Effect of Constant Heat Flux on Forced Convective Micropolar Fluid Flow over a Surface of Another Quiescent Fluid

Due to the many applications of micropolar fluid such as blood, paint, body fluid, p olymers, c olloidal fl uid and suspension fluid, it has become a prominent subject among the researchers. However, the characteristics of micropolar fluid flow over a surface of another quiescent fluid with heavier density of micropolar fluid under the effect of constant heat flux is still unknown. Therefore, the objective of the present work is to investigate numerically the forced convection of micropolar fluid fl ow over a su rface of an other qu iescent flu id usi ng constant heat flux boundary c ondition. In this study, the similarity transformation is used to reduce the boundary layer governing equations for mass, momentum, angular momentum and energy from partial differential equations to a system of nonlinear ordinary differential equations. This problem is solved numerically using shooting technique with Runge-Kutta-Gill method and implemented in Jupyter Notebook using Python 3 language. The behaviour of micropolar fluid in terms of velocity, skin friction, microrotation and temperature are analyzed and discussed. It is found that, the temperature is higher in constant wall temperature (CWT) compared to constant heat flux (CHF) at stretching or shrinking parameter λ = 0.5 and various micropolar parameter K. Furthermore, as Prandtl num-ber increases, the temperature is decreasing in both CHF and CWT.


Introduction
Eringen [1] was the first who propose the theory of micropolar fluid to extend the widely known equation of viscous fluid which is Navier-Stokes equation. Micropolar fluid can be defined as a non-Newtonian fluid that consist of microstructure which undergo translational motion and rotational motion. In addition, this theory also capable to describe some fluid phenomena that contain suspended fluid additives such as animal blood, body fluid, colloidal fluid, magnetic fluids, cloud with dust, muddy fluids and polymer that cannot be interpreted by Navier-Stokes equation. A comprehensive information about micropolar fluid can be found in papers by Ariman et. al [2] - [3], Willson [4] and books written by Lukaszewicz [5] and Eremeyev et. al [6]. Due to the extensive applications, micropolar fluid has been studied on different geometry and system by various researchers such as Rees and Bassom [7], Guram and Smith [8], Peddieson [9], Takhar and Soundalgekar [10], Ishak et. al [11] and Das & Guha [12].
The problem of heat transfer in boundary layer flow is one of the growing research field as it has numerous applications such as transpiration cooling, drag reduction, bearing and radial diffusers, thermal recovery of oil, the design of thrust bearings and radial diffusers, material drying, laser pulse heating and more [13]. Eringen extended the problem of micropolar fluid with addition of thermal effect [14] . Gorla et al. [15] studied the heat transfer in micropolar boundary layer flow over a flat plate for both constant wall temperature and constant heat flux. Ishak et al. [16] discussed the effect of constant surface heat flux on boundary layer flow of a micropolar fluid on a continuous flat plate moving in a parallel stream. They concluded Universal Journal of Mechanical Engineering 7(4): 198-205, 2019 199 micropolar fluid exhibit drag reduction compared to Newtonian fluid and delayed the boundary layer separation.
However, there are some systems that exist with two fluids involved. Thus, several attempts have been made in the case of another quiescent fluid such as Wang [17] who investigated the stagnation flow on the surface of a quiescent fluid. Next, Liu [18] analyzed the nonorthogonal stagnation flow on the surface of a quiescent fluid. Reza et al. [19] carried out the numerical study of stagnation point flow and heat transfer for viscoelastic fluid impinging on a quiescent fluid. The magnetic effect on the electrically conducting fluid in the surface of another quiescent fluid was attempted by Reza & Gupta [20]. Rohini et al. [21] examined stagnation-point flow of a fluid on a shrinking surface of another quiescent fluid. More recently, Isa and Mohammad [22] performed numerical analysis of boundary layer flow on a stretching sheet of another quiescent fluid. Although some research were carried out on another quiescent fluid, there is no study involve micropolar fluid in context of convective flow within our knowledge. Therefore, the aim of this paper is to investigate the effect of constant heat flux (CHF) and constant wall temperature (CWT) on forced convective micropolar fluid flow over a surface of another quiescent fluid.

