4. Numerical simulations and discussion

This section presents the numerical simulations for dimensionless velocity, temperature, skin friction, Nusselt numbers and streamlines under the influence of the flow parameters which are illustrated through the (Figures 2–11). Table 1 is also given for thermophysical properties of base fluids and CNTs (MWCNT and SWCNT).

Figures 2 and 3 illustrate the velocity profiles for assisting and opposing flows where multi walled CNT is considered as nanoparticle and Kerosene is taken as base fluid. The Grashof number is fixed at 0.5. From all the figures, it is depicted that f<sup>0</sup> (η) attains maximum values at <sup>η</sup> = 0 i.e. near the stretching sheet. It is clear from the transformation <sup>η</sup> <sup>¼</sup> ffiffiffi a νf q r ð Þ n�1 <sup>2</sup> z that η = 0

at z = 0. The curves show asymptotic nature long the η-axis. It is physically interpreted that the radial velocity diminishes when fluid is flowing away from the sheet. In Figure 2 (a and b), the effects of nanoparticle volume fraction (ϕ) and viscosity parameter (α) on the velocity profiles for assisting flow and opposing flows are depicted and the value of power index is fixed at 0.5. It is inferred that with increasing the volume fraction parameter, the velocity slightly enhances for η < 4 for both types of flows. It is also revealed that with increasing the magnitude of viscosity parameter, the radial velocity goes up η < 2 for both types of flows. This is very clear

Figure 4. Temperature profile for different values of nanoparticle volume fraction and viscosity parameter: (a) assisting

Figure 3. Velocity profile for different values of nanoparticle volume fraction and power law index: (a) assisting flow and

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), that with increasing the magnitude of α, the

from equation, μ<sup>f</sup> (θ) = e

flow and (b) opposing flow.

(b) opposing flow.

(αθ)

= 1 (αθ) + <sup>O</sup>(α<sup>2</sup>

Figure 2. Velocity profile for different values of nanoparticle volume fraction and viscosity parameter: (a) assisting flow and (b) opposing flow.

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3. Numerical scheme

278 Numerical Simulations in Engineering and Science

compared the calculated values of f

4. Numerical simulations and discussion

of base fluids and CNTs (MWCNT and SWCNT).

number is fixed at 0.5. From all the figures, it is depicted that f<sup>0</sup>

at <sup>η</sup> = 0 i.e. near the stretching sheet. It is clear from the transformation <sup>η</sup> <sup>¼</sup> ffiffiffi

and θ<sup>0</sup>

and (b) opposing flow.

Numerical solutions of ordinary differential Eqs. (8) and (9) subject to boundary conditions (10) are 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 f00(0) and θ<sup>0</sup>

needed but no such values are given prior to the computation. The initial guess values of f00(0)

Secant method for better approximation. The step-size is taken as Δη = 0.01 and accuracy to the fifth decimal place as the criterion of convergence. It is important to note that the dual

This section presents the numerical simulations for dimensionless velocity, temperature, skin friction, Nusselt numbers and streamlines under the influence of the flow parameters which are illustrated through the (Figures 2–11). Table 1 is also given for thermophysical properties

Figures 2 and 3 illustrate the velocity profiles for assisting and opposing flows where multi walled CNT is considered as nanoparticle and Kerosene is taken as base fluid. The Grashof

Figure 2. Velocity profile for different values of nanoparticle volume fraction and viscosity parameter: (a) assisting flow

solutions are obtained by setting two different initial guesses for the values of f00(0).

0

with the given boundary conditions (10) and the values of f00(0) and θ<sup>0</sup>

(0) are chosen and fourth order Runge-Kutta method is applied to obtain a solution. We

(η) and θ(η) at the far field boundary condition η∞(=20)

(0) are

(0), are adjusted using

(η) attains maximum values

<sup>2</sup> z that η = 0

a νf q r ð Þ n�1

Figure 3. Velocity profile for different values of nanoparticle volume fraction and power law index: (a) assisting flow and (b) opposing flow.

Figure 4. Temperature profile for different values of nanoparticle volume fraction and viscosity parameter: (a) assisting flow and (b) opposing flow.

at z = 0. The curves show asymptotic nature long the η-axis. It is physically interpreted that the radial velocity diminishes when fluid is flowing away from the sheet. In Figure 2 (a and b), the effects of nanoparticle volume fraction (ϕ) and viscosity parameter (α) on the velocity profiles for assisting flow and opposing flows are depicted and the value of power index is fixed at 0.5. It is inferred that with increasing the volume fraction parameter, the velocity slightly enhances for η < 4 for both types of flows. It is also revealed that with increasing the magnitude of viscosity parameter, the radial velocity goes up η < 2 for both types of flows. This is very clear from equation, μ<sup>f</sup> (θ) = e (αθ) = 1 (αθ) + <sup>O</sup>(α<sup>2</sup> ), that with increasing the magnitude of α, the

Figure 5. Temperature profile for different values of nanoparticle volume fraction and power law index: (a) assisting flow and (b) opposing flow.

