Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium

Abstract This article presents the effect of nonlinear thermal radiation on three dimensional flow and heat transfer of fluid particle suspension over a stretching sheet. The combined effects of non-uniform source/sink and convective boundary condition are taken into consideration. The governing partial differential equations are transformed into ordinary differential equations using similarity variables, which are then solved numerically by using Runge Kutta Fehlberg-45 method with shooting technique. The influence of various parameters on velocity and temperature profiles are illustrated graphically, and discussed in detail. The results indicate that the fluid phase velocity is greater than that of the particle phase for various existing parameters.


Introduction
Heat transfer phenomenon due to suspended particles into the uid has important role in recent and advanced processes of industrial and engineering problems concerned with powder technology, sedimentation, rain erosion in guided missiles, combustion, atmospheric fallout, uidization, nuclear reactor cooling, electrostatic precipi- tation of dust, waste water treatment, acoustics batch settling and so forth. A study on fundamentals of dusty uid was made by Sa man [1]. Vajravelu and Nayfeh [2] have investigated the hydromagnetic ow of a dusty uid over a porous stretching sheet. Some more investigations on heat transfer process with uid particle suspension can be seen in the Refs [3][4][5][6][7][8][9]. Al-Rashed et al. [10] report the in uence of surface waviness on natural convection boundary layer ow of the two-phase dusty uid having compressible nature. Recently the e ect of thermal strati cation on MHD ow and heat transfer of dusty uid over a vertical stretching sheet embedded in a thermally strati ed porous medium in the presence of uniform heat source and thermal radiation has been numerically investigated by Gireesha et al. [11]. Thermal radiation plays an important role in manufacturing industries for the design of nuclear power plants and several engineering applications. Due to its vital applications, numerous researchers have paid their attention to thermal radiation e ect [12][13][14][15][16][17]. Mahanthesh et al. [18] investigated the Marangoni transport of dissipating SWCNT and MWCNT nano uids under the in uence of magnetic force and radiation. Further, it is worth to notice that the linear radiation is valid for small temperature di erence. But, for the larger temperature di erence nonlinearized Rosseland approximation is to be considered. Hayat et al. [19] initiated the tangent hyperbolic nano uid ow in the presence of nonlinear thermal radiation. The idea of nonlinear thermal radiation along with heat transfer phenomenon has recently been presented by so many researchers (see [20][21][22][23]). Recently, Prasannakumara et al. [24] studied the e ect of nonlinear thermal radiation on slip ow and heat transfer of uid particle suspension with nanoparticles over a nonlinear stretching sheet immersed in a porous medium.
Three dimensional ow has many applications in solar collectors, aeronautical engineering, science and technology, crude oil puri cation, magnetic material processing, geophysics and controlling of cooling rate, insulation engineering, grain storage devices, ground water pollution, puri cation process and petroleum reservoirs.
Wang [25] proposed three dimensional boundary layer ow induced by a stretching surface. The unsteady laminar boundary-layer ow of a viscous electrically conducting uid induced by the impulsive stretching of a at surface in two lateral directions through an otherwise quiescent uid has been studied by [26] Takhar et al. The three dimensional ow and heat transfer over a stretching surface has been carried out by Ahmad et al. [27]. Ahmad and Nazar [28] studied the hydromagnetic ow and heat transfer over a bidirectional stretching surface. Choudhury and Das [29] studied the viscoelastic e ect on free convective three-dimensional ow along with the phenomenon of heat and mass transfer. Nadeem et al. [30] examined the MHD three dimensional Casson uid ow past a porous linearly stretching sheet. MHD three dimensional ow of couple stress uid was studied by Ramzan et al. [31].
Convective ow in porous media has been widely studied in the recent years due to its wide applications in engineering as geophysical thermal and insulation engineering, the modeling of packed sphere beds, the cooling of electronic systems, groundwater hydrology, chemical catalytic reactors, ceramic processes, grain storage devices, ber and granular insulation, petroleum reservoirs, coal combustors, ground water pollution and ltration processes, to name just a few of these applications. Chamkha et al. [32,33] studied the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nano uid. Chamkha et al. [34] have analyzed the boundary layer analysis for the mixed convection past a vertical wedge in a porous medium saturated with a power law type non-Newtonian nano uid. Natural convection boundary-layer ow over a permeable vertical cone embedded in a porous medium saturated with a nano uid in the presence of uniform lateral mass ux was presented by Chamkha et al. [35,36].
To the best of author knowledge, until now, the ow and heat transfer of dusty uid past a stretching sheet along with porous media in the presence of nonlinear thermal radiation and non-uniform heat source/sink has never been studied. The numerical solutions are obtained by applying RKF 45 method along with shooting technique.

