Effects of Viscous Dissipation on Unsteady MHD Thermo Bioconvection Boundary Layer Flow of a Nanofluid Containing Gyrotactic Microorganisms along a Stretching Sheet

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1. Introduction

Nanofluids are produced by suspending nanoparticles made up of metal carbides, nitrides and carbon nanotubes in heat transferring fluids (e.g. water, oil, ethylene glycol, polymer solution and Bio-fluids), the thermal conductivity of these fluids plays a crucial role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. Nanoparticles are the particles (that have the random movement in the nanofluid called Brownian motion) with diameter less than 100 nm. By pointing out that the use of solid particles as an additive suspended into the base fluids is a manner for increasing the heat transfer rate. Thus, nanofluid is a highly effective method of enhancing heat transfer. Gorla and Hossain [1] , Makinde [2] and Mustafa et al. [3] published papers on nanofluids. Kuznetsov and Nield [4] studied analytically the free convective boundary-layer flow of a nanoﬂuid past a vertical plate. They found that the reduced Nusselt number is a decreasing function of each of bouncy force parameter, Brownian motion parameter and thermophoresis parameter. The problem of boundary-layer flow of a nanofluid past a stretching sheet is investigated numerically [5] . They observed that the reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of higher and a decreasing function of lower number for each, and numbers. Bachok et al. [6] examined boundary layer flow of nanofluids over a moving surface in a flowing fluid. Their results indicate that dual solutions exist when the plate and the free stream move in the opposite directions. Applications of nanofluid: electronics cooling (Nanofluids have been considered as the working fluid for heat pipes in electronic cooling applications), transportation (The addition of nanoparticles to the standard engine coolant has the potential to improve automotive and heavy-duty engine cooling rates), surface coating, biomedical, etc. Sun and Pop [7] studied numerically the steady-state natural convection heat transfer behavior of nanoﬂuid inside a right-angle triangular enclosure saturated by a porous media. It is found that the heat transfer in the cavity is improved with the increasing of solid volume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when the Rayleigh number is high. Ferdows et al. [8] presented the problem of transient mixed convective laminar boundary layer flow of an incompressible, viscous, dissipative, electrically conducting nanofluid from a continuously stretching permeable surface in the presence of magnetic field and thermal radiation flux. The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically by Sohail and Changhoon [9] . Na and Pop [10] analyzed an unsteady flow due to a stretching sheet.

MHD (Magneto-hydrodynamics) is the science of the motion of electrically conducting fluids under the influence of applied magnetic forces [11] . MHD boundary layer flow problem of a nanofluid through a porous medium over an exponentially stretching sheet was studied by Ferdows et al. [12] . Heat and mass transfer in the boundary-layer flow of unsteady viscous nanofluid along a vertical stretching sheet in the presence of magnetic field, thermal radiation, heat generation, and chemical reaction are presented by Eshetu and Shankar [13] . They found that the velocity, temperature, and concentration profiles of the unsteady flow are less than the corresponding parts of the steady state flow scenario. Recently, Magnetic field effect in three-dimen- sional flow of an Oldroyd-B nanofluid over a radiative surface is presented by Shehzad et al. [14] .

Bioconvection is a phenomenon in which physical laws that govern smaller scales lead to a phenomenon visible on a larger scale [15] . The problem of natural convection boundary layer flow about a vertical cone in porous media saturated by a nanofluid due to gyrotactic microorganisms is presented by Mahdy [16] . A numerical study of a natural convection about a vertical cone embedded in a non Darcian nanofluid containing gyrotactic microorganisms saturated porous medium is studied by Hady et al. [17] . Bioconvection concerns with suspensions of self-propelled microorganisms [18] . Bioconvection in a suspension of gyrotactic motile microorganisms is investigated by Kuznetsov and Avramenko [19] . The effect of small particles (that are heavier than water) on the stability of a suspension of motile gyrotactic microorganisms in a horizontal fluid layer of finite depth is investigated by Kuznetsov and Avramenko [16] . The stability of thermo-bioconvection of oxytactic bacteria in a porous medium is investigated numerically using a Galerkin method by Kuznetsov [20] . Nanofluid bioconvection is generated by the combined effects of buoyancy forces and magnetic field on the interaction of motile microorganisms and nanoparticles [21] . Khan and Makinde [22] used Oberbeck-Boussinesq approximation and similarity transformations to investigate MHD laminar boundary layer flow with heat and mass transfer of an electrically conducting water-based nanofluid containing gyrotactic microorganisms along a convectively heated stretching sheet. Recently, Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms was studied by Jashim Uddin et al. [23] . The unsteady flow of liquid containing nanoparticles and motile gyrotactic microorganisms between two parallel plates while keeping one moving and other fixed is described by Ammarah et al. [24] . Finally, the main objective of the present paper is to study the effect of viscous dissipation on unsteady thermo bioconvection boundary layer flow of water based nanofluid containing gyrotactic microorganisms along a stretching surface with the influence of magnetic field.

