Similarity analysis of MHD flow field and heat transfer of a second grade convection flow over an unsteady stretching sheet
- Rehan Ali Shah^{1},
- Sajid Rehman^{2}Email author,
- M Idrees^{2},
- M Ullah^{2} and
- Tariq Abbas^{3}
Received: 28 August 2017
Accepted: 27 October 2017
Published: 7 November 2017
Abstract
Unsteady magnetohydrodynamic (MHD) flow of a second grade fluid over a stretching sheet is a focus of this steady. Surface tension is considered to be varies linearly with temperature. The stretching velocity is defined in (Liu and Andersson in Int. J. Therm. Sci. 47(6):766-772, 2008). Similarity transformation reported by Abbas et al. (Math. Comput. Model. 48:518-526, 2008) are used to develop nonlinear system of differential equations coupled in velocity and temperature fields. The system is solved by the homotopy-analysis method (HAM), while the effects of different parameters such as the unsteadiness parameter S, film thickness, Hartmann number Ma, Prandtl number Pr, Thermocapillary number M, heat flux \(-\theta'(0)\), surface skin-friction coefficient \(f''(0)\), free surface temperature \(\theta(1)\) for flow field, and heat transfer are studied in this article.
Keywords
1 Introduction
In many manufacturing processes the flows of non-Newtonian fluids have acquired special attention because boundary layer behavior over the flow and heat transfer phenomena of an unsteady two-dimensional free surface flow of a viscous incompressible conducting fluid have promising applications, such as the performance of lubricants, metal and polymer extrusion, application of paints, drawing of plastic sheets, fiber and wire coating, transpiration cooling, foodstuff processing and movement of biological fluids, chemical equipments, reactor fluidization and microchip production, continuous casting, and the process of designing various heat exchangers. The rate of heat transfer of the stretching sheet determines the best quality product for the coating process. Much research has been carried out on the non-Newtonian boundary layer equations in Cartesian coordinates both theoretically and experimentally. However, the non-Newtonian fluids cannot be described simply like Newtonian fluids. Therefore several researchees proposed their respective models for non-Newtonian fluids. Among these, viscoelastic fluids have a high status for the researchers due to its special characteristics. The simplest subclass of viscoelastic fluids is the second grade fluid, for which an analytic solution is possible.
Sakiadis [3, 4] in 1961 was first to present various aspects on boundary layer behavior on continuous solid surface of the stretching problem involving Newtonian and non-Newtonian fluids; these have been extensively studied by several authors. Crane [5] in 1970 was first to study the hydrodynamics of a steady stretching of a flat elastic sheet in a two-dimensional boundary layer flow by reducing the steady Navier-Stokes equations to a nonlinear ordinary differential equations by means of a similarity transformation. Wang [6] in 1990 first studied the hydrodynamics of an unsteady stretching surface in a thin liquid film of a flow by converting the unsteady Navier-Stokes equations to a nonlinear ordinary differential equations by means of a similarity transformation. But Lai and Kulacki [7], in 1991, assumed that viscosity and thermal conductivity vary as inverse functions of the temperature and then solved the equations numerically by using the Runge-Kutta shooting method. Anderson et al. [8] in 2000 extended the work of Wang [6] by studying heat transfer and an analysis has been performed by shooting method. Liao [9] in 2004 was first to introduce the homotopy-analysis method (HAM). The problem studied by Anderson et al. [8] was considered by Wang [10] in 2006, he presented an analytical solution using HAM [9] and found good agreement with that of the multiple shooting method. Wang et al. [11] in 2006 presented HAM solutions for the non-Newtonian problem studied by Anderson et al. [12]. Furthermore, in 2007 thermocapillary effects were discussed by Dandapat et al. [13] and Chen [14], and viscous dissipation in the presence of a magnetical effect was discussed by Abel et al. [15] in 2008. A more extended form considered by Liu et al. [1] in 2008 for the stretching sheet of the prescribed temperature variation was considered by Anderson et al. [8]. Noor et al. [16] in 2010 introduced a magnetic field as considered in [15] and thermocapillary effect as used in [13] to extend the model in [1]. Further Noor et al. [16] in 2010 took a similarity transformation from [10, 11] and used it for the purpose of reducing the range of independent variables to 0-1. A more realistic approach was used by Yasir et al. [17] in 2011 by studying the flow over a stretching sheet by taking variable physical properties. For solution purposes they used the homotopy-perturbation method (HPM). Hazarika and Konch [18] in 2014 investigated the effects of varying thermal conductivity and viscosity, variable heat flux and constant suction on the magnetic hydrodynamics (MHD) boundary layer flow forced by convection past a stretching/shrinking sheet.
