Computational optimization for the deposition of bioconvection thin Oldroyd-B nanofluid with entropy generation

The behavior of an Oldroyd-B nanoliquid film sprayed on a stretching cylinder is investigated. The system also contains gyrotactic microorganisms with heat and mass transfer flow. Similarity transformations are used to make the governing equations non-dimensional ordinary differential equations and subsequently are solved through an efficient and powerful analytic technique namely homotopy analysis method (HAM). The roles of all dimensionless profiles and spray rate have been investigated. Velocity decreases with the magnetic field strength and Oldroyd-B nanofluid parameter. Temperature is increased with increasing the Brownian motion parameter while it is decreased with the increasing values of Prandtl and Reynolds numbers. Nanoparticle’s concentration is enhanced with the higher values of Reynolds number and activation energy parameter. Gyrotactic microorganism density increases with bioconvection Rayleigh number while it decreases with Peclet number. The film size naturally increases with the spray rate in a nonlinear way. A close agreement is achieved by comparing the present results with the published results.


D m
Mass diffusivity (m 2 s −1 ) D T  Deborah number on behalf of retardation time γ 1 Chemical reaction rate σ Electrical conductivity (S m −1 ) σ * * Stefan-Boltzmann constant (J K −1 ) µ f Dynamic viscosity (N s m −2 ) v f Kinematic viscosity (m 2 s −1 ) α 1 Thermal diffusivity(m 2 s −1 ) β 1 Nondimensional film thickness parameter χ Dimensionless motile microorganism's concentration φ Dimensionless nanoparticles concentration θ Dimensionless temperature ρc p f Heat capacity of nanofluid (J K −1 ) ρc p p Heat capacity of nanoparticle (J K −1 ) τ Ratio of heat capacity τ w Wall shear stress The progress in non-Newtonian liquids has a great deal of importance in projects and emerging developments. Magnetohydrodynamics (MHD) applied to electrically conductive fluids primarily concerned with the results that can be obtained from the connection between fluid motion with any external magnetic field current. Albano  68 presented the unsteady bioconvection thermal boundary layer nanofluid flow in the presence of gyrotactic microorganisms on a stretching plate and a vertical cone in porous medium. More studies on bioconvection can be found in the references [69][70][71][72][73] . It is observed that due to stretching cylinder the flow receives adequate attention. Wang 74,75 was the first to study the steady-state incompressible viscous fluid across the growing hollow cylinder. Bachok and Ishak 76 examined and reported the numerical flow and thermal transfer solution for the stretching cylinder. Chuhan et al. 77 investigated the effects of magnetohydrodynamics and thermal radiation on the movement of fluid past a porous stretching cylinder. Irfan et al. 78 studied the motion of a nanofluid past a stretching cylinder with heat transfer and magnetic field.
Literature has several interesting studies on stretching cylinder like references 79,80 which are followed by the present study. Spraying phenomena occurs in the analysis and design of coating processes. This paper is unique in the sense that it investigates the film deposition of a bioconvection Oldroy-B nanofluid containing motile gyrotactic microorganisms on a stretching cylinder. In the present article, the steady two-dimensional, incompressible radiative flow of the Oldroy-B axisymmetric sprayed thin film nanofluid past a stretching cylinder is analyzed. The fluid flow problem is governed by the partial differential equations and are converted into ordinary ones by means of suitable similarity variables. Initially, Liao presented HAM 81-83 in 1992. The solution of this method is fast convergent. Due to its rapid convergence, various researchers [84][85][86][87][88] have used HAM to solve their fluid flow problems. The computed results concerning the effects of all the related parameters on all the profiles are presented graphically.

Problem formulation
The steady, two-dimensional, and incompressible radiative Oldroyd-B and axisymmetric sprayed thin film nanofluid flow is considered past a stretching cylinder at r = 0 . The flow is in the domain r > 0 . The z − axis is taken along the axis of cylinder and r − axis is measured along the radial direction. The effects of the magnetic field are used in the direction of r − axis . Assuming induced magnetic field effects to be negligible. The expression 2cz is the surface velocity, where z is the axial coordinate and c is a proportional constant. As the material stretches, the cylinder's thickness decreases, but the cylinder's outer radius a remains relatively constant. A radial axisymmetric spray with a V velocity condenses as a film and is drawn in by the cylinder's outer surface (see Fig. 1).
The basic governing equations for the fluid flow are as [56][57][58][59]74,75,79,80 : The dimensionless parameters are defined as ∂r , www.nature.com/scientificreports/ The shear stress on the surface of the outer film is zero i.e.

And the shear stress on the cylinder is
The deposition velocity V in terms of film thickness β 1 is given by Mass flux m 1 is another interesting quantity which in connection with the deposition per axial length is The normalized mass flux m 2 is Physical quantities. The physical quantities of interests are given as following.

