Bioconvection flow in accelerated couple stress nanoparticles with activation energy: bio-fuel applications

On the account of significance of bioconvection in biotechnology and several biological systems, valuable contributions have been performed by scientists in current decade. In current framework, a theoretical bioconvection model is constituted to examine the analyzed the thermally developed magnetized couple stress nanoparticles flow by involving narrative flow characteristics namely activation energy, chemical reaction and radiation features. The accelerated flow is organized on the periodically porous stretched configuration. The heat performances are evaluated via famous Buongiorno’s model which successfully reflects the important features of thermophoretic and Brownian motion. The composed fluid model is based on the governing equations of momentum, energy, nanoparticles concentration and motile microorganisms. The dimensionless problem has been solved analytically via homotopic procedure where the convergence of results is carefully examined. The interesting graphical description for the distribution of velocity, heat transfer of nanoparticles, concentration pattern and gyrotactic microorganism significance are presented with relevant physical significance. The variation in wall shear stress is also graphically underlined which shows an interesting periodic oscillation near the flow domain. The numerical interpretation for examining the heat mass and motile density transfer rate is presented in tubular form.

Owing to the convinced thermal importance of nanoparticles and their novel importance in industries, biomedical and engineering sciences, much focus is elaborated by dynamic scientists in current century. Nanoparticles are mixture of small sized metallic particles (1-100 nm) with enhanced thermo-physical properties. Nanoparticles obey a number of interrelationships that may contribute to creation of a specific turbidity patterns or density delimitation, formation of nanoparticles and buoyant forces. The nanofluids have been reported to have more thermal conductivity values as compared with other ordinary fluids. The basic idea of such metallic nanoparticles was conveyed by Choi 1 . The experimental based investigation presented by Choi 1 showed that low performances of traditional base liquids can by enhanced up to extraordinary level with utilization of such nanoparticles. Later on, Buongiorno 2 developed a non-homogeneous equilibrium model to explain the slip mechanism of nanoparticles by introducing Brownian movement and thermophoresis features. Owing to higher thermal conductivity content of nanofluids and optical activity values, these fluids are also having applications in the fields of power and heat generations, biotechnology, light based chemical sensor, nano-medicines and cooling systems. Recently many numerous investigators reported their analysis with utilization of such nanoparticles. For instance, Sheikholeslami and Bhatti 3 spotted that change of shape and velocity of nanofluid is increased with increasing Reynold and Darcy numbers and the platelet shape was reported the best for maximum heat transfer. Babu and Sandeep 4 discussed nano-material properties subject to the slip mechanism confined by a slandering surface. They employed the Buongiorno's nanofluid model to search out the Brownian and thermophoretic assessment. Hamid et al. 5 reported rotating flow MoS 2 nanoparticles with shape effects and variable thermal conductivity. Imtiaz et al. 6 claimed that single wall carbon nanotubes (SWCNTs) are potentially more feasible www.nature.com/scientificreports/ et al. 33 claimed the bioconvection aspects in order to analyze the thermal assessment of water-based nanofluid in a porous space. In another investigation Shaw et al. 34 utilized the soret features in magnetized flow of nanofluid in presence of gyrotactic microorganisms. Magagula et al. 35 inspected the double dispersed flow of Casson nano-material containing gyrotactic microorganisms with additional features of first order chemical reaction and nonlinear thermal radiation. Due to immense industrial and significance of non-Newtonian material, a supreme interest has been developed by scientists to examine the complex rheology of such non-Newtonian fluids. The interesting applications of such complex materials includes wire coating, food manufacturing, oils, greases, blood, petroleum industries etc. To confirm the discriminative aspects of non-Newtonian fluid materials, these materials are characterized into various models based on complex constitutive relations. Couple stress fluid is one which encountered size dependent features which cannot be explain by using classical viscous presumption. The speculation of couple stress is basically extension of traditional viscous mathematical model which encountered couple stress and body couples. The primly work on the rheology of couple stress fluid was proposed by Stokes 36 which was further extended by many investigators 12,[37][38][39][40][41] .
Inspired by above listed literature and applications, the central accent of current exploration is to investigate the bioconvection phenomenon of couple stress nanofluid with contains gyrotactic microorganisms over periodically accelerated surface. The nonlinear mathematical expression for radiative flux is captured in the heat equation. The activation energy applications are encountered in the concentration equation. The fundamental idea and work on flow induced by periodically accelerated geometry was induced by Wang 42 and later on extensive investigations were led by numerous researchers [43][44][45][46] . Till now, no such study is reported in the literature with these flow features. The analytical solution is based on homotopic technique and variation of each flow parameter is graphically underlined.

