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Unsteady stagnation-point flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet.

Naganthran K, Nazar R, Pop I - Sci Rep (2016)

Bottom Line: The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab.Dual solutions exist for both cases of stretching and shrinking sheet.Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematical Sciences, Faculty of Science &Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia.

ABSTRACT
In this paper, the unsteady stagnation-point boundary layer flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet has been studied. Similarity transformation is used to transform the system of boundary layer equations which is in the form of partial differential equations into a system of ordinary differential equations. The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab. The numerical solutions for the skin friction coefficient and heat transfer coefficient as well as the velocity and temperature profiles are presented in the forms of tables and graphs. Dual solutions exist for both cases of stretching and shrinking sheet. Stability analysis is performed to determine which solution is stable and valid physically. Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

No MeSH data available.


Related in: MedlinePlus

Velocity profiles, f ′(η) for different values of β when λ = −1, s = 3, and K = 1.
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f8: Velocity profiles, f ′(η) for different values of β when λ = −1, s = 3, and K = 1.

Mentions: The velocity and temperature profiles which have been shown in Figs 6, 7, 8, 9, 10, 11, 12, 13 satisfy the far field boundary conditions (10) asymptotically, which support the validity of the numerical results obtained and the existence of the dual solutions. For example, Figs 6, 8, 10 and 12 display the identified converged solutions when the plot of velocity profile f ′(η) approaches 1 as the boundary layer thickness value is less than or equals to 8 and they show the relationship f(0) = s. Figures 7, 9, 11 and 13 also able to reflect the boundary conditions θ(0) = 1 and θ(∞) → 0. In Figs 6 and 7, an increase in the rate of suction reduces the boundary layer thickness. Besides, the slower motion of the flow lowers the boundary layer thickness and this is shown in Figs 8 and 9. The high effects of non-Newtonian characteristic in the special third grade fluid will increase the boundary layer thickness and are well portrayed in Figs 10 and 11. Meanwhile, in Figs 12 and 13, the higher shrinking rate increases the boundary layer thickness, but the second solution opposes the trend where the higher shrinking rate decreases the boundary layer thickness compared to the stretching case. As mentioned earlier in this paper, the existence of dual solutions when the sheet is stretched and shrunk as the value of λ lies in between λc < λ < 4 have been noticed. Therefore, there is a necessity to conduct the stability analysis and we found that the first solution (upper branch) is stable and physically applicable while the second solution (lower branch) is unstable. The stable solution is identified based on the positive smallest eigenvalue whereas the unstable solution is recognized based on the negative smallest eigenvalue. Table 7 illustrates the smallest eigenvalue, γ1 for some values of λ when s = 3, K = 3, and β = −1.


Unsteady stagnation-point flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet.

Naganthran K, Nazar R, Pop I - Sci Rep (2016)

Velocity profiles, f ′(η) for different values of β when λ = −1, s = 3, and K = 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4835756&req=5

f8: Velocity profiles, f ′(η) for different values of β when λ = −1, s = 3, and K = 1.
Mentions: The velocity and temperature profiles which have been shown in Figs 6, 7, 8, 9, 10, 11, 12, 13 satisfy the far field boundary conditions (10) asymptotically, which support the validity of the numerical results obtained and the existence of the dual solutions. For example, Figs 6, 8, 10 and 12 display the identified converged solutions when the plot of velocity profile f ′(η) approaches 1 as the boundary layer thickness value is less than or equals to 8 and they show the relationship f(0) = s. Figures 7, 9, 11 and 13 also able to reflect the boundary conditions θ(0) = 1 and θ(∞) → 0. In Figs 6 and 7, an increase in the rate of suction reduces the boundary layer thickness. Besides, the slower motion of the flow lowers the boundary layer thickness and this is shown in Figs 8 and 9. The high effects of non-Newtonian characteristic in the special third grade fluid will increase the boundary layer thickness and are well portrayed in Figs 10 and 11. Meanwhile, in Figs 12 and 13, the higher shrinking rate increases the boundary layer thickness, but the second solution opposes the trend where the higher shrinking rate decreases the boundary layer thickness compared to the stretching case. As mentioned earlier in this paper, the existence of dual solutions when the sheet is stretched and shrunk as the value of λ lies in between λc < λ < 4 have been noticed. Therefore, there is a necessity to conduct the stability analysis and we found that the first solution (upper branch) is stable and physically applicable while the second solution (lower branch) is unstable. The stable solution is identified based on the positive smallest eigenvalue whereas the unstable solution is recognized based on the negative smallest eigenvalue. Table 7 illustrates the smallest eigenvalue, γ1 for some values of λ when s = 3, K = 3, and β = −1.

Bottom Line: The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab.Dual solutions exist for both cases of stretching and shrinking sheet.Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematical Sciences, Faculty of Science &Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia.

ABSTRACT
In this paper, the unsteady stagnation-point boundary layer flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet has been studied. Similarity transformation is used to transform the system of boundary layer equations which is in the form of partial differential equations into a system of ordinary differential equations. The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab. The numerical solutions for the skin friction coefficient and heat transfer coefficient as well as the velocity and temperature profiles are presented in the forms of tables and graphs. Dual solutions exist for both cases of stretching and shrinking sheet. Stability analysis is performed to determine which solution is stable and valid physically. Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

No MeSH data available.


Related in: MedlinePlus