2. Mathematical Analysis

Consider a tow-dimensional unsteady thermo bioconvection laminar boundary layer flow of a viscous incompressible, electrically conducting nanofluid containing gyrotactic microorganisms past a vertical stretching sheet under the influence of a uniform transversely applied magnetic felid and viscous dissipation. Bioconvection induced flow only take place in a dilute suspension of nanoparticles. The presence of nanoparticles is assumed to have no effect on the direction in which microorganisms swim and on their swimming velocity. We choose the coordinate system with the -axis measured along the stretching sheet in the upward direction and the -axis measured in the normal direction to the stretching sheet. The origin 0 of the coordinate system is placed fixed. The physical model and coordinate system is shown in Figure 1. From the figure, the sheet is stretched due to two equal and opposite forces are introduced along

Figure 1. Flow model and physical coordinate system.

the -axis. Also, the figure shows that a strong magnetic field of strength is applied in the direction. and are the temperature and the density of the motile microorganisms at the wall, respectively, which are kept constant thereafter, and the nanoﬂuid particle fraction on the boundary is passively rather than actively controlled., and are the ambient values of the temperature, nanoparticle volume fraction, and density of motile microorganisms, respectively, far away from the plate. With knowing that, the plate temperature and the density of motile microorganisms are raised to and, at.

Using the Oberbeck?Boussinesq approximation, the boundary-layer approximations of the continuity, momentum, energy, nanoparticle concentration and conservation for microorganisms equations are:

(1)

(2)

(3)

(4)

(5)

where is time, and are the velocity components, and are the Cartesian coordinates, is the temperature, is the nanoparticle volume fraction, is the density of motile microorganisms, is the kinematic viscosity, is the density of the fluid, is the density of the nanoparticles, is the microorganism density, is the fluid electrical conductivity, is the strength of magnetic field, is the acceleration due to gravity, is the volumetric expansion coefficient, is the average volume of a micro-organism, is the thermal conductivity, is the specific heat at constant pressure, is the thermophoresis diffusion coefficient, is the Brownian diffusion coefficient, is the diffusivity of microorganisms, is the chemotaxis constant, and is maximum cell swimming speed.

The initial and boundary conditions are:

(6)

(7)

By substituting the following dimensionless quantities:

into Equations (1)-(5) gives the following dimensionless equations:

(8)

(9)

(10)

(11)

(12)

where the Grashof number is defined as, the bouncy ratio parameter is defined as, the bioconvection Rayleigh number is defined as, the magnetic parameter is defined as, the Prandtl number is defined as, the Eckert number is defined as, the Brownian motion parameter is defined as, the thermophoresis parameter is defined as, the nanoparticle Lewis number is defined as, the bioconvection Lewis number is defined as, the bioconvection Péclet number is defined as, the bioconvection constant is defined as .

The dimensionless form of the initial and boundary conditions are:

(13)

(14)

The dimensionless form of the local Nusselt number, density number of the motile microorganisms and Skin-friction coefficient, respectively is defined as:

(15)

where, the local Reynolds number is defined as.

3. Numerical Method

The explicit finite difference method is used to solve the unsteady nonlinear coupled partial differential Equations (8)-(12) with initial and boundary conditions (13) and (14). The dimensionless equations are solved for the dependent variables, , , and as functions of, and. Here, is the height of the plate and is the boundary layer thickness. It is assumed that and are constant mesh sizes along the and directions, respectively and

, these values gives accurate results. In the case of, , and approach zero in the unsteady state problem, the steady state condition is ob-

tained. Let, , , and denote the values of, , , and at the end of a time step. The finite difference equations corresponding to Equations (8)-(12) are:

(16)

(17)

(18)

(19)

(20)

subject to initial and boundary conditions

(21)

(22)

where,.

where represent the grid points with and coordinates, respectively, and n represents the value of time. The value of time which represent the steady state condition in the figures is. In order to check the accuracy of the present method, the obtained numerical results are compared with previously published works by Khan et al. [25] .

Table 1 shows that the present results are in excellent agreement with the results presented by Khan et al. [25] .