Problem description
We have assumed 2-D velocity pattern for couple stress nanofluid flow which is confined by an accelerated moving surface where magnetic field effects are utilized normally. Following to the coordinate system, u (velocity component) has been taken in x-directions while v is considered along y-axis. The stretched surface has been accelerated with uniform velocity u = u ω = bx sin ̟ t, where ̟ being frequency and b is stretching rate. The energy equation retained thermal radiation via theory of Rosseland approximations while activation energy expressions are entertained in equation of concentration. Let the convectively heated surface occupied the surface temperature T w , surface concentration C w and motile microorganisms N w . Further, T ∞ , C ∞ and N ∞ are respectively free stream temperature, free stream concentration and free stream motile microorganisms as shown in Fig. 1. The magnetized couple stress fluid is subjected to a uniform magnetic force which is employed perpendicular to the accelerated and stretched surface. The utilization of these features, the developed governing representing the bioconvection flow of couple stress nanofluid are 12,38,50,51 : www.nature.com/scientificreports/ The physical quantities appeared in the above equations are couple stress fluid constant η 0 , electrical conductivity σ e ,gravity g, nanoparticles density ρ p , motile microorganism particles density ρ m , fluid density ρ f , temperature T, thermal diffusivity α 1 , Stefan-Boltzmann constant σ * , mean absorption constant k * , diffusion constant D B , concentration C, reaction rate K r , activation energy E a , magnetic field strength B 0 , Boltzmann constant κ, chemotaxis constant b 1 and swimming cells speed W e .
The boundary assumptions for the current flow analysis are 12,38,50,51 : Before analyze the rheological features of various flow parameters from constituted flow equations, we need achieve the non-dimensional form of these equations by inserting following dimensionless quantities 12,38,50,51 : The utilization of above suggested quantities in Eqs. (1)(2)(3)(4)(5) yield The boundary conditions in transformed form are: where K (couple stress parameter), M Hartmann number, S ( ratio between oscillating frequency to rate of stretching), (mixed convection parameter), E (activation energy parameter), N r (buoyancy ratio constant), Pr (Prandtl number), R b (bioconvection Rayleigh number), Nb (Brownian motion constant), Rd (radiation www.nature.com/scientificreports/ parameter), θ w (surface heating parameter) Nt (thermophoresis constant), σ (chemical reaction parameter),Le (Lewis number), Pe (Peclet number), δ (temperature difference), Lb (Lewis number) are elaborated as It is emphasized that Eq. (11) contains two important parameters namely radiation constant Rd and Prandtl number Pr and some diverse values may be assigned while performing the graphical analysis.
The executed local Nusselt number (Nu x ), local Sherwood number (Sh x ) and motile density number (Nn x ) are mathematically detected as 12,50,51 :

Solution methodology
In order to develop analytical expressions for the constituted partial differential Eqs. (10)-(14) with boundary conditions (Eqs. 15, 16), we employ the most interesting technique homotopy analysis technique. This solution technique was basically intended by Liao 47 and later on many investigations were reported for which the solution procedure was pursued with appliances of homotopic procedure [48][49][50][51] . As a first step, we suggest following initial guesses for constituted problem where a i (i = 1, 2, ..., 9) are arbitrary constants.

Convergence of solution
The solution computed via analytical procedure contains most imperative parameters namely auxiliary parameters h f , h θ , h φ and h χ for which specific rang can be defined to better accuracy of solution. For this purpose, we have sketched h-curves for velocity, temperature, concentration and gyrotactic microorganism distribution in Fig. 2. While sketching this figure all the flow parameters have assign fixed values like K = 0, 2, = 0.1,

Validation of solution
The solution procedure and accuracy of obtained results as a limiting case has been verified by comparing current results with already reported investigations in Tables 1 and 2. Table 1 shows the comparison of results with Abbas et al. 43 and Zheng et al. 44 against different values of τ . An excellent accuracy between both numerical computations is noted. The obtained results are further verified with the results claimed by Hayat 49 and Turkyilmazoglu 49 in Table 2. Again a favorable accuracy of both results is founded.

