Rheological Analysis of CNT Suspended Nanofluid with Variable Viscosity: Numerical Solution

Transcription

1 Commun. Theor. Phys Vol. 67, No. 6, June 1, 2017 Rheological Analysis of CNT Suspended Nanofluid with Variable Viscosity: Numerical Solution Noreen Sher Akbar 1, and Zafar Hayat Khan 2 1 DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan 2 Department of Mathematics, University of Malakand Dir lower Khyber Pakhtunkhwa, Pakistan Received October 9, 2016; revised manuscript received March 2, 2017 Abstract In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold s model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically. PACS numbers: b DOI: / /67/6/681 Key words: two-dimensional flow, variable nanofluid viscosity, stagnation-point flow, stretching sheet, carbon nanotubes, numerical solution 1 Introduction In recent years, the research on flow over a stretching sheet has spawned important interest because of its copious manufacturing applications such as in the fabrication of canvas stuff over an extrusion progression, the conservation of bath, the boundary layer along corporeal administration conveyers, the sweptback extrusion of soft sheets crystal and polymer industries, fiber mechanized etc. Boundary layer deportment above a poignant unremitting solid exterior is a momentous sort of flow arising in abundant manufacturing processes. Sakiadis [1 2] was the first to examine the boundary layer flow over an incessant stretching surface. In this stare, Crane [3] premeditated flow over a stretching plate. Stagnation-point flow has been originated in frequent applications in industrialized and apparatus. It can be positioned in the stagnation region of flow fleeting with any shape of frame, i.e., wharf and aerofoil. Hiemenz [4] showed that stagnation-point flow can be examined by the Navier Stokes equations concluded with similarity solution in which the number of variables can be reduced by one or supplementary by a complement redecoration. For more detail see Refs. [5 10]. Fluids in association with heat transfer used in preeminence compeers, substance production, microelectronics cooling, air-conditioning, refrigeration, transportation, and several other applications. It is required to enhance effective thermal conductivity of the fluids to increase heat transfer rate. After Choi, [11] it has been proved experimentally and theoretically by many researchers [12 18] Corresponding author, noreensher1@gmail.com c 2017 Chinese Physical Society and IOP Publishing Ltd that flush with small solid volume fraction of nanoparticles i.e., less than 5 percent, the thermal conductivity of heat transfer fluids can be enhanced by %. Carbon nanotubes, a form of fullerene, obligate imminent in fields such as nanotechnology, optics, electronics, architecture, and materials science. In recent years new solicitations have taken benefit of their sole electrical properties, unexpected strength, and competence in heat conduction. An experimental investigation was conducted by Kim and Peterson [19] to explore the effect of the morphology of carbon nanotubes on the thermal conductivity of suspensions. According to them the tremendous enrichment for the SWNT rebellious to a volume fraction of 1.0% approached 10%, which was sensible to be redundant than twofold that of the other values, 3.5%, attained in the case of aluminum oxide nanofluids. The heat transfer nanofluids encompass carbon nanotubes CNT s and magnetic-field delicate nanoparticles of Fe2O3 reported by Hong et al. [20] They pragmatic that on lengthier possessions in compelling field, the particles slowly move and form large bunches of particles, producing clomping of CNT s, and then declining the thermal conductivity. Kamali and Binesh [21] examined numerically the convective heat transfer of multi-wall carbon nanotubes MWCNT with constant wall heat flux stipulation. They deciphered Navier stokes equations by the finite volume method using CNT-based nanofluids using power law model. They pragmatic that the heat transfer coefficient is subjugated by the wall region because of non-newtonian behavior of

