Chebyshev spectral method for studying the viscoelastic slip flow due to a permeable stretching surface embedded in a porous medium with viscous dissipation and non-uniform heat generation
- MM Khader^{1, 2}Email author and
- Samy Mziou^{1}
Received: 3 September 2016
Accepted: 2 March 2017
Published: 16 March 2017
Abstract
Herein, we study the numerical solution with the help of Chebyshev spectral collocation method for the ordinary differential equations which describe the flow of viscoelastic fluid over a stretching sheet embedded in a porous medium with viscous dissipation and slip velocity. The novel effects for the parameters which affect the flow and heat transfer, such as the Eckert number coupled with a porous medium and the velocity slip parameter, are studied. Also, the convergence analysis for the proposed method is addressed.
Keywords
viscoelastic fluid Chebyshev spectral collocation method porous medium slip velocity1 Introduction
Owing to the importance of the fluid flow over a stretching surface because of its practical applications such as hot rolling, fiber plating, and lubrication processes, Crane [1] was the first researcher to investigate an analytical solution to the problem of Newtonian boundary layer equations for the flow due to a stretching surface. In the same context, similar problems of Newtonian flow at different situations past a stretching surface have been extended by many authors [2–5].
The viscoelastic fluid belongs to a very important class of non-Newtonian fluids which is often found in many fields of engineering fluid mechanics because of its immense applications, such as inks, paints, and jet fuels. From this standpoint and because of this great importance to this type of fluid, many researchers turned to the study of this type under different conditions. From these researchers, for example, but not limited [6–14].
Motivated by the above mentioned studies, the effects of velocity slip is absence. However, this phenomenon is very interesting in fluid mechanics. The earliest slip boundary condition was proposed by Navier [15]. After the pioneering work of Navier [15], many papers dealing with this aspect are presented [16–18]. In the same context, velocity slip effect on the viscoelastic fluid flow and heat transfer was introduced by many authors [19–21].
The prediction of heat transfer characteristics for the non-Newtonian viscoelastic fluids in porous media is very important due to its practical engineering applications, such as oil recovery, flow through filtering media, and food processing. So, in this work a new visualization for the effects of viscous dissipation, non-uniform heat generation/absorption and velocity slip on the flow and heat transfer of a viscoelastic fluid over a stretching sheet embedded in porous medium is presented.
The Chebyshev collocation methods are used to solve many problems of ODEs and PDEs [22, 23]. We can solve ODEs or PDEs to high accuracy on a simple domain using these methods. They can often achieve 10 digits of accuracy where the finite difference method or finite element method would get two or three digits of accuracy [24]. At lower accuracies, they demand less computational time and computer memory than the alternatives. For a recent efficient use of spectral methods, in physical and engineering problems, see [25, 26]. So, in this work, we use the properties of the Chebyshev polynomials to derive an approximate formula of the integer derivative \(D^{(n)}y(x)\) and estimate an error upper bound of this formula, then we use this formula to solve numerically the proposed problem.
2 Formulation of the problem
The sheet is assumed to have the velocity \(U=cx\) where x is the coordinate measured along the stretching surface and \(c(>0)\) is a constant for a stretching sheet. Likewise, the temperature distribution for the sheet is assumed to be in the form \(T_{w}=T_{\infty}+Ax^{r}\) where \(T_{w}\) is the temperature of the sheet, \(T_{\infty}\) is the temperature of the ambient, A and r are constants. Also, the sheet is assumed to be porous with the suction velocity \(v_{w}>0\).
3 Procedure solution using Chebyshev spectral collocation method
3.1 An approximate formula of the derivative for Chebyshev polynomials expansion
In order to use these polynomials on the interval \([0,1]\) for example, we define the so-called shifted Chebyshev polynomials by introducing the change of variable \(z=2x-1\). The shifted Chebyshev polynomials are denoted by \(ST_{n}(x)=T_{n}(2x-1)=T_{2n}(\sqrt{x})\). We can generalize this formula in any interval \([0,\eta_{\infty}]\).
