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Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid.

Yacob NA, Ishak A, Pop I, Vajravelu K - Nanoscale Res Lett (2011)

Bottom Line: The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique.Two types of nanofluids, namely, Cu-water and Ag-water are used.It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter.

View Article: PubMed Central - HTML - PubMed

Affiliation: Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania. popm.ioan@yahoo.co.uk.

ABSTRACT
The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied numerically. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique. Two types of nanofluids, namely, Cu-water and Ag-water are used. The effects of nanoparticle volume fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at the surface of Cu-water nanofluid is higher than that at the surface of Ag-water nanofluid even though the thermal conductivity of Ag is higher than that of Cu.

No MeSH data available.


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Variation of the local Nusselt number with λ for different values of φ when γ = 0.5 for Ag-water nanofluid.
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Figure 3: Variation of the local Nusselt number with λ for different values of φ when γ = 0.5 for Ag-water nanofluid.

Mentions: The nonlinear ordinary differential equations (7) and (8) subject to the boundary conditions (9) were solved numerically by the Runge-Kutta-Fehlberg method with shooting technique. We consider two different types of nanoparticles, namely, Cu and Ag with water as the base fluid. Table 1 shows the thermophysical properties of water and the elements Cu and Ag. The Prandtl number of the base fluid (water) is kept constant at 6.2. It is worth mentioning that this study reduces to those of a viscous or regular fluid when φ = 0. Figure 2 shows the variation of the skin friction coefficient (1/(1-φ)2.5)f"(0) with λ of Ag-water nanofluid when γ = 0.5 for different nanoparticle volume fraction φ, while the respective local Nusselt number -(knf/kf) θ' (0) is displayed in Figure 3. It can be seen that for a particular value of λ, the skin friction coefficient and the local Nusselt number increase with increasing φ. Dual solutions are found to exist when λ < 0 (shrinking case) as displayed in Figures 2 and 3. Moreover, the solution can be obtained up to a critical value of λ (say λc), and /λc/ decreases with increasing φ. The similar pattern is observed for Cu-water nanofluid, which is not presented here, for the sake of brevity. It is observed that, the solution is unique for λ ≥ 0, dual solutions exist for λc < λ < 0, and no solution for λ <λc. The values of λc for Ag-water nanofluid and Cu-water nanofluid for different values of φ are presented in Table 2. It is seen that for φ = 0.1 and φ = 0.2, the value of /λc/ for Cu-water nanofluid is greater than those of Ag-water nanofluid. The temperature profiles of Ag-water and Cu-water nanofluids for different values of φ when γ = 0.5 and λ = -0.53 are presented in Figures 4 and 5, respectively. These profiles show that, there exist two different profiles satisfying the far field boundary condition (9) asymptotically, thus supporting the dual nature of the solutions presented in Figures 2 and 3. Both Figures 4 and 5 show that the boundary layer thickness is higher for the second solution compared to the first solution, which in turn produces higher surface temperature θ(0) for the former.


Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid.

Yacob NA, Ishak A, Pop I, Vajravelu K - Nanoscale Res Lett (2011)

Variation of the local Nusselt number with λ for different values of φ when γ = 0.5 for Ag-water nanofluid.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3211401&req=5

Figure 3: Variation of the local Nusselt number with λ for different values of φ when γ = 0.5 for Ag-water nanofluid.
Mentions: The nonlinear ordinary differential equations (7) and (8) subject to the boundary conditions (9) were solved numerically by the Runge-Kutta-Fehlberg method with shooting technique. We consider two different types of nanoparticles, namely, Cu and Ag with water as the base fluid. Table 1 shows the thermophysical properties of water and the elements Cu and Ag. The Prandtl number of the base fluid (water) is kept constant at 6.2. It is worth mentioning that this study reduces to those of a viscous or regular fluid when φ = 0. Figure 2 shows the variation of the skin friction coefficient (1/(1-φ)2.5)f"(0) with λ of Ag-water nanofluid when γ = 0.5 for different nanoparticle volume fraction φ, while the respective local Nusselt number -(knf/kf) θ' (0) is displayed in Figure 3. It can be seen that for a particular value of λ, the skin friction coefficient and the local Nusselt number increase with increasing φ. Dual solutions are found to exist when λ < 0 (shrinking case) as displayed in Figures 2 and 3. Moreover, the solution can be obtained up to a critical value of λ (say λc), and /λc/ decreases with increasing φ. The similar pattern is observed for Cu-water nanofluid, which is not presented here, for the sake of brevity. It is observed that, the solution is unique for λ ≥ 0, dual solutions exist for λc < λ < 0, and no solution for λ <λc. The values of λc for Ag-water nanofluid and Cu-water nanofluid for different values of φ are presented in Table 2. It is seen that for φ = 0.1 and φ = 0.2, the value of /λc/ for Cu-water nanofluid is greater than those of Ag-water nanofluid. The temperature profiles of Ag-water and Cu-water nanofluids for different values of φ when γ = 0.5 and λ = -0.53 are presented in Figures 4 and 5, respectively. These profiles show that, there exist two different profiles satisfying the far field boundary condition (9) asymptotically, thus supporting the dual nature of the solutions presented in Figures 2 and 3. Both Figures 4 and 5 show that the boundary layer thickness is higher for the second solution compared to the first solution, which in turn produces higher surface temperature θ(0) for the former.

Bottom Line: The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique.Two types of nanofluids, namely, Cu-water and Ag-water are used.It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter.

View Article: PubMed Central - HTML - PubMed

Affiliation: Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania. popm.ioan@yahoo.co.uk.

ABSTRACT
The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied numerically. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique. Two types of nanofluids, namely, Cu-water and Ag-water are used. The effects of nanoparticle volume fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at the surface of Cu-water nanofluid is higher than that at the surface of Ag-water nanofluid even though the thermal conductivity of Ag is higher than that of Cu.

No MeSH data available.


Related in: MedlinePlus