Numerical Study of the Influence of Heat Source on Stagnation Point Flow towards a Stretching Surface of a Jeffrey Nanoliquid

An analysis is carried out to study the flow of Jeffrey fluid near a stagnation point towards a permeable stretching sheet. In particular, we investigate the effect of temperature dependent internal heat generation or absorption in the presence of nanoparticles. The governing system of partial differential equations is transformed into ordinary differential equations, which are then solved numerically using the fourth-fifth-order Runge-Kutta-Fehlberg method. Comparisons with previously published work on special cases of the problem are performed and found to be in excellent agreement.The results of the governing parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.


Introduction
Investigations on boundary layer flow and heat transfer of non-Newtonian fluids are increasing substantially due to the large number of practical applications in industrial and manufacturing processes.Examples of such applications are drilling muds, plastic polymers, optical fibers, hot rolling paper production, metal spinning, cooling of metallic plates in cooling baths, and many others.In the past, investigators proposed different non-Newtonian models because a single model cannot predict all the features of non-Newtonian materials.There is one subclass of non-Newtonian fluids known as Jeffrey fluid which has attracted much attention from the researchers in view of its simplicity.This fluid model is capable of describing the characteristics of relaxation and retardation times [1,2].The investigation of flow due to a stretching sheet has been intentional because of this flow's various industrial applications, such as in the manufacturing of polymer sheets, filaments, and wires.During the manufacturing process, the moving sheet is assumed to stretch on its own plane, and the stretched surface interacts with the ambient fluid both mechanically and thermally.Stretching and shrinking can occur in a variety of materials each having a different strength, stretching transparency, and luster.Initially, Sakiadis [3] introduced the concept of a boundary layer flow over a stretching surface.Crane [4] modified the idea introduced by Sakiadis and extended this idea for both linear and exponentially stretching sheets.
Flow in the neighborhood of a stagnation point in a plane was first studied by Hiemenz [5] and Mahapatra et al. [6][7][8] investigated the magnetohydrodynamic stagnation point flow towards a stretching sheet; they have shown that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity.Also they have studied the temperature distribution when the surface at constant temperature and constant heat fluxes.Further they have extended their work on power-law fluid and discussed the unique solutions of stagnation point flow of a power-law fluid towards a stretching surface.The study of heat source/sink effects on heat transfer is very important because its effects are crucial in controlling the heat transfer.Postelnicu et al. [9] examined the effect of variable viscosity on forced convection flow past a horizontal flat plate in a porous medium with internal heat generation, but in heat generation part they considered only space dependent heat source.Abel et al. [10] analyzed the non-Newtonian viscoelastic boundary layer flow of Walter's liquid B past a stretching sheet, taking account of nonuniform heat source.
Aforementioned studies were primarily concerned with the laminar flow of a clear fluid.In the recent past a new class of fluids, namely, nanofluids, has attracted the attention of the science and engineering community because of the many possible industrial applications of these fluids.Nanotechnology is an emerging science that is finding extensive use in industry due to the unique chemical and physical properties that the nanosized materials possess.These fluids are colloidal suspensions, typically metals, oxides, carbides, or carbon nanotubes in a base fluid.The term nanofluid was coined by Choi [11] in his seminal paper presented in 1995 at the ASME Winter Annual Meeting.It refers to fluids containing a dispersion of submicronic solid particles (nanoparticles) with typical length on the order of 1-50 nm.Kuznetsov and Nield [12] analytically studied the natural convective boundary layer flow of a nanofluid past a vertical plate.In a recent paper Khan and Pop [13] studied for the first time the problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid.Mustafa et al. [14] investigated the stagnation point flow of a nanofluid towards a stretching surface using homotopy analysis method.Alsaedi et al. [15] examined the influence of heat generation/absorption on the stagnation point flow of nanofluid towards a linear stretching surface.Rahman et al. [16] investigated the dynamics of the natural convection boundary layer flow of water based nanofluids over a wedge in the presence of a transverse magnetic field with internal heat generation or absorption with the help of Matlab software.Nandy and Mahapatra [17] analyzed the effects of velocity slip and heat generation/absorption on magnetohydrodynamic stagnation point flow and heat transfer over a stretching/shrinking surface and obtained the solution numerically using fourth-order Runge-Kutta method with the help of shooting technique.Different from a stretching sheet, it was found that the solutions for a shrinking sheet are nonunique.Makinde et al. [18] studied the combined effects of buoyancy force, convective heating, Brownian motion, and thermophoresis on the stagnation point flow and heat transfer of an electrically conducting nanofluid towards a stretching sheet under the influence of magnetic field.Effect of magnetic field on stagnation point flow and heat transfer due to nanofluid towards a stretching sheet have been investigated by Ibrahim et al. [19].Recently Ramesh and Gireesha [20] investigated the boundary layer flow of Maxwell fluid over a stretching surface with convective boundary condition in the presence of nanoparticles and heat source/sink effect.Nadeem et al. [21,22] reported the numerical solutions of non-Newtonian nanofluid flow over a stretching sheet using the Jeffrey fluid model.Further they obtained the analytic solution for nonorthogonal stagnation point flow of a non-Newtonian nanofluid towards a stretching surface with heat transfer; here they use the second-grade model.Nadeem et al. [22] studied the natural convection boundary layer flow over a downward-pointing vertical cone in a porous medium saturated with a non-Newtonian nanofluid in the presence of heat generation or absorption and they used power-law model.Some interesting recent investigations related to the topic are presented in [23][24][25].
In this paper, we study the behaviour of the stagnation point flow towards a stretching sheet with the effects of heat source/sink and suction in the presence of nanoparticles.Similarity transforms are presented for this problem, and nondimensionalized equations are addressed numerically.Graphical results for various values of the parameters are presented to gain thorough insight towards the physics of the problem.To the best of my knowledge, this problem has not been studied before.

