- Research
- Open Access
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
- Casson fluid
- oscillating flows
- free convection
- velocity field
- exact solutions
- slip effect
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
References
- Anglin, JR: Handbook of Lubrication and Tribology. Taylor & Francis, London (2006) Google Scholar
- Chhabra, RP, Richardson, JF: Non-Newtonian Flow and Applied Rheology: Engineering Application, 2nd edn. Butterworth-Heinemann, Burlington (2008) Google Scholar
- Fetecau, C, Fetecau, C: Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int. J. Eng. Sci. 43, 781-789 (2005) View ArticleMathSciNetMATHGoogle Scholar
- Fetecau, C, Hayat, T, Fetecau, C, Ali, N: Unsteady flow of a second grade fluid between two side walls perpendicular to a plate. Nonlinear Anal., Real World Appl. 9, 1236-1252 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Fetecau, C, Fetecau, C, Rana, M: General solutions for the unsteady flow of second grade fluid over an infinite plate that applies arbitrary shear to the fluid. Z. Naturforsch. 66a, 753-759 (2011) Google Scholar
- Pop, I, Soundalgekar, VM: Effects of Hall current on hydromagnetics flow near a porous plate. Acta Mech. 20, 315-318 (1974) View ArticleMATHGoogle Scholar
- Hayat, T, Asghar, S, Siddiqui, AM: Periodic flows of a non-Newtonian fluid. Acta Mech. 131, 169-175 (1998) View ArticleMathSciNetMATHGoogle Scholar
- Hayat, T, Kara, AH, Momoniat, E: Exact flow of a third grade fluid on a porous wall. Int. J. Non-Linear Mech. 38, 1533-1537 (2003) View ArticleMathSciNetMATHGoogle Scholar
- Hayat, T, Fetecau, C, Sajid, M: Analytic solution for MHD transient rotating flow of a second grade fluid in a porous space. Nonlinear Anal., Real World Appl. 9, 1619-1627 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Imram, MA, Vieru, D, Rauf, A: Slip effect of second grade on free convection flow with ramped wall temperature. Heat Transf. Res. 46(8), 713-724 (2015) View ArticleGoogle Scholar
- Imran, MA, Imran, M, Fetecau, C: MHD oscillating flows of rotating second grade fluids in a porous medium. Commun. Numer. Anal. 2014, Article ID 00196 (2014) MathSciNetGoogle Scholar
- Khan, M, Ali, SH, Hayat, T, Fetecau, C: MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium. Int. J. Non-Linear Mech. 43(4), 302-319 (2007) View ArticleGoogle Scholar
- Ali, F, Norzieha, M, Sharidan, S, Khan, I, Hayat, T: New exact solutions of Stokes’ second problem for an MHD second grade fluid in a porous space. Int. J. Non-Linear Mech. 47, 521-525 (2012) View ArticleGoogle Scholar
- Roberts, GE, Kaufman, H: Table of Laplace Transform. W. B. Saunders Company, Philadelphia (1966) Google Scholar
- Elahi, R: Effects of the slip boundary condition on non-Newtonian flows in a channel. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1377-1384 (2009) View ArticleGoogle Scholar
- Elahi, R, Hussain, F: Simultaneous effects of MHD and partial slip on peristaltic flow of Jeffery fluid in a rectangular duct. J. Magn. Magn. Mater. 393, 284-292 (2015) View ArticleGoogle Scholar
- Riaz, A, Nadeem, S, Ellahi, R, Zeeshan, A: Exact solution for peristaltic flow of Jeffrey fluid model in a three dimensional rectangular duct having slip at the walls. Appl. Bionics Biomech. 11, 81-90 (2014) View ArticleGoogle Scholar
- Ellahi, R, Wang, X, Hameed, M: Effects of heat transfer and nonlinear slip on the steady flow of Couette fluid by means of Chebyshev spectral method. Z. Naturforsch. A 69a, 1-8 (2014) View ArticleGoogle Scholar
- Khan, AA, Elahi, R, Usman, M: The effects of variable viscosity on the peristaltic flow of non-Newtonian fluid through a porous medium in an inclined channel with slip boundary conditions. J. Porous Media 16(1), 59-67 (2013) View ArticleGoogle Scholar
- Zeeshan, A, Elahi, R: Series solutions of nonlinear partial differential equations with slip boundary conditions for non-Newtonian MHD fluid in porous space. Appl. Math. Inf. Sci. 7(1), 253-261 (2013) View ArticleGoogle Scholar
