Radiative Effects on Boundary-Layer Flow of a Nanofluid on a Continuously Moving or Fixed Permeable Surface

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3 178 Recent Patents on Mechanical Engineering 01, Vol. 5, No. 3 Chamkha et al. hich are the combinations of the traditional ones:, y Re f B = B = (7 Re x for the Blasius problem (stationary all and uniform free stream velocity and f, y = = Re (8 s s Re x from the Sakiadis (uniformly moving all ith stagnant free stream problem. The Reynolds numbers are defined as: u x ux Re =, Re = (9 A velocity ratio parameter is defined as u u 1 Re 1 = = (1 + = (1 + (10 u + u u Re Note that form the Blasius problem u = 0, therefore = 0. On the other hand, for the Sakiadis problem u = 0 and thus = 1. In addition, e also define dimensionless temperature and concentration function as T T C C =, = (11 qx / (Re mx / (Re and the radiation * 4 4 T q r = * 3k y T = T + 4 T( T T + 6 T ( T T +... ( T + 4TT 3 * qr 16T T = * y 3k y Using the transformation variables defined in equations (6-1, the governing transformed equations may be ritten as: 1 f + ff = 0 ( (1 + Rd + f + Nb + Nt = 0 Pr 3 ( Nt f = 0 Le Le Nb (15 The transformed initial and boundary conditions become: f (0 = f, f (0 = ±, (0 = 1, (0 = 1, (16 f ( = 1, ( = ( = 0 here Eq. (1 is identically satisfied. In Eqs. (13-16, a prime indicates differentiation ith respect to and the parameters Dq T (Re Nt = T( u + u * 3 4 T R d = *, Pr =, Le k k DBm (Re, Nb =, ( u + u (Re f = v. ( u + u (17 =, D Here, Pr, Rd, f, Le, Nb and Nt denote the Prandtl number, the radiation parameter, suction/injection parameter, the Leis number, the Bronian motion parameter and the thermophoresis parameter respectively. It is important to note that this boundary value problem reduces to the classical problem of flo and heat and mass transfer due to a the Blasius problem hen Nb and Nt are zero. Most nanofluids examined to date have large values for the Leis number Le > 1 (see Nield and Kuznetsov [7]. For ater nanofluids at room temperature ith nanoparticles of nm diameters, the Bronian diffusion coefficient D B ranges from to m - /s. Furthermore, the ratio of the Bronian diffusivity coefficient to the thermophoresis coefficient for particles ith diameters of nm can be varied in the ranges of - for alumina, and from to 0 for copper nanoparticles (see Buongiorno [8] for details. Hence, the variation of the non-dimensional parameters of nanofluids in the present study is considered to vary in the mentioned range. Of special significance for this type of flo and heat transfer situation are the local skin-friction coefficient, local Nusselt number and the local Sherood number. These physical parameters can be defined in dimensionless form as: 3/ Cf Re = (1 f (0, (18 3/ Cf Re = f (0 4 (1 + Rd Nu /Re 3 x = (19 (0 Sh /Re = x (0 ( RESULTS AND DISCUSSIONS In order to get a clear insight of the physical problem, numerical results are displayed ith the help of graphical and tabulated illustrations. The system of equations (13-15 ith the boundary conditions (16 ere solved numerically by means of an efficient, iterative, tri-diagonal implicit finite-difference method discussed previously by Blottner [9]. Since the changes in the dependent variables are large in the immediate vicinity of the plate hile these changes decrease greatly as the distance aay from the plate increases, variable step sizes in the direction ere used. The used initial step size in the direction as 1 = 01 and the groth factor as K = 375 such that n = K n-1. This indicated that the edge of the boundary layer = 35. The convergence criterion used as based on the relative difference beteen the current and the previous iterations hich as set to 10-5 in the present ork. It is possible to compare the results obtained by this numerical method ith the previously published ork of El-Kabeir et al. [18]. Table 1 shos that excellent agreement beteen the results exists. This lends confidence in the numerical results to be reported subsequently. Computations ere carried out for various values of parameters at Pr = 0.7 and Le = 10. The results of this parametric study are shon in Figs. (-15.

4 Boundary-Layer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No Table 1. Comparison of Skin Friction f (0 for Various Values of Velocity Ratio. EL-Kabeir et al. [18] Present Results Figures -4 display the effect of velocity ratio on the velocity, temperature and volume fraction profiles, respectively. It can be seen that, the velocity of the fluid increases as velocity ratio increases in cases < 0.5 and decreases in case > 0.5, on the other hand the temperature of the fluid increases as velocity ratio increases, hile nanoparticle volume fraction decreases as velocity ratio increases. Figures 5 & 6 present the effect of thermophoresis parameter N t and Bronian motion parameter N b on the temperature and nano-particle volume fraction, respectively. It can be seen that, temperature and nanoparticle volume fraction of the fluid increase as thermophoresis parameter N t and Bronian motion parameter N b increase. 6 Fig. (1. Physical model and coordinates system. f'( Nb=0.3,Nt=0.3,,,, = =0. = = = = Fig. (. Velocity profiles ith various values of velocity ratio parameter. θ( ,0.,,,, Nb=0.3 Nt=0.3 = Fig. (3. Temperature profiles ith various values of velocity ratio parameter. Figures 7 & 8 display the effect of thermophoresis parameter N t and Bronian motion parameter N b on local Nusselt number and local Sherood number respectively. It can be seen that the thermophoresis parameter Nt appears in the

