- Original research
- Open Access
Williamson fluid flow due to a nonlinearly stretching sheet with viscous dissipation and thermal radiation
- Ahmed M. Megahed^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s42787-019-0016-y
© The Author(s) 2019
- Received: 3 August 2018
- Accepted: 5 November 2018
- Published: 7 June 2019
Abstract
Williamson boundary layer fluid flow and heat transfer due to a nonlinearly stretching sheet is undertaken in this research. An important aspect of this study is that the thermal radiation and viscous dissipation phenomena are also included in this model. Because there is still much need for more accurate results, the viscosity and the fluid conductivity are assumed to vary with temperature. Our main purpose is to achieve a similar solution in this physical model. Here, we come across a highly nonlinear differential constitutive equations which are solved numerically after utilizing the shooting method. In this respect, it is noteworthy that the main characteristic developed here is that the study is made for the governing parameters and their effects on both the drag velocity and the heat transfer rate. Further, a significant effect of both radiation parameter and viscosity parameter on heating process is observed.
Keywords
- Williamson fluid
- Nonlinearly stretching sheet
- Variable properties
- Viscous dissipation
Mathematics subject classification
- 76A05
- 76D10
- 65L10
- 65L06
Introduction
During the last two decades, different immense attempts have been performed in fluid flow due to stretching sheet operation to give precious descriptions of standard manufacturing and industrial processes, such as polymer manufacturing, paper production, food preserving processes, crystal manufacturing, and petroleum filtering operation. For this sake and owing to these actual importance on this topic, many researchers have compelled to search about the novel findings which serve this field. The first contributions in this topic were inaugurated in discussion the fluid flow due to a stretched surface by Sakiadis [1]. The problem of fluid flow of Blasius type due to a stretching sheet, which has the application in the field of the drawing of plastic films has been studied by Crane [2]. Furthermore, we can resort to Chen and Char [3] to explore the mechanism of wall heatflux and its influence on an impermeable linearly stretching plate. Mohammadein and Gorla [4] investigated the heat transfer characteristics of a steady micropolar boundary layer fluid flow over a linearly stretching, continuous sheet. Shortly after the authors [4], Liu [5] perceived the general importance of hydromagnetic flow over a stretching sheet by applying it in a heat and mass transfer. Two years elapsed until Cortell [6] again presented steady flow and mass transfer approach to a second-grade fluid behavior based upon the concept of impermeable stretching sheet. In a later study, Chen [7] attempted to complete his research on some important mechanisms of MHD non-Newtonian power-law fluid over a stretching sheet. Recently, an increasing number of studies which concerning in exponentially stretching sheet can be introduced in Ref. [8–10] and in nonlinearly stretching sheet can be adequately described in Ref. [11–16].
In this work, we use the Williamson model, which was firstly studied since in 1929 by Williamson [17]. Due to the usefulness of this type of fluid, more of accurate researches are introduced in this topic [18–22]. In the framework of all previous studies and in an effort to analyze the Williamson fluid flow due to nonlinear stretching sheet with variable properties, thermal radiation, and viscous dissipation, this paper is introduced.
Mathematical description for the physical model
where the constant σ^{∗} is the Stefan-Boltzmann and k^{∗} is the absorption coefficient. In this work, the coefficient k^{∗} must be coincide with the Rosseland approximation. As we observed from Eq. (4), the highly nonlinearity of the term T^{4} can be simplified by using Taylor expansion about the constant value T_{∞} as \(T^{4}\cong 4T_{\infty }^{3}T-3T_{\infty }^{4}\), after ignoring all higher-order terms [24].
The final relations present the coefficient μ_{∞} and κ_{∞} which represent the viscosity and the thermal conductivity at the ambient, respectively. Also, from the same relations, the parameter α is the viscosity but the parameter ε is the thermal conductivity.
where \(\delta =\left (\frac {\sqrt {2}c^{\frac {3}{2}}x^{\frac {3m-1}{2}}}{\sqrt {\nu _{\infty }}}\right)\Gamma \) is the local Williamson fluid parameter, \(\text {Pr}=\frac {\mu _{\infty } c_{p}}{\kappa _{\infty }}\) is the Prandtl number, \(R=\frac {16\sigma ^{*}T_{\infty }^{3}}{3\kappa _{\infty }k^{*}}\) is the radiation parameter, and \(\text {Ec}=\frac {{U^{2}_{w}} }{c_{p}(T_{w}-T_{\infty })}=\frac {c^{2}x^{2m-r}}{\text {Ac}_{p}}\) is the local Eckert number. After finishing this analyzing, we observe that both the δ parameter and the Ec number are dependent of x. To overcome this dilemma which results in a non-similar solution for our problem, we should take \(r=2m=\frac {2}{3}\). So, these parameters take the form, \(\delta =\left (\frac {\sqrt {2}c^{\frac {3}{2}}}{\sqrt {\nu _{\infty }}}\right)\Gamma \) is the Williamson fluid parameter and \(\text {Ec}=\frac {{U^{2}_{w}} }{c_{p}(T_{w}-T_{\infty })}=\frac {c^{2}}{\text {Ac}_{p}}\) is the Eckert number.
