- Research
- Open Access
Heat transfer over a steady stretching surface in the presence of suction
- Zailan Siri^{1}Email authorView ORCID ID profile,
- Nor Artisham Che Ghani^{1} and
- Ruhaila Md. Kasmani^{2}
- Received: 23 January 2018
- Accepted: 19 June 2018
- Published: 15 August 2018
Abstract
The purpose of this paper is to present the Cattaneo–Christov heat flux model for Maxwell fluid past a stretching surface where the presence of suction/injection is taken into account. The governing system of equations is reduced to the ordinary differential equations with the boundary conditions by similarity transformation. These equations are then solved numerically by two approaches, Haar wavelet quasilinearization method (HWQM) and Runge–Kutta–Gill method (RK Gill). The behavior of various pertinent parameters on velocity and temperature distributions is analyzed and discussed. Comparison of the obtained numerical results is made between both methods and with the existing numerical solutions found in the literature, and reasonable agreement is noted.
Keywords
- Cattaneo–Christov heat flux
- Maxwell fluid
- Haar wavelet quasilinearization method
- Runge–Kutta–Gill method
- Suction/injection
1 Introduction
The phenomenon of heat transfer exists due to the difference in temperature between objects or between different parts of the same object. The well-known heat conduction law, known as Fourier’s law, proposed by Fourier [1] provides an insight into the heat transfer analysis. However, this law causes a parabolic energy equation, which means that any initial disturbance is instantly felt through the medium under consideration. Due to this obstacle, Cattaneo [2] revised this law by adding a relaxation time term. Later, Christov [3] made some modification on the Cattaneo model by replacing the ordinary derivative with Oldroyd’s upper-convected derivative. This model is recognized as Cattaneo–Christov heat flux model in the literature. Straughan [4] studied thermal convection in a horizontal layer of incompressible Newtonian fluid by using the Cattaneo–Christov model. Ciarletta and Straughan [5] proved the uniqueness and stability of solutions for the Cattaneo–Christov equations. By using the Cattaneo–Christov model, Tibullo and Zampoli [6] studied the uniqueness of solutions for incompressible fluid.
The Maxwell fluid model is one of the simplest viscoelastic models that can address the influence of fluid relaxation time. Due to these reasons, this model has received remarkable attention of researchers. Han et al. [7] employed the upper-convected Maxwell (UCM) model and Cattaneo–Christov heat flux model to investigate the heat transfer and boundary layer flow of viscoelastic fluid above a stretching plate with velocity slip boundary by using homotopy analysis method (HAM). Mustafa [8] also used HAM to investigate the rotating flow of UCM fluid through the Cattaneo–Christov heat flux model. Khan et al. [9] studied the boundary layer flow of UCM fluid induced by an exponentially stretching sheet using the Cattaneo–Christov model. Hayat et al. [10] discussed the impact of Cattaneo–Christov heat flux in the flow over a stretching sheet with variable thickness.
Abbasi et al. [11] investigated the Cattaneo–Christov heat flux model for a two-dimensional laminar boundary layer flow of incompressible Oldroyd-B fluid over a linearly stretching sheet, where the dimensionless velocity and temperature profiles are obtained through the optimal homotopy analysis method (OHAM). Mushtaq et al. [12] studied the Sakiadis flow of Maxwell fluid along a moving plate in calm fluid by considering the Cattaneo–Christov model. Abbasi and Shehzad [13] proposed a mathematical model to study the Cattaneo–Christov heat flux model for the three-dimensional flow of Maxwell fluid over a bi-directional stretching surface by employing the homotopic procedure. Rubab and Mustafa [14] used HAM to investigate the magnetohydrodynamic (MHD) three-dimensional flow of UCM fluid over a bi-directional stretching surface.
