Numerical Solution of Stagnation Point Flow Of Nano Fluid Due To an Inclined Stretching Sheet.

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1 ISSN (e): Volume, 08 Issue, February 018 International Journal o Computational Engineering Research (IJCER) Numerical Solution o Stagnation Point Flow O Nano Fluid Due To an Inclined Stretching Sheet. Yasin Abdela 1, Bandari Shankar, T.Srinivasulu 3 1 Department o Mathematics, O.U, Hyderabad, , India.. Proessor o Mathematics, University College o Science, O.U, Hyderabad, , 3 Departments o Mathematics, O.U, Hyderabad, , India. Corresponding author: Yasin Abdela ABSTRACT This paper deals with numerical solution o stagnation point low o nanoluid due to an inclined stretching sheet. The governing partial dierential equations are transormed in to a system o coupled ordinary dierential equations using similarity transormations. The resulted ordinary dierential equations were solved using the ourth order Runge-Kutta method with shooting technique. The various parameters in the system such as the Grasho number Gr, the solutal Grasho number Gc, the angle o inclination, the Prandtle number Pr, the Lewis number Le, the Brownian motion parameter Nb, the thermophoresis parameter Nt, the skin riction C,the local Sherwood number Sh, the local Nusselt number Nu were determined or dierent cases and analyzed. On the analysis it was ound that the roles o the parameters, Gr, Gc and Pr on the dynamics o the cooling liquid due to an inclined stretching sheet were qualitatively similar except or the Pr that has an opposite eect to that o the above parameters on the low and The temperature thermal boundary layer thickness appreciably increased in line with the Brownian motion and thermophoresis eects strengthened. Apart rom that the volume raction o the nonparticles was increased while the thermophoresis eect intensiied. Key words: Inclined stretching sheet, Stagnation point low, Nanoluid, Runge-Kutta method, Brownian motion parameter, Nusselt Number Skin riction. Nomenclature a Velocity o the stretching surace b Free Stream velocity T Temperature o the luid in the boundary layer C Concentration o the luid in the boundary layer T w Stretching sheet temperature C w Stretching sheet concentration T Ambient luid temperature C Ambient luid concentration u Velocity component along the x- axis v Velocity component along the y- axis u w Velocity component at the wall v w Velocity component at the wall Kinematic viscosity The eective heat capacity o the Nano particle to the heat capacity o the luid ( ) Eective heat capacity o Nano particle material c p ( ) Eective heat capacity o the Fluid D B c Thermal diusivity Brownian diusion coeicient Open Access Journal Page 1