Similarly Hayat et al. [19] in 2007 considered a steady second grade fluid. Magnetic field is applied normal to the flow of electrically conducted fluid in a porous channel while solution is possible by using HAM. Abbas et al. [2] in 2008 investigated the flow of an unsteady second grade fluid over a stretching surface, where HAM gives the analytical solution for the model problem. Meanwhile Abel and Mahesha [20] in 2008 studied the MHD boundary layer flow of a non-Newtonian viscoelastic fluid in the presence of non-uniform heat source and thermal radiation. Moreover, the thermal conductivity may vary linearly with temperature and the regular perturbation technique is used for solution. Further Hayat et al. [21] in 2011 used convective boundary conditions for the second grade fluid and HAM has been used for the series result. Hussnain et al. [22] in 2012 used HAM for the analytic solution of second grade fluid in the rotating system between two horizontal plates in the presence of a transverse magnetic field. Recently Temitope and Samuel [23] in 2015 worked out on the variable physical properties in the steady second grade fluid, solution is establish by numerical Runge-Kutta shooting technique. Meanwhile Gital et al. [24] in 2015 proposed a problem of unsteady second grade fluid due to an oscillating porous wall and modified version of the variable separation technique is used for the solution. Very recently Das and Sharma [25] in 2016 investigated a second grade MHD fluid past a semi-infinite stretching sheet which is electrically conducting, while there is convective surface heat flux along them. Furthermore, the influence of MHD on the fluid flow in various geometries was studied in [26–32].
Motivated by these analyses, the aim of the present investigation is to observe the case of a non-Newtonian fluid for thin film two-dimensional flow satisfying the constitutive equations of second grade fluid with heat transfer over an unsteady stretching sheet under the influence of a transverse magnetic field with surface tension in the boundary conditions. The surface tension varies linearly with temperature. The model boundary layer non-linear partial differential equations transform to ODEs by means of proper transformations concerning the geometry of the problem under consideration. Analyses are made for skin friction, heat transfer and for the flow speed’s various natural parameters by using the well-known analytical method HAM. Different effects of non-dimensional values such as unsteadiness parameter, film thickness, Hartmann number, surface skin-friction coefficient, Prandtl number, Thermocapillary number, heat flux, and free surface temperature are discussed and sketched for the effects of various pertinent parameters and meaningful results have been pointed out.
2 Problem formulation
2.1 Governing equations
2.2 Similarity transformation
3 Problem approach
3.1 Skin-friction coefficient and Nusselt number
3.2 Solution approach
The model given in equations (19)-(22), are solved by the homotopy-analysis method (HAM) [9, 35]. HAM is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy-analysis method entails the concept of homotopy from topology to develop a convergent series solution for nonlinear systems. This is implemented by applying a homotopy-Maclaurin series to compromise with the nonlinearities in the system. It is a series development method that is not precisely dependent on small or large natural parameters. Thus, it is suitable for not only weakly but also strongly nonlinear models, addressing some of the fundamental conditions of the basic perturbation methods. Further, the HAM is a cooperative method for the delta expansion method, the Lyapunov artificial small parameter method, the homotopy perturbation method and the Adomian decomposition method. The higher generalization of the method usually takes for granted the strong convergence of the solution over larger spatial and parameter domains. Furthermore, the HAM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. It provides great freedom to choose the basis functions of the desired solution and the corresponding auxiliary linear operator of the homotopy. Finally, unlike the other analytic approximation techniques, HAM provides a simple way to ensure the convergence of the solution series. Free software based on the homotopy-analysis method for nonlinear boundary-value and eigenvalue problems is available called Mathematica package BVPh2.0.