Related to this presence, Vajravelu [15] analyzed the convection flow and heat transfer of viscous fluid near an infinite, porous, and vertical stretching surface by using variable size finite difference method. Muthucumaraswamy [16] studied the effects of suction on heat and mass transfer along a moving vertical surface in the presence of chemical reaction. El-Arabawy [17] investigated the effects of suction/injection and chemical reaction on mass transfer over a stretching surface. Elbashbeshy and Bazid [18] analyzed the effect of internal heat generation and suction or injection on the heat transfer in a porous medium over a stretching surface. Sultana et al. [19] discussed the effects of internal heat generation, radiation, and suction or injection on the heat transfer in a porous medium over a stretching surface. Rajeswari et al. [20] studied the effect of chemical reaction, heat, and mass transfer on a nonlinear MHD boundary layer flow through a vertical porous surface with heat source in the presence of suction. Elbashbeshy et al. [21] used the Runge–Kutta technique to study the effects of suction/injection and variable chemical reaction on mass transfer characteristics over the unsteady stretching surface embedded in a porous medium.
In view of all the above mentioned literature and to the best of our knowledge, no attempt has been made so far to study suction/injection on a Maxwell fluid flow past a steady stretching surface. Motivated by this, our aim here is to analyze the effects of a suction/injection parameter on Maxwell fluid by considering the Cattaneo–Christov heat flux model. The governing system of partial differential equations is non-dimensionalized [22–24] and transformed into the ordinary differential equations and solved numerically by employing the HWQM and RK Gill method. Comparisons are made with the available results in a limiting manner. The effects of involved parameters on the velocity and temperature fields are analyzed and discussed.
The rest this paper is organized as follows. Section 2 is devoted to the mathematical formulation of the flow problem including the reduced ordinary differential equations. In Sect. 3, the numerical solutions of HWQM and RK Gill are shown. Numerical results and discussion are presented in Sect. 4. Finally, the conclusion is given in Sect. 5.
2 Mathematical formulation
In the above equation, \(v_{0}\) represents the velocity of suction/injection at the wall, \(T_{w}\) is the temperature at the wall, and \(T_{\infty} \) is the ambient fluid temperature.
3 Numerical solutions
- (a)
Haar wavelet quasilinearization method (HWQM) and
- (b)
Runge–Kutta–Gill method (RK Gill)
3.1 Haar wavelet quasilinearization method (HWQM)
This subsection presents the Haar wavelet and quasilinearization approach based scheme for two coupled ordinary differential equations (6) and (7) with boundary conditions (8). Wavelet methods are one of the relatively new techniques for obtaining approximate solutions of differential equations. Commonly used wavelet schemes are Haar wavelets, Legendre wavelets, and Chebyshev wavelets. Among them, we are more interested in Haar wavelet because it is the simplest possible wavelet with a compact support, which means that it vanishes outside of a finite interval. In numerical analysis, the discovery of compactly supported wavelets has proven to be a useful tool for the approximation of functions.
The generalized Haar wavelet and its integration are derived, which could cater the Haar series expansion domain greater than one. This is because the boundary layer fluid flow problem deals with a sufficiently large number of infinite intervals.
The above series terminates at finite terms if \(f(\eta )\) is a piecewise constant or can be approximated as a piecewise constant during each subinterval.
Then we put the Haar coefficients in equations (20) and (22) to find the approximate solutions.
As our work is based on the quasilinearization technique and Haar wavelet method, so the convergence for both schemes can be seen in literature [31, 32].
3.2 Runge–Kutta–Gill method (RK Gill)
4 Results and discussion
The transformed momentum equation (6) and energy equation (7) subjected to the boundary conditions of equation (8) were numerically solved by means of HWQM and RK Gill method. The computations for HWQM and RK Gill were performed by using MATLAB.
The elasticity number is important for viscoelastic materials. If \(\beta < 1\), it corresponds to the fluids, thus a smaller elasticity number (\(\beta < 1\)) characterizes purely viscous behavior of fluids. On the contrary, for \(\beta > 1\), the fluid behaves like elastically solid material. Due to this, the magnitude of velocity is larger in smaller β fluid. By using HWQM, the value of \(f''(0)\) in the case of \(\beta = 1\) is −1.24178, while \(\theta '(0)\) is −0.55244 at \(\gamma = 0.6\).