- Ellahia, R, Shivanian, E, Abbasbandy, S, Rahman, SU, Hayat, T: Analysis of steady flows in viscous fluid with heat transfer and slip effects. Int. J. Heat Mass Transf. 55, 6384-6390 (2012) View ArticleGoogle Scholar
- Elahi, R, Hameed, M: Numerical analysis of steady non-Newtonian flows with heat transfer analysis, MHD and nonlinear slip effects. Int. J. Numer. Methods Heat Fluid Flow 22, 24-38 (2012) View ArticleGoogle Scholar
- Sultan, Q, Nazar, M, Imran, M, Ali, U: Flow of generalized Burgers fluid between parallel walls induced by rectified sine pulses stress. Bound. Value Probl. 2014, Article ID 152 (2014) View ArticleMathSciNetGoogle Scholar
- Ellahi, R, Hayat, T, Mahomed, FM, Zeeshan, A: Exact solutions of flows of an Oldroyd 8-constant fluid with nonlinear slip conditions. Z. Naturforsch. A 65a, 1081-1086 (2010) Google Scholar
- Zhao, H, Yao, Z: Optimal boundary conditions for the Navier-Stokes fluid in a bounded domain with a thin layer. Bound. Value Probl. 2015, Article ID 221 (2015) View ArticleMathSciNetGoogle Scholar
- Ahmed, N, Dutta, M: Heat transfer in an unsteady MHD flow through an infinite annulus with radiation. Bound. Value Probl. 2015, Article ID 11 (2015) View ArticleMathSciNetGoogle Scholar
- Casson, N: A flow equation for the pigment oil suspension of the printing ink type. In: Rheology of Disperse Systems, pp. 84-102. Pergamon, New York (1959) Google Scholar
- McDonald, DA: Blood Flows in Arteries, 2nd edn., Chapter 2. Arnold, London (1974) Google Scholar
- Mustafa, M, Hayat, T, Pop, I, Aziz, A: Unsteady boundary layer flow of a Casson fluid due to impulsively started moving flat plate. Heat Transf. Asian Res. 40(6), 563-576 (2011) View ArticleGoogle Scholar
- Hayat, T, Shehzad, SA, Alsaedi, A, Alhothuali, MS: Mixed convection stagnation point flow of Casson fluid with convective boundary conditions. Chin. Phys. Lett. 29(11), Article ID 114704 (2012) View ArticleGoogle Scholar
- Mukhopadhyay, S: Effects of thermal radiation on Casson fluid flow and heat transfer over an unsteady stretching surface subject to suction/blowing. Chin. Phys. B 22(11), Article ID 114702 (2013) View ArticleGoogle Scholar
- Mukhopadhyay, S, De, PR, Bhattacharyya, K, Layek, GC: Casson fluid flow over an unsteady stretching surface. Ain Shams Eng. J. 4, 933-938 (2013) View ArticleGoogle Scholar
- Mabood, F, Shateyi, S, Khanm, WA: Effects of thermal radiation on Casson flow heat and mass transfer around a circular cylinder in porous medium. Eur. Phys. J. Plus 130, 188 (2015) View ArticleGoogle Scholar
- Bhattacharyya, K: Boundary layer stagnation-point flow of Casson fluid and heat transfer towards a shrinking/stretching sheet. Front. Heat Mass Transf. 4, Article ID 023003 (2013) View ArticleGoogle Scholar
- Pramanik, S: Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation. Ain Shams Eng. J. 5, 205-212 (2014) View ArticleGoogle Scholar
- Shateyi, S: A new numerical approach to MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction. Bound. Value Probl. 2013, Article ID 196 (2013) View ArticleMathSciNetGoogle Scholar
- Makanda, G, Shaw, S, Sibanda, P: Effects of radiation on MHD free convection of a Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium with viscous dissipation. Bound. Value Probl. 2015, Article ID 75 (2015) View ArticleMathSciNetGoogle Scholar
- Kim, S: Study of non-Newtonian viscosity and yield stress of blood in a scanning capillary-tube rheometer. Ph.D. thesis, Mechanical Engineering and Mechanics (2002) Google Scholar
- Khalid, A, Khan, I, Shafie, S: Exact solutions for unsteady free convection flow of Casson fluid over an oscillating vertical plate with constant wall temperature. Abstr. Appl. Anal. 2014, Article ID 946350 (2014) MathSciNetGoogle Scholar
- Hetnarski, RB: An algorithm for generating some inverse Laplace transforms of exponential form. Z. Angew. Math. Phys. 26, 249-253 (1975) View ArticleMathSciNetMATHGoogle Scholar