Effects of slip on free convection flow of Casson fluid over an oscillating vertical plate
- Muhammad A Imran^{1}Email author,
- Shakila Sarwar^{2} and
- Muhammad Imran^{3}
Received: 9 November 2015
Accepted: 21 January 2016
Published: 2 February 2016
Abstract
The slip effect on free convection of a Casson fluid past an infinite oscillating vertical plate with constant wall temperature is investigated. It is used to characterize the non-Newtonian fluid behavior. By introducing appropriate non-dimensional variables, the resulting equations are solved analytically by using the Laplace transform technique. The corresponding solutions for a Casson fluid without slip at the boundary for \(\lambda\rightarrow0\), a Newtonian fluid with slip for \(\gamma \rightarrow\infty\), and a Newtonian fluid in the absence of slip for \(\lambda\rightarrow0\) and \(\gamma\rightarrow\infty\) are obtained as limiting cases. The effect of the Casson parameter is seen to suppress the velocity field. Also, the influence of the slip parameter causes a decrease in the velocity field. Numerical results for velocity, temperature, and Nusselt number are shown in various graphs and discussed for the embedded flow parameters.
Keywords
1 Introduction
There has been great deal of interest in understanding the behavior of non-Newtonian fluids [1, 2]. Examples of such rheological complex fluids are blood plasma, chocolate, mustard mayonnaise, tooth paste, shampoo, food stuffs, mud, polymer melts, clay coatings, oils and greases, paints etc. These kinds of fluids offer special challenges to the engineers, modelers, mathematicians, and physicists. The study of non-Newtonian fluids is very important in view of its applications in various branches of engineering and technology, therefore the flow analysis of these fluids is very important in theory and practice. From a theoretical point of view, flows of this type are fundamental in fluid mechanics. Practically speaking, these flows have applications in many manufacturing processes in industry. Due to the great diversity in the physical structures of non-Newtonian fluids, it is not possible to establish a single constitutive equation. Thus, many non-Newtonian fluid models have been proposed of which most are empirical or semi-empirical. The equations of motion for non-Newtonian fluids are much more complicated and are of higher order than the Navier-Stokes equations. The solutions of most of the problems in the real world are usually even numerical, on computers. However, analytical solutions, even if they may not be accurate, can provide some penetrating insight into the physics of a problem which manages a maze of numbers crunched on a computer. For that reason, researchers still look for analytical solutions of the known and unknown problems, particularly the former, as they can act as a bench mark for the latter. Various analytical and numerical approaches/methods by a number of people with and without slip condition [3–26] have been done.
Different models are suggested to express the constitutive equations of non-Newtonian fluids. Amongst these fluid models, there is one known as Casson fluid which was originally introduced by Casson [27]. We can define a Casson fluid as a shear thinning liquid which is assumed to have infinite viscosity at zero rate of shear, and a yield stress below which no flow occurs and a zero viscosity at an infinite rate of shear. The non-linear Casson’s constitutive equation has been found to describe accurately the flow curves of suspensions of pigments in lithographic varnishes used for preparation of printing inks. In particular, the Casson fluid model describes the flow characteristics of blood more accurately at low shear rates and when it flows through small blood vessels [28]. Some famous examples of the Casson fluid include jelly, tomato sauce, honey, soup, and concentrated fruit juices etc. Many researchers [29–38] studied the Casson fluid under different boundary conditions. Some find the solutions by using either approximate methods or numerical schemes and some find its exact analytical solutions. The solutions when the Casson fluids are in free convection flow with constant wall temperature are also determined. On the other hand the flow of the Casson fluid in the presence of heat transfer is also an important research area. Motivated by Khalid et al. [39] we focused on the unsteady flow of a Casson fluid past an oscillating vertical plate with constant wall temperature under the non-slip conditions. In the present paper, we extended the work of Khalid et al. by applying the slip condition at the boundary. Exact solutions are obtained by applying the Laplace transform technique and it is found that the results in the absence of slip are fully agreed with that of [39].
2 Statement of the problem
Let us consider the heat transfer effect on unsteady boundary layer flow in a Casson fluid past an infinite oscillating vertical plate fixed at \(y=0\), the flow being confined to \(y>0\), where y is the coordinate axis normal to the plate. Initially, for time \(t=0\), both plate and fluid are under stationary conditions with the temperature \(T_{\infty}\). At time \(t=0^{+}\), the plate started an oscillatory motion in its plane and slip is considered at the boundary.
3 Solution of the problem
4 Limiting cases
4.1 Motion without slip when \(\eta\rightarrow0\), implies \(\lambda\rightarrow0\)
4.2 Motion corresponding to viscous fluid with slip when \(\gamma \rightarrow\infty\)
4.3 Motion corresponding to viscous fluid without slip when \(\lambda\rightarrow0\)
5 Graphical results and discussion
6 Conclusion
- 1.
The velocity increases with increasing t and Pr, whereas it decreases with increasing values of Pr, ω, γ and λ.
- 2.
The temperature increases with increasing t, whereas it decreases when Pr is increased.
- 3.
Solution (4.1) corresponding to the no slip condition is found to be in excellent agreement with the result obtained by Khalid et al. [39], equation (14).
- 4.
Solution (4.3) corresponding to a viscous fluid without slip is found to be in excellent agreement with the result obtained by Khalid et al. [39], equation (21).
Declarations
Acknowledgements
The authors would like to thank the editors and the anonymous reviewers, whose insightful comments and constructive suggestions helped us to significantly improve the quality of this paper. The authors would also like to acknowledge the University of Management and Technology, Lahore, for the financial support for this research.
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
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