3.3 Optimal convergence control parameters
Optimal value of convergence control parameters versus different orders of approximation
Order of approximation | \(\boldsymbol{\hbar_{f}}\) | \(\boldsymbol{\hbar_{\theta}}\) | \(\boldsymbol{\varepsilon_{m}^{t}}\) | CPU time |
---|---|---|---|---|
2 | −0.816110 | −0.570098 | 7.37847 × 10^{−3} | 11.3260 seconds |
3 | −0.830974 | −0.761688 | 1.30091 × 10^{−4} | 18.5925 seconds |
4 | −0.840993 | −0.617020 | 1.99436 × 10^{−6} | 37.0145 seconds |
5 | −0.846134 | −0.687839 | 3.39372 × 10^{−8} | 63.8482 seconds |
6 | −0.850443 | −0.690781 | 5.04199 × 10^{−10} | 108.937 seconds |
7 | −0.858637 | −0.731049 | −1.1845 × 10^{−10} | 196.690 seconds |
Individual averaged squared residual errors using optimal values of auxiliary parameters
m | \(\boldsymbol{\varepsilon_{m}^{f}}\) | \(\boldsymbol{\varepsilon_{m}^{\theta}}\) | CPU time |
---|---|---|---|
2 | 2.54396 × 10^{−3} | 2.50182 × 10^{−7} | 2.46878 seconds |
4 | 3.93474 × 10^{−7} | 5.39818 × 10^{−9} | 6.66087 seconds |
6 | 1.08366 × 10^{−10} | 1.16081 × 10^{−10} | 13.5679 seconds |
8 | 6.10692 × 10^{−13} | 2.49521 × 10^{−12} | 22.4765 seconds |
10 | 2.17244 × 10^{−14} | 5.36337 × 10^{−14} | 34.5937 seconds |
12 | 5.23527 × 10^{−16} | 1.15284 × 10^{−15} | 49.7814 seconds |
14 | 1.14949 × 10^{−17} | 2.47801 × 10^{−17} | 67.1306 seconds |
16 | 2.47258 × 10^{−19} | 5.32646 × 10^{−19} | 86.7832 seconds |
18 | 5.26476 × 10^{−21} | 1.14492 × 10^{−20} | 110.962 seconds |
20 | 1.00927 × 10^{−22} | 2.46099 × 10^{−22} | 138.456 seconds |
Convergence of HAM on the basis of skin friction \(\pmb{f''(0)}\) and heat flux \(\pmb{-\theta'(0)}\) for selected values of \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\Upsilon=0.127013}\) , \(\pmb{\mathit {Pr}=0.2}\) , \(\pmb{\mathit {Gr}=5}\) , \(\pmb{S=0.2}\) and \(\pmb{K=0.1}\)
m | \(\boldsymbol{f''(0)}\) | \(\boldsymbol {-\theta'(0)}\) |
---|---|---|
1 | −2.987594767874287 | 0.001666755677540 |
5 | −2.988314378094173 | 0.002678333954988 |
10 | −2.988310366379021 | 0.002700353358962 |
15 | −2.988310346982944 | 0.002700534393216 |
18 | −2.988310346828816 | 0.002700535809751 |
20 | −2.988310346820995 | 0.002700535881606 |
25 | −2.988310346820995 | 0.002700535881606 |
30 | −2.988310346820995 | 0.002700535881606 |
Hence, HAM Mathematica package BVPh2.0 is a choice of selection to the set of local convergence control parameters to get convergent results.
