Flow and heat transfer in a Maxwell liquid film over an unsteady stretching sheet in a porous medium with radiation
- Shimaa E. Waheed^{1, 2}Email author
Received: 11 March 2016
Accepted: 23 June 2016
Published: 12 July 2016
Abstract
A problem of flow and heat transfer in a non-Newtonian Maxwell liquid film over an unsteady stretching sheet embedded in a porous medium in the presence of a thermal radiation is investigated. The unsteady boundary layer equations describing the problem are transformed to a system of non-linear ordinary differential equations which is solved numerically using the shooting method. The effects of various parameters like the Darcy parameter, the radiation parameter, the Deborah number and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. It is observed that increasing values of the Darcy parameter and the Deborah number cause an increase of the local skin-friction coefficient values and decrease in the values of the local Nusselt number. Also, it is noticed that the local Nusselt number increases as the Prandtl number increases and it decreases with increasing the radiation parameter. However, it is found that the free surface temperature increases by increasing the Darcy parameter, the radiation parameter and the Deborah number whereas it decreases by increasing the Prandtl number.
Keywords
Maxwell fluid Liquid film Thermal radiation Unsteady stretching sheet Porous mediumBackground
The flow and heat transfer within a thin liquid film due to the stretching surface in otherwise quiescent fluid are important because of their wide applications in a number of industrial engineering processes. Examples may be found in the cooling of a large metallic plate in a cooling path, design of various heat exchangers, wire and fiber coating, manufacturing plastic films, continuous casting, crystal growing,artificial fiber, reactor fluidization, chemical processing equipment, a polymer sheet, polymer extrusion, annealing and tinning of copper wires, etc. The flow problem within a liquid film of Newtonian fluid on an unsteady stretching surface where the similarity transformation was used to transform the governing partial differential equations describing the problem to a non-linear ordinary differential equation with an unsteadiness parameter first are studied by Wang (1990). Many authors (Usha and Sridharan 1995; Andersson et al. 2000; Dandapat et al. 2003, 2007; Wang 2006; Dandapat and Maity 2006; Liu and Andersson 2008; Noor and Hashim 2010; Mahmoud 2010; Ray and Mazumder 2001; Abel et al. 2009) investigated thin liquid film under different situations.
Many important fluids, however, such as molten plastics, polymers, etc., are non-Newtonian in their flow characteristics. The flow of non-Newtonian fluids are finding increasing applications in several manufacturing processes. Flow of a thin liquid film of a power-law fluid caused by the unsteady stretching of a surface studied numerically by Andersson et al. (1996) and analytically by Wang and Pop (2006). The thin film flow problem with a third grade fluid on an inclined plane hase beed investigated by Siddiqui et al. (2008). Chen (2007) examined the effect of Marangoni convection of the flow and heat transfer within a power-law liquid film on unsteady stretching sheet. Siddiqui et al. (2007) presented the thin film flow of two non-Newtonian fluids namely, Sisko and an Oldroyd 6-constant fluid on a vertical moving belt. Hayat et al. (2008) presented an exact solution for the thin film flow problem of a third grade on an inclined plane. The problem of the flow and heat transfer in a thin film of power-law fluid on an unsteady stretching surface has been investigated by Chen (2003, 2006) where he also studied the effect of viscous dissipation on heat transfer in a non-Newtonian thin liquid film over an unsteady stretching sheet. The flow and heat transfer problem of a second grade fluid film over an unsteady stretching sheet has been presented by Hayat et al. (2008). Abel et al. (2009) investigated the effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet. Sajid et al. (2009) presented exact solutions for thin film flows of a micropolar fluid down an inclined plane on moving belt and down a vertical cylinder. Mahmoud and Megahed (2009) investigated the effects of variable viscosity and thermal conductivity on the flow and heat transfer of an electrically conducting non-Newtonian power-law fluid within a thin liquid film over an unsteady stretching sheet in the presence of a transverse magnetic field.