Comparison of local Nusselt number \(- \theta '(0)\) in the case of Newtonian fluid (\(\beta = \gamma = b = f_{w} = 0\)) for different values of Pr
The values of \(- \theta '(0)\) and \(- f''(0)\) when \(\Pr = 1\) and \(s = 0\)
γ | \(- f''(0)\) | \(- \theta '(0)\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
β = 0.1 | β = 0.15 | β = 0.2 | β = 0.1 | β = 0.15 | β = 0.2 | |||||||
RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | |
0.1 | 1.02654 | 1.02653 | 1.03940 | 1.03939 | 1.05215 | 1.05214 | 0.58379 | 0.58379 | 0.57983 | 0.57983 | 0.57593 | 0.57593 |
0.4 | 0.61014 | 0.61014 | 0.60553 | 0.60554 | 0.60101 | 0.60101 | ||||||
0.5 | 0.61998 | 0.61998 | 0.61516 | 0.61516 | 0.61042 | 0.61042 | ||||||
0.6 | 0.63029 | 0.63029 | 0.62526 | 0.62526 | 0.62031 | 0.62031 | ||||||
0.8 | 0.65215 | 0.65215 | 0.64673 | 0.64673 | 0.64138 | 0.64138 | ||||||
1.0 | 0.67551 | 0.67551 | 0.66972 | 0.66972 | 0.66400 | 0.66400 |
The values of \(- f''(0)\) and \(- \theta '(0)\) for different values of β and s when \(\Pr = 1\) and \(\gamma = 0.5\)
s | \(- f''(0)\) | \(- \theta '(0)\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
β = 0.1 | β = 0.15 | β = 0.2 | β = 0.1 | β = 0.15 | β = 0.2 | |||||||
RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | RK Gill | HWQM | |
−1.0 | 0.59681 | 0.59764 | – | 0.58640 | – | 0.57485 | 0.14996 | 0.16047 | – | 0.16116 | – | 0.16186 |
−0.6 | 0.73250 | 0.73351 | 0.72619 | 0.72733 | 0.72004 | 0.72106 | 0.29747 | 0.29864 | 0.29721 | 0.29825 | 0.29673 | 0.29788 |
−0.3 | 0.86492 | 0.86440 | 0.86510 | 0.86508 | 0.86570 | 0.86568 | 0.43848 | 0.43362 | 0.43370 | 0.43161 | 0.42215 | 0.42964 |
0 | 1.02654 | 1.02653 | 1.03940 | 1.04003 | 1.05215 | 1.05271 | 0.61998 | 0.61998 | 0.61516 | 0.61516 | 0.61042 | 0.61061 |
0.2 | 1.15770 | 1.15770 | 1.18362 | 1.18361 | 1.21115 | 1.20962 | 0.79129 | 0.79129 | 0.78417 | 0.78417 | 0.77714 | 0.77715 |
0.3 | 1.23124 | 1.23064 | 1.26593 | 1.26542 | 1.30242 | 1.30056 | 0.89857 | 0.89891 | 0.88998 | 0.89024 | 0.88119 | 0.88164 |
0.6 | 1.48751 | 1.48644 | 1.56384 | 1.56195 | – | 1.64070 | 1.37559 | 1.37604 | 1.35999 | 1.36049 | – | 1.34471 |
5 Conclusions
- (a)
The elasticity number β has opposite effects on the velocity field and temperature field;
- (b)
Temperature profile decreases with an increase in Pr and the temperature boundary layer becomes thinner;
- (c)
Variation of suction/injection parameter s affects both velocity and temperature fields. The larger number of s leads to reduction in velocity and temperature distributions;
- (d)
\(f''(0)\) is found to decrease upon increasing the suction/injection parameter;
- (e)
No effect of changing the value of heat flux relaxation γ on the surface friction coefficient \(f''(0)\) is noticed;
- (f)
\(\theta '(0)\) decreases with the increase in heat flux relaxation, but it tends to increase with the enhanced elasticity number.