Mathematical Formulation
Consider a steady three dimensional boundary layer ow of an incompressible dusty uid over a horizontal stretching sheet embedded in a porous medium. The sheet is aligned with the xy-plane (z = ) and the ow takes place in the domain z > . Let uw = cλx and vw = cy be the velocities of the stretching sheet along the x− and y−directions respectively where c is the stretching rate and λ is the coe cient which indicates the di erence between the sheet velocity components in x and y directions. The particles are taken to be small enough and of sucient number and are treated as a continuum which allow concepts such as density and velocity to have physical meaning. The dust particles are assumed to be spherical in shape, uniform in size and mass, and are undeformable. The coordinate system and ow regime are illustrated as in Figure 1. The boundary layer equations of three dimensional incompressible dusty uid are stated as [8]; with boundary conditions as [8]; where (u, v, w) and (up , vp , wp) denote the respective velocity components of the uid and dust phases along the x, y and z−directions. ρ and ρp are the density of the uid and dust phase respectively. k, ν, cp and cm are thermal conductivity, kinematic viscosity, the speci c heat of uid and dust phase, respectively. τ T is the thermal equilibrium time i.e., the time required by the dust cloud to adjust its temperature to the uid, τv is the relaxation time of the of dust particle i.e., the time required by a dust particle to adjust its velocity relative to the uid. T and Tp represents the temperatures of the uid and dust particles inside the boundary layer respectively. Throughout the study, it is assumed that, cp = cm. In deriving these equations, the drag force is considered for the interaction between the uid and particle phases. h f is the convective heat transfer coe cient, T f is the convective uid temperature below the moving sheet. Further, q is the space and temperature dependent internal heat generation/absorption (nonuniform heat source/sink) which can be expressed as; where A * and B * are the parameters of the space and temperature dependent internal heat generation/absorption. It is to be noted that, A * and B * are positive for internal heat source and negative for internal heat sink.
Introduce the following similarity transformations to reduce the partial di erential equations in to set of ordinary ones; Here prime denotes di erentiation with respect to η. Making use of (12) in equations (1) to (7), continuity equations (1) and (4) are identically satis ed and the remaining momentum equations take the following form: Corresponding boundary conditions becomes; At η = : f = , f = , g = g = .
A η → ∞ : where, β = /cτv is the uid-particle interaction parameter, H = ρp /ρ is the relative density and kp = v k c is the permeability parameter and ω is the density ratio and is considered as 0.2 in this present study.

. Heat transfer solution
To transform the energy equations into a non-dimensional form, dimensionless temperature pro le for the clean uid and dusty uid are introduced as follows: where, T∞ denotes the temperature at larger distance from the wall with T = T∞( + (θw − ) θ) and θw = T f T∞ being the temperature ratio parameter. Making use of equations (12) and (20) in equations (8) and (9) the energy equations takes the following form: [G + (λ + ) where, Rd = σ * T ∞ kk * is the radiation parameter, Pr = are the Eckert numbers and βτ = τ T c is the uid-particle interaction parameter for temperature. The corresponding boundary conditions take the following form; η → ∞ : θp = θ = .
where B i = ν c h f k is the Biot number. The wall shear stress is given by, The friction factor is written as, Cfx Re x = −f ( ) , Cfy Re y = −g ( ) .
The surface heat transfer rate is given by, The local Nusselt number is written as

Numerical solution
The set of non-linear di erential equations (13)(14)(15)(16)(17)(18) and (21)(22) with boundary conditions (19) and (23) have been solved using Runge-Kutta-Fehlberg fourth-fth order method along with shooting technique. In the rst step, equations of higher order are discretized to a system of simultaneous di erential equations of rst order by introducing new dependent variables. Missed initial conditions are obtained with the help of shooting technique. Afterward, a nite value for η∞ is chosen in such a way that all the far eld boundary conditions are satis ed asymptotically. Our bulk computations are considered with the value at η∞= 5, which is su cient to achieve the far eld boundary conditions asymptotically for all values of the parameters considered.