5 180 Recent Patents on Mechanical Engineering 01, Vol. 5, No. 3 Chamkha et al ,0.,,,, =,,, Nb,Nt=0.1,0.3,0.5,0.7 φ( Nb=0.3 Nt= = (1+4 /3/θ( Fig. (4. Volume fraction profiles ith various values of velocity ratio parameter. Fig. (7. Local Nusselt number ith various values of Nb and Nt =,,, Nb,Nt=0.1,0.3,0.5, θ( 3 1/φ(0 0.9 =0.5 1 Nb,Nt=0.1,0.3,0.5,0.7,0.9 = Fig. (5. Temperature profiles ith various values of Nb and Nt. φ( Nb,Nt=0.1,0.3,0.5,0.7,0.9 =0.5 = Fig. (6. Volume fraction profiles ith various values of Nb and Nt. thermal and concentration boundary layer equations. As e note, it is coupled ith the temperature function and plays a strong role in determining the diffusion of heat and nanoparticles concentration in the boundary layer. Moreover, In nanofluid systems, oing to the size of the nanoparticles, Fig. (8. Local Sherood number ith various values of Nb and Nt. θ( Fig. (9. Temperature profiles ith various values of and. Bronian motion takes place, and this can enhance the heat transfer properties. This is due to the fact that the Bronian diffusion promotes heat conduction. The nanoparticles increase the surface area for heat transfer. A nanofluid is a to phase fluid here the nanoparticles move randomly and increase the energy exchange rates. Hoever, the Bronian motion reduces nanoparticles diffusion. The increase in the =-0.,-0.1,,0.1,0. =.0 =0.5 Nb=0.3 Nt=0.3

6 Boundary-Layer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No local Sherood number as Nb changes is relatively small. Therefore, as indicated before, increasing the Bronian motion parameter N b and the thermophoresis parameter N t cause increasing the temperature and volume fraction profiles. This yields reduction in the local Nusselt number and local Sherood number. Figures 9 & 10 sho the representative of temperature and nanoparticles volume fraction profiles for different values of suction/injection and radiation R parameters, respectively. It can be observed that the increasing the value of suction/injection parameter causes decreasing in both temperature and volume fraction profiles, hile radiation R leads to a significant change in temperature and slight change in volume fraction, as radiation parameter increases the temperature of the fluid increases and volume fraction decreases. Cf Re =,Nb=0.3,Nt=0.3,, =-0.,-0.1,,0.1, Fig. (11. Local skin-friction coefficient ith various values of. C Re φ( =-0.,-0.1,,0.1,0. =.0 =0.5 Nb=0.3 Nt= (1+4 /3/θ( =,Nb=0.3,Nt=0.3,, =-0.,-0.1,,0.1,0. Fig. (10. Volume fraction profiles ith various values of and. Figure 11 displays the effect of velocity ratio and suction/injection parameters on the values of local skin friction coefficient, It can be seen that, local skin friction decreases as velocity ratio increases in cases < 0.5 and increases in case > 0.5, hile local skin friction coefficient increases as suction/injection parameter increases. Finally, Figs. (1-15 sho the effect of suction/injection and radiation parameters on local Nusselt number and local Sherood number, respectively. It can be seen that increasing the value of suction/injection and radiation parameters led to increasing in both local Nusselt number and local Sherood number. 4. CURRENT & FUTURE DEVELOPMENTS The objective of the present ork as to study the effect of thermal radiation on boundary layer flo of an incompressible nanofluid floing over a permeable uniform heat flux surface moving continuously. Some related patents for the preparation of heat transfer nanofluids are highlighted. The model used for the nanofluid incorporated the effects of Bronian motion and thermophoresis. The governing boundary layer equations ere solved numerically using an efficient, implicit finite-difference method. The local skinfriction coefficient, local Nusselt number and the local Fig. (1. Local Nusselt number ith various values of. 1/φ( =,Nb=0.3,Nt=0.3,, =-0.,-0.1,,0.1, Fig. (13. Local Sherood number ith various values of. Sherood number as ell as the temperature, concentration and velocity distributions for various values of the velocity ratio, suction/injection parameter, radiation parameter and nanofluid parameters ere illustrated graphically and discussed. Based on the obtained results of the above numerical investigation, the folloing conclusions could be dran:

8 Boundary-Layer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No [3] Hijikata, Y., Torigo, E., Kaaguchi, T., Morishita, T. Heat transport fluid, heat transport structure, and heat transport method. US (008. [4] Hong, H., Wensel, J. Carbon nanoparticle-containing hydrophilic nanofluid ith enhanced thermal conductivity. US (011. [5] Hayes, A.M., McCants, D.A. Nanofluids for thermal management systems. US (01. [6] Quintero, L., Cardenas, A.E., Clark, D.E. Nanofluids and methods of use for drilling and completion fluids. US (01. [7] Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transfer 009; 5: [8] Buongiorno J. Convective transport in nanofluids. J Heat Trans-T, ASME 006;18: [9] Blottner FG. Finite-difference methods of solution of the boundarylayer equations. AIAA J 1970; 8:

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