The local skin-friction coefficient and the local Nusselt number
\(\text {Re}_{x}=\frac {U_{w} x}{\nu _{\infty }}\) is the local Reynolds number. It is clear from Eq. (14) that the local skin-friction coefficient Cf_{x} is directly affected with both the viscosity parameter α and the Williamson fluid parameter δ, while noted from Eq. (15) that the local Nusselt number is influenced by both the thermal conductivity parameter ε and the radiation parameter R. Also, an implicit effects for another parameters on both the local Nusselt number and the local skin-friction coefficient can be occurred.
Solution methodology
Comparison of Nusselt number \(\left (\!\text {Nu}_{x}\text {Re}_{x}^{\frac {-1}{2}}\!\right)\) for various values of Pr when δ = α = ε = R = Ec =r = 0 and m=1
Pr | Gorla and Sidawi [26] | Present study |
---|---|---|
0.07 | 0.06562 | 0.065531 |
0.20 | 0.16912 | 0.169117 |
2.0 | 0.91142 | 0.911358 |
7.0 | 1.89546 | 1.895453 |
20.0 | 3.35391 | 3.353902 |
Results and discussion
Values of \(\frac {1}{2}\text {Re}_{x}^{\frac {1}{2}}Cf_{x}\) and \(\text {Re}_{x}^{\frac {-1}{2}}\text {Nu}_{x} \) for various values of α, δ, ε, R, and Ec with \(m=\frac {1}{3}, r=\frac {2}{3}, \text {Pr}=2.0\)
α | δ | ε | R | Ec | \(\frac {1}{2}\text {Re}_{x}^{\frac {1}{2}}\text {Cf}_{x}\) | \(\text {Re}_{x}^{\frac {-1}{2}}\text {Nu}_{x} \) |
---|---|---|---|---|---|---|
0.0 | 0.2 | 0.2 | 0.2 | 0.2 | 0.659037 | 1.33148 |
0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.546574 | 1.27392 |
1.0 | 0.2 | 0.2 | 0.2 | 0.2 | 0.439826 | 1.20470 |
0.5 | 0.0 | 0.2 | 0.2 | 0.2 | 0.567823 | 1.28576 |
0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.546574 | 1.27392 |
0.5 | 0.5 | 0.2 | 0.2 | 0.2 | 0.506503 | 1.24864 |
0.5 | 0.2 | 0.0 | 0.2 | 0.2 | 0.549231 | 1.21855 |
0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.546574 | 1.27392 |
0.5 | 0.2 | 0.5 | 0.2 | 0.2 | 0.543210 | 1.34926 |
0.5 | 0.2 | 0.2 | 0.0 | 0.2 | 0.549629 | 1.19309 |
0.5 | 0.2 | 0.2 | 0.5 | 0.2 | 0.542856 | 1.37819 |
0.5 | 0.2 | 0.2 | 1.0 | 0.2 | 0.538196 | 1.51929 |
0.5 | 0.2 | 0.2 | 0.2 | 0.0 | 0.548239 | 1.36147 |
0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.546574 | 1.27392 |
0.5 | 0.2 | 0.2 | 0.2 | 0.5 | 0.544103 | 1.14333 |
Concluding remarks
According to the yielded analysis for the proposed Williamson fluid flow, some of main interesting results which are drawn from this study are established below in an elaborated form as follows: (i) Both the thermal radiation parameter and the Eckert number have the influence to enhance the temperature distribution, thicken the thermal region, thus increase the local Nusselt number and decrease the local skin-friction coefficient. (ii) Increasing both the viscosity parameter and the Williamson parameter will result in a rise in the temperature distribution, and hence a diminishing behavior for both the rate of heat transfer and the local skin-friction coefficient. (iii) The thermal conductivity parameter has an impact in enhancing the temperature distribution, hence an increasing behavior for the local Nusselt number.
Declarations
Acknowledgements
The helpful criticism of honorable editor and the anonymous reviewers in the preparation of this paper is gratefully acknowledged.
Funding
Not applicable.
Availability of data and materials
Not applicable.