Sherwood number.
Local density motile flux.
Analysis of entropy generation. For the bio-nanofluid system, the irreversibility formulation is www.nature.com/scientificreports/ where R denotes the ideal gas constant and D represents the diffusivity. In Eq. (26), the first term represents the irreversibility due to heat transfer, the second term is entropy generation due to viscous dissipation and third to six terms are irreversibility due to diffusion effect. The seventh term stands for the entropy generation due to magnetic field. The characteristic entropy generation rate is Notice that irreversibility N G (ς) in scaled form is Using Eqs. (9, 10), dimensional Eq. (28) converted into the following dimensionless form where N G represents the entropy generation rate, are respectively the Brinkman number, diffusivity constant parameters due to nanoparticle and gyrotactic microorganism concentration and magnetic field parameter.
are respectively the dimensionless heat, nanoparticle concentration and microorganism concentration ratio variables.

Solution of the problem by homotopy analysis method (HAM)
Taking the initial guesses and the linear operators as satisfying the properties as given below are the arbitrary constants. The zeroth order form of the problems are given as www.nature.com/scientificreports/ where p is an embedding parameter in this case and f , θ , φ , χ are the non-zero auxiliary parameters. N f , N θ , N φ , N χ represent the none-linear operators and can be obtained through Eqs. (11)-(14) as follows For p = 0 and p = 1 , the following results are obtained Obviously, when p is increased from 0 to 1 , then . Through Taylor's series expansion, the expressions in Eq. (45) become as the following www.nature.com/scientificreports/ The convergence of the series in Eqs. (46)- (49) depend strongly upon f , θ , φ , χ . By considering that f , θ , φ , χ are selected properly so that the series in Eqs. (46)-(49) converge at p = 1 , then the following simplifications are achieved The result of the problems at order m deformation can be constructed as follow where R m f (ζ ), R m θ (ζ ), R m φ (ζ ) and R m χ (ζ ) can be calculated as The general solutions are in which f * m (ζ ), θ * m (ζ ), φ * m (ζ ), χ * m (ζ ) are the special solutions.