Figure 6. Temperature profile for different values of nanoparticle volume fraction and Grashof number: (a) assisting flow and (b) opposing flow.

The effects of viscosity parameter (α), power law index (n) and Grashof number (Gr) with various values of nanoparticle volume fraction (ϕ) on temperature profiles are depicted through Figures 4–6. For all computations, the nanoparticles and base fluid are considered as MWCNT and kerosene respectively. It is noticed that temperature is maximum at the wall of stretching sheet (i.e. η = 0 or z = 0). And when we see the temperature surrounding the sheet, it goes down significantly up to η = 0.5 and then after it is constant which is very close to zero. The patterns for temperature profile for assisting and opposing flows are similar. In Figure 4, the values of power index and Grashof number are considered as 1 and 0.5 respectively. Figure 4 depicts that temperature falls with increasing the magnitude of viscosity parameter

Figure 8. Variation of skin-friction coefficient against nanoparticle volume fraction for different base fluids and different

Figure 7. Variation of skin-friction coefficient against nanoparticle volume fraction for different values of (a) viscosity

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(Figure 4(a and b)). The values of viscosity parameter and Grashof number are fixed as 0.5 in

(αθ)

= 1 (αθ) + <sup>O</sup>(α<sup>2</sup>

) for both type flows

which is also very clear from the relation μ<sup>f</sup> (θ) = e

CNTs: (a) assisting flow and (b) opposing flow.

parameter, (b) power law index, and (c) Grashof number.

viscosity will reduce. It is obvious that with increasing the viscosity of fluids, the shearing resistance will enhance and then velocity will go down. Figure 3(a, b) show that the effects of power law index (n) on velocity profile for both types of flows at fixed value of viscosity parameter 0.5. It is observed that the radial velocity rises with increasing the magnitude of power law index parameter. It is further noted that the velocity profile enlarges with increasing the nanoparticle volume fraction for all values of power law index and also viscosity parameter for both types of flows.

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Figure 7. Variation of skin-friction coefficient against nanoparticle volume fraction for different values of (a) viscosity parameter, (b) power law index, and (c) Grashof number.

Figure 8. Variation of skin-friction coefficient against nanoparticle volume fraction for different base fluids and different CNTs: (a) assisting flow and (b) opposing flow.

The effects of viscosity parameter (α), power law index (n) and Grashof number (Gr) with various values of nanoparticle volume fraction (ϕ) on temperature profiles are depicted through Figures 4–6. For all computations, the nanoparticles and base fluid are considered as MWCNT and kerosene respectively. It is noticed that temperature is maximum at the wall of stretching sheet (i.e. η = 0 or z = 0). And when we see the temperature surrounding the sheet, it goes down significantly up to η = 0.5 and then after it is constant which is very close to zero. The patterns for temperature profile for assisting and opposing flows are similar. In Figure 4, the values of power index and Grashof number are considered as 1 and 0.5 respectively. Figure 4 depicts that temperature falls with increasing the magnitude of viscosity parameter which is also very clear from the relation μ<sup>f</sup> (θ) = e (αθ) = 1 (αθ) + <sup>O</sup>(α<sup>2</sup> ) for both type flows (Figure 4(a and b)). The values of viscosity parameter and Grashof number are fixed as 0.5 in

viscosity will reduce. It is obvious that with increasing the viscosity of fluids, the shearing resistance will enhance and then velocity will go down. Figure 3(a, b) show that the effects of power law index (n) on velocity profile for both types of flows at fixed value of viscosity parameter 0.5. It is observed that the radial velocity rises with increasing the magnitude of power law index parameter. It is further noted that the velocity profile enlarges with increasing the nanoparticle volume fraction for all values of power law index and also viscosity parame-

Figure 6. Temperature profile for different values of nanoparticle volume fraction and Grashof number: (a) assisting flow

Figure 5. Temperature profile for different values of nanoparticle volume fraction and power law index: (a) assisting flow

ter for both types of flows.

and (b) opposing flow.

and (b) opposing flow.

280 Numerical Simulations in Engineering and Science

Figure 9. Variation of Nusselt number against nanoparticle volume fraction for (a) viscosity parameter, (b) power law index, and (c) Grashof number.