2 D T k Gr Gc Pr Nb Nt Le C x Nu x Sh x Thermophoresis diusion coeicient Thermal conductivity The velocity ratio parameter The angle o inclination Local Grasho number Local Solutal Grasho number Prandtl number Brownian motion parameter Thermophoresis parameter Lewis number Local skin riction Local Nusselt number Local Sherwood number I. INTRODUCTION The study o stagnation point low o nanoluid over stretching sheet has various applications in manuacturing industries and technological process. Such as the polymer processing o chemical engineering plant metallurgy or the metal processing o glass iber, paper production, plastic sheets, liquid ilms in the condensation process ilament extrusion rom a dye, artiicial ibers, hot rolling in metal and polymer processing industries and others. Based on the importance and applicability o the low many researchers have been involving in the investigation o the behaviors o the low. Hazem [1] has analyzed the eect o the strength o the uniorm magnetic ield, the surace stretching velocity and the heat generation/absorption coeicient on both the low and heat transer. Sin Wei Wong et al. [] have investigated the behaviors o the low and characteristics o heat transer and witnessed that dual solution exists or the shrinking case where as the solution is unique or the stretching case. Noriah and Anuar [3] the variation o the skin riction coeicient and the heat transer rate at the surace with the governing parameters have been studied. Suali e al. [4] have investigated The unsteady two dimensional stagnation point ad heat transer over a stretching/shrinking sheet with prescribed surace and the eect o the governing parameters have been examined. Muhammad et al. [5] the stagnation point low over stretching have been investigated and the eatures o the low and heat transer characteristics or dierent parameters with various values have been considered. Aliereza et al. [6] have Studied that the steady two dimensional MHD stagnation point low due to permeable stretching sheet with chemical reaction and have examined the eect o the dierent parameters involved in the three proiles Krishnendu [7] has examined the role o non uniorm heat lux on the heat transer in boundary layer stagnation point low over shrinking sheet and has concluded that the direct and inverse variations o heat lux along the sheet have completely dierent eects on the temperature distribution. Aurangzaid and Sharidan [8] the unsteady MHD low, heat and mass transer in micro polar luid near the orward stagnation point having thermophoresis has been investigated with suction or injection eects. Krishnendu [9] has presented the eatures o the low and characteristics o heat transer o MHD stagnation point low o electrically conducting casson luid and heat transer towards a stretching sheet. Meraj et al. [10Have studied that stagnation point low o nanoluid past an exponentially stretching sheet the dierent parameters have been examined and results have been presented. Krishnendu [11] has witnessed that the range o velocity ratio parameter or which similarity solution exists is unaltered or any change in casson parameter though the skin riction changes with casson parameter and concluded that the possibility o similarity solution or casson luid low is the same as that o the Newtonian luid low. Sin Wei Wong et al. [1] have investigated the low and heat transer characteristics o stagnation point low o an incompressible viscous luid towards a vertical stretching sheet and or dierent values o the governing parameters have been studied and analyzed. Sarajimma and Vasundhara [13] the low,heat and mass transer characteristics o MHD casson luid in parallel plate channel with stretching walls subject to a uniorm transverse magnetic ield have been studied and the dierent parameters have been examined. Emmanuel et al. [14The low over a vertical porous surace o casson luid having chemical reaction along with the existence o magnetic ield has been investigated and they have examined and witnessed the eects o the governing parameters involved. Hayat et al. [15] have reported that the low and heat transer o the third grade luid due to exponentially stretching surace along with the eects o the inclined magnetic ield. NurFatihah et al. [16] The eect o slip on the low o stagnation point due to permeable shrinking sheet has been studied and the eect o the dierent parameters have been checked and reported. Subhas and Jayashree [17] have reported the heat transer rom a warm laminar liquid to a melting stretching sheet o the steady two dimensional stagnation point low and concluded the eect o the dierent Open Access Journal Page