Results and discussion
Numerical computation has been carried out in order to study the in uence of di erent parameters that describes the ow and heat transfer characteristics of dusty uid. The results are presented and discussed in detail through graphical representation and tables.  The dimensionless velocity pro les for di erent values of kp proportional to u and v velocity components are depicted in Figure 2 and 3, respectively for both uid and dust phase. Here f (η) represents the velocity in x−direction while f (η) + g (η) is the velocity along  y−direction for uid phase. Further, F(η) represents the velocity in x−direction while [F (η) + G (η)] is the velocity in y−direction for dust phase respectively. From these gures, it is observed that both the velocities are retarded for the case of increasing kp. This is because, an increase in permeability parameter causes the resistance to the uid motion and hence velocity decreases for both uid and dust phase. The e ect of increasing values of permeable parameter contributes to the thickening of thermal boundary layer, and is shown in Figure 4. This is evident from the fact that, the porous medium opposes the uid motion. The resistance o ered to the ow is responsible in enhancing the temperature.
The e ect of various values of λ for dimensionless velocity pro les are depicted in Figure 5 and 6 for both uid and dust phase. These gures portraits that, the behavior  of the f (η) , F(η), f (η) + g (η) and F(η) + G(η) are the same and it decreases with increase in values of λ. It can also be seen that, uid phase velocity is greater than dust phase velocity. The variation of dimensionless temperature pro le for both phases along with di erent values of λ, are illustrated in Figure 7. From this gure it is seen that, increase of λ causes decrease of the temperature pro le of both dusty and uid phase. Furthermore, one can observe from this gure that values of the temperature are higher for clean uid than for the dusty uid at all points, as excepted. Figures 8 and 9 are plotted to view the e ect of uid particle interaction parameter β on velocity pro le respectively in both directions for uid and dust phase. From these gures it is examined that, the velocity u, v decays while up , vp is enhanced for larger values of β.  The variation of dimensionless temperature proles for di erent values of space-dependent heat source/sink parameter A * and temperature-dependent heat source/sink parameter B * are plotted in Figures 10  and 11 respectively. It is evident from these graphs that, increasing A * and B * results in the enhancement of both uid phase and dust phase temperature. Figure 12 and 13 characterize the temperature pro les for distinct values of uid and thermal particle interaction parameter β and βτ respectively. Figure 12 indicates that the temperature of both phases increases with increases in β and it reveals that, e ect of variation of β is more sensible on dusty phase than for the uid phase. This is because of the direct e ect of β on velocity and since the temperature depends on velocity then the temperature varies with  variation of β. E ect of βτ on uid and dust phase temperature pro les for both linear and nonlinear thermal radiation cases are shown in Figure 13. It can be seen that uid phase temperature decreases and dust phase temperature increases with increase in βτ for both the cases. It can also be seen that nonlinear radiation is more in uential than linear radiation. Figures 14 and 15 illustrate the e ect of the Eckert number (Ec) on temperature distribution. These gures show that the increasing values of Eckert number is to increase the temperature distribution. This is due to the fact that, heat energy is stored in the liquid due to frictional heating which results into increasing in its temperature and this is true in both the cases. Figure 16 resemble the change in temperature pro le for di erent values of Biot number (Bi). It describes that        the increasing values of Biot number lead to elevated temperature and thicker thermal boundary layer. This is due to the fact that, the convective heat exchange at the surface leads to enhance the thermal boundary layer thickness. Figure 17 is sketched for the temperature distribution against the Prandtl number (Pr). From this gure, it reveals that the temperature decreases with increase in the value of Pr. This is because, as the Prandtl number increases, thermal di usivity decreases there by decreases the temperature. Hence Prandtl number can be used to increase the rate of cooling. Figure 18 exhibits the variation of temperature pro le for the radiation parameter. As, the radiation parameter releases the heat energy into the ow, with an increase of radiation parameter, temperature pro le increases.
The e ect of temperature ratio parameter (θw) over the dimensionless temperature is shown in Figure 19. It is observed that, the temperature pro le increases for both phases for the increasing the values of θw. This is because, uid temperature is much higher than the ambient temperature for increasing values of θw , which increases the thermal state of the uid. From the obtained result it can be concluded that, non linear radiation has more in uence as compared to linear radiation.

Conclusion
In the present study, three-dimensional boundary layer ow and heat transfer of a dusty uid towards a stretching sheet embedded in a porous medium in presence of non-uniform source/sink with convective boundary condition is investigated. By using the appropriate transformation for the velocity and temperature, the basic equations governing the ow and heat transfer were reduced to a set of ordinary di erential equations. These equations are solved numerically using the fourth-fth-order Runge-Kutta-Fehlberg method. Some of the conclusion obtained from this investigation are summarized as follows: 1. Fluid phase velocity is always greater than that of the particle phase. 2. Velocity of uid and dust phases decrease with increases in permeability parameter. 3. Velocity and temperature pro le of uid phase and dust phases decrease with increases in velocity ratio parameter λ. 4. Increase of β will decrease uid phase velocity and increases dust phase velocity. 5. Increase of β T will decrease uid phase and increases the dust phase of temperature pro le. 6. Fluid phase temperature is higher than the dust phase temperature. 7. Temperature pro les of uid and dust phases increases with the increase of the Ec, A * , B * , Bi, Rd and θw.