Competing interests
The author declares that he has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This 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
References
- Sakiadis, B. C.: Boundary layer behaviour on continuous moving solid surfaces. I. Boundary layer equations for two-dimensional and axisymmetric flow. II. Boundary layer on a continuous flat surface. III. Boundary layer on a continuous cylindrical surface. AIChE J. 7, 26–28 (1961).View ArticleGoogle Scholar
- Crane, L. J., Angew, Z.: Flow past a stretching plate. Math. Phys. 21, 645–647 (1970).Google Scholar
- Chen, C. K., Char, M. I.: Heat transfer of a continuous stretching surface with suction or blowing. J. Math. Anal. Appl. 135, 568–580 (1988).MathSciNetView ArticleGoogle Scholar
- Mohammadein, A. A., Gorla, R. S. R.: Heat transfer in a micropolar fluid over a stretching sheet with viscous dissipation and internal heat generation. Int. J. Numer. Meth. Heat Fluid Flow. 11, 50–58 (2001).View ArticleGoogle Scholar
- Liu, I. C.: A note on heat and mass transfer for a hydromagnetic flow over a stretching sheet. Int. Commun. Heat Mass Transf. 32, 1075–1084 (2005).View ArticleGoogle Scholar
- Cortell, R.: MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species. Chem. Eng. Process. 46, 721–728 (2007).View ArticleGoogle Scholar
- Chen, C.: Magneto-hydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption. Int. J. Non-Linear Mech. 44, 596–603 (2009).View ArticleGoogle Scholar
- Rehman, F. U., Nadeem, S., Haq, R. U.: Heat transfer analysis for three-dimensional stagnation-point flow over an exponentially stretching surface. Chinese J. Phys. 55, 1552–1560 (2017).View ArticleGoogle Scholar
- Rehman, F. U., Nadeem, S.: Heat transfer tnalysis for three-dimensional stagnation-point flow of water-based nanofluid over an exponentially stretching surface. J Heat Transf. 140(5), 0524011–0524017 (2018).View ArticleGoogle Scholar
- Hayat, T., Nadeem, S.: Flow of 3D Eyring-Powell fluid by utilizing Cattaneo-Christov heat flux model and chemical processes over an exponentially stretching surface. Results Phys. 8, 397–403 (2018).View ArticleGoogle Scholar
- Megahed, A. M.: Variable viscosity and slip velocity effects on the flow and heat transfer of a power-law fluid over a non-linearly stretching surface with heat flux and thermal radiation. Rheol. Acta. 51, 841–874 (2012).View ArticleGoogle Scholar
- Mahmoud, M. A. A., Megahed, A. M.: Non-uniform heat generation effects on heat transfer of a non-Newtonian fluid over a non-linearly stretching sheet. Meccanica. 47, 1131–1139 (2012).MathSciNetView ArticleGoogle Scholar
- Megahed, A. M.: Flow and heat transfer of a non-Newtonian power-law fluid over a non-linearly stretching vertical surface with heat flux and thermal radiation. Meccanica. 50, 1693–1700 (2015).MathSciNetView ArticleGoogle Scholar
- Saif, R. S., Hayat, T., Ellahi, R., Muhammad, T., Alsaedi, A.: Stagnation-point flow of second grade nanofluid towards a nonlinear stretching surface with variable thickness. Results Phys. 7, 2821–2830 (2017).View ArticleGoogle Scholar
- Hayat, T., Sajjad, R., Muhammad, T., Alsaedi, A., Ellahi, R.: On MHD nonlinear stretching flow of Powell-Eyring nanomaterial. Results Phys. 7, 535–543 (2017).View ArticleGoogle Scholar
- Megahed, A. M.: Flow and heat transfer of non-Newtonian Sisko fluid past a nonlinearly stretching sheet with heat generation and viscous dissipation. J. Braz. Soc. Mech. Sci. Eng. 40, 492 (2018). https://doi.org/doi.org/10.1007/s40430-018-1410-3.
- Williamson, W. W.: The flow of pseudoplastic materials. Ind. Eng. Chem. 21, 1108–1111 (1929).View ArticleGoogle Scholar
- Nadeem, S., Hussain, S. T., Lee, C.: Flow of a Williamson fluid over a stretching sheet. Braz. J. Chem. Eng. 30, 619–625 (2013).View ArticleGoogle Scholar
- Khan, N. A., Khan, H. A.: A Boundary layer flows of non-Newtonian Williamson fluid. Nonlinear Eng. 3, 107–115 (2014).Google Scholar
- Malik, M. Y., Salahuddin, T.: Numerical solution of MHD stagnation point flow of Williamson fluid model over a stretching cylinder. Int. J. Nonlin. Sci. Num. 16, 161–164 (2015).MathSciNetView ArticleGoogle Scholar
- Malik, M. Y., Bibi, M., Khan, F., Salahuddin, T.: Numerical solution of Williamson fluid flow past a stretching cylinder and heat transfer with variable thermal conductivity and heat generation/absorption. AIP Adv. 6, 035101 (2016).View ArticleGoogle Scholar
- Vittal, C., Reddy, M. C. K., Vijayalaxmi, T.: MHD stagnation point flow and heat transfer of Williamson fluid over exponential stretching sheet embedded in a thermally stratified medium. Glob. J. Pur. Appl. Math. 13, 2033–2056 (2017).Google Scholar
- Raptis, A.: Flow of a micropolar fluid past a continuously moving plate by the presence of radiation. Int. J. Heat Mass Tran. 41, 2865–2866 (1998).View ArticleGoogle Scholar
- Raptis, A.: Radiation and viscoelastic flow. Int. Commun. Heat Mass Tran. 26, 889–895 (1999).View ArticleGoogle Scholar
- Mahmoud, M. A. A., Megahed, A. M.: MHD flow and heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet with variable fluid properties. Can. J. Phys. 87, 1065–1071 (2009).View ArticleGoogle Scholar
- Gorla, R. S. R., Sidawi, I.: Free convection on a vertical stretching surface with suction and blowing. Appl. Sci. Res. 52, 247–257 (1994).View ArticleGoogle Scholar