2 682 Communications in Theoretical Physics Vol. 67 CNT nanofluid. Very recently Khan et al. [22] studied homogeneous fluid model to analyze the flow and heat transfer of carbon nanotubes CNT s with Navier slip and constant heat flux boundary conditions. Further recent literature can be viewed through Refs. [23 30]. The aim of the present article is to discuss the twodimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically. 2 Formulation of the Problem We consider the two-dimensional stagnation-point flow over a stretching sheet with water as based fluids encompassing single-wall CNT s. The flow is presumed to be laminar, steady, and incompressible. The base fluid and the CNT s are expected to be in updraft equilibrium. Sheet is assumed to be stretched with the different velocity u w, v w along the x-axis and y-axis respectively. Further, we have taken the constant temperature T w at wall and the ambient temperature T. Fluid viscosity is considered to be temperature dependent. see Fig. 1 Fig. 1 Geometry of the problem a Shrinking case b Stretching case. With the above analysis the boundary layer equations for the proposed model can be written as follows u x + v = 0, 1 u u x + v u u e = u e x + 1 ρ nf µ nf T u u T x + v T σb2 o u u e ± gβ T T, ρ nf 2 2 T = α nf 2. 3 The relevant boundary conditions are of the form u = u w x = cx, v = v w, T = T w, at y = 0, u u e x = ax, v 0, T T, as y. 4 In above equations u and v are the velocity components along the x- and y-axes, respectively, a, c > 0 the constant, u w is velocity at wall, T is the temperature, ρ nf is the nanofluid density, µ nf is the viscosity of nanofluid and α nf is the thermal diffiusivity of nanofluid defined as [28] µ nf θ = µ f θ 1 ϕ 2.5, ρc p nf = 1 ϕρc p f + ϕρc p CNT, α nf = k nf ρ nf C p nf, k nf k f = 1 ϕ+2ϕk CNT/k CNT k f lnk CNT + k f /2k f 1 ϕ + 2ϕk f /k CNT k f lnk CNT + k f /2k f, 5 where µ f is the viscosity of base fluid, ϕ is the nanoparticle fraction, ρc p nf is the effective heat capacity of a nanoparticle, k nf is the thermal conductivity of nanofluid, k f and k CNT are the thermal conductivities of the base fluid and carbon nano tubes, respectively, ρ f and ρ CNT are the thermal conductivities of the base fluid and carbon nano tubes, respectively. Introducing the following similarity transformations c η = y, u = cxf η, ν f v = cν f fη, θ = T T. 6 T w T Making use of Eqs. 5 6 in Eq. 1 to Eq. 4, we have µ f θ 1 ϕ 2.5 f + µ f θ 1 ϕ 2.5 f + [1 ϕ + ϕρ CNT /ρ f {ff f 2 + S 2 } + M 2 S f ± G r θ] = 0, 7 knf θ + Pr 1 ϕ + ϕ ρc p CNT [fθ ] = 0, k f ρc p f 8 f0 = 0, f 0 = 1, f = S, 9a θ0 = 1, θ = 0, 9b where P r = µc p f /k f is the Prandtl number and S = a/c stagnation parameter. Reynolds model of viscosity expression can be taken as [18] µ f θ = e αθ = 1 αθ + Oα 2, 10 where α is the viscosity parameter. Expressions for the skin-friction coefficient and the local Nusselt number Nu are c f = µ nft ρ f u 2 w u y=0,

3 No. 6 Communications in Theoretical Physics 683 xk nf T Nu x = k f T w T. 11 y=0 Dimensionless form of Eq. 11 takes the form Re x 1/2 c f = µ f θ0f 0 1 ϕ 2.5, Re x 1/2 Nu x = k nf k f θ Numerical Illustration Numerical solutions to the governing ordinary differential equations 7 8 with the boundary conditions 9 were obtained using a shooting method. First we have converted the boundary value problem BVP into initial value problem IVP and assumed a suitable finite value for the far field boundary condition, i.e. η, say η. To solve the IVP, the values for f 0 and θ 0 are needed but no such values are given prior to the computation. The initial guess values of f 0 and θ 0 are chosen and fourth order Runge Kutta method is applied to obtain a solution. We compare the calculated values of f η and θη at the far field boundary condition η = 20 with the given boundary conditions 9b and the values of f 0 and θ 0 are adjusted using Secant method for better approximation. The step-size is taken as η = 0.01 and accuracy to the fifth decimal place as the criterion of convergence. It is important to note that the dual solutions are obtained by setting two different initial guesses for the values of f 0. 4 Graphical Results and Discussion The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, Nusselt numbers and streamlines are presented in Figs Figures 2a 2c show the variation of velocity profile for different values of nanoparticle volume fraction with Hartmann number M, Viscosity parameter α, Grashof number Gr. Since Hartmann number M is the ratio of electromagnetic force to the viscous force and magnetic field is applied in the opposite direction of the fluid so with the increase in Hartmann number causes increase in electromagnetic force that increases velocity profile and boundary layer thickness for assisting flow but decreases velocity profile for opposing flow. Viscosity parameter α shows the same behavior on velocity profile as we are considering temperature dependent viscosity so when rises temperature dependent viscosity, fluid resistance becomes slow and the fluid moves speedily so when we increases viscosity parameter α velocity field increases rapidly and boundary layer thickness also increases see Fig. 2b. Figure 2c shows that when we increase Grashof number Gr the ratio of the buoyancy to viscous force acting on a fluid, then there will be more buoyancy forces, that causes increase in velocity field and boundary layer thickness. It is also seen that for each case with the increase in solid volume fraction of nanoparticles velocity profile increases for assisting flow but decreases for opposing flow. Fig. 2 Variation of velocity profile for different values of nanoparticle volume fraction with a Hartmann number M. b Viscosity parameter α. c Grashof number Gr.