Discussion
This section involves the interesting physical significance of flow parameters governed by the dimensionless equations. Since this study is based on theoretical model therefore which need to allocate some arbitrary assigned values to flow involved constants like , , .  Fig. 3a convey that the distribution of velocity periodically truncated without any phase shift. Here K = 0 corresponds to viscous case for which variation in velocity is quite minimum. However, a dominant velocity distribution is noted for non-Newtonian case due to high viscosity. Physically, the decay in the velocity profile is due to presence of couple stresses. From Fig. 3b, we examined again an increasing velocity distribution by increasing mixed convection constant. The physical interpolation of such trend is justified as mixed convection constant is associated with Grashoff number which yields an impressive enhancement in velocity distribution. The results reported in Fig. 3c deals with the variation of Nr on f ξ . Here, again the velocity distribution shows a sinusoidal behavior as the surface is assumed to be oscillatory. A declined profile of f ξ is observed for Nr. The justified fact for such swelled distribution of velocity is defended as Nr involves buoyancy ratio forces which offer more resistance and fluid particles are not allowed to move freely. Similar observation has been searched out in Fig. 3d.    Table 2. Numerical values of f yy (0, τ ) when τ = π/2, Nr = Rb = S = 0. www.nature.com/scientificreports/ increases the nanoparticles temperature effectively. However, opposite trend has been reported in case of K. A declining distribution of θ is figured out with K due to presence of couple stresses. From Fig. 4b, an improved and more thermal boundary layer is found due to variation of Nr and Rb. The variation in both physical parameters is attributed to the buoyancy forces which help to enhanced the nanoparticles temperature. Further, the thickness of thermal boundary layer is quite stable. The truncation in θ due to Nt and Nb is inspected in Fig. 4c. Thermophoresis phenomenon is based on the migration of accelerated particles towards cool region. The fluctuation in temperature is resulted from migration of fluid particles from hot region. Similarly temperature distribution increases due to increment in Nb. Brownian movement is random molecular movement of fluid particles due to which the temperature distribution reached at maximum level. Figure 4d capture the graphical results for radiation parameter Rd and surface heating constant θ w on θ . The curve of temperature θ get arises with increment of both Rd and θ w . Here θ w = 1 corresponds to the linear radiation while θ w = 1.0, 1.5 are associated with the nonlinear radiative case. With utilization of nonlinear thermal radiation, a more complicated and nonlinear heat equation is resulted. Unlike linear thermal radiation, the change in temperature is more progressive by using nonlinear thermal radiation consequences. Figure 5a-c characterizes the physical consequences of couple stress parameter K, Hartmann number M, thermophoresis parameter Nt and Brownian motion constant Nb, buoyancy ratio constant Nr and activation energy parameter E on concentration φ. The analysis worked out in Fig. 5a reports that a lower change in φ is observed as K assign maximum values. However, concentration distribution enhanced due to interaction of magnetic field. The utilization of magnetic force results Lorentz force which increases the nanofluid concentration effectively. The graphical observations in order to notice the change in concentration profile www.nature.com/scientificreports/ due to variation of thermophoresis parameter Nt and Brownian motion constant Nb is determined in Fig. 5b. An enrich concentration distribution is noted when Nt assigned maximum values. On contrary the concentration distribution declined with Nb. The decrement in concentration profile due to Brownian motion Nb is justified as Nb attain reverse relation which dimensionless concentration Eq. (12). From Fig. 5c it is visualized that presence of activation energy E and buoyancy ratio constant Nr increases the concentration distribution efficiently. The roll of activation energy in various chemical processes is important as it enhanced the reaction rate. Similarly, enrollment of buoyancy forces also play sufficient role to improve the nanoparticles concentration.