3.4 Results and discussion
Variation of \(\pmb{\beta^{2}=\Upsilon}\) , \(\pmb{f''(0)}\) , \(\pmb{\theta(1)}\) and \(\pmb{-\theta'(0)}\) using 20th-order (HAM) via Mathematica package BVPh2.0 approximation when \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\mathit {Pr}=0.2}\) , \(\pmb{\mathit {Gr}=5}\) and \(\pmb{S=0.2}\)
\(\boldsymbol{\hbar_{f}}\) | \(\boldsymbol{\hbar_{\theta}}\) | \(\boldsymbol{\beta^{2}}\) | \(\boldsymbol {f''(0)}\) | θ (1) | \(\boldsymbol{-\theta'(0)}\) |
---|---|---|---|---|---|
K = 0.1 | |||||
−0.832780 | −0.941760 | 0.727013 | −3.16449 | 0.977408 | 0.0721367 |
−0.836000 | −0.857229 | 0.527013 | −3.11339 | 0.983654 | 0.0524063 |
−0.838367 | −0.795225 | 0.327013 | −3.06291 | 0.989878 | 0.0325884 |
−0.840192 | −0.717392 | 0.127013 | −3.01303 | 0.996077 | 0.0126844 |
−0.840993 | −0.617020 | 0.027013 | −2.98831 | 0.999166 | 0.0027005 |
K = 0.2 | |||||
−0.720505 | −0.669723 | 0.727013 | −2.98309 | 0.977904 | 0.0721543 |
−0.72187 | −0.614013 | 0.527013 | −2.93775 | 0.984014 | 0.0524153 |
−0.72276 | −0.555679 | 0.327013 | −2.89292 | 0.990101 | 0.0325917 |
−0.723643 | −0.475951 | 0.127013 | −2.84858 | 0.996164 | 0.0126848 |
−0.724163 | −0.372249 | 0.027013 | −2.82660 | 0.999185 | 0.0027003 |
K = 0.3 | |||||
−0.632928 | −0.491987 | 0.727013 | −2.84154 | 0.978332 | 0.0721680 |
−0.633377 | −0.453117 | 0.527013 | −2.80042 | 0.984325 | 0.0524221 |
−0.633868 | −0.413418 | 0.327013 | −2.75968 | 0.990294 | 0.0325937 |
−0.634495 | −0.359594 | 0.127013 | −2.71935 | 0.996239 | 0.0126836 |
−0.634866 | −0.309759 | 0.027013 | −2.69935 | 0.999201 | 0.0026989 |
Variation of \(\pmb{\beta^{2}=\Upsilon}\) , \(\pmb{f''(0)}\) , \(\pmb{\theta(1)}\) and \(\pmb{-\theta'(0)}\) using 20th-order HAM via Mathematica package BVPh2.0 approximation when \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\mathit {Pr}=0.2}\) , \(\pmb{S=0.2}\) and K is varied
K | \(\boldsymbol{\beta^{2}}\) | \(\boldsymbol {f''(0)}\) | θ (1) | \(\boldsymbol{-\theta'(0)}\) |
---|---|---|---|---|
Gr = 5 | ||||
0.10 | 0.727013 | −3.16449 | 0.977408 | 0.0721367 |
0.15 | 0.527013 | −3.01960 | 0.983841 | 0.0524111 |
0.20 | 0.327013 | −2.89292 | 0.990101 | 0.0325917 |
0.25 | 0.127013 | −2.78049 | 0.996202 | 0.0126846 |
0.30 | 0.027013 | −2.69935 | 0.999201 | 0.0026989 |
Gr = 10 | ||||
0.10 | 0.727013 | −3.17265 | 0.977385 | 0.0721359 |
0.15 | 0.527013 | −3.02354 | 0.983833 | 0.0524109 |
0.20 | 0.327013 | −2.89432 | 0.990099 | 0.0325917 |
0.25 | 0.127013 | −2.78069 | 0.996202 | 0.0126846 |
0.30 | 0.027013 | −2.69935 | 0.999201 | 0.0026989 |