Theorem 1
Proof
The proof of this theorem can be done directly with the help of equation (19) and some properties of the shifted Chebyshev polynomials. □
Also in this subsection, special attention is given to study the convergence analysis and evaluating the upper bound of the error of the proposed formula.
Theorem 2
Proof
Theorem 3
Proof
3.2 Procedure solution
Equations (29)-(30), together with the six equations of the boundary conditions (31), give a system of \((2m+2)\) algebraic equations which can be solved, for the unknowns \(c_{i}\), \(d_{i}, i =0,1,\ldots,m\), using Newton iteration method.
4 Results and discussion
Comparison of \(\pmb{-(1-K)f^{\prime\prime}(0)}\) with \(\pmb{f_{w}=\beta=\lambda=0}\) using the previous work and the Chebyshev spectral collocation method
K | Rajagopal et al . [ 33 ] | Present work |
---|---|---|
0.005 | 0.9975 | 0.99739823 |
0.010 | 0.9949 | 0.99479907 |
0.030 | 0.9846 | 0.98462934 |
0.050 | 0.9738 | 0.97216181 |
5 Conclusions
The Chebyshev spectral collocation method is used to solve the problem of flow and heat transfer of a viscoelastic fluid over a stretching sheet which is embedded in a porous medium with viscous dissipation, internal heat generation/absorption, and slip velocity. The convergence analysis and the upper bound of the error for the proposed method are presented. From the study, it was found that increasing the suction parameter, the porous parameter, and the slip velocity parameter lowers the momentum boundary layer thickness. Furthermore, an increasing Eckert number causes the thermal boundary layer and temperature distribution to increase.
Declarations
Acknowledgements
The authors thank the Deanship of Academic Research, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA, for the financial support of the project number (361203).
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
- Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645-647 (1970) View ArticleGoogle Scholar
- Vleggar, J: Laminar boundary layer behavior on continuous accelerating surfaces. Chem. Eng. Sci. 32, 1517-1525 (1977) View ArticleGoogle Scholar
- Dutta, BK, Roy, P, Gupta, AS: Temperature field in flow over a stretching sheet with uniform heat flux. Int. Commun. Heat Mass Transf. 12, 89-94 (1985) View ArticleGoogle Scholar
- Pop, I, Na, TY: A note on MHD flow over a stretching permeable surface. Mech. Res. Commun. 25, 263-269 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Ali, ME: On thermal boundary layer on a power law stretched surface with suction or injection. Int. J. Heat Fluid Flow 16, 280-290 (1995) View ArticleGoogle Scholar
- Vajravelu, K, Rollins, D: Heat transfer in a viscoelastic fluid over a stretching sheet. J. Math. Anal. Appl. 158, 241-255 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Andersson, HI: MHD flow of a viscoelastic fluid past a stretching surface. Acta Mech. 95, 227-230 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Sarma, MS, Nageswara Rao, B: Heat transfer in a viscoelastic fluid over a stretching sheet. J. Math. Anal. Appl. 222, 268-275 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Subhas, A, Veena, P: Viscoelastic fluid flow and heat transfer in a porous medium over a stretching sheet. Int. J. Non-Linear Mech. 33, 531-540 (1998) View ArticleMATHGoogle Scholar
- Sanjayanand, E, Khan, SK: On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet. Int. J. Therm. Sci. 45, 819-828 (2006) View ArticleGoogle Scholar
- Nandeppanavar, MM, Abel, MS, Vajravelu, K: Flow and heat transfer characteristics of a viscoelastic fluid in a porous medium over an impermeable stretching sheet with viscous dissipation. Int. J. Heat Mass Transf. 53, 4707-4713 (2010) View ArticleMATHGoogle Scholar