Declarations
Acknowledgements
The authors would like to thank the Ministry of Higher Education Malaysia and University of Malaya for the financial support (RG397-17AFR).
Availability of data and materials
Not applicable.
Funding
University of Malaya Research Grant (RG397-17AFR).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
- Fourier, J.B.J.: Théorie Analytic. De La Chaleur, Paris (1822) MATHGoogle Scholar
- Cattaneo, C.: Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 3, 83–101 (1948) MathSciNetMATHGoogle Scholar
- Christov, C.I.: On frame indifferent formulation of the Maxwell–Cattaneo model of finite speed heat conduction. Mech. Res. Commun. 36(4), 481–486 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Straughan, B.: Thermal convection with the Cattaneo–Christov model. Int. J. Heat Mass Transf. 53(1–3), 95–98 (2010) View ArticleMATHGoogle Scholar
- Ciarletta, M., Straughan, B.: Uniqueness and structural stability for the Cattaneo–Christov equations. Mech. Res. Commun. 37(5), 445–447 (2010) View ArticleMATHGoogle Scholar
- Tibullo, V., Zampoli, V.: A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids. Mech. Res. Commun. 38(1), 77–79 (2011) View ArticleMATHGoogle Scholar
- Han, S., Zheng, L., Li, C., Zhang, X.: Coupled flow and heat transfer in viscoleastic fluid with Cattaneo–Christov heat flux model. Appl. Math. Lett. 38, 87–93 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Mustafa, M.: Cattaneo–Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. AIP Adv. (2015) https://doi.org/10.1063/1.4917306 Google Scholar
- Khan, J.A., Mustafa, M., Hayat, T., Alsaedi, A.: Numerical study of Cattaneo–Christov heat flux model for viscoelastic flow due to an exponentially stretching sheet. PLoS ONE (2015) https://doi.org/10.1371/journal.pone.0137363 Google Scholar
- Hayat, T., Farooq, M., Alsaedi, A., Al-Solamy, F.: Impact of Cattaneo–Christov heat flux in the flow over a stretching sheet with variable thickness. AIP Adv. (2015) https://doi.org/10.1063/1.4929523 Google Scholar
- Abbasi, F.M., Mustafa, M., Shehzad, S.A., Alhuthali, M.S., Hayat, T.: Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid. Chin. Phys. B (2016) https://doi.org/10.1088/1674-1056/25/1/014701 Google Scholar
- Mushtaq, A., Abbasbandy, S., Mustafa, M., Hayat, T., Alsaedi, A.: Numerical solution for Sakiadis flow of upper-convected Maxwell fluid using Cattaneo–Christov heat flux model. AIP Adv. (2016) https://doi.org/10.1063/1.4940133 Google Scholar
- Abbasi, F.M., Shehzad, S.A.: Heat transfer analysis for three-dimensional flow of Maxwell fluid with temperature dependent thermal conductivity: application of Cattaneo–Christov heat flux model. J. Mol. Liq. 220, 848–854 (2016) View ArticleGoogle Scholar
- Rubab, K., Mustafa, M.: Cattaneo–Christov heat flux model for MHD three-dimensional flow of Maxwell fluid over a stretching sheet. PLoS ONE (2016) https://doi.org/10.1371/journal.pone.0153481 Google Scholar
- Vajravelu, K.: Convection heat transfer at a stretching sheet with suction or blowing. J. Math. Anal. Appl. 188(3), 1002–1011 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Muthucumaraswamy, R.: Effects of suction on heat and mass transfer along a moving vertical surface in the presence of a chemical reaction. Forsch. Ingenieurwes. 67, 129–132 (2002) https://doi.org/10.1007/s10010-002-0083-2 View ArticleGoogle Scholar
- El-Arabawy, H.A.M.: Exact solution of mass transfer over a stretching surface with chemical reaction and suction/injection. J. Math. Stat. 5(3), 159–166 (2009) View ArticleMATHGoogle Scholar
- Elbashbeshy, E.M.A., Bazid, M.A.A.: Heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection. Appl. Math. Comput. 158(3), 799–807 (2004) MathSciNetMATHGoogle Scholar