Variation of \(\pmb{\beta^{2}=\Upsilon}\) , \(\pmb{f''(0)}\) , \(\pmb{\theta(1)}\) and \(\pmb{-\theta'(0)}\) using 20th-order HAM via Mathematica package BVPh2.0 approximation when \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\mathit {Gr}=5}\) , \(\pmb{S=0.2}\) and Pr is varied
Pr | \(\boldsymbol{\beta^{2}}\) | \(\boldsymbol {f''(0)}\) | θ (1) | \(\boldsymbol{-\theta'(0)}\) |
---|---|---|---|---|
K = 0.1 | ||||
0.4 | 0.527013 | −3.11042 | 0.967667 | 0.1042320 |
0.6 | 0.327013 | −3.05718 | 0.970051 | 0.0970981 |
0.8 | 0.127013 | −3.00845 | 0.984435 | 0.0505359 |
1.0 | 0.027013 | −2.98683 | 0.995842 | 0.0134874 |
K = 0.3 | ||||
0.4 | 0.527013 | −2.79721 | 0.968979 | 0.1042950 |
0.6 | 0.327013 | −2.75420 | 0.971266 | 0.0971506 |
0.8 | 0.127013 | −2.71525 | 0.985072 | 0.0505494 |
1.0 | 0.027013 | −2.69804 | 0.996014 | 0.0134869 |
Variation of \(\pmb{\beta^{2}=\Upsilon}\) , \(\pmb{f''(0)}\) , \(\pmb{\theta(1)}\) and \(\pmb{-\theta'(0)}\) using 20th-order HAM via Mathematica package BVPh2.0 approximation when \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\mathit {Pr}=0.2}\) , \(\pmb{S=0.2}\) and Gr is varied
Gr | \(\boldsymbol{\beta^{2}}\) | \(\boldsymbol {f''(0)}\) | θ (1) | \(\boldsymbol{-\theta'(0)}\) |
---|---|---|---|---|
K = 0.1 | ||||
1 | 0.727013 | −3.15798 | 0.977426 | 0.0721373 |
3 | 0.527013 | −3.11168 | 0.983657 | 0.0524064 |
5 | 0.327013 | −3.06291 | 0.989878 | 0.0325884 |
10 | 0.127013 | −3.01328 | 0.996077 | 0.0126844 |
20 | 0.027013 | −2.98834 | 0.999166 | 0.0027005 |
K = 0.3 | ||||
1 | 0.727013 | −2.83675 | 0.978347 | 0.0721684 |
3 | 0.527013 | −2.79914 | 0.984328 | 0.0524222 |
5 | 0.327013 | −2.75968 | 0.990294 | 0.0325937 |
10 | 0.127013 | −2.71954 | 0.996239 | 0.0126833 |
20 | 0.027013 | −2.69937 | 0.999201 | 0.0026989 |
Variation of \(\pmb{\beta^{2}=\Upsilon}\) , \(\pmb{f''(0)}\) , \(\pmb{\theta(1)}\) and \(\pmb{-\theta'(0)}\) using 20th-order HAM via Mathematica package BVPh2.0 approximation when \(\pmb{\mathit {Ma}=1}\) , \(\pmb{M=1}\) , \(\pmb{\mathit {Pr}=0.2}\) , \(\pmb{\mathit {Gr}=5}\) and S is varied
S | \(\boldsymbol{\beta^{2}}\) | \(\boldsymbol {f''(0)}\) | θ (1) | \(\boldsymbol{-\theta'(0)}\) |
---|---|---|---|---|
K = 0.1 | ||||
0.2 | 0.727013 | −3.16449 | 0.977408 | 0.0721367 |
0.3 | 0.527013 | −2.98620 | 0.970255 | 0.0779990 |
0.4 | 0.327013 | −2.80557 | 0.973148 | 0.0645101 |
0.5 | 0.127013 | −2.62791 | 0.986181 | 0.0315136 |
0.6 | 0.027013 | −2.47627 | 0.996330 | 0.0080868 |
K = 0.3 | ||||
0.2 | 0.727013 | −2.84154 | 0.978332 | 0.0721680 |
0.3 | 0.527013 | −2.71539 | 0.970797 | 0.0780346 |
0.4 | 0.327013 | −2.58351 | 0.973421 | 0.0645281 |
0.5 | 0.127013 | −2.45124 | 0.986266 | 0.0315166 |
0.6 | 0.027013 | −2.33454 | 0.996345 | 0.0080868 |
4 Concluding remarks
- 1.