Mathematical Analysis
Consider the flow of an incompressible Jeffrey fluid in the region  > 0 driven by a stretching surface located at  = 0 with a fixed stagnation point at  = 0.The stretching velocity   () and the ambient fluid velocity  ∞ () are assumed to vary linearly from the stagnation point; that is,   () =  and  ∞ () = , where  and  are constant as shown in Figure 1.The problem under consideration is governed by the following boundary layer equations of Jeffrey fluid and nanoparticles and heat generation or absorption are given by Nadeem et al. [21]: where  and V are the velocity components along the and -axes, respectively.Further, ,   ,   , ], , and  ∞ are, respectively, the thermal diffusivity, density of the base fluid, density of the particles, kinematic viscosity of the fluid, fluid temperature, and ambient fluid temperature. and  1 are ratios of relaxation to retardation times and retardation time,  0 is the dimensional heat generation/absorption coefficient,   is the Brownian diffusion coefficient,   is the thermophoresis diffusion coefficient, and   is the specific heat at constant pressure.Here  is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the ordinary fluid and  is the nanoparticle volume fraction.The associated boundary conditions for the present problem are where   () =  is the stretching sheet velocity,  > 0; this is known as stretching rate.  and   are the temperature of fluid and nanoparticles fraction at wall, and  ∞ is the ambient nanoparticle volume fraction.The specific forms of the stretching velocity and the surface temperature and concentration are chosen to allow the coupled nonlinear partial differential equations ( 1)-( 4) to be converted to a set of coupled, nonlinear ordinary differential equations by the similarity transformation where  is the similarity variable; , , and  are the dimensionless stream function, temperature, and concentration, respectively.The velocity components  and V in ( 6) automatically satisfy the continuity equation (1).In terms of (), (), and () the momentum equation ( 2), energy equation ( 3), and concentration equation ( 4) can be written as Here   () represents an ordinary derivative with respect to  and the corresponding boundary conditions in the nondimensional form are  =  0 ,   = 1,  = 1,  = 1 at  = 0,   → ,   → 0,  → 0,  → 0 as  → ∞. ( The dimensionless parameters in (7), (8), and ( 9) are  = / is the velocity ratio,  =  1 is the Deborah number,  0 =   /(]) 1/2 is suction parameter,  =   (  −  ∞ )/] is the Brownian motion,  =   (  −  ∞ )/] ∞ is the thermophoresis parameter, and Le = ]/  is the Lewis number.Pr = ]/ is the Prandtl number;  =  0 /    is the heat generation ( > 0) or absorption ( < 0) parameter.

Solutions for Some Special Cases
In the limiting case of  = 0,  = 0,  = 0, and  = 0 (i.e., in the absence of velocity ratio, Deborah number, ratio of relaxation to retardation times, and heat source/sink parameter ) our system of ( 7) and ( 9) reduces to those of Khan and Pop [13] (Newtonian nanofluid).
In the presence of velocity ratio parameter, when there is no Deborah number, ratio of relaxation to retardation times, and heat source/sink parameter, system of ( 7) and ( 9) reduces to those of Mustafa et al. [14] ( ̸ = 0,  = 0,  = 0, and  = 0).Further in the absence of melting effect, heat source/sink, and nanoparticles volume fraction, the equations are similar to the ones studied by Hayat [27].In the absence of nanoparticles, the stagnation point flow of Jeffrey fluid and heat transfer problem degenerates.In this case, the approximate numerical solutions for the velocity field, temperature, and concentration fields are obtained.

