Scaling Group Analysis of Mixed Bioconvective Flow in Nanofluid with Presence of Slips, MHD and Chemical Reactions

Bioconvective flows have attracted attention in recent years due to actual and potential applications. In this paper, we consider a steady and laminar convective MHD flow of a nanofluid with heat, mass and microorganism transfer with a heat source/sink present. In addition, we assume there exists a first order chemical reaction. The governing partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) using the scaling group transformation and the associated boundary value problem is then solved. The influences of selected governing parameters on the dimensionless velocity, temperature, nanoparticle concentration, density of motile microorganisms, skin friction, heat transfer, mass transfer, and motile microorganism density rates are computed and discussed.


Introduction
Bioconvection is a process where microorganisms (denser than water) swim in an upward manner on average causing the upper surface of the suspension to be very dense (due to the accumulation of microorganisms), become unstable and sink. Resilient microorganisms swimming upwards maintain this bioconvection pattern [1][2][3]. Several studies have been conducted to explain the mechanism of directional motion of the different types of microorganisms [4][5][6][7]. It has been found that the resultant large-scale motion of fluid containing self-propelling motile microorganisms enhances mixing, preventing nanoparticle agglomeration in nanofluids [8][9][10]. Bioconvection control is essential for certain biological and biotechnological processes. For instance, controlling populations of plankton communities in the ocean and separating vigorous swimming subpopulation of microorganisms in laboratory experiments [11]. "Taxes" are the response to the movements. "Chemotaxis" means that the swimming is due to chemical gradients whereas "phototaxis" means the movement along or opposite the direction of light, "gravitaxis" refers to swimming under gravitational field, "gyrotaxis" refers to swimming which results from the balance between the physical torques generated by viscous drag and gravity operating on an asymmetrical distribution of mass within the microorganism [12]. Finally, "oxytaxis" refers to swimming due to oxygen gradient.
Kuznetsov [13] developed a theory of nanofluid bioconvection in a horizontal porous layer heated from below. Non-oscillatory and oscillatory convections were investigated. The obtained results indicate that the effect of microorganisms on the stability of the suspension may depend on the value of bioconvection Péclet number. Water-based nanofluid with heat and mass transfer along a wavy surface with the presence of motile gyrotactic microorganisms were theoretically investigated by [14,15]. The Buongiorno model with no-slip boundary conditions was employed.
MHD mixed convective heat transfer flow with chemical reaction has several applications. Example includes polymer production and manufacture of ceramics [16], autocatalytic reactions in biochemical reactions involving enzyme systems [17], dyeing technologies, moisture over agricultural food processing [18], atmospheric and oceanic circulation [19]. This study investigates the MHD flow of water-based nanofluid in the presence of microorganisms. The influence of chemical reaction and the combined effects of heat generation/absorption and multiple slip boundary conditions are considered. To the best our knowledge, no work thus far has dealt with the combined effect of chemical reaction and heat source along with multiple slip boundary conditions on MHD bioconvective nanofluid flow involving heat and mass transfer. This paper investigates the effect of magnetic field and chemical reaction, heat generation/ absorption over a vertical plate in the presence of microorganisms and multiple slip effect. It is an extension of the work of Mutuku and Makinde [20] which did not consider chemical reaction, multiple slip boundary conditions and heat source/sink. Further, in our paper, the coupled nonlinear governing PDEs governing the flow, heat, mass and microorganism transfer have been reduced to a set of coupled nonlinear ODEs using similarity transformation developed using the Lie group method (Scaling group of transformation).