Figure 10. Variation of Nusselt number against nanoparticle volume fraction for different base fluids and also for different CNTs: (a) assisting flow and (b) opposing flow.

least at φ = 0 and it starts to enhance nonlinearly with increasing the values of nanoparticle volume fraction. In Figure 7, the nanoparticle and base fluids are considered as SWCNT and Kerosene respectively. The effect of viscosity parameter on skin-friction coefficient at fixed values n = Gr = 0.5 for both type of flows is presented in Figure 7a. It is observed that the magnitude of skin friction coefficient diminishes with increasing the viscosity parameter for both type of flows. The influence of power law index on skin friction coefficient is examined in Figure 7b at fixed values α = Gr = 0.5 for both type of flows. It is depicted that skin friction coefficient extends for more values of power law index. Figure 7c is plotted for effects of Grashof number on skin friction coefficient at fixed values α = n = 0.5 for both type of flows. It is revealed that magnitude of skin friction coefficient diminishes with increasing the magnitude of Grashof number for assisting flow and it increases with increasing the Grashof number

) 997 783 884 1115 1600 2600

cp (J/kg-K) 4179 2090 1910 2430 796 425 k (W/m-K) 0.613 0.15 0.144 0.253 3000 6600

Water Kerosene Engine oil Ethylene glycol MWCNT SWCNT

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Figure 11. Stream lines for different values of power law index for α = 0.4, Gr = 3,φ= 0.3.

Pr 6.2 21 6450 2.0363

Table 1. Thermophysical properties of different base fluid and CNTs.

Physical properties Base fluid Nanoparticles

Figure 8(a and b) are illustrated for the effects of different types of base fluids (kerosene, ethylene glycol and engine oil) and also effects of different CNTs on skin friction coefficient at fixed values α = Gr = 0.5, n = 2 for assisting flow (Figure 8a) and opposing flow (Figure 8b). It is found that the magnitude of skin fraction is more for SWCNT in compare to MWCNT for all type of base fluids

for opposing flow.

ρ (kg/m<sup>3</sup>

Figure 5. It is inferred that the temperature rises with increasing the power law index for different values of φ= 0, 0.1, 0.2 for both type flows (Figure 5(a and b)). The values of power law index and nanoparticle volume fraction are taken as 0.2 and 0.5 respectively in Figure 6. It is revealed that the temperature slightly increases with increasing the values of Grashof number for assisting flow, see Figure 6a however the temperature decreases with increasing the Grashof number for opposing flow, see Figure 6b. From Figures 4–6, it is also pointed out that the temperature goes up with increasing the nanoparticle volume fraction for all values of viscosity parameter, power law index and Grashof number, and also for both types of flows.

The variations in skin-friction coefficient against nanoparticle volume fraction are computed in Figures 7 and 8 for assisting and opposing flows. It is found that the skin friction coefficient is

Numerical Simulation of Nanoparticles with Variable Viscosity over a Stretching Sheet http://dx.doi.org/10.5772/intechopen.71224 283

Figure 11. Stream lines for different values of power law index for α = 0.4, Gr = 3,φ= 0.3.


Table 1. Thermophysical properties of different base fluid and CNTs.

Figure 5. It is inferred that the temperature rises with increasing the power law index for different values of φ= 0, 0.1, 0.2 for both type flows (Figure 5(a and b)). The values of power law index and nanoparticle volume fraction are taken as 0.2 and 0.5 respectively in Figure 6. It is revealed that the temperature slightly increases with increasing the values of Grashof number for assisting flow, see Figure 6a however the temperature decreases with increasing the Grashof number for opposing flow, see Figure 6b. From Figures 4–6, it is also pointed out that the temperature goes up with increasing the nanoparticle volume fraction for all values of viscosity parameter, power law index and Grashof number, and also for both types of flows. The variations in skin-friction coefficient against nanoparticle volume fraction are computed in Figures 7 and 8 for assisting and opposing flows. It is found that the skin friction coefficient is

Figure 10. Variation of Nusselt number against nanoparticle volume fraction for different base fluids and also for different

Figure 9. Variation of Nusselt number against nanoparticle volume fraction for (a) viscosity parameter, (b) power law index,

and (c) Grashof number.

282 Numerical Simulations in Engineering and Science

CNTs: (a) assisting flow and (b) opposing flow.

least at φ = 0 and it starts to enhance nonlinearly with increasing the values of nanoparticle volume fraction. In Figure 7, the nanoparticle and base fluids are considered as SWCNT and Kerosene respectively. The effect of viscosity parameter on skin-friction coefficient at fixed values n = Gr = 0.5 for both type of flows is presented in Figure 7a. It is observed that the magnitude of skin friction coefficient diminishes with increasing the viscosity parameter for both type of flows. The influence of power law index on skin friction coefficient is examined in Figure 7b at fixed values α = Gr = 0.5 for both type of flows. It is depicted that skin friction coefficient extends for more values of power law index. Figure 7c is plotted for effects of Grashof number on skin friction coefficient at fixed values α = n = 0.5 for both type of flows. It is revealed that magnitude of skin friction coefficient diminishes with increasing the magnitude of Grashof number for assisting flow and it increases with increasing the Grashof number for opposing flow.