It is concluded that as the second grade parameter K increases the flow velocity decreases slightly up to some extent and then increases, it means swing impact is detectable, while temperature consistently increases.
- 2.
It is also concluded that by increasing magnetic parameter Ma, the thin film flow swings from slight deceleration to higher velocity and temperature is lowered.
- 3.
It is found that increasing the thermocapillary number M, the flow velocity form a parabolic profile while temperature is lowered.
- 4.
Moreover, increasing the film thickness ϒ, the flow swings from lower velocity to higher velocity and temperature is lowered consistently.
- 5.
Furthermore increasing the Prandtl number Pr, the flow temperature decreases and velocity remains unchanged for fixed values of parameter.
- 6.
It is also investigated that as the magnitude of stretching parameter S rise the velocity increases and the temperature is lowered.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive comments and valuable suggestions to improve the quality of the paper.
Funding
This paper is self-supported by the authors in respect of funding and technically supported by Islamia Collogue University, Khyber Pakhtunkhwa, Peshawar, Pakistan.
Authors’ contributions
All authors participated in the analysis of the results and manuscript coordination. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no conflict of 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
- Liu, IC, Andersson, HI: Heat transfer in a liquid film on an unsteady stretching sheet. Int. J. Therm. Sci. 47(6), 766-772 (2008) View ArticleGoogle Scholar
- Abbas, Z, Hayat, T, Sajid, M, Asghar, S: Unsteady flow of a second grade fluid film over an unsteady stretching sheet. Math. Comput. Model. 48, 518-526 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Sakiadis, BC: Boundary layer behavior on continuous solid surface I: boundary layer equations for two dimensional and axisymmetric flow. AIChE J. 7, 26-28 (1961) View ArticleGoogle Scholar
- Sakiadis, BC: Boundary layer behaviour on continuous solid surface II: boundary layer on a continuous flat surface. AIChE J. 7, 221-225 (1961) View ArticleGoogle Scholar
- Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645-647 (1970) View ArticleGoogle Scholar
- Wang, CY: Liquid film on an unsteady stretching sheet. Q. Appl. Math. 48, 601-610 (1990) View ArticleMATHGoogle Scholar
- Lai, FC, Kulacki, FA: The effect of variable viscosity on convective heat and mass transfer along a vertical surface in saturated porous media. Int. J. Heat Mass Transf. 33, 1028-1031 (1991) View ArticleGoogle Scholar
- Andersson, HI, Aarseth, JB, Dandapat, BS: Heat transfer in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 43, 69-74 (2000) View ArticleMATHGoogle Scholar
- Liao, SJ: Beyond Perturbation: Introduction to Homotopy Analysis Method, pp. 17-18. Chapman & Hall, Boca Raton (2004) Google Scholar
- Wang, C: Analytic solutions for a liquid thin film on an unsteady stretching surface. Heat Mass Transf. 42(8), 759-766 (2006) View ArticleGoogle Scholar
- Wang, C, Pop, I: Analysis of the flow of a power-law fluid film on an unsteady stretching surface by means of homotopy analysis method. J. Non-Newton. Fluid Mech. 138(2), 161-172 (2006) View ArticleMATHGoogle Scholar
- Andersson, HI, Aarseth, JB, Braud, N, Dandapat, BS: Flow of a power-law fluid film on an unsteady stretching surface. J. Non-Newton. Fluid Mech. 62(1), 1-8 (1996) View ArticleGoogle Scholar
- Dandapat, BS, Santra, B, Vajravelu, K: The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet. Int. J. Heat Mass Transf. 50(5), 991-996 (2007) View ArticleMATHGoogle Scholar
- Chen, CH: Marangoni effects on forced convection of power-law liquids in a thin film over a stretching surface. Phys. Lett. A 370(1), 51-57 (2007) View ArticleGoogle Scholar
- Abel, MS, Mahesha, N, Tawade, J: Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field. Appl. Math. Model. 33(8), 3430-3441 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Noor, NFM, Hashim, I: Thermocapillarity and magnetic field effects in a thin liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 53(9), 2044-2051 (2010) View ArticleMATHGoogle Scholar