Mathematical Formulations of the Problem
Consider a two-dimensional, steady, viscous incompressible, hydromagnetic laminar mixed convective boundary layer slip flow of a nanofluid over a stationary plate with microorganism. The effect of heat generation/absorption and chemical reaction are considered. The configuration of the problem is shown in Fig. 1. Within the boundary layer, the fluid temperature, the nanoparticle volume friction and the density of motile microorganisms are denoted by T, C and n respectively. At the wall, the temperature, nanoparticle volume friction and density of motile microorganisms are represented by w T , w C and , w n (respectively) and far away from the wall they are denoted by T  , C  and , n  (respectively). In addition, there are four different boundary layers formed near the plate includes (i) velocity boundary layer, (ii) concentration boundary layer, temperature boundary layer and microorganism's boundary layer (In general in Figure 1, they are not the same). The surface of the plate is subjected to boundary conditions with multiple slip conditions. The velocity components along the x and y directions are denoted by u and v respectively. A magnetic field of variable field is applied in the positive direction of the y -axis. We have incorporated first order chemical reaction in the concentration equation. We also included heat generation/absorption in the energy equation. By using these assumptions, the following five field equations representing the conservation of mass, momentum, thermal energy, nanoparticle volume friction and density of motile microorganisms are [20].
By applying boundary layer approximations, we obtain: The relevant amended boundary conditions are [21], 1 where e u is the dimensional external fluid velocity, f   is the ambient fluid density,  is the volumetric thermal expansion coefficient of the base fluid,  is the kinematic viscosity,  is the dynamic viscosity, g is the acceleration due to gravity, p  is the nanoparticles mass density,  is the fluid density,  is the average volume of a microorganism, m  is the microorganisms density,  is the electric conductivity, L is the plate characteristics length, p c is the specific heat at constant where 3 ( and the corresponding boundary conditions becomes: The parameters involved in the above dimensionless Eqns. (1 )

Applications of scaling group of transformations
It is often computationally expensive and complicated to directly solve the PDEs numerically [22]. A common approach is to convert the PDEs to ODEs and then solve the ODEs numerically. There are several types of group methods including (a) group method followed by boundary layer concepts,  [23,24]. Therefore, to carry out scaling group of transformations, we scale all independent and dependent variables as: with corresponding boundary conditions: Apply the scaling group transformation Eqn. (18) into Eqns. (19)(20)(21)(22)(23), the following relationship among the exponents are found to be invariants if 's i  are related as follows: and for boundary conditions: Relationship among  are: Using Eqn. (26), transformations in Eqn. (18) become: Expanding using the Taylor's series in power of ε, by neglecting the higher power of ε: , Differential form of expression in (23) is: From Eqn. (29), by solving one by one, the following transformations are obtained: where  is the similarity independent variable, ( ) and the boundary conditions become: Here, prime ( ) ' denotes ordinary differentiation with respect to ,    It is interesting to note that in the absence of the microorganism equation (Eqn. 34), magnetic field M, heat generation/absorption Q and chemical reaction K, our problem reduce to Uddin et al. [25] with the same similarity equations and boundary conditions.

Numerical solutions and validation
The set of nonlinear ODEs, Eqs. (31)

Physical quantities
The quantities of practical interest in this study are the skin friction ,

6.1
Effect of velocity slip (a)  [38]. Rising of slip factor may be viewed as a miscommunication between the stationary plate and the moving fluid.
In Figure 2(a), the velocity gradient decreases as the velocity slip increases due to the increase in the slip conditions at the stationary surface plate. This is because as slip conditions increase, the existence of the stationary plate will go unnoticed by the flow and hence the rate of heat transfer decreases. In Figure 2(b), dimensionless temperature is found to decrease with the growing of the velocity slip parameter. The physics behind it is that as the velocity slip parameter increases; more flow will infiltrate the boundary layer due to the slipping effect. As a result, hot plate heats more volumes of fluid and this pointers to the shrinkage in the temperature profiles. However, as velocity slip increases, effect on the dimensionless concentration and microorganism also slightly decreases. parameter. The fluid axial microorganism at the plate surface is maximum when the microorganism slip parameter is at d=0 (conventional no-slip case). It can be seen that for the no-slip case, the boundary condition and the microorganism parameter is between zero to one. The microorganism parameter approaches to zero asymptotically which means that the solution satisfies the graph obtained. In Figure 5(b) microorganism transfer rate however increases with increasing mass slip. A decrease in microorganism in the boundary layer results in enhanced migration of microorganism species to the sheet surface, resulting in a boost of microorganism transfer rates at the wall.

6.5
Effect of the magnetic field parameter (M) Figure 6 displays the dimensionless velocity, temperature, velocity gradient and temperature gradient distribution for different values of magnetic field strength, . M The effect of M on the dimensionless nanoparticles volume fraction and microorganism profiles is insignificant. In this study, the presence of M introduces a Lorentz force which gives a drag-like force on the velocity when M is applied perpendicular to the wall. Therefore, in Figure 6, application of M tends to increase the movement of the fluid which increases the velocity near the surface and hence velocity gradient (shear stress) also increases. Meanwhile, temperature gradient decreases with the increasing of M. Also, notice that the value of ' 1 f  and therefore magnetic field term will always enhance the velocity in momentum equation.

6.6
Effect of heat generation/absorption parameter (Q) Next, Table 2