4 684 Communications in Theoretical Physics Vol. 67 Fig. 3 Variation of temperature profile for different values of nano particle volume fraction for assisting and opposing flow with Hartmann number M. Fig. 4 Variation of temperature profile for different values of nano particle volume fraction for assisting and opposing flow with viscosity parameter α. Figure 3a to Fig. 5b show the temperature profile for different values of nanoparticle volume fraction with Hartmann number M, Viscosity parameter α, and Grashof number Gr, it is seen that when we increase Hartmann number causes increase in electromagnetic force that increases temperature profile and thermal boundary layer thickness also increases for assisting as well as for opposing flow see Figs. 3a and 3b. Viscosity parameter α shows the same behavior on temperature profile as we are considering temperature dependent viscosity so when rises temperature dependent viscosity, fluid resistance becomes slow and the fluid moves speedily so when we increase viscosity parameter α temperature field increases rapidly and thermal boundary layer thickness also increases see Figs. 4a and 4b. Figures 5a and 5b show that when we increase Grashof number Gr the ratio of the buoyancy to viscous force acting on a fluid, then there will be more buoyancy forces, that cause increase in temperature profile and thermal boundary layer thickness also increases. It is also observed that for each case with the increase in solid volume fraction of nanoparticles temperature profile increases for both assisting and opposing flow. Variation of skin-friction coefficient for assisting and opposing flow with Grashof number Gr, Hartmann number M, Viscosity parameter α are presented in Figs. 6a 6c. Figure 6a shows that when we increase Grashof number Gr the ratio of the buoyancy to viscous force acting on a fluid, then there will be more buoyancy forces it causes increase in skin friction coefficient for SWCNT for assisting flow but decreases for opposing flow. It is seen that with the increase in M electromagnetic force are high as compare to viscous force skin friction coefficient decreases for SWCNT for assisting flow but increases for opposing flow, similar behavior is observed for viscosity parameter α, rise in viscosity parameter α skin friction

5 No. 6 Communications in Theoretical Physics 685 coefficient decreases for SWCNT for assisting flow but increases for opposing flow. Variation of Nusselt number for assisting and opposing flow with Grashof number Gr, Hartmann number M, Viscosity parameter α are presented in Figs. 7a 7c. Figure 7a shows that when we increase Grashof number Gr the ratio of the buoyancy to viscous force acting on a fluid, then there will be more buoyancy forces it causes decrease in Nusselt number for SWCNT for assisting flow but increases for opposing flow. It is seen that with the increase in M electromagnetic force are high as compare to viscous force Nusselt number increases for SWCNT for assisting flow but decreases for opposing flow, opposite behavior is observed for viscosity parameter α, rise in viscosity parameter α, Nusselt number decreases for SWCNT for assisting flow but increases for opposing flow. Table 1 presents thermophysical properties of different base fluid and CNT s. Table 2 gives the numerical values of skin friction assisting flow for water functionalized SWCNT nanoparticle with the various values of flow parameters. Table 3 gives numerical values of Nusselt number assisting flow for water functionalized SWCNT nanoparticle with the various values of flow parameter. Table 4 gives the comparison of present results with the existing literature. Fig. 5 Variation of temperature profile for different values of nano particle volume fraction for assisting and opposing flow with Grashof number Gr. Fig. 6 Variation of skin-friction coefficient for assisting and opposing flow with a Grashof number Gr. b Hartmann number M. c Viscosity parameter α. Table 1 Thermal properties of base fluid water and nanoparticles. Physical properties Fluid Phase Water SWCNT c p /J/kg K ρ/kg/m k/w/mk

6 686 Communications in Theoretical Physics Vol. 67 Fig. 7 Variation of local Nusselt number for assisting and opposing flow with a Grashof number Gr. b Hartmann number M. c Viscosity parameter α. Table 2 Numerical values of skin friction assisting flow for water functionalized SWCNT nanoparticle with the various values of M, α, and Gr with S = 1. ϕ M = 0 M = 1 α = 0.0 α = 0.5 α = 0.0 α = 0.5 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = Table 3 Numerical values of Nusselt number opposing flow for water functionalized SWCNT nanoparticle with the various values of M, α, and Gr with S = 1.0. ϕ M = 0 M = 1 α = 0.0 α = 0.5 α = 0.0 α = 0.5 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = 1.0 Gr = 0.5 Gr = Table 4 Comparison of results for the reduced Nusselt number for pure fluid. P r Present results Khan and Pop [24] Wang [25] Gorla and Sidawi [26] Reddy et al. [27]