Figure 8(a and b) are illustrated for the effects of different types of base fluids (kerosene, ethylene glycol and engine oil) and also effects of different CNTs on skin friction coefficient at fixed values α = Gr = 0.5, n = 2 for assisting flow (Figure 8a) and opposing flow (Figure 8b). It is found that the magnitude of skin fraction is more for SWCNT in compare to MWCNT for all type of base fluids and also for both type of flows. It is further depicted that skin friction is maximum for kerosene and minimum for engine oil for both types of CNTs and also for both types of flows.

• The sequence for skin friction coefficient of different nanoparticles is observed as: Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Cfx

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• The sequence for skin friction coefficient of different base fluids is also noted as: Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Cfx

• The Nusselt number diminishes with increasing the viscosity parameter and it also diminishes with decreasing the power law index and nanoparticle volume fraction for both type

• The sequence for Nusselt number of kerosene, ethylene glycol and engine oil is noted as:

\*, Dharmendra Tripathi2 and Zafar Hayat Khan<sup>3</sup>

1 DBS&H CEME, National University of Sciences and Technology, Islamabad, Pakistan 2 Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan, India

3 Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa,

[1] Choi, S. U. S. (1995). Enhancing thermal conductivity of fluids with nanoparticles. ASME,

[2] Wang XQ, Mujumdar AS. Heat transfer characteristics of nanofluids: A review. Interna-

[3] Das SK, Choi SU, Patel HE. Heat transfer in nanofluids—A review. Heat Transfer Engi-

[4] Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. International Journal of Heat andMass Transfer. 2013;57(2):582-594

[5] He Y, Jin Y, Chen H, Ding Y, Cang D, Lu H. Heat transfer and flow behaviour of aqueous suspensions of TiO2 nanoparticles (nanofluids) flowing upward through a vertical pipe.

[6] Kim SJ, McKrell T, Buongiorno J, Hu LW. Experimental study of flow critical heat flux in alumina-water, zinc-oxide-water, and diamond-water nanofluids. Journal of Heat Trans-

International Journal of Heat and Mass Transfer. 2007;50(11):2272-2281

Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Nux (kerosene) < Re1=<sup>2</sup> <sup>x</sup> Nux (ethylene glycol) < Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Nux (engine oil).

(kerosene) > Re1=<sup>2</sup> <sup>x</sup> Cfx (ethylene glycol)> Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Cfx (engine oil).

\*Address all correspondence to: noreensher@yahoo.com

tional Journal of Thermal Sciences. 2007;46(1):1-19

(SWCNT) > Re<sup>1</sup>=<sup>2</sup> <sup>x</sup> Cfx (MWCNT).

of flows.

Author details

Noreen Sher Akbar<sup>1</sup>

Pakistan

References

Publications-Fed, 231, 99-106

neering. 2006;27(10):3-19

fer. 2009;131(4):043204

Figures 9 and 10 are plotted to see the variations in Nusselt number against nanoparticle volume fraction both types of flows.

It is observed that the magnitude of Nusselt number is minimum at φ= 0. And it significantly increases with increasing the magnitude of nanoparticle volume fraction. The nanoparticle and base fluids are considered as SWCNT and kerosene respectively in Figure 9. Figure 9a illustrates that Nusselt number diminishes with increasing the magnitude of viscosity parameter for both type of flows at fixed values n = Gr = 0.5. Figure 9b shows that Nusselt number increases with increasing the power index both type of flows at fixed values α = Gr = 0.5. Figure 9c reveals that Nusselt number increases with increasing the Grashof number for assisting flow however it reduces with increasing the Grashof number for opposing flow at fixed values α = n = 0.5. Figure 10(a and b) are plotted to see the effects of Kerosene, Ethylene glycol and Engine oil and also effects of SWCNT and MWCNT on Nusselt number at fixed values α = Gr = 0.5, n = 2 for both types of flows (Figure 10a and b). A very minor variation in Nusselt number for SWCNT and MWCNT nanoparticles is noted for assisting and opposing flows. A significant variation in Nusselt number for Kerosene, Ethylene glycol and Engine oil is pointed out where the Nusselt number is largest for Engine oil and smallest for Kerosene for both type of flows.

The stream lines are plotted through the Figure 11(a–c) to see the effects of power law index at fixed values α = 0.4, Gr = 3,φ= 0.3 for assisting flow where SWCNT as nanoparticle and Kerosene as base fluid are considered. It is noted that stream lines go closer to each other with increasing the power law index.