3 parameters included. Rashidi et al.[18] The stagnation point low in porous medium over a permeable stretching surace along with the heat generation/absorption having convective boundary condition has been examined and the results has been reported. Santosh [19] The low o steady two dimensional and laminar due to a stretching sheet having heat generation in electrically conducting incompressible viscous luid with porous medium has been studied. Syahira et al.[0] have examined the MHD stagnation point low o nanoluid due to a permeable stretching/shrinking sheet and have reported the eect o the governing parameters. El-Sayed et al. [1] The steady two dimensional boundary layer MHD stagnation point low along with heat transer o an incompressible micro polar luid due to a stretching sheet with the existence o radiation, heat generation and absorption has been investigated. Dulapal et al.[] have studied the bouncy-driven radiative non-isothermal heat transer in nanoluid stagnation point low due to stretching/shrinking sheet with a porous medium and eects o thermal radiation and internal heat generation/absorption along with suction/injection at the boundary layer are also analyzed. Khairy and Anuar [3] The stagnation point low and heat transer over a stretching vertical sheet along with the eect o partial slip and buoyancy parameters on velocity, temperature, skin riction coeicient and local Nusselt number has been investigated and the results have been reported. Mageswari and Nirmala [4] have investigated the stagnation point low due to a stretching sheet with Newtonian heating and shown the power o the Laplace Adomain Decomposition method in approximating the solution o non linear dierential equations. Majeed et al.[5] the unsteady two dimensional mixed convection stagnation point low due to a vertical stretching cylinder with sinusoidal wall temperature has been studied and the governing parameters has been examined. Monica et al. [6] have investigated the stagnation point low o Williamson luid due to a non linear stretching sheet with thermal radiation and reported the eects o the parameters involved. FerozAhmed et al. [7] the eect o velocity slip on the stagnation point low due to porous stretching sheet has been examined and the inluence o the parameters included has been presented. Madasi et al. [8] the eect o slip boundary condition and chemical reaction on a steady MHD stagnation point low o a nanoluid over a non linear permeable stretching sheet has been studied and the results o the eects o the governing parameters has been reported. Sirinivasulu et al. [9] the eect o viscous dissipation on MHD stagnation point low o casson luid due to a linear stretching sheet has been investigated and the eects o the parameters involved has been presented. Abuzar et al. [30] have studied the eect o radiation and convective boundary condition on the enhancement o thermal conductivity o elastic viscous luid illed with nanoparticles and the low is considered impinging obliquely in the region o oblique stagnation point on the stretching surace. Vajravelu et al. [31] the mixed convective low o casson luid due to a vertical stretching sheet with variable conductivity has been studied and the governing parameters have been examined as well as the results have been reported. Eswara and Srenadh [3] the steady two dimensional MHD convective boundary layer low o casson luid due to a permeable stretching surace in the existence o thermal radiation and chemical reaction has been investigated along with the eects o velocity, thermal, solutal slips, thermal radiation chemical reaction and suction/blowing. The authors know that the problem o stagnation point low o nano luid due to an inclined stretching sheet is not yet investigated. II. MATHEMATICAL FORMULATION We consider the laminar boundary layer low o nanoluid in the region o the stagnation point towards an inclined stretching sheet driven by a pressure gradient. The sheet stretched along the x-axis by the action o two L equal and opposite orces as shown in Fig1 below, with velocityu w U0e. The sheet velocity varies as an exponential unction o the distance rom the origin where the slit is situated. L and U 0 are the reerences or w x L length and velocity respectively. Let T T Ae, C C Be be the temperature and the nanoparticle concentration at the sheet where T and C are the ambient temperature and concentration respectively. The boundary layer equations governing the conservation o mass, momentum, and energy and nanoparticle volume raction are: w x x L Fig.1: Schematic o the inclined stretching sheet problem. Open Access Journal Page 3

4 u v 0 x y u u u u v g ( )sin ( )sin T T T gc C C x y y D u v D T T T C T T T [ ( ) ] B x y y y y T y C C C DT T u v DB (4) x y y T y With boundary conditions x x x L L L u U w U0e, v 0, T Tw T Ae, C Cw C Be at y 0 (5) u 0, T T C C as y, Where u and v are the velocity components along x and y directions respectively, is the kinematic viscosity, g is the acceleration due to gravitation, is the coeicient o thermal expansion, T (1) () (3) C is the solutal expansion coeicient is the angle o inclination, is the thermal diusivity, is the eective heat capacity o nanoparticle to the heat capacity o the luid i.e. ( c) p, B ( c) thermophoresis diusion coeicient. Now let us introduce the ollowing dimensionless variables 0 D is the Brownian motion coeicient, D is the 1 1 ( x, y Re ) ( u, v Re T T C C ( X, Y), ( U, V ) T, C (6) L U T T C C w w T UL 0 Re is the Reynolds number. With the help o the above dimensionless variables, the governing equations take the orm U V 0 X Y U U U X X U V e TGr sin e CGc sin X Y Y T T 1 T T C T U V Nb Nt( ) X Y Pr Y Y Y Y C C C Nt T U V Le X Y Y Nb Y Where g ( ) ( w ) T AL c p DB C C Gr, Pr, Le, Nb U D ( ) 0 ( ) D ( T T ) g BL Nt, c p T w C GC ( c ) U T 0 And the boundary conditions take the ollowing orm B c (7) Open Access Journal Page 4

