Effect of Thermal Radiation on a Three-dimensional Stagnation Point Region in Nanofluid under Microgravity Environment

The unsteady three dimensional boundary layer flow near a stagnation point region is studied numerically under the influence of microgravity environment. The boundary layer plate was embedded by the nanofluid with nanosized copper particles and water as a based fluid together with thermal radiation effect. The problem was mathematically formulated in term of coupled governing equations consisting of continuity, momentum and energy equations derived from the fundamental physical principles with Tiwari and Das nanofluid model. Boundary layer and Boussinesq approximation were then applied to the coupled equations and then reduced into non-dimensional equations to lessen the complexity of the problem using semi-similar transformation technique. Implicit finite different method known as Keller box method was used in this problem. The problem was then analyzed in terms of physical quantities of principal interest known as skin frictions and Nusselt number which explained the flow behavior and heat transfer analysis. From the outcome of the analysis, it was found that the parameter values for curvature ratio lead to the different cases of the stagnation point flow which is either plane stagnation flow or asymmetry stagnation flow. On the other hand, by increasing the nanoparticles volume fraction which is one of the nanofluid parameter may increase the skin frictions on both x − and y − directions. The presence of thermal radiation parameter was found to have increased the rate of change of heat transfer at the boundary layer flow.


Introduction
A part of fluid dynamic discipline which is concerned with the mechanic of the fluid and the extended force on them; boundary layer flow is a slice of bigger picture which studies transportation phenomena happen to the flow such as flow behavior, heat transfer and concentration distribution. A bunch of studies has been conducted to discover and get more understanding on this research field either experimentally [1][2][3][4][5][6] or theoretically [7][8][9][10][11]. Hence, boundary layer concept is the most successful in achieving simplification of the equation of motion and energy and has been applied to a large variety of practical situation importantly in engineering applications such as hot rolling, skin friction drag reduction, grain storage, glass fiber and paper production [12]. In general, the boundary layer concept provides a good description of the velocity and temperature field.
Stagnation point is a part of boundary layer flow which holds important properties as proven by Bernoulli principal is having the higher pressure on the flow [13]. For a past few decades, stagnation point flow has engrossed the attention of many researchers due to its growing industry and scientific applications such as cooling of electronic devices by fan, cooling of nuclear reactor during emergency shutdown and hydrodynamic process. Hiemenz [14] was the first person who has studied the two-dimensional forward stagnation point and reduced the Navier-Stokes equation to nonlinear ordinary differential equations. Then, Mahapatra and Gupta [15] focus on the heat transfer on steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet. The problem is then extended using non-Newtonian fluid by Nazar et al. [16] by studying steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching sheet.
Generally, there are three different ways of heat transfer, namely conduction, convection and radiation. Radiation is a form of electromagnetic energy transmission and it is independent from any medium between the emitter and the receiver. In addition, the studies of radiative heat transfer in a fluid flow is currently undergoing great enlargement of many researcher. Raptis,Perdikis and Takhar [17] have done an analysis on the steady MHD asymmetric flow of an electrically conducting fluid past a semi-infinite stationary plate with presence of thermal radiation. For the case of heat and mass transfer, Pal [18] analyses two-dimensional stagnation point flow of an incompressible viscous fluid over a stretching vertical sheet in the presence of buoyancy force and thermal radiation. Next, Bhattacharyya and Layek [19] add effect of suction/blowing on steady boundary layer stagnation-point flow over a porous shrinking sheet with the presence of thermal radiation.
Nanofluid is engineered by dispersing a small quantity of nanosized particles usually less than 100nm, which are uniformly and stably suspended into a conventional fluid called base fluid such as ethylene glycol, oil, water, bio-fluids and polymer solution [20]. Choi [21] was the first person who has developed this special class of fluid by elaborated it using Maxwell equation. For the boundary layer problem, there are two famous mathematical models which are proposed by Tiwari and Das [22] and Buongiorno [23] in studying the effect of nanofluid at boundary layer. Mustafa et al. [24] focus on the effect of Brownian motion and thermophoresis for two-dimensional stagnation-point flow of a nanofluid towards a stretching sheet. Haq et al. [25] on the other hand present a model for the stagnation point flow of nanofluid with magnetohydrodynamics (MHD) and thermal radiation effects passed over a stretching sheet.
On the other hand, g jitter effect can be defined as inertia effect due to quasi-steady, oscillatory or transient accelerations arising from crew motions and machinery vibrations in parabolic aircrafts, space shuttles or other microgravity environments. Theoretical studies on g-jitter indicated convection and temperature gradient on microgravity environment at boundary layer flow has also been conducted previously [26] and [27]. Theoretical study on the boundary layer flow induced by g-jitter has been conducted by Sharidan et al. [28] by considering g-jitter in a three-dimensional stagnation point flow problem with free convection flow. As for the nanofluid with g-jitter effect, Rawi et al. [29] studies the unsteady two-dimensional convective boundary layer flow of nanofluids past a vertical permeable stretching sheet associated with the effect of g-jitter.
Based on this motivation, a part of the present work focusses on the conjugate study of a flow and heat transfer with several physical effects. The nanofluid model used in this study is proposed by Tiwari and Das [22] and effects such as g-jitter and thermal radiation that is considered with boundary layer flow is studied near a three-dimensional stagnation point region. The mathematical model is solved using Keller box method and the effects considered in this problem will be analyzed in terms of skin frictions and Nusselt number graphically.