Concentration distribution.
Motile microorganism distribution. The graphical analysis for microorganism distribution χ for various values of Nr, Rayleigh number Rb, buoyancy parameter, bioconvection Lewis number Lb and Peclet number Pe is presented in Fig. 6a,b. Figure 6a presents variation in χ for Nr and Rb. A remarkable enhancement in χ is observed for both parameters which is referred to buoyancy forces. The graphical results for Lb and Pe are notified in Fig. 6b. The microorganism distribution declined slightly for both flow parameters. In fact, Pe allows reverse relations with motile diffusivity due to which microorganism becomes weaker.
Skin friction coefficient. In order to examine the variation in most interesting physical quantity namely wall shear force for assigned values of K, and , Fig. 7a,b is utilized. From both figure, it is noted that skin friction coefficient oscillates periodically and the magnitude of oscillation increases with both K and . The periodical fluctuation in wall shear force is due to fact that considered stretched configuration is assumed to be periodically accelerated. However, magnitude of oscillation for K is relatively larger as compared to .   x Nu x for Pr, (b) variation in Re 1/2 x Sh x for Sc and (c) variation in Re 1/2 x Nn x for Pe. www.nature.com/scientificreports/ Physical quantities. Figure 8a represents the change in local Nusselt number Re 1/2 x Nu x against time τ due to change in Pr . The local Nusselt number vacillate periodically with τ and rate of oscillation get maximum amplitude for privilege values of Pr . Figure 8b reports the influence of Sc on local Sherwood number Re 1/2 x Sh x which is varies against τ . An increasing periodically variation of local Sherwood number is noticed with Sc. Figure 8c signifies that motile density number Re 1/2 x Nn x also shows increasing trend with variation of Pe. The numerical illustration of Re 1/2 x Nu x , Re 1/2 x Sh x and Re 1/2 x Nn x against different physical parameters are shown in Table 3. It is noted that all these quantities increases for couple stress parameter K while these quantities decreases for Hartmann number M and effective Prandtl number Pr and thermophoresis parameter Nt.

Conclusions
A theoretical model based on the flow couple stress nanofluid containing gyrotactic microorganism over an accelerated stretched surface has been evaluated analytically. The interesting features of activation are also entertained in the concentration equation. The radiation effects are simplified via one parametric approach by utilizing the Prandtl number and radiation constant. Main results reported from current contribution are listed as: • The velocity distribution periodically accelerated and magnitude of oscillation increases with couple stress.
• The temperature of nanoparticles reduces with couple stress parameter while it improves with Brownian constant and thermophoresis parameter. • The activation energy parameter increases the concentration distribution while Brownian motion constant and couple stress parameters declined the concentration profile. • The gyrotactic microorganism distribution becomes weaker for Peclet number and bioconvection Lewis number in contrast to buoyancy ratio constant and Rayleigh number. • The skin friction coefficient increases with couple stress parameter and mixed convection constant.
• The reported scientific contribution may helpful for improvement of extrusion processes, enhancement of heat transfer, biotechnology and bio-fuels. • The simulations presented in the work can be further extended for three-dimensional flows in presence of various flow features like nanofluids, activation energy, Joule heating, thermal radiation and entropy generation features. Moreover, various numerical schemes can also be employed for such formulated accelerated surfaces problems. Table 3. Numerical iteration for −θ ξ (0, τ ), −φ ξ (0, τ ) and −χ ξ (0, τ ) for various parameters at τ = 0.5π.

Appendix
Governing equation for couple stress fluid. The equations of motion governing the flow of a couple stress fluid in presence of magnetic field and porous media are given by 24 : In above equations V is the velocity vector, J is the current density, B is the magnetic flux vector , ρ is the density of the fluid, d/dt represents the material derivative, is Laplacian operator, p is the pressure, and µ are the viscosity coefficients, η is couple stresscoefficients. f is a body force and I is a body couple moment. For couple-stress fluid, shear stress tensor is not symmetric. The mathematical expressions for force stress tensor τ and the couple-stress tensor M 1 that arises in the couple-stress fluids theory are expressed by where η ′ are couple stress coefficients. Further, the materials constants and µ, η and η ′ satisfy the following inequalities: In the absence of body force and body couple moment (Eq. 25) reduce to Further, we have used the generalized Ohm's law to write in which σ e is the electrical conductivity.
For flow under consideration, the appropriate velocity field is By employing boundary layer approximations, above set of equations yield following governing equation for couple stress fluid