Magnetohydrodynamic Boundary Layer Flow of a Viscoelastic Fluid Past a Nonlinear Stretching Sheet in the Presence of Viscous Dissipation Effect
Abstract
:1. Introduction
2. Constitutive Equation
3. Flow Problem Formulation
4. Heat Transfer Analysis
- (i)
- Prescribed surface temperature (PST)
- (ii)
- Prescribed heat flux (PHF)
4.1. Case I: Prescribed Surface Temperature
4.2. Case II: Prescribed Heat Flux
5. Method of the Solution
- (i)
- the transformed ordinary equations are expressed in the system of first order equations in .
- (ii)
- the resulting first order system of equations is written as finite difference equations using the central difference method about the mid-point.
- (iii)
- the resulting finite difference equations are linearized by Newton’s method.
- (iv)
- the linearized system of equations is written in matrix vector form and then solved using the block tri-diagonal elimination method.
6. Results and Discussion
7. Conclusions
- The effect of nonlinear stretching parameter n was to reduce the thickness of the momentum boundary layer and increase the thermal boundary layer thickness in the PST and PHF cases. However, the rate of increase was higher in the PST case.
- The increase in the value of the magnetic parameter resulted in decreasing the velocity profile and increasing the temperature profiles in both cases.
- The influence of n, M, and K was to increase the coefficient of skin friction with its increase.
- The magnitude of the rate of heat transfer reduced with an increase in , and , thereby increasing the temperature of the fluid for the PST case. However, the decrease was more pronounced with a higher value of . Similarly, the temperature at the wall also decreased with the increase in these governing parameters for the PHF case.
- The effect of was to reduce the thickness of the thermal boundary layer and the rate of heat transfer at the surface for the PST case.
- The surface temperature reduced with the increase in for the PHF case. This implies that an increase the Prandtl number had a cooling effect on the surface.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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n | K | ||||
---|---|---|---|---|---|
Vajravelu [7] | Present Results | Arnold et al. [43] | Present Results | ||
0 | – | 1.0000 | 0.9999 | – | – |
5 | – | 1.1945 | 1.1946 | – | – |
10 | – | 1.2348 | 1.2346 | – | – |
– | 0 | – | – | −1.0000 | −1.0004 |
– | 0.1 | – | – | −1.0041 | −1.0052 |
– | 0.3 | – | – | −1.1952 | −1.1971 |
– | 0.5 | – | – | −1.4142 | −1.4147 |
Arnolds et al. [43] | Present Results | Arnolds et al. [43] | Present Results | |
---|---|---|---|---|
1 | 0.988 | 1.000 | 1.009 | 1.000 |
5 | 2.236 | 2.229 | 0.624 | 0.622 |
10 | 3.080 | 3.056 | 0.564 | 0.567 |
100 | 8.787 | 8.786 | 0.503 | 0.506 |
M | n | K | |
---|---|---|---|
0 | 1 | 0.4 | 1.0590 |
0 | 2 | 0.4 | 2.1084 |
0 | 3 | 0.4 | 6.1704 |
5 | 1 | 0.4 | 2.4888 |
5 | 2 | 0.4 | 3.3021 |
5 | 3 | 0.4 | 6.1704 |
0 | 2 | 0.4 | 2.1084 |
0 | 2 | 2.0 | 10.2509 |
0 | 2 | 4.0 | 12.6666 |
5 | 2 | 0.4 | 3.3021 |
5 | 2 | 2.0 | 10.1751 |
5 | 2 | 4.0 | 12.3794 |
2 | 0 | 0.4 | 0.1 | 0.968208 | 3.388100 | 0.828806 | 0.201817 |
2 | 0 | 0.4 | 0.5 | 0.846103 | 2.595794 | 0.893924 | 0.295620 |
2 | 0 | 0.4 | 1.0 | 0.693472 | 1.605411 | 0.975321 | 0.412874 |
2 | 0 | 2.0 | 0.1 | 0.772763 | 2.402988 | 0.760442 | 0.183771 |
2 | 0 | 2.0 | 0.5 | 0.451428 | −0.365513 | 0.791460 | 0.224805 |
2 | 0 | 2.0 | 1.0 | 0.049759 | −3.826139 | 0.830232 | 0.276098 |
2 | 5 | 0.4 | 0.1 | 0.681828 | 2.842824 | 1.078284 | 0.266512 |
2 | 5 | 0.4 | 0.5 | 0.418812 | 0.959411 | 1.281700 | 0.563798 |
2 | 5 | 0.4 | 1.0 | 0.090042 | −1.345851 | 1.535970 | 0.935405 |
2 | 5 | 2.0 | 0.1 | 0.563905 | 2.067077 | 0.875843 | 0.213427 |
2 | 5 | 2.0 | 0.5 | 0.160478 | −1.345851 | 0.965558 | 0.343080 |
2 | 5 | 2.0 | 1.0 | −3.343807 | −5.612011 | 1.077701 | 0.505146 |
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Jafar, A.B.; Shafie, S.; Ullah, I. Magnetohydrodynamic Boundary Layer Flow of a Viscoelastic Fluid Past a Nonlinear Stretching Sheet in the Presence of Viscous Dissipation Effect. Coatings 2019, 9, 490. https://doi.org/10.3390/coatings9080490
Jafar AB, Shafie S, Ullah I. Magnetohydrodynamic Boundary Layer Flow of a Viscoelastic Fluid Past a Nonlinear Stretching Sheet in the Presence of Viscous Dissipation Effect. Coatings. 2019; 9(8):490. https://doi.org/10.3390/coatings9080490
Chicago/Turabian StyleJafar, Ahmad Banji, Sharidan Shafie, and Imran Ullah. 2019. "Magnetohydrodynamic Boundary Layer Flow of a Viscoelastic Fluid Past a Nonlinear Stretching Sheet in the Presence of Viscous Dissipation Effect" Coatings 9, no. 8: 490. https://doi.org/10.3390/coatings9080490