Mathematical Formulations
Consider an incompressible viscous nanofluid which unsteadily flows at the boundary layer near a three-dimensional stagnation point region with uniform temperatures at the wall. Thermal radiation effect is considered in this boundary layer problem and the flow is conducted under microgravity environment. Copper nanoparticles and water that carries their own thermophysical properties are considered in representing the presence of nanofluid in this flow problem. The temperature for both; boundary body, w T and ambient nanofluid, T ∞ are assume be in a uniform temperature initially. Three dimensional stationary orthogonal where the gravitational field represents in term of mean gravitational acceleration 0 g , ε is the amplitude of the gravitational modulation and ω is the frequency of oscillation driven from the g-jitter effect. From the fundamental of physical principle, the nature of the fluid flow near the boundary layer can be presented mathematically in terms of coupled equations. Under the boundary and Boussinesq approximations, the coupled partial differential equations that represent the flow are as follows: (2) 274 Effect of Thermal Radiation on a Three-dimensional Stagnation Point Region in Nanofluid under Microgravity Environment (4) ( ) The appropriate initial and boundary conditions are given by (6) where , u v and w are the velocity components along the directions , , x y z axes and T is the temperature of where * σ is Stefan-Boltzman constant and * k is the mean absorption coefficient. By using Taylor's series expansion and neglecting the higher order, the temperature difference within the flow 4 T about T ∞ can be expressed as 4 3 4   4 3 .
Substituting equation (8) into equation (7), then equation (5) Since water is used as the based fluid, which is a type of viscous Newtonian fluid, the expression of thermophysical characteristic of the nanofluid derived from Maxwell equation defined in Rawi et al. [29] in term of nanoparticles and the based fluid are (10) where φ is an important parameter which represents nanofluid properties stressed out by Tiwari and Das nanofluid model known as nanoparticles volume fraction and k hold the thermal conductivity parameter. The subscript f and s represent the fluid and solid characteristics carried by the nanofluid chosen in this research and the thermophysical properties are shown in Table 1. Equations (1)- (3) and (9) are then undertake semi similar transformation technique together with the initial and boundary conditions equation (6) to reduce the complexity of the problem with the following semi-similar solution (11) where υ is the kinematic viscosity of the base fluid and Grashof number is denoted by Gr and defined as where is the Prandtl number, 1 , 1 .

Results and Findings
The system of non-dimensional partial differential equations (12)- (14) together with boundary equations (15) are solved numerically using implicit finite different methods known as Keller box method which is firstly developed by Keller [32]. This method is widely used in solving non-linear parabolic problem either for two-dimensional [33][34] or three-dimensional problem [35][36]. This method is chosen since it seems to be most flexible of the common methods, being easily adaptable and unconditionally stable while achieves the exceptional accuracy. All the results are obtained by using a uniform On the other hand, since stagnation point parameter is presented by parameter , the different values chosen for parameter will lead to a different case flow near a stagnation point region. It is interesting to discuss that in Figure 1B; there are no changes in terms of magnitude for the skin friction on y − direction as parameter ε increases. The value 0 c = represents the specific geometry for the plane which is the sphere which leads to the plane stagnation case flow. In addition, there is another type of flow happening on Figure 3A and 3B when the magnitude of the skin friction for both directions area found the same as the parameter increases. Cylindrical geometry on Figure 3 caused by the chosen parameter value leads to the axisymmetric stagnation case flow. . It is found that,from the physical quantities of principal interest, there are some significant differences for different size of frequency of oscillation as parameter increases. The bigger size of provided a decreasing value of the peak variation response for skin frictions and Nusselt number. Generally, for the larger size frequency of oscillation, the convergence rate occurs faster than the smaller size of frequency of oscillation in the microgravity environment.
The effect of nanofluid is analyzed on Figure 5. Increasing values of parameter , both skin frictions and Nusselt number are also found to have increased. The purpose of nanofluid which is to enhance the thermal conductivity of conventional fluid is proven in this analysis when the Nusselt number results which represent the heat transfer at the boundary layer are found to increase as parameter increases. Corresponding with that, by adding nano size particles on the fluid, the frictions on the boundary layer are found to be increased due to the increases of resistance which disturb the flow. It will lead to the increases of skin friction on both directions and will decrease the velocity of the flow.
Thermal radiation Nr, effect is analyzed in this boundary layer flow and from Figure 6, skin friction for both directions together with Nusselt number are increased as Nr parameter increases. Physically, increases of Nr parameter will increase the thermal conductivity of the fluid and decreases viscosity of the fluid. As a result, better heat transfer in the fluid can be achieved by considering thermal radiation effect because of the increases of rate of energy transported in the fluid.

Conclusions
In this article, three-dimensional boundary layer nanofluid flow near a stagnation point region with thermal radiation induced by g-jitter was investigated numerically. The flow problem was govern into mathematical formulation based on physical principal together with considered effects and non-dimensional partial differential equations solved using implicit finite difference method known as Keller box method. The effect considered were then analyzed in terms of physical quantities of principal interest such as skin frictions and Nusselt number which were then presented graphically. It can be concluded that