Magneto-hydrodynamics (MHD) Bioconvection Nanofluid Slip Flow over a Stretching Sheet with Thermophoresis, Viscous Dissipation and Brownian Motion

The bioconvection Magneto-Hydrodynamics (MHD) flow of nanofluid over a stretching sheet with velocity slip and viscous dissipation is studied. The governing nonlinear partial differential equations of the flow are transformed into a system of coupled nonlinear ordinary differential equations using similarity transformation. These coupled ordinary differential equations are solved using fourth order Runge Kutta-Fehlberg integration method along with shooting technique. Solutions showing the effects of pertinent parameters on the velocity temperature, nanoparticles concentration, skin friction, Nusselt number and microorganism density are illustrated graphically and discussed. It is observed that there is enhancement of the motile microorganism density as thermal slip and Eckert number increase but microorganism density slip parameter have the opposite effect on the microorganism density. It is also found that an increase in Lewis number results in reduction of the volume fraction of nanoparticles and concentration boundary-layer thickness. Brownian motion, Nb and Eckert number, Ec decrease both local Nusselt number and local motile microorganism density but increases local Sherwood number. In addition, as the values of radiation parameter R increase, the thermal boundary layer thickness increases. Finally, thermophoresis parameter, Nt decreases both local Sherwood number, local Nuseselt number and local motile microorganism density. Comparisons of the present result with the previously published results show good agreement.


Introduction
The concept of boundary layer flow over a stretching sheet has many applications in mechanical, chemical engineering, etc. Quite a number of researches [1][2][3][4] have focused on flow on a stretching sheet with different properties in the presence of additional effects. The flow and heat transfer characteristics over a stretching sheet have important industrial applications, for instance, in the aerodynamic extrusion of polymer sheet from a die, in metallurgy, cooling of an infinite metallic plate in a cooling bath, cooling or drying of papers and in textile and glass fiber production. Many subsequent studies also appeared examining heat and mass transfer in stretching boundary layer flows which includes Sreedevi et al, Shafiq et al. and Togun et al. [5][6][7] Recently nanofluid stretching boundary layer flows have also been considered with representative works of Uddin et al., Rana and Bhargava, Nadeem et al., and Rana et al. [8][9][10][11] among others. In the manufacturing of such sheets, the melt issuing from a slit is subsequently stretched. The rates of stretching and cooling have a significant influence on the quality of the final product with desired characteristics. The aforementioned processes involve cooling of a molten liquid by drawing into a cooling system. The properties desired for the product of such process usually depend on two characteristics which are the cooling liquid used and the rate of stretching. Liquids of non-Newtonian characteristics with weak electrical conductivity can be chosen as a cooling Sheet with Thermophoresis, Viscous Dissipation and Brownian Motion liquid as their flow and the heat transfer rate can be regulated through some external means.
Furthermore, the MHD fluid flow and heat transfer over a surface which stretches linearly or nonlinearly have been extensive studied because of its several practical applications in different chemical and mechanical industries. After the pioneering work of Sakiadis [12] a large number of research papers on a stretching sheet have been published by considering various governing parameters such as suction/injection, porosity, magnetic field parameter, and radiation with different types of fluids such as Newtonian, non-Newtonian, micropolar, and couple stress fluids. However, more literatures show that boundary layer flow over a stretching plate is within the scope of Newtonian and non-Newtonian fluids flow with no slip boundary condition. In addition to the past work, Nazar et al. [13] studied the unsteady boundary-layer flow of a nanofluid over a stretching sheet caused by an impulsive motion or a suddenly stretched surface using the Keller-box method. Abel et al and Bakier A. Y [14][15] investigated the Effects of non-uniform heat source on viscoelastic fluid flow and heat transfer over a stretching sheet. Makinde et al [16] studied the effect of buoyancy force on MHD flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet and they found that both the skin friction and the local Sherwood number decrease while the local Nusselt number increases with increasing buoyancy force. In addition, the theory of bio-thermal convection due to temperature gradient and swimming of the microorganisms was introduced by Kuznetsov [17] and verified that the bioconvection parameters influence mass, heat, and motile microorganism transport rates.
In this study, our objective is to investigate the effect of thermophoresis and Brownian motion on the rate of heat transfer of a nanofluid over a stretching sheet with temperature and velocity slip.