Non of the above authors deals with the problem involving the thermal radiation on the flow and heat transfer in a liquid film on unsteady stretching surface. Thermal radiation effects may play an important role in controlling heat transfer in industry where the desired product with a sought characteristics depends on the heat controlling factors to some extent. The effect of thermal radiation on the flow and heat transfer of a non-Newtonian fluids has been studied by several authors (Aliakbar et al. 2009; Mahmoud 2007; Raptis 1998, 1999; Siddheshwar and Mahabaleswar 2005; Hayat and Qasim 2010). The transfer of heat due to the missing electromagnetic waves (thermal radiation) has been presented by Baleanu et al. (2015). Available literature shows that the effect of thermal radiation on Maxwell liquid film over an unsteady stretching sheet immersed in a porous medium is not being carried out. Therefore, the aim of this study is to investigate the influence of thermal radiation on heat transfer in an upper-convected Maxwell liquid film over an unsteady stretching surface embedded in a porous medium.
Formulation of the problem
Results and discussion
Comparison of \(\beta\) and \(-f^{''}\)(0) with \(De=D=0\)
S | Abel et al. (2009) | Present result | ||
---|---|---|---|---|
\(\beta\) | \(-f^{''}\)(0) | \(\beta\) | \(-f^{''}\)(0) | |
0.4 | 4.981455 | 1.134098 | 4.981455 | 1.134096 |
0.6 | 3.131710 | 1.195128 | 3.131711 | 1.195125 |
0.8 | 2.151990 | 1.245805 | 2.151992 | 1.245805 |
1.0 | 1.543617 | 1.277769 | 1.543616 | 1.277769 |
1.2 | 1.127780 | 1.279171 | 1.127781 | 1.279171 |
1.4 | 0.821033 | 1.233545 | 0.821032 | 1.233545 |
1.6 | 0.576176 | 1.114941 | 0.576175 | 1.114939 |
1.8 | 0.356390 | 0.867416 | 0.356389 | 0.867416 |
Comparison of \(\theta (\beta )\) and \(-\theta ^{'}\)(0) with \(De=D=R=0\)
Pr | S | \(\beta\) | Abel et al. (2009) | Present result | ||
---|---|---|---|---|---|---|
\(\theta (\beta )\) | \(-\theta ^{'}\)(0) | \(\theta (\beta )\) | \(-\theta ^{'}\)(0) | |||
0.01 | 0.8 | 2.151990 | 0.960438 | 0.042120 | 0.960480 | 0.042042 |
0.1 | 0.8 | 2.151990 | 0.692269 | 0.351920 | 0.692533 | 0.351377 |
1.0 | 0.8 | 2.151990 | 0.097825 | 1.671919 | 0.097884 | 1.671003 |
2.0 | 0.8 | 2.151990 | 0.024869 | 2.443914 | 0.024862 | 2.443866 |
3.0 | 0.8 | 2.151990 | 0.008324 | 3.034915 | 0.008311 | 3.036115 |
0.01 | 1.2 | 1.127780 | 0.982312 | 0.033515 | 0.982331 | 0.033459 |
0.1 | 1.2 | 1.127780 | 0.843485 | 0.305409 | 0.843622 | 0.304963 |
1.0 | 1.2 | 1.127780 | 2.86634 | 1.773772 | 2.86718 | 1.773030 |
2.0 | 1.2 | 1.127780 | 0.128174 | 2.638431 | 0.128123 | 2.638725 |
3.0 | 1.2 | 1.127780 | 0.067737 | 3.280329 | 0.067645 | 3.281949 |
Values of \(-f^{''}\)(0) and \(-\theta ^{'}\)(0) for various values of D, R, De, S and Pr
D | R | De | Pr | S | \(-f^{''}\)(0) | \(\theta (\beta )\) | \(-\theta ^{'}\)(0) |
---|---|---|---|---|---|---|---|
0.0 | 1 | 0.2 | 5 | 0.8 | 1.2827 | 0.0187 | 2.7469 |
0.5 | 1 | 0.2 | 5 | 0.8 | 1.4606 | 0.0378 | 2.7177 |
1 | 1 | 0.2 | 5 | 0.8 | 1.6187 | 0.0600 | 2.6910 |
0.0 | 1 | 0.2 | 5 | 1.2 | 1.3056 | 0.1023 | 2.9708 |
0.5 | 1 | 0.2 | 5 | 1.2 | 1.4548 | 0.1427 | 2.9404 |