- Sultana, T., Saha, S., Rahman, M.M., Saha, G.: Heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection in the presence of radiation. J. Mech. Eng. 40(1), 22–28 (2009) View ArticleGoogle Scholar
- Rajeswari, R., Jothiram, B., Nelson, V.K.: Chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through a vertical porous surface in the presence of suction. Appl. Math. Sci. 3(50), 2469–2480 (2009) MathSciNetMATHGoogle Scholar
- Elbashbeshy, E.M.A., Emam, T.G., Abdel-wahed, M.S.: Mass transfer over unsteady stretching surface embedded in porous medium in the presence of variable chemical reaction and suction/injection. Appl. Math. Sci. 5(12), 557–571 (2011) MATHGoogle Scholar
- Autuori, G., Cluni, F., Gusella, V., Pucci, P.: Mathematical models for nonlocal elastic composite materials. Adv. Nonlinear Anal. 6(4), 355–382 (2017) MathSciNetMATHGoogle Scholar
- Ghergu, M., Radulescu, V.: Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics. Springer Monographs in Mathematics. Springer, Heidelberg (2012) MATHGoogle Scholar
- Vallée, C., Radulescu, V., Atchonouglo, K.: New variational principles for solving extended Dirichlet–Neumann problems. J. Elast. 123(1), 1–18 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Ghani, C.N.A.: Numerical solution of elliptic partial differential equations by Haar wavelet operational matrix method. Dissertation, University of Malaya (2012) Google Scholar
- Mandelzweig, V.B., Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput. Phys. Commun. 141, 268–281 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Marin, M., Craciun, E.M.: Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials. Composites, Part B, Eng. 126, 27–37 (2017) View ArticleGoogle Scholar
- Marin, M., Baleanu, D.: On vibrations in thermoelasticity without energy dissipation for micropolar bodies. Bound. Value Probl. 2016, 111 (2016) https://doi.org/10.1186/s13661-016-0620-9 MathSciNetView ArticleMATHGoogle Scholar
- Lepik, Ü.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68(2), 127–143 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Lepik, Ü.: Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput. 185(1), 695–704 (2007) MathSciNetMATHGoogle Scholar
- Lee, E.S.: Quasilinearization and Invariant Imbedding with Applications to Chemical Engineering and Adaptive Control. Academic Press, New York (1968) MATHGoogle Scholar
- Saeed, U., Rehman, M.U.: Haar wavelet-quasilinearization technique for fractional nonlinear differential equations. Appl. Math. Comput. 220, 630–648 (2013) MathSciNetMATHGoogle Scholar
- Cebici, T., Keller, H.: Shooting and parallel shooting methods for solving the Falkner–Skan boundary-layer equation. J. Comput. Phys. 7(2), 289–300 (1971) View ArticleMATHGoogle Scholar
- Gill, S.: A process for the step-by-step integration of differential equations in an automatic digital computing machine. Math. Proc. Camb. Philos. Soc. 47(1), 96–108 (1951) MathSciNetView ArticleMATHGoogle Scholar
- Wang, C.Y.: Free convection on a vertical stretching surface. J. Appl. Math. Mech. 69(11), 418–420 (1989) MATHGoogle Scholar
- Gorla, R.S.R., Sidawi, I.: Free convection on a vertical stretching surface with suction and blowing. Appl. Sci. Res. 52(3), 247–257 (1994) View ArticleMATHGoogle Scholar
- Khan, W.A., Pop, I.: Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53(11–12), 2477–2483 (2010) View ArticleMATHGoogle Scholar
- Malik, R., Khan, M., Shafiq, A., Mushtaq, M., Hussain, M.: An analysis of Cattaneo–Christov double-diffusion model for sisko fluid flow with velocity slip. Results Phys. 7, 1232–1237 (2017) View ArticleGoogle Scholar