nanoparticles. The sphere is revolving with a constant angular velocity . At time t = 0, the sphere is at rest and the surface temperature, concentration and motile microorganisms are T ∞ , C ∞ and N ∞ in an ambient fluid, respectively. The flow of motile microorganisms remains constant along sphere surface and zero mass flux condition is applied at the surface. The sphere surface is warmed due to convection from a warm nanofluid at the temperature T ∞ which provides a heat transfer factor h f , is to be strengthen or weaken to the value T ∞ , where T w > T ∞ leads to aiding flow and T w < T ∞ leads to reversing flow, respectively. Apart from nanofluid properties, which are chosen to be constant, density variation is based on Boussinesq approximation. Both Joule heating and viscous Magnetic dipole. The characteristics of the magnetic field have an effect on the flow of ferrofluid due to the magnetic dipole 2,4,6 . Magnetic dipole effects are recognized by the magnetic scalar potential shown as in Eq. (1) where γ stands for the magnetic field strength at the source. Taking H x and H y as the components of magnetic field as shown in Eqs. (2) and (3), Since the magnetic body strength is usually proportional to H x and H y gradient, c is the distance of the line currents from the leading edge, it is therefore given as in (4) Equation (5) can approximate the linear shape of magnetization M by temperature T as The value of K 1 is identified as a ferromagnetic coefficient. Figure 1 shows the physical diagram of the problem about magnetic dipole. The problem equations for second grade nanofluid with dipole effect are given as [21][22][23]29,30,[43][44][45] (1)  www.nature.com/scientificreports/ The boundary conditions are as in 29,30 where α 1 (>0) is the material parameter and u w is the stretching velocity.
Using the following transformations as in 29,30 the above governing equations provide the following non-dimensional form with non-dimensional boundary conditions given as Here c 1 is the constant such that c 1 > 0, η designates the dimensionless time parameter, ζ indicates the converted variable, f ′ and g stand for the velocity components in the x− and y− directions, T means the dimensionless temperature, φ is the dimensionless nanoparticles concentration, N is density of motile microorganisms, γ 1 denotes the mixed convection parameter, Nr points out the buoyancy ratio, β is the ferrohydrodynamic interaction parameter, Pr is the Prandtl number, Sc is the traditional Schmidt number, Sb is the bioconvection Schmidt number, 1 is the rotation parameter, is the heat dissipation parameter, ε is the curie temperature, d is the dimensionless distance, Nt is the thermophoresis, Nb is the Brownian motion parameter, N δ denotes the microorganisms concentration difference parameter, Rb gives bioconvection Rayleigh number, Pe is the bioconvection Peclet number, Re is the Reynolds number, A 1 is the viscoelastic parameter, A 2 is the dimensionless parameter, Bi is the thermal Biot number and ′ indicates the derivatives with respect to ζ . These parameters are given in mathematical expressions by www.nature.com/scientificreports/ According to the dimensionless variables, the significant designed physical quantities termed as the skin friction factor C f , Nusselt number Nu, and the density of the motile microorganisms number Nn are given in the form where shear stress, surface heat and motile surface microorganisms fluxes are mathematically expressed as Using the similarity transformations and Eq. (21) through Eq. (20), the simplified forms are where Re x = cx 2 ν f is the Reynolds number.

Solution by homotopy analysis method
For nonlinear systems of partial or ordinary differential equations, Homotopy Analysis Method (HAM) is recognized as an important alternative to the conventional numerical methods. Liao [67][68][69] proposed the Homotopy Analysis Method, which uses the basic concepts of homotopy in topology to develop an alternative and general analytical-numerical method for nonlinear problems. The validity of HAM is independent of whether or not the considered equation contains small parameter(s). As a result, HAM overcomes the limitations of perturbation methods.
Taking the initial guesses and the linear operators as Equation (23)  The non-linear operators are given by (24)