1 | 1 | 0.2 | 5 | 1.2 | 1.5901 | 0.1813 | 2.9087 |
0.5 | 0 | 0.2 | 5 | 0.8 | 1.4606 | 0.0059 | 3.9397 |
0.5 | 1 | 0.2 | 5 | 0.8 | 1.4606 | 0.0378 | 2.7177 |
0.5 | 2 | 0.2 | 5 | 0.8 | 1.4606 | 0.0826 | 2.1753 |
0.5 | 5 | 0.2 | 5 | 0.8 | 1.4606 | 0.2187 | 1.4581 |
0.5 | 0 | 0.2 | 5 | 1.2 | 1.4548 | 0.0458 | 4.2638 |
0.5 | 1 | 0.2 | 5 | 1.2 | 1.4548 | 0.1426 | 2.9404 |
0.5 | 2 | 0.2 | 5 | 1.2 | 1.4548 | 0.2323 | 2.3364 |
0.5 | 5 | 0.2 | 5 | 1.2 | 1.4548 | 0.4247 | 1.5150 |
0.5 | 1 | 0.0 | 5 | 0.8 | 1.4280 | 0.0316 | 2.7249 |
0.5 | 1 | 0.2 | 5 | 0.8 | 1.4606 | 0.0378 | 2.7177 |
0.5 | 1 | 0.5 | 5 | 0.8 | 1.5088 | 0.0479 | 2.7067 |
0.5 | 1 | 0.0 | 5 | 1.2 | 1.4311 | 0.1322 | 2.9478 |
0.5 | 1 | 0.2 | 5 | 1.2 | 1.4548 | 0.1426 | 2.9404 |
0.5 | 1 | 0.5 | 5 | 1.2 | 1.4904 | 0.1584 | 2.9286 |
0.5 | 1 | 0.2 | 3 | 0.8 | 1.4606 | 0.0984 | 2.0508 |
0.5 | 1 | 0.2 | 5 | 0.8 | 1.4606 | 0.0378 | 2.7177 |
0.5 | 1 | 0.2 | 10 | 0.8 | 1.4606 | 0.0059 | 3.9397 |
0.5 | 1 | 0.2 | 3 | 1.2 | 1.4548 | 0.2591 | 2.1955 |
0.5 | 1 | 0.2 | 5 | 1.2 | 1.4548 | 0.1427 | 2.9404 |
0.5 | 1 | 0.2 | 10 | 1.2 | 1.4548 | 0.0458 | 4.2638 |
The numerical values of the local skin-friction and the local Nusselt number in terms of \(-\theta ^{'}(0)\) for various values of the Darcy parameter D, the radiation parameter R, the Deborah number De and the Prandtl number Pr for both cases \(S=0.8\) and \(S=1.2\) are tabulated in Table 3. It can be seen that the local skin-friction coefficient increased by increasing D or De, whereas the local Nusselt number decreases with the increasing the Darcy parameter and the Deborah number. Also, it is noticed that the local Nusselt number decreases as the radiation parameter increases and increases with the increase of the Prandtl number. However it is found that the local Nusselt number decreases as S increases whereas the local skin-friction coefficient decreases with the unsteadiness parameter. Moreover, it is observed that the free surface temperature \(\theta (\beta )\) increases by increasing D, R, S and De whereas it decreased with increasing the Prandtl number.
Conclusions
- 1.
The velocity and the film thickness decreases with increasing the Darcy parameter and the Deborah number.
- 2.
Increasing values of the Darcy parameter, radiation parameter and Deborah number leads to an increase in the temperature.
- 3.
The temperature decreases with increasing the Prandtl number.
- 4.
The Darcy parameter and the Deborah number have the effect of enhancing the local skin-friction coefficient.
- 5.
The local Nusselt number decreases by increasing the radiation parameter, the Darcy parameter and the Deborah number and increases with increasing the Prandtl number.
- 6.
The free surface temperature decreases as the Darcy parameter, the radiation parameter and the Deborah number increase while it decreases as the Prandtl number increases.
Declarations
Acknowledgements
The author would like to thank the reviewers for their valuable comments, which led to the improvement of the work.
Competing interests
The author declares that she has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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