MHD Boundary Layer Flow of a Power-Law Nanofluid Containing Gyrotactic Microorganisms Over an Exponentially Stretching Surface

: This study focusses on the numerical investigations of boundary layer flow for magnetohydrodynamic (MHD) and a power-law nanofluid containing gyrotactic microorganisms on an exponentially stretching surface with zero nanoparticle mass flux and convective heating. The nonlinear system of the governing equations is transformed and solved by Runge-Kutta-Fehlberg method. The impacts of the transverse magnetic field, bioconvection parameters, Lewis number, nanofluid parameters, Prandtl number and power-law index on the velocity, temperature, nanoparticle volume fraction, density of motile microorganism profiles is explored. In addition, the impacts of these parameters on local skin-friction coefficient, local Nusselt, local Sherwood numbers and local density number of the motile microorganisms are discussed. The results reveal that the power law index is considered an important factor in this study. Due to neglecting the buoyancy force term, the bioconvection and nanofluid parameters have slight effects on the velocity profiles. The resultant Lorentz force, from increasing the magnetic field parameter, try to decrease the velocity profiles and increase the rescaled density of motile microorganisms, temperature and nanoparticle volume fraction profiles. Physically, an augmentation of power-law index drops the reduced local skin-friction and reduced Sherwood number, while it increases reduced Nusselt number and reduced local density number of motile microorganisms.


Introduction
After nanofluids in 1995 [Choi and Eastman (1995)], the study of enhancement heat transfer by adding suitable nanoparticles has received numerous attentions due to its wide engineering applications. The review articles for enhancement heat transfer using nanofluids and their applications can be found in Daungthongsuk et al. [Daungthongsuk and Wongwises (2007); Trisaksri and Wongwises (2007); Wang and Mujumdar (2007); Kakaç and Pramuanjaroenkij (2009) The study of boundary-layer MHD flow which is resulting from the presence of magnetic fields controls many systems using electrically conducting fluids. Moreover, there are several applications for MHD flow including nuclear reactors, MHD generators and geothermal energy extractions. Sparrow et al. [Sparrow and Cess (1961)] firstly introduced the effects of magnetic field on natural convection flow. Chamkha et al. [Chamkha and Aly (2010)] studied the MHD natural convection flow on boundary layer flow of a nanofluid along a permeable plate. Moreover, Chamkha et al. [Chamkha, Mansour and Aly (2011)] investigated the presence of transverse magnetic field and Hall current with the effects of chemical reaction and heat generation on unsteady free convective along a porous plate. Uddin et al. [Uddin, Khan and Ismail (2012)] studied numerically MHD laminar boundary layer flows of an electrically conducting Newtonian nanofluid over a solid stationary plate. Mabood et al. [Mabood, Khan and Ismail (2015)] adopted Runge-Kutta Fehlberg fourth-fifth order method to study the MHD laminar boundary layer flow of a nanofluid over a nonlinear stretching sheet. The macroscopic convection motion of a fluid caused by the density gradient created by collective swimming of motile microorganisms is called bioconvection [Avramenko and Kuznetsov (2004);Hill and Pedley (2005);Kuznetsov (2006Kuznetsov ( , 2011; Nield and Kuznetsov (2006)]. Kuznetsov et al. [Kuznetsov (2010)] first introduced the bioconvection term for nanofluid. Siddiqa et al. [Siddiqa, Gul and Begum et al. (2016)] studied the bioconvection flow of a nanofluid and gyrotactic microorganisms along a wavy cone. Khan [Khan (2018) . Runge-Kutta-Fehlberg method will be used to study the MHD boundary layer flow of a power-law nanofluid containing gyrotatic microorganisms over an exponentially stretching surface. It is found that, the power law index is considered an important factor in this study. The rescaled density of motile microorganisms increases as the bioconvection Péclet number, bioconvection constant and magnetic field parameter are increase. The nanoparticle volume fraction increases as thermophoresis parameter, generalized Biot number and magnetic field parameter are increase. The reduced local skin-friction coefficient has the higher values at lower magnetic field parameter and power law index. The reduced Nusselt number is decreasing with an increase on the magnetic field parameter and thermophoresis parameter. The reduced Sherwood number is increasing according to an increase on the magnetic field parameter, Lewis number and Brownian motion parameter. The reduced local density number of the motile microorganisms increases as the Prandtl number, magnetic field parameter and power law index are increase.

Problem formulation
The two-dimensional steady MHD boundary layer flow of a power-law nanofluid containing gyrotactic microorganisms over an exponentially stretching surface is considered. The flow is originated by virtue of exponentially stretching of the sheet. At a lower surface, the sheet is heated convectively with temperature T f and a heat transfer coefficient hf. The ambient temperature and concentration are T∞ and C∞. Fig. 1 presents the initial schematic diagram of the current problem. Here, the x-axis is taken along the exponentially stretching surface and y-axis is normal to it. As shown in this figure, the transverse non-uniform magnetic field with strength is taken as parallel to the y-axis. The governing equation are: (1) (3) The imposed boundary conditions are: (6) at Introducing the following similarity transformations [Abd El-Aziz and Afify (2016); Afify and Abd El-Aziz (2017)]: Then, the velocity components are: The dimensionless forms of the governing equations are: With the boundary conditions: where, is applied magnetic field, is magnetic field parameter, where, is surface heat flux, is the surface mass flux and is the surface motile microorganisms flux. is shear stress.

