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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

Variations of −θ′(0) with λ for some values of s when β = −1 and K = 1.
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f3: Variations of −θ′(0) with λ for some values of s when β = −1 and K = 1.

Mentions: On the other hand, the effects of suction towards the critical values, λc which have been displayed in Table 5, express that the higher rate of suction s lowers the critical point values. However, this trend is in contrast with the impact of the non-Newtonian parameter K over the critical values. The higher non-Newtonian characteristics of the special third grade fluid seem to increase the critical point values and this is shown in Table 6. There is no solution when λ < λc and this statement is clearly illustrated in Figs 2, 3, 4, 5. The existence of dual solutions, namely first (upper branch) solution and second (lower branch) solution has been noticed from Figs 2, 3, 4, 5. It is seen that for λc < λ < 4 (see Figs 2 and 3), and for λc < λ < 3 (see Figs 4 and 5), the equations have two solutions, while for λ < λc < 4 (see Figs 2 and 3), and for λc < λ < 3 (see Figs 4 and 5), there is no solution, respectively. In this region, the full Navier-Stokes equations should be solved where λc is the critical value of λ. Moreover, Figs 2 and 3 indicate that the reduced skin friction coefficient, f ″(0) and the reduced local Nusselt number, −θ ′(0) will increase when the rate of suction increases. Figures 4 and 5 interpret that high non-Newtonian characteristics, (K = 3) on a special third grade fluid has small reduced skin friction coefficient and lower rate of heat transfer at the surface of the sheet compared to the case when K = 1 and K = 2.


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)

Variations of −θ′(0) with λ for some values of s when β = −1 and K = 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Variations of −θ′(0) with λ for some values of s when β = −1 and K = 1.
Mentions: On the other hand, the effects of suction towards the critical values, λc which have been displayed in Table 5, express that the higher rate of suction s lowers the critical point values. However, this trend is in contrast with the impact of the non-Newtonian parameter K over the critical values. The higher non-Newtonian characteristics of the special third grade fluid seem to increase the critical point values and this is shown in Table 6. There is no solution when λ < λc and this statement is clearly illustrated in Figs 2, 3, 4, 5. The existence of dual solutions, namely first (upper branch) solution and second (lower branch) solution has been noticed from Figs 2, 3, 4, 5. It is seen that for λc < λ < 4 (see Figs 2 and 3), and for λc < λ < 3 (see Figs 4 and 5), the equations have two solutions, while for λ < λc < 4 (see Figs 2 and 3), and for λc < λ < 3 (see Figs 4 and 5), there is no solution, respectively. In this region, the full Navier-Stokes equations should be solved where λc is the critical value of λ. Moreover, Figs 2 and 3 indicate that the reduced skin friction coefficient, f ″(0) and the reduced local Nusselt number, −θ ′(0) will increase when the rate of suction increases. Figures 4 and 5 interpret that high non-Newtonian characteristics, (K = 3) on a special third grade fluid has small reduced skin friction coefficient and lower rate of heat transfer at the surface of the sheet compared to the case when K = 1 and K = 2.

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