Table 1. Comparison of for different values of and when with and.

4. Results and Discussion

In the following section, the numerical solutions of our problem are discussed and displayed by graphic. The effects of the governing physical parameters on the dimensionlesss velocity, temperature, nanoparticle volume fraction and density of motile microorganisms are illustrated in Figures 2-13.

Figure 2. Effect of magnetic parameter on velocity profiles.

Figure 3. Effect of thermophoresis parameter on velocity profiles.

Figure 4. Effect of Prandtl number on velocity profiles.

Figure 5. Effect of Eckert number on temperature profiles.

Figure 6. Effect of Prandtl number on temperature profiles.

Figure 7. Effect of Lewis number on nanoparticle volume fraction profiles.

Figure 8. Effect of thermophoresis parameter on nanoparticle volume fraction profiles.

Figure 9. Effect of Brownian motion parameter on nanoparticle volume fraction profiles.

Figure 10. Effect of Eckert number on nanoparticle volume fraction profiles.

Figure 11. Effect of bouncy ratio parameter on nanoparticle volume fraction profiles.

Figure 12. Effect of bioconvection Péclet number on density motile microorganisms profiles.

Figure 13. Effect of bioconvection Lewis number on density motile microorganisms profiles.

Figures 2-4 present the effect of magnetic parameter, thermophoresis parameter and Prandtl number on the velocity distribution. It is found that an increase in and leads to reduction of the velocity profiles and the velocity profiles increase with increasing. The physical meaning of the behavior in Figure 2 is due to the Lorentz force which results from the presence of a magnetic filed in the nanofluid and it works to slow down the velocity. Figure 5 and Figure 6 display the temperature distribution for different values of Eckert number and Prandtl number.It is clear that the increase in enhancement the temperature profiles and the increase in decreases the temperature. The effects of Lewis number, thermophoresis parameter, Brownian motion parameter, Eckert number and bouncy ratio parameter on the nanoparticles volume fraction profiles are presented in Figures 7-11. It is observed that, increases both and decrease the nanoparticles volume fraction profiles and increase, and leads to increases the concentration profiles within the boundary layer. For different values of bioconvection Péclet number and bioconvection Lewis number, the profiles of the rescaled density of motile microorganisms are plotted in Figure 12 and Figure 13. It is noted that the density of motile microorganisms increases with increasing, while it deceases with increasing. Figure 14 and Figure 15 show the influence of magnetic parameter

Figure 14. Effect of magnetic parameter on the dimensionless heat transfer rate.

Figure 15. Effect of bioconvection Rayleigh number Rb on the dimensionless heat transfer rate.

and bioconvection Rayleigh number on the axial distributions of the local Nusselt number. It can be seen that the local Nusselt number increases with increasing, and the axial distance.

The effects of Grashof number and Eckert number on the axial distributions of the local Nusselt number are offered in Figure 16 and Figure 17. It is found that the heat transfer rate decreases with increasing and but the opposite behavior became clear in the case of the axial distance. Figures 18-21 depict the axial distributions of the local density number of the motile microorganisms at various values of bioconvection Péclet number , bioconvection constant, Grashof number and Eckert number. It is observed that when the values of, , and increase, then the local density number of the motile microorganisms decreases. Also theses figures show that the local density number of the motile microorganisms increases as the axial distributions increase. Figure 22 and Figure 23 illustrate the influence of bioconvection Rayleigh number and magnetic parameter on the axial distributions of the local density number of the motile microorganisms. It is obvious that the dimensionless motile microorganisms transfer rate increases with increasing, and the axial distance. Finally, it is observed that the velocity profiles increases for different values of dimensionless time in Figure 24. The velocity profiles do not show any difference

Figure 16. Effect of Grashof number Gr on the dimensionless heat transfer rate.

Figure 17. Effect of Eckert number on the dimensionless heat transfer rate.

Figure 18. Effect of bioconvection Péclet number on the dimensionless motile microorganisms transfer rate.

Figure 19. Effect of microorganisms concentration difference parameter

Figure 20. Effect of Grashof number on the dimensionless motile microorganisms transfer rate.

Figure 21. Effect of Eckert number Ec on the dimensionless motile microorganisms transfer rate.

Figure 22. Effect of bioconvection Rayleigh number on the dimensionless motile microorganisms transfer rate.

Figure 23. Effect of magnetic parameter on the dimensionless motile microorganisms transfer rate.

Figure 24. Effect of dimensionless time on velocity profiles.

after which means that the steady state conditions are essentially reached at.

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