5 X U e, V 0, T 1, C 1 at Y 0 U 0, T 0, C 0, as Y (8) Using the stream unction ( XY, ) such that U, V, that satisies the continuity equation in Y X the dimensionless equations (7) and the boundary conditions (8), we have 3 (, ) Y X X e TGr sin e CGc sin 0 3 Y ( X, Y) 1 T (, T) T C T Nb Nt( ) 0 (9) Pr Y ( X, Y) Y Y Y Le C (, C) Nt T 0 Y ( X, Y ) Nb Y X e, 0, T 1, C 1 at Y 0 Y X (10) 0, T 0, C 0, as Y Y By applying the similarity transormation below, the coupled partial dierential equations will be changed into a system o ordinary dierential equations. X Ye ( X, Y) e ( ), T( X, Y) ( ), C( X, Y) ( ) and (11) Substituting equations (11) into equations (9) and (10), we get ''' '' ' Gr sin Gc sin 0 1 '' ' Nb ' ' Nt ' 0 Pr Nt Le '' ' '' 0 Nb (1) With boundary conditions (0) 0, '(0) 1, (0) 1, (0) 1 at 0 '( ) 0, ( ) 0, ( ) 0, as Pr is the Prandtle number, Le is the Lewis Number, Gr is the local Grasho number, Gc is the local solutal Grasho number, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter. Since we know that Nb and Nt are unctions o x, the x-coordinate can t be deleted rom the second and third equations o (1), thereore we search or the variability o the local similarity solution that allows us to investigate the behavior o these parameters at aixed location above the sheet. The skin riction coeicient C, the local Nusselt number Nu and the local Sherwood number Sh are given by u T C x x y y y C, Nu, Sh u T T C C y0 y0 y0 w w w Substituting equations (11) into equations (14), the reduced orm will be X (13) (14) Open Access Journal Page 5

6 L L Nu Sh Re C ''(0), x '(0) Nur, x '(0) shr (15) Re 1 1 x Rex III. METHOD OF SOLUTION Using the ourth order Runge-Kutta method with shooting technique, we solve the coupled ordinary dierential equations (1), with their respective boundary condition equations (13) numerically. The system o the ordinary dierential equations (1) is highly non linear higher order dierential equations. To solve these equations, we transorm them into a system o irst order dierential equations by applying the ollowing denotations. Let (1), ' (), '' (3) (4), ' (5) (16) (6), ' (7) Using equations 16 in equations (1), we have ''' '' ' Gr sin Gc sin (1)* (3) * ()* () * Gr * (4)*sin * Gc* (6)*sin '' Pr[ ' Nb ' ' Nt ' ] Pr[(1)*(5) Nb*(7)*(5) Nt*(5)*(5)] 1 Nt '' [ ' ''] Le Nb 1 Nt [ (1)* (7) *[ Pr[ (1)*(5) Nb*(7)*(5) Nt*(5)*(5)]] Le Nb The boundary conditions are a (1) 0, a () 1, a(4) 1, a(6) 1 b() 0, b(4) 0 b(6) 0 where a 0and b (18) The boundary conditions ''(0), '(0) and '(0) equations. Thereore it is must or us to evaluate the approximate values o ''(0), '(0) and '(0) help o shooting technique and then we apply the ourth order Runge-Kutta method to obtain, and. We choose the step size to be h 0.01 with accuracy o (17) are not given in the system o ordinary dierential with the IV. RESULTS AND DISCUSSIONS The inluence o the dierent parameters included in the couple o ordinary dierential equations such as local Grasho number Gr, local solutal Grasho number Gc, the angle o inclination, the prandtl number Pr, the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis number Le on the stagnation point low o nanoluid due to an inclined exponentially stretching sheet has been tabulated and their graphs sketched as well as discussed. Open Access Journal Page 6