Mathematical Formulation
Consider a two-dimensional steady boundary layer flow of a nanofluid over a stretching sheet with surface temperature T w and concentration C w . The stretching velocity of the sheet is u w =ax, with a, being a constant. Let the wall mass transfer be V w . The flow is assumed to be generated by stretching sheet issuing from a thin slit at the origin. The sheet is then stretched in such a way that the speed at any point on the sheet becomes proportional to the distance from the origin. The ambient temperature and concentration are T ∞ and C ∞ respectively. The flow is subjected to the combined effects of thermal radiation R and a transverse magnetic field of strength B 0 , which is assumed to be applied in the positive y direction.
It is assumed that the induced magnetic field, the external electric field and the electric field due to the polarization of charges are negligible. Using the bosea boundary layer approximations, the governing equations can be written thus: -.
In the above expressions u and v are the nanofluid velocity components, T is the temperature, n is the density of motile microorganisms, p , C, B 0 , D b , D t and D n are the fluid pressure, the density of the base fluid, electrical conductivity, magnetic field, the Brownian diffusion, thermophoresis diffusion coefficient and the diffusivity of microorganisms respectively.
The given boundary conditions are: Where u w =L , T w = Q + Y( Z ) , Where L, J , K and O are the velocity, the thermal, concentration and density slip factors respectively, and when L=J = K = O=0, the no-slip condition is recovered, [ is the reference length of the sheet. The above boundary condition is valid when x << [ which occurs very near to the slit.
We introduce similarity functions as: Where ψ is the stream function, the continuity equation is The radiative heat flux q r is given by Where C * and K * are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature difference with in the flow is sufficiently small such that the term T 4 may be expressed as a linear function of temperature. This is done by expanding T 4 in a Taylor series about a free stream temperature T ∞ as follows: If we neglect higher-order terms in the above Eq. (14) beyond the first order in (T -Q ), we have: Thus, substituting Eq. (15) into Eq. (13), we have: The transformed governing conservation equations and boundary conditions are then obtained as, as η → 0.

Numerical Solution of the Governing Equation
The governing equations with the associated boundary Sheet with Thermophoresis, Viscous Dissipation and Brownian Motion conditions equations. (17) - (20) are numerically integrated using the fourth order Runge-Kutta method and a shooting technique with Matlab package. Firstly, they were converted into initial value differential equations using shooting technique and Runge-Kutta method to solve first order differential equations. We assumed the unspecified initial conditions for unknown variables, the transformed first order differential equations are integrated numerically as an initial valued problem until the given boundary conditions are satisfied. We define new variables as: Equations (17)- (20) are then reduced to systems of first order differential equation.

Results and Discussion
The present work is compared it with the works of Wubshet and Bandari, Anderson and Hayat et al [18][19][20]. (MHD flow and heat transfer over permeable stretching sheet with slip conditions). The results from this work with regards to good was found to be in agreement with theirs comparison of Results for the skin friction `′′(0) and reduced Nusselt number -u b (0) are presented in tables 1 and 2. Numerical values of skin friction coefficient, local Nusselt number, and local Sherwood number and motile microorganism local density are presented in Table 3 against all the pertinent parameters.  In Figures 1-2, it is observed that as N b and N t increase, the boundary layer thickens and surface temperature increases which results in reduction of heat transfer. Also, as thermal radiation R increases, the thermal boundary layer thickness increases due to the reduction of rate of heat transfer at the surface, therefore there is increasing surface temperature. In figure 3, it is found that the increase in thermophoresis parameter Nt makes the nanoparticles concentration boundary layer increase. In figure 4 It is also observed that the increase in Brownian motion Nb makes the nanoparticle concentration profile diminish. This indicates that an increase in thermophoresis parameter induce resistance to the diffusion of mass. Figure 6 illustrates the effect of the prandlt number on the concentration profile. It is found that an increase in prandlt number results in reduction of the volume fraction of nanoparticles and concentration boundary-layer thickness. Figure 6 shows different values of Lewis number. It is found that an increase in Lewis number results in reduction of the volume fraction of nanoparticles and concentration boundary-layer thickness. Also, figure 7, illustrates the variation of radiation on concentration profile. As the values of radiation parameter R increase, the thermal boundary layer thickness increases. This may be due to the reduction of rate of heat transfer at the surface.
We investigated the bioconvection in nanofluid flow over a stretching sheet surface with microorganism. The effect of various slip parameters on the microorganism concentration of the fluid and other pertinent parameters are shown in Figures 8-11.
In Figures 8-9 we notice that increase in both Eckert number and temperature slip lead to increase in microorganism density in the boundary layer but microorganism density slip parameter have the opposite effect on the microorganism density as shown in Figure 10, ‡ decreases the microorganism concentration in the boundary surface. In figure 11, it is found that the increase in thermal radition parameter increases the motile microorganism profile.

Conclusion
In this present study, we have investigated the effect of Thermophoresis parameter, Eckert number, Prandlt number Brownian motion and slip parameters. The main findings of the study shows nanoparticle concentration decreases with increase Brownian motion Nb and Prandlt number Pr inside the boundary layer but increases with increase Nt. Also there is enhancement of the motile microorganism density profiles as thermal slip and Eckert number increase. Brownian motion Nb and Eckert number Ec decrease both local Nusselt number and local motile microorganism density but increases local Sherwood number. Thermophoresis parameter Nt suppresses both local Sherwood number, local Nuseselt number and local motile microorganism density.