Mathematical Model
Consider an incompressible micropolar fluid of density ρ 1 , dynamic viscosity µ 1 , vortex viscosity κ 1 , spin-gradient viscosity γ 1 and microinertia density j 1 impinging orthogonally on a surface of another quiescent, heavier incompressible micropolar fluid of density ρ 2 , dynamic viscosity µ 2 , vortex viscosity κ 2 , spin-gradient viscosity γ 2 and microinertia density j 2 . T 1,w , T 2,w and T ∞ are temperature of surface for upper fluid, lower fluid and free stream temperature respectively. A sketch of the physical problem is shown in Figure 1. Let (x,y 1 ) denote the Cartesian coordinates for the upper fluid with x = 0 as the symmetry plane, and x-axis is taken along the interface between the two fluids. It is assumed that the surface is stretched or shrinked with velocity u w (x) = cx, where c is a constant that indicates c > 0 for a stretching sheet and c < 0 for a shrinking sheet, respectively. The coordinate system for the lower fluid is (x,y 2 ) as shown in Figure 1. Note that the z-axis is normal to the (x,y 1 ) plane.
Under the boundary layer approximations, the governing equations of continuity, momentum, angular momentum and energy are, subject to boundary conditions Plate y 1 for temperature, we have or where u i and v i are the velocity along x and y i axes, U i is free stream velocity, T i is the temperature and N i is the angular velocity. i = 1 represents upper fluid while i = 2 represents lower fluid. Moreover, q w is constant heat flux, k 0 is thermal conductivity, c p is specific heat capacity and n is a constant such that 0 ≤ n ≤ 1. From Jena and Mathur [23], the strong concentration case (n = 0) represents the concentrated particle flows in which the microelements close to the wall surface are unable to rotate. According to Ahmadi [24], the weak concentration case (n = 1/2) indicates the vanishing of the anti-symmetrical part of the stress tensor. The case n = 1, as suggested by Peddieson [9], is used for the modeling of turbulent boundary layer flows. In this paper, we consider the cases of n = 0 (strong concentration) only and it is worth to mention case K = 0 represent the classical Navier-Stokes equations for a viscous and incompresible fluid. Further, we assume that spin-gradient viscosity γ i is defined as [25,26,27] Following Attia [13] for the upper fluid, the similarity variables are or and for lower fluid for temperature distribution, or where prime denotes differentiation with respect to η and ξ respectively. Clearly with u 1 and v 1 given in (10), the equation of continuity (1) is satisfied. Similarly, for the lower fluid, with u 2 and v 2 given as in (13), it is readily seen that the continuity equation (1) is identically satisfied. Using (10) - (15), equations (2) -(8) are transformed into ordinary differential equations for the upper fluid flow with the boundary conditions On the other hand, for the lower fluid, we obtain with the boundary conditions where Pr is Prandlt number, λ is stretching or shrinking parameter, K 1 and K 2 are micropolar parameters. These quantities can be written as In the case of constant heat flux, the initial conditions for temperature are whilst, in the case of constant wall temperature, the initial conditions for temperature are In addition, the local Nusselt number N u x can be denoted in non-dimensional form as and q w can be defined as Next using (11) - (12), (27) and (28), for upper fluid we attained Furthermore, using equations (14), (15), (27) and (28), for lower fluid we obtained

Results and Discussion
In this study, the system of non-linear ordinary differential equations with boundary conditions (16) - (23) and (25) -(26) were solved using shooting technique with Runge-Kutta-Gill method. The numerical procedure was implemented in Jupyter Notebook using Python 3 language. The results presented are in term of velocity, skin friction, microrotation and temperature of the micropolar fluids in strong concentration (n = 0) where the micropolar parameter for both upper and lower fluids are the same (K = K 1 = K 2 ). The solutions obtained involve initial guess of f (0), g (0) and θ(0) for CHF case and f (0), g (0) and θ (0) for CWT case. Then the Newton-Raphson method is used to find the correct initial conditions that satisfy the boundary conditions. The comparison of the solutions obtained in current study with existing work for wall temperature θ(0) and heat transfer coefficient θ (0) are depicted in table 1 and table 2 respectively. In these tables, the temperature at surface and the heat transfer at surface from current study are found in a good agreement with Lok et al. [28] and Salleh et al. [29]. Practically, small values of Pr (< 1) resemble liquid metal that have low viscosity but high thermal conductivity, meanwhile large values of Pr (> 1) equivalent to high viscosity. In addition, Prandtl number Pr = 0.71, 1, 7 represent air, electrolyte solution and water, respectively [29].
The graphical results for velocity, temperature and microrotation profiles for upper and lower fluids are illustrated in figures 2 -7 for various micropolar parameter K with Pr = 7 and  figure 4, the microrotation of upper fluid is decreasing with the addition of micropolar parameter K. It is also has a profile that achieving the maximum value nearby the wall which later declining to zero as boundary layer increases. In figure 5, the microrotation of lower fluid is increasing as micropolar parameter K increases. Meanwhile figure 6 shows the temperature of upper fluid is decreasing when micropolar parameter K increases. Also, the temperature of upper fluid is higher under the influence of CWT boundary condition compared to CHF. On the other hand, the temperature of lower fluid is increasing with the increament of micropolar parameter K as displayed in figure 7. It seem that upper fluid and lower fluid display opposite characteristic.  (16) - (17) and (18) as mentioned before. In figure 12, we found that the temperature of upper fluid is higher in CWT compared to CHF but as Pr increases the values of temperature decrease. Increasing Pr means increasing the viscosity of the fluid which   lead to less heat and the fluid is highly conductive when Pr number is small. It is seen that thermal boundary layer thickness is greater for CHF compared to CWT and has decreasing trend as Pr number increase. The thermal diffusivity decreases as Pr number increase consequently lead to the decreasing of energy transfer capability that abbreviates the thermal boundary layer thickness. Figure 13 also potrays similar trend. It is seen that for CHF, lower fluid has greater temperature at surface and thermal boundary layer thickness.

Conclusion
This study analyzed the effect of CHF on micropolar flow on a surface of another quiescent fluid. This problem was solved numerically using shooting method with Runge-Kutta-Gill in Python 3 software. The main points are summerized as follows: • It is shown that, the temperature is higher in CHF compared to CWT at strecthing or shrinking parameter λ = 0.5.
• Meanwhile, as Pr increases, the temperature is decreasing for both CHF and CWT.
• Upper fluid and lower fluid has opposite trend with variation of micropolar parameter K.