- Subhas Abel, M, Siddheshwar, PG, Nandeppanavar, MM: Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. Int. J. Heat Mass Transf. 50(5), 960-966 (2007) View ArticleMATHGoogle Scholar
- Nandeppanavar, MM, Subhas Abel, M, Siddalingappa, MN: Heat transfer through a porous medium over a stretching sheet with effect of viscous dissipation. Chem. Eng. Commun. 200, 1513-1529 (2013) View ArticleGoogle Scholar
- Hongjun, Y, Yang, Z: A class of compressible non-Newtonian fluids with external force and vacuum under no compatibility conditions. Bound. Value Probl. 2016, 201 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Navier, CLMH: Mémoire sur les lois du mouvement des fluids. Mém. Acad. Sci. Inst. Fr. 6, 389-416 (1823) Google Scholar
- Thompson, PA, Troian, SM: A general boundary condition for liquid flow at solid surfaces. Nature 389, 360-362 (1997) View ArticleGoogle Scholar
- Abbas, Z, Wang, Y, Hayat, T, Oberlack, M: Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int. J. Numer. Methods Fluids 59, 443-458 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Turkyilmazoglu, M: Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet. Int. J. Therm. Sci. 50, 2264-2276 (2011) View ArticleGoogle Scholar
- Ferras, LL, Afonso, AM, Alves, MA, Nobrega, JM, Carneiro, OS, Pinho, FT: Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction. J. Non-Newton. Fluid Mech. 214, 28-37 (2014) View ArticleGoogle Scholar
- Singh, KD: Effect of slip condition on viscoelastic MHD oscillatory forced convection flow in a vertical channel with heat radiation. Int. J. Appl. Mech. Eng. 18(4), 1-10 (2013) Google Scholar
- Krishan, DS: MHD mixed convection visco-elastic slip flow through a porous medium in a vertical porous channel with thermal radiation. Kragujevac J. Sci. 35, 27-40 (2013) Google Scholar
- Khader, MM, Sweilam, NH: On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method. Appl. Math. Model. 37, 9819-9828 (2013) MathSciNetView ArticleGoogle Scholar
- Khader, MM, Sweilam, NH, Mahdy, AMS: Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM. Appl. Math. Inf. Sci. 7(5), 2013-2020 (2013) MathSciNetView ArticleGoogle Scholar
- Emran, T: Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs. Ain Shams Eng. J. 6(1), 373-379 (2015) View ArticleGoogle Scholar
- Rakib, FE, Hamzaga, DO, Zaki, FAE-R: Spectral analysis of wave propagation on branching strings. Bound. Value Probl. 2016, 215 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Khader, MM: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 16, 2535-2542 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Ingham, DB, Pop, I, Cheng, P: Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation. Transp. Porous Media 5, 381-398 (1990) View ArticleGoogle Scholar
- Rees, D, Lage, JL: The effect of thermal stratification of natural convection in a vertical porous insulation layer. Int. J. Heat Mass Transf. 40, 111-121 (1997) View ArticleMATHGoogle Scholar
- Beckett, PM: Combined natural and forced convection between parallel vertical walls. SIAM J. Appl. Math. 39, 372-384 (1980) MathSciNetView ArticleMATHGoogle Scholar
- Beckett, PM, Friend, IE: Combined natural and forced convection between parallel walls: developing flow at higher Rayleigh numbers. Int. J. Heat Mass Transf. 27, 611-621 (1984) View ArticleMATHGoogle Scholar
- Chamkha, AJ, Khaled, AA: Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption. Heat Mass Transf. 37, 117-123 (2001) View ArticleGoogle Scholar
- Snyder, MA: Chebyshev Methods in Numerical Approximation. Prentice-Hall, Englewood Cliffs (1966) MATHGoogle Scholar
- Rajagopal, KR, Na, TY, Gupta, AS: Flow of a viscoelastic fluid over a stretching sheet. Rheol. Acta 23, 213-215 (1984) View ArticleGoogle Scholar