7 No. 6 Communications in Theoretical Physics Conclusion Two-dimensional stagnation point flow for single wall carbon nanotubes suspended water towards a stretching sheet under the influence of temperature dependent viscosity is discussed. Main conclusion is drawn as follows: i The increase in Hartmann number M causes increase in electromagnetic force that increases velocity profile and boundary layer thickness for assisting flow but decreases velocity profile for opposing flow. ii When rises temperature dependent viscosity, fluid resistance becomes slow and the fluid moves speedily so when we increase viscosity parameter α velocity field increases rapidly and boundary layer thickness also increases. iii Increase in Grashof number Gr causes increase in velocity field and boundary layer thickness. iv It is also seen that for each case with the increase in solid volume fraction of nanoparticles velocity profile increases for assisting flow but decreases for opposing flow. v Increase in viscosity parameter α temperature field increases rapidly and thermal boundary layer thickness also increases. vi It is also observed that for each case with the increase in solid volume fraction of nanoparticles temperature profile increases for both assisting and opposing flow. vii It is seen that with the increase in M electromagnetic force are high as compare to viscous force skin friction coefficient decreases for SWCNT for assisting flow but increases for opposing flow. viii When we increase Grashof number Gr causes decrease in Nusselt number for SWCNT for assisting flow but increases for opposing flow. ix It is seen that with the increase in M electromagnetic force are high as compare to viscous force Nusselt number increases for SWCNT for assisting flow. References [1] B. C. Sakiadis, J. American Instit. Chem. Eng [2] B. C. Sakiadis, J. American Instit. Chem. Eng [3] L. Crane, Zeitschrift Für Angewandte Mathematik und Physik [4] K. Hiemenz, Dinglers Polytechnisches Journal [5] A. Ishak, R. Nazar, and I. Pop, Comput. Math. Appl [6] A. Ishak, R. Nazar, and I. Pop, Nonlinear Anal. RWA [7] F. Labropulua and I. Pop, Int. J. Thermal Sci [8] T. R. Mahapatra, S. K. Nandy, K. Vajravelu, and R. A. Van Gorder, Meccanica [9] Noreen Sher Akbar, S. Nadeem, Rizwan Ul Haq, and Z. H. Khan, Indian J. Phys [10] S. Nadeem, Int. J. of Heat and Mass Transfer [11] S.U.S. Choi, ASME Fluids Engng. Div [12] A. Ebaid, Hasan A. El-arabawy, and Y. Nader, Int. J. Differential Equations Volume 2013, Article ID , 1-8 pages. [13] E.H. Aly and A. Ebaid, J. Comput. Theor. Nanosci [14] Noreen Sher Akbar, S. Nadeem, Rizwan Ul Haq, and Z. H. Khan, Chinese Journal of Aeronautics [15] E. H. Aly and A. Ebaid, J. Comput. Theor. Nanosci [16] S. Nadeem and S. T. Hussain, Appl. Math. Mech [17] M. Sheikholeslami, R. Ellahi, H. R. Ashorynejad, and G. Domairry, J. Comput. Theor. Nanosci [18] R. Ellahi, M. Raza, and K. Vafai, Math. Comput. Mode [19] B. H. Kim and G. P. Peterson, J. Therm. Phys. Heat Transf [20] H. Hong, B. Wright, J. Wensel, S. Jin, X. Rong Ye, and W. Roy, Synthetic Metals [21] R. Kamali and A. Binesh, Int. Commun. Heat Mass Transf [22] W. A. Khan, Z. H. Khan, and M. Rahi, Appl Nanosci [23] A. Ebaid, Emad H. Aly, and N. Y. Abdelazem, J. Appl. Math. Inf. Sci [24] W. A. Khan and I. Pop, Int. J. Heat Mass. Transfer [25] C. Y. Wang, J. Appl. Math. Mech. ZAMM [26] R. S. R. Gorla and I. Sidawi, Appl. Sci. Res [27] N. Bhaskar Reddy, T. Poornima, and P. Sreenivasulu, International Journal of Engineering Mathematics Volume 2014 Article ID , 1-10 pages. [28] S. Nadeem and S. Ijaz, AIP Adv [29] S. T. Hussain, R. Haq, and S. Nadeem, J. Mol. Liq [30] R. Ellahi, IEEE Trans. Nanotechnol