Numerical method
In this section, the numerical procedure for solving the similar nonlinear Eqs. (9)-(12) with boundary conditions (13) (2019)] is applied to solve the nonlinear equations as followings: The nonlinear Eqs. (9)-(12) are transformed into set of first-order ordinary differential equations as: Using Eq. (18) into the system (9)-(12), hence the nonlinear equations are converted to the first order differential equations as: with the boundary conditions: Finally, the shooting technique is used to estimate disappeared initial conditions and by a stepwise process. The step size in Runge-Kutta-Fehlberg method for solving initial value problem (Eqs. (19)- (22)) is .
The computed values at with boundary conditions at , are fixed by Newton-Raphson method to give a superior estimation for the required solution. The iterative process is repeated until getting the results with correction up to 10 −6 .

Results and discussion
For getting clear insight of the current physical problem, the graphical illustrations were displayed for the numerical results. The computations of the physical parameters were carried out including magnetic field parameter , bioconvection Péclet number , bioconvection constant , Brownian motion parameter , thermophoresis parameter , bioconvection Lewis number , generalized Biot number , Prandtl number , Lewis number , and power-law index .
The profiles of the rescaled density of motile microorganisms under the effects of bioconvection Péclet number, and magnetic field parameter at two values of a power law index n=0.7 and n=1.2 have been shown in Figs. 2 (a) and 2(b). It is observed that, the rescaled density of motile microorganisms within the boundary layer increases as both of the bioconvection Péclet number and magnetic field parameter are increase. The rescaled density of motile microorganisms is lower at higher value of power law index n=1.2. Moreover, the boundary layer thickness is shrinking as power index n is increasing from 0.7 to 1.2. Fig. 3 presents the rescaled density of motile microorganisms under the impacts of bioconvection Lewis number , and bioconvection constant at two values of power law index n=0.5 and n=1.2. The rescaled density of motile microorganisms decreases as increases from 0.5 to 2. In addition, the rescaled density of motile microorganisms increases as bioconvection constant increases. Moreover, the boundary layer thickness is shrinking as power law index n is increasing from 0.5 to 1.2.     9 shows the velocity profiles under the effects of magnetic field parameter at two values of power law index n=0.7 and n=1.2. Due to Lorentz force, that suppresses the velocity, then velocity profiles are decrease with an increase on the magnetic field parameter. Moreover, the velocity profiles decrease slightly as power law index increases. In addition, due to neglecting the buoyancy force terms in the momentum equation, the bioconvection parameters and nanofluid parameters have slight effects on the velocity profiles. Fig. 10 depicts the profiles of the rescaled density of motile microorganisms under the effects of Prandtl number at two values of power law index n=0.5 and n=1.2. The rescaled density of motile microorganisms reduces as Prandtl number increases and it has the lowest values at higher power law index n=1.2.  Fig. 11 shows the variations of the reduced local skin-friction coefficient with magnetic field, power law index along thermophoresis parameter. It is observed that, the reduced local skin-friction coefficient has the higher values at lower magnetic field parameter and lower power law index n=0.7. The reduced local skin-friction coefficient is slightly change according to an increase on thermophoresis parameter. Figure 11: Variation of reduced local skin-friction coefficient with magnetic field, power law index along thermophoresis parameter The variations on the reduced Nusselt numbers under the effects of several parameters are shown in Fig. 12. In Fig. 12(a), the reduced Nusselt number is decreasing with an increase on the magnetic field parameter and thermophoresis parameter. This is relevant to the fact that magnetic field parameter reduce the velocity and increase the temperature within the boundary layer and then the reduced Nusselt number decreases as magnetic field parameter increases. In addition, the reduced Nusselt number has the higher values at higher power law index n=1.2. In Fig. 2(b), the reduced Nusselt number is increasing as the Prandtl number, generalized Biot number and power law index are increase. The variations on the reduced Sherwood numbers under the effects of several parameters are shown in Fig. 13. In this figure, the reduced Sherwood numbers are increasing according to an increase on the magnetic field parameter, Lewis number and Brownian motion parameter. Moreover, the reduced Sherwood numbers are decreasing, as the Parndtl number, power law index and generalized Biot number are increase. It is important to observe that, an increment rate on the reduced Sherwood number is higher at Brownian motion parameter (0.1-0.5) compare to Lewis number (1-5).

Conclusion
This study investigated the steady natural convection MHD boundary layer flow of a power-law nanofluid containing gyrotactic microorganisms over an exponentially stretching surface. The nonlinear system of the governing equations is transformed into dimensionless similar equations, which are solved numerically using Runge-Kutta-Fehlberg method. The effects of the governing parameters such as transverse magnetic field, bioconvection parameters, nanofluid parameters, Prandtl number, Lewis number and power-law index on the velocity, temperature, nanoparticle volume fraction, density of motile microorganism's profiles as well as the local skin-friction coefficient, local Nusselt, local Sherwood numbers and local density number of the motile microorganisms are explored. The main findings of this work are reported: • The rescaled density of motile microorganisms increases as the bioconvection Péclet number, bioconvection constant and magnetic field parameter are increase. • The nanoparticle volume fraction increases as thermophoresis parameter, generalized Biot number and magnetic field parameter are increase. • Due to Lorentz force, that suppresses the velocity, velocity profiles are decrease with an increase on the magnetic field parameter. • The reduced local skin-friction coefficient has higher values at a lower magnetic field parameter and a lower power law index. • The reduced Nusselt number is decreasing with an increase on the magnetic field parameter and thermophoresis parameter.
• The reduced Sherwood number is increasing according to an increase on the magnetic field parameter, Lewis number and Brownian motion parameter. • The reduced local density number of the motile microorganisms increases as the Prandtl number, magnetic field parameter and power law index are increase.