Results and discussion
Solution authentication has an important role in the evaluations of the problems. Therefore, the present solution is compared with the published work. Order of approximation of the present work in Table 1 is given which presents the close agreement with the published paper results 28 . Table 1. Comparison of the current work.
Order of approximation f ′′ (0) 28 f ′′ (0) (Present) g ′ (0) 28 g ′ (0) (Present) www.nature.com/scientificreports/ Velocity profiles. Figures 2 and 3 show the impact of the rotation parameter on the velocity profiles. For elevating values of the rotation parameter 1 , the velocity f ′ is weakened and the velocity g is enhanced by higher values of rotation parameter 1 . The physical interpretation for this feature is attributed to the reduction of momentum and thermal boundary layers, which lead to an increase in the gradients of nanofluid velocity. Figure 4 shows a declining trend in velocity f ′ for the higher estimation of the ferromagnetic parameter β .
Physically higher values develop more resistance to fluid flow. At the end, it reduces the velocity profile. Figure 5 shows a growing trend in velocity f ′ for the larger values of γ 1 . This can be perceived as buoyancy aspects understanding, the convection cooling effects are enhanced by a strong acceleration of the flow. Figure 6 shows that the dimensionless velocity decreases with an increase in the buoyancy ratio parameter Nr leading to an increase in the negative buoyant force caused by the presence of nanoparticles. The effect of increasing the bioconvection Rayleigh number Rb is that the convection power caused by bioconvection is enhanced against the convection of the buoyancy force. As a result, it could be noted that the flow velocity decreases with increasing the bioconvection Rayleigh number Rb values as shown in Fig. 7. Figure 8 is plotted here to measure the velocity variance against the viscoelastic parameter A 1 . The increasing velocity trend is aligned with the rising viscoelastic parameter A 1 . This behavior is rationalized by the mathematical representation of A 1 = α 1 c 1 /µ f ∞ that, by increasing  www.nature.com/scientificreports/ the magnitude of A 1 , the viscosity decreases due to the velocity of the fluid. Mounts and the chaotic behavior of the uplifters are seen. It is efficient to note that the present problem reduces to the Newtonian case when A 1 = 0 . From the boundary layer point of view, the thickness of the fluid increases with an increase in the viscoelastic parameter A 1 . The opposite trend of velocity g for the viscoelastic parameter A 1 is observed as shown in Fig. 9.
Temperature profile. Figure 10 is used to measure the effect of the ferromagnetic parameter β on temperature. The temperature here rises with the greater values of ferromagnetic parameter β . The effect of the heat dissipation parameter on temperature is shown in Fig. 11. The temperature here increases with an increase in the the heat dissipation parameter values. Physically thermal conductivity of the fluid decays with the higher values of heat dissipation parameter . The temperature characteristics with the curie temperature parameter ε are shown in Fig. 12. The temperature decreases in the estimation of the curie temperature parameter ε . Nanofluid thermal conductivity increases with the increase in the curie temperature ε . In effect, the added heat is removed and the temperature rises from the surface to the nanofluid. Figure 13 shows variations in temperature for the   Figure 14 shows the response of the thermal Biot number Bi to the temperature profile. The temperature is indicated to rise as a result of the increase in the thermal Biot number values. Thermal Biot number Bi depends on the coefficient of heat transfer or is directly proportional to the coefficient of heat transfer. Figure 15 shows that the non-dimensional temperature and thermal boundary layer thickness decrease as the thermophoresis parameter Nt increases. The extra energy produced by the interaction of nanoparticles with the fluid due to the thermophoresis effect reduces the temperature. As a result, the thickness of the thermal boundary layer is reduced by higher values of the thermophoresis parameter Nt. As shown in Fig. 16, the dimensionless temperature and thermal boundary layer thickness are increased with an increase in Brownian motion parameter Nb. The additional heat generated by the interaction between nanoparticles and the fluid due to Brownian motion increases the temperature. As a result, the thermal boundary layer thickness enhances with the positive values of the Brownian motion parameter Nb.
Nanoparticles concentration profile. The Schmidt number Sc is attributed to mass diffusion and therefore increases the mass diffusivity leading to a lower concentration of nanoparticles due to less mass diffusion transport, as shown in Fig. 17. Figure 18 shows that the concentration of nanoparticles and hence the thickness of concentration boundary layer increase with the increase of the Brownian motion parameter Nb. Figure 19    Motile microoganisms concentration profile. Figure 20 shows the effect of the rotation parameter 1 on the motile microorganism density for the elevated values which increases the density of the motile microorganisms. Due to the strong relation of parameter in the governing equation 18 for the microorganisms, the dimensionless density of the motile microorganisms is highly influenced by the bioconvection Schmidt number Sb. It can be seen from Fig. 21 that the rising values of Sb reduces the dimensionless density of the motile microorganisms profile. This is due to the bioconvection Schmidt number Sb which aids in the weakening of microorganisms concentration layer thickness, as indicated. The effect of the bioconvection Rayleigh number Rb on the density of motile microorganisms fluctuations is shown in Fig. 22. It is clear that the bioconvection Rayleigh number Rb is enhancing the motile microorganisms. Of course, the Biot number Bi strengthens the convective

Conclusion
Analysis of the themodynamics with the dipole effect and microorganisms shows that flow and heat transfer are enhanced with rotation parameter while for the magnetic dipole effect they have the opposite trend. Prandtl number has the cooling effect and the heat transfer increases with convective conditions. Nanoparticles concentration is improved with the Brownian motion parameter while the motile microorganisms motion is also enhanced with the rotation parameter, bioconvection Rayleigh number and convective condition.        www.nature.com/scientificreports/

Data availability
Availability exists for whole of the data.  www.nature.com/scientificreports/