Result and Discussion
The nonlinear coupled ordinary differential equations ( 7)-( 9) that are subject to the boundary conditions (10) have been solved numerically using a fourth-fifth-order Runge-Kutta-Fehlberg method to obtain the missing values of   (0), −  (0), and −  (0) for some values of the governing parameters, namely, the velocity ratio parameter , heat source/sink parameter , Deborah number , ratio of relaxation to retardation times parameter , Brownian parameter , thermophoresis parameter , Lewis number Le, and Prandtl number Pr.In order to validate the numerical results obtained, we compare our results with those reported by Mahapatra and Gupta [7], Hayat et al. [28], and Ibrahim et al. [19] for various values of  as shown in Tables 1 and 2, and they are found to be in a favorable agreement.
The velocity profiles for different values of  are presented in Figure 2. It is found that when the stretching velocity is less than the free stream velocity ( > 1), the flow has a boundary layer structure, physically saying that the straining motion near the stagnation region increases so the acceleration of the external stream increases which leads to decrease in the thickness of the boundary layer with increase in .When the stretching velocity  of the surface exceeds the free stream velocity ( < 1) inverted boundary layer structure is formed and  = 1; there is no boundary layer formation because the stretching velocity is equal to the free stream velocity.The temperature and concentration profiles for different values of  with other parameters being fixed are presented in Figures 3 and 4, respectively.It can be seen from these figures that both () and () decrease with increase in .  of , whereas when  > 1 the velocity decreases with the increase of .Similar effects can be seen in Figure 6 for various values of  when  > 1, while the opposite effect can be seen when  < 1.This is because  is dependent upon  (retardation time); physically larger retardation time of any material makes it less viscous resulting in an increase in its motion.It is worth mentioning that the liquid-like behavior is associated with small Deborah number.However, large Deborah number signifies the solid-like behavior.The graph of velocity profile   versus  for different values of  0 is plotted in Figure 7.It is found that for a fixed value of  < 1 the velocity decreases with the increase of  0 .The velocity profiles tend asymptotically to the horizontal axis; the nondimensional velocities absorb maximum at the wall.It is a fact that suction stabilizes the boundary layer growth.At  > 1 velocity increases with the increase of  0 .Figure 8 presents temperature profiles for different values of .For  > 0 (heat source), it can be observed that the thermal boundary layer generates the energy, and this causes the temperature in the thermal boundary layer to increase   with increase in , whereas  < 0 (heat sink) leads to a decrease in the thermal boundary layer.
Figures 9 and 10 illustrate the variation of ,  with  for various values of .It is found that the increase in the value of  is to increase the () in the boundary layer, whereas in concentration boundary layer reduces as  increases which thereby enhances the nanoparticles concentration at the sheet.The graphs of  on the  and  profile are shown in Figures 11 and 12. From these plots, it is observed that the effect of increasing values of  is to increase the temperature and concentration profiles.Figure 13 displays the effect of Le on concentration profiles.It is noted that the concentration of the fluid decreases with increase of Le, physically due to the fact that mass transfer rate increases as Le increases.It also reveals that the concentration gradient at surface of the plate increases.
Temperature and concentration profiles for the selected values of Pr are plotted in Figures 14 and 15.The graph depicts that the temperature decreases when the values of Pr increase.This is due to the fact that a higher Pr fluid has relatively      low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness, and as a result temperature decreases.We note that , , Pr, and  have no influence on the flow field, which is clear from (7).From  Table 3 one can see that the skin friction coefficient   (0) is negative at  < 1. Physically, negative value of   (0) means the surface exerts a drag force on the fluid, and positive value means the opposite.This is not surprising since, in the present problem, we consider the case of a stretching sheet, which induces the flow.Negative value of   (0) means the heat flows   from the fluid to the solid surface.This is not surprising since the fluid is hotter than the solid surface.

Conclusions
In the present investigation, the influence of the different parameters on the velocity, temperature, and concentration profiles is illustrated and discussed.The numerical results  give a view towards understanding the response characteristics of the Jeffrey fluid in the presence of nanoparticles and heat generation/absorption.It is found that boundary layer is formed when  > 1; on the other hand inverted boundary layer is formed when  < 1.Some results of thermal characteristics at the wall are usually analyzed from the numerical results and the same are documented in Tables 3 and 4. Analyzing this table, it reveals that the effect

Figure 1 :
Figure 1: Schematic diagram of the flow.

Figure 2 :
Figure 2: Velocity profile for different values of .

Figure 5 Figure 3 :
Figure 3: Temperature profile for different values of .

Figure 4 :
Figure 4: Concentration profile for different values of .

Figure 5 :
Figure 5: Velocity profile for different values of .

Figure 6 :
Figure 6: Velocity profile for different values of .

Figure 7 :
Figure 7: Velocity profile for different values of  0 .

Figure 8 :
Figure 8: Temperature profile for different values of .

Figure 9 :
Figure 9: Temperature profile for different values of .

Figure 10 :
Figure 10: Concentration profile for different values of .

Figure 11 :
Figure 11: Temperature profile for different values of .

Table 1 :
Comparison of the values of skin friction coefficient   (0) for various values of velocity ratio .

Table 2 :
Comparison of the values of Nusselt number −  (0) for various values of velocity ratio .