7 Table 1: Comparison table between the skin rictions o present and literature results or Nt=GC=Gr=A=0, Pr=Le=1 Skin riction or - ''(0) o Meraj [10] Present skin riction or - ''(0) Table : Comparison table o Nusselt and Shrwood Numbers where Nb=Nt=0.1, λ=0. Nu or Present Nu or -θ (0) Sh or -ϕ (0) o Meraj Present Sh or -ϕ (0) Pr -θ (0) o Meraj[10] In table 1 and above it is clearly indicated that the present results have an excellent agreement with the results reported by Meraj [10]. In table 1, the skin riction decreases, the velocity o the luid raises and the boundary layer thicken with the increment o as <1 that is the ree stream velocity is smaller than the velocity o the stretching sheet. Table above reveals that increasing the Prandtl number increases the reduced Nusselt number and decreases the reduced Sherwood number; it is due to the reduction o the thermal diusivity which has an impact on the reduction o the thermal boundary layer and the Sherwood number but an increment on the Nusselt number. Gr Gc Pr Nb Nt Le Table 3 Skin riction - "(0) Nusselt '(0) Number Sherwood Number '(0) Open Access Journal Page 7

8 The irst and ith parts o table 3 above have the same results as table 1 and respectively. In the second part o table 3 we observe that as Gr increases both the Nusselt and Sherwood numbers increase, it is because o, increasing Gr increases the buoyancy orce and the thermal boundary layer thickness decays. The third part o table 3 illustrates that both the Nusselt number and Sherwood numbers enhanced with the increment o Gc. The Nusselt number and the Sherwood number increase as is enlarged, this is because o at =0, the the sheet is erecting to the vertical sheet is horizontal and the buoyancy orce is decreased and as direction increased gravitational orce act on the low that enlarges the buoyancy orce and reduces the thermal boundary layer thickness is described in the ourth part o table 3. As Nb increases in the sixth part o table both Nusselt and Sherwood numbers decrease by the increment o the Brownian motion a large extent o luid enorced to become together so that the thermal boundary layer thickness increase. The seventh part o table 3 shows that with the increase o Nt the Nusselt number decreases and the Sherwood number increases, it is due to the act that the thermophoresis diusion penetrates deeper in to the luid with the increase o Nt. The last part o table 3 illustrates that increasing Lewis number reduces Nusselt number and increases Sherwood number. Figure : Variation o '( ) with the velocity ratio. Figure 3: The eect o Gr on '( ). Open Access Journal Page 8

9 Figure 4: The inluence o Gc on '( ). Figure 5: Variation o '( ) with. Open Access Journal Page 9

10 Figure 6: The eect o Gr on ( ) Figure 7: The inluence o on ( ). Figure 8: Variation o ( ) with Pr. Figure 9: Inluence o Nb on ( ). Open Access Journal Page 30

11 Figure 10: Eect o Nt on ( ). Figure 11: Variation o ( ) with Gc. Figure 1: The eect o on ( ). Open Access Journal Page 31

12 Figure 13: Inluence o Nb on ( ).. Figure 14: Eect o Nt on ( ). Figure 15: Variation o ( ) with Le. Figure illustrates that as increases in the interval (0,1) the velocity o the luid and the boundary layer b thickness increase as the velocity o the stretching sheet is greater than the ree stream velocity(i.e. <1), a or >1, as increases the low velocity increases and the boundary layer thickness decreases that is when Open Access Journal Page 3