- Khan, Y, Qingbiao, W, Faraz, N, Yildirim, A: The effect of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Comput. Math. Appl. 61, 3391-3399 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Hazarika, GC, Konch, J: Effects of variable viscosity and thermal conductivity on magnetohydrodynamic forced convective boundary layer flow past a stretching/shrinking sheet prescribed with variable heat flux in the presence of heat source and constant suction. Int. J. Comput. Appl. 107(1), 50-58 (2014) Google Scholar
- Hayat, T, Ahmed, N, Sajid, M, Asghar, S: On the MHD flow of a second grade fluid in a porous channel. Comput. Math. Appl. 54, 407-414 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Abel, MS, Mahesha, N: Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl. Math. Model. 32, 1965-1983 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Hayat, T, Shehzad, SA, Qasim, M, Obaidat, S: Flow of a second grade fluid with convective boundary conditions. Therm. Sci. 15, 253-261 (2011) View ArticleGoogle Scholar
- Hussnain, S, Mehmood, A, Ali, A: Three-dimensional channel flow of second grade fluid in rotating frame. Appl. Math. Mech. 33(3), 289-302 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Akinbobola, TE, Okoya, SS: The flow of second grade fluid over a stretching sheet with variable thermal conductivity and viscosity in the presence of heat source/sink. J. Niger. Math. Soc. 34, 331-342 (2015) View ArticleMATHMathSciNetGoogle Scholar
- Gital, AY, Abdulhameed, M, Abdulhamid, BM, Aliyu, MS, Hina, AD: Analytical solution of unsteady flow of a viscoelastic fluid due to an oscillating porous wall. Appl. Math. 5(4), 88-92 (2015) Google Scholar
- Das, K, Sharma, RP, Sarkar, A: Heat and mass transfer of a second grade magnetohydrodynamic fluid over a convectively heated stretching sheet. J. Comput. Des. Eng. 3(4), 330-336 (2016) Google Scholar
- Hayat, T, Aziz, A, Muhammad, T, Ahmad, B: On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet. J. Magn. Magn. Mater. 408, 99-106 (2016) View ArticleGoogle Scholar
- Dinarvand, S, Doosthoseini, A, Doosthoseini, E, Rashidi, MM: Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces. Nonlinear Anal., Real World Appl. 11(2), 1159-1169 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Rashidi, MM, Momoniat, E, Rostami, B: Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving stretching surface by homotopy analysis method with two auxiliary parameters. J. Appl. Math. 2012, Article ID 780415 (2012) MATHMathSciNetGoogle Scholar
- Mabood, F, Ibrahim, SM, Rashidi, MM, Shadloo, MS, Lorenzini, G: Non-uniform heat source/sink and Soret effects on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation. Int. J. Heat Mass Transf. 93, 674-682 (2016) View ArticleGoogle Scholar
- Qing, J, Bhatti, MM, Abbas, MA, Rashidi, MM, Ali, ME: Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface. Entropy 18(4), 123-128 (2016) View ArticleGoogle Scholar
- Mabood, F, Shateyi, S, Rashidi, MM, Momoniat, E, Freidoonimehr, N: MHD stagnation point flow heat and mass transfer of nanofluids in porous medium with radiation, viscous dissipation and chemical reaction. Adv. Powder Technol. 27(2), 742-749 (2016) View ArticleGoogle Scholar
- Abbasbandy, S, Hayat, T, Alsaedi, A, Rashidi, MM: Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid. Int. J. Numer. Methods Heat Fluid Flow 24(2), 390-401 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Rajagopal, KR, Gupta, AS: An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate. Meccanica 19, 156-160 (1984) View ArticleMATHMathSciNetGoogle Scholar
- Criminale, WO, Ericksen, JL, Filbey, GL: Steady shear flow of non-Newtonian fluid. Arch. Ration. Mech. Anal. 1, 410-417 (1957) View ArticleMATHMathSciNetGoogle Scholar
- Liao, SJ: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15(8), 2003-2016 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Zhao, Y: http://numericaltank.sjtu.edu.cn/BVPh2_0.htm (retrieved 30.08.14)