Results and discussion
The dynamics of an Oldroyd-B nanoliquid coolant and shielding paint or film sprayed on a stretching cylinder is studied. The normalized spray rate m 2 which is functionally correlated with the film size is shown in Fig. 2. The film size naturally increases with the spray rate at once, but not in a linear fashion. If the spray is not uniform, the film's outer surface may be affected. It's interesting to note that the spray rate increases the thickness of the film in a non-linear way. The spray deposits an Oldroyd-B nanoliquid film on the stretching cylinder, which can be used to cool the extruded material to promote solidification via a water bath or coolant spraying. Spraying also improves cooling because it creates a thinner boundary layer. Figures 3 and 4 depict the effect of the magnetic field M and Oldroyd-B nanofluid parameter 1 on velocity profile. Figure 3 shows that the velocity decreases as the magnetic field parameter increases. In general, when a magnetic field is applied to a conduction-capable fluid flow, the momentum boundary layer becomes thin. The reason for this is that during this process, resistance forces known as Lorentz forces are produced, which have a negative impact on fluid flow. This force tends to slow the velocity of the nanofluid as it passes through the vertical surface. Figure 4 demonstrates that increasing the value of 1 decreases the velocity and hence momentum boundary layer thickness decreases. Thermal Grashof number Gr and solutal Grashof number Gm effects on the velocity profile are shown in Figs. 5 and 6. The graphs show that the velocity is increased with Gr and Gm due to the dominant effects of the buoyancy force in the central region and generates changes in the velocity and high viscous effects across the walls. As a result, when Gm increases, the concentration of the liquid film increases directly and hence the viscosity increases. Figure 7 shows the effects of Reynolds number Re on the velocity profile. The velocity is enhanced with the Reynolds number. The reason is that as the Reynolds number increases, www.nature.com/scientificreports/ the inertial force overcomes the flow regarding the viscous forces. High viscous forces are highly resistive to the fluid flow and with strong inertial forces, the flow of the boundary layer decreases. When Re is small, then it means there exists small inertial effect compared to that of viscous effect. Since Re = ca 2 ν f so for Re = 0, the stretch-   www.nature.com/scientificreports/ ing rate c tends to vanishing since the cylinder radius a cannot be zero in the present case. Also, the thickness is made infinite for finite deposition rate and the steady form cannot exist. Figures 8 and 9 depict the effects of the magnetic field and Prandtl number on the temperature profile. Figure 8 reveals that increasing the values of the magnetic parameter M , increases the temperature of the nanofluid. The magnetic field produces a resistive force that opposes the flow field and increases the thickness of the thermal boundary layer, consequently heat transfer increases. Figure 9 shows that the nanofluid temperature drops when the values of Pr increases, thus the thermal boundary layer decreases for higher quantities of Pr which shows that the effective cooling for nanofluid is achieved quickly. Given the relatively small size of the motion layer, the influence of a high Prandtl number is even clearer. The liquid retains a low thermal boundary layer for larger amounts of Pr which leads to a thinner thermal boundary layer resulting in an increase in heat transfer rate on the surface. Figures 10 and 11 show the effects of the Brownian motion parameter Nb and the thermophoresis parameter Nt on the temperature profile. Figure 10 shows that the enhancement in temperature of the fluid is observed with the increasing values of Nb which results in decrease in the friction of the free surface of nanoparticles. Figure 11 shows that the temperature of nanofluid decreases as the Nt values increase. Thermophoresis is a phenomenon of the diffusion of particles because of a temperature gradient effect. The force that transfers nanoparticles to the ambient fluid due to the temperature gradient is called thermophoretic force. Increased thermophoretic force results in a wider transfer of nanoparticles to the fluid layer. Figures 12 and 13 show the impacts of thermal radiation parameter Rd and film thickness parameter β 1 respectively on the temperature profile. As shown in Fig. 12, the radiation parameter is used to add heat to the temperature of the nanoparticles as the temperature of the nanofluid rises. The analysis of thermal radiation is essential in the cooling of the cylinder. The thin film parameter β 1 has a special role in the temperature distribution. The temperature of the thermal boundary surface is high and small along with the transverse distance. The film thickness parameter, as shown in Fig. 13, reduces the temperature for greater values. The heat transfer rate is improved by thinning the nanofluid. In the present case, however, it is depreciating. The reason for this is that as the thickness of the fluid film increases, so does the mass of the fluid, which exhausts the temperature. As a result, heat enters the fluid and the environment cools. Thick film fluid requires more heat than thin film fluid.         www.nature.com/scientificreports/ increases the mass diffusivity values leading to lessen down the nanoparticle's concentration due to the less mass diffusion transportation as observed in Fig. 16. Figure 17 manifests the influence of Peclet number Pe . It shows a decrement in the boundary layer thickness of the motile microorganisms. The maximum values of Pe result a fall in the diffusivity of the microorganisms. Figure 18 portrays the influence of Rb on motile microorganism's density. It shows that χ(ς) increases with increasing the bioconvection Rayleigh number. The density of motile microorganisms is higher than that of liquid (water) and generally swims upwards to the outside (wall) of the cylinders.
The streamlines are the tangent curves to the local instantaneous velocity field. The formation of an inner mixing bolus within a fluid surrounded by streamlines is referred to as trapping. Figure 19 depicts the effect of the magnetic field parameter on the streamlines. It is shown that the number of the trapped boluses increases when the value of magnetic field parameter M is 0.30 which shows that the flow velocity is highly influenced by the magnetic field. The compression of streamlines is high at the lower portion compared to that of upper portion at the surface of stretching cylinder. Figure 20 shows that the entropy generation increases as the magnetic field parameter increases. In general, increasing the magnetic field parameter causes a slight increase in entropy generation. Because the magnetic parameter has little influence on entropy generation, a wide difference in the magnetic field parameter results in a small variation in entropy.

Comparison of the present work with published work
The present work is compared with the published work 79 in Table 1 for various values of Oldroyd-B nanofluid parameter which shows the close agreement. In Tables 2, 3, 4    www.nature.com/scientificreports/

Conclusions
The heat and mass transfer flow of an Oldroyd-B nanoliquid film sprayed on a stretching cylinder containing gyrotactic microorganisms is investigated using similarity transformations. Thermodynamics and spraying phenomena are mathematically modeled and then analyzed using HAM solution with profiles such as spray rate, velocity, heat and mass transfer, and gyrotactic microorganism's motion.    www.nature.com/scientificreports/ The summary of findings are as follows: • Spray rate increases with the film thickness nonlinearly.
• The velocity profile shows decreasing behavior for magnetic field parameter, bioconvection Rayleigh number and Oldroyd-B nanofluid parameter while increases with thermal Grashof, solutal Grashof and Reynolds numbers. • The temperature increases with increasing the magnetic field, Brownian motion and thermal radiation parameters while it is decreased with the positive values of Prandtl number, film thickness and thermophoresis parameters. • The concentration profile shows an increasing behavior with the activation energy parameter while it decreases with increasing the thermal radiation, chemical reaction parameter and Schmidt number as well. • The gyrotactic microorganisms motion increases with increasing the bioconvection Rayleigh number while it is decreased with the Peclet and Lewis numbers. • The entropy generation increases with the magnetic field parameter.
• Skin friction coefficient, heat and mass transfer rate, and motile density number consistently decrease with the different parameters.