13 b the ree stream velocity is larger than the velocity o the stretching sheet(i.e. >1). In the case where the a velocity o the stretching sheet is equal to the ree stream velocity, there is no boundary layer thickness o the nano luid near the sheet. The velocity increases with the enlargement o the Grasho and the solutal Grasho numbers due to the eect o enhanced buoyancy orce as indicated in Figure 3 and 4. An enlargement in the angle o inclination enhances the velocity ield as shown in Figure 5 because at =0, the sheet is horizontal and the buoyancy orce is decreased and as the sheet is erecting to the vertical direction increased gravitational orce act on the low that enlarges the buoyancy orce and increases the velocity boundary layer thickness.the temperature proile decrease with the increase o Grasho number as given in Figure 6 and it is because o the enhancement o buoyancy orce. In Figure 7 as the angle o inclination enlarged the thermal boundary layer diminished. Figure 8 reveals that the thermal boundary layer thickness reduces as the Prandtl number increases because the prandtl number increases as the thermal diusivity decreases. The increase in Brownian motion parameter increases the thermal boundary layer thickness as shown in Figure 9, This is due to the increased Brownian motion increases the collision o the luid molecules with the nano particles and this enorce a large extent o luid molecules to wedge together make the thermal boundary layer thicken. Figure10 shows that as the thermophoresis parameter increases the thermal boundary layer thicken since the increased thermophoresis diusion enorce the nano particles to shit rom hot to cold area which increases the thermal boundary layer thickness. It is observed rom Figure 11 that as solutal Grasho number increases the concentration boundary layer decreases because o the increased buoyancy orce. The concentration boundary layer thickness reduces with the enlargement o the angle o inclination as it is shown in Figure 1, this is because o at =0, the sheet is horizontal and the buoyancy orce is decreased and as the sheet is erecting to the vertical direction increased gravitational orce act on the low that enlarges the buoyancy orce and reduces the concentration boundary layer thickness. The concentration boundary layer diminishes as the Brownian motion parameter enhanced obtained rom Figure 13, the increased Brownian motion enable the nano particles to transer the surace heat to the luid and the nano particles gain higher kinetic energy that contributes to the thermal energy o the luid that cause the movement o the nano particles rom hot to the cold region. However the concentration boundary layer thickens with the increase o Nt as it is revealed in Figure 14, the nano particles move rom a region o high temperature to a region o low temperature with the enlargement o thermophoresis diusion. The molecular diusivity decays as the Lewis number increases and the concentration boundary layer becomes thinner as it is illustrated in Figure 15. V. CONCLUSIONS Stagnation point low o nanoluid due to an inclined stretching sheet is studied. The governing equations are numerically solved and the results obtained permit the computation o the low, heat and mass transer behaviors or various values o the velocity ratio, the Grasho number Gr, the solutal expansion parameter Gc, the Prandtl number Pr, the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis Number Le. The results have been ound that 1. The increase o the velocity ratio parameter, the Grasho number Gr, the solutal expansion parameter Gc and the angle o inclination increase the velocity boundary layer thickness.. The thermal boundary layer thickness increases with the increase o the Brownian motion parameter Nb and the thermophoresis parameter Nt but reduces with the increase o Prandtl number. 3. As the Prandtl number Pr and the thermophoresis parameter Nt increase the concentration boundary layer thickness increases. 4. Increasing the Brownian motion parameter and Lewis number reduces the concentration boundary layer thickness. REFERENCES [1]. Hazem Ali Attia (010): Steady Three Dimensional Hydro magnetic Stagnation Point Flow towards a stretching Sheet with Heat Generation. Italian Journal o Pure and Applied Mathematics. N (9-18). []. Sin Wei Wong, Md.Abdu Omar andanuar Ishak (011): Stagnation Point Flow over an Exponentially Stretching/shrinking sheet.z. Naturorsch. 66a, ,DOI: /ZNA [3]. Noriah Bachok and Anuar Ishak (011): Similarity Solution or Stagnation Point Flow and Heat Transer over a non-linearly Stretching /Shrinking sheet. Sains Malayisana 40(11) (011): [4]. M.Suali, N.M.Anik Long and A.Ishak (01): Unsteady Stagnation Point Flow and Heat Transer over a stretching /shrinking sheet with prescribed surace heat lux. App. Math. and Comp. Intel, Vol.1 (01) Open Access Journal Page 33

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