Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity

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1 Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity Taza Gul 1, Saed Islam 1, Rehan Ali Shah, Ilyas Khan, Sharidan Shafie 3 * 1 Department of mathematics, Abl Wali Khan University Mardan, Mardan, KPK, Pakistan, Department of mathematics, U.E.T Peshawar, Peshawar, KPK, Pakistan, 3 College of Engineering Majmaah University, Majmaah, Saudi Arabia, Department of mathematical Sciences, Faculty of science, University Teknology Malaysia, UTM Johor Bahru, Johor, Malaysia Abstract In this work, we have carried out the influence of temperature dependent viscosity on thin film flow of a magnetohydrodynamic (MHD) third grade fluid past a vertical belt. The governing coupled non-linear differential equations with appropriate boundary conditions are solved analytically by using Adomian Decomposition Method (ADM). In order to make comparison, the governing problem has also been solved by using Optimal Homotopy Asymptotic Method (OHAM). The physical characteristics of the problem have been well discussed in graphs for several parameter of interest. Citation: Gul T, Islam S, Shah RA, Khan I, Shafie S (1) Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity. PLoS ONE 9(6): e9755. doi:1.1371/journal.pone.9755 Editor: Victor M. Ugaz, Texas A&M University, United States of America Received November 6, 13; Accepted April 15, 1; Published June, 1 Copyright: ß 1 Gul et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproction in any medium, provided the original author and source are credited. Funding: The authors would like to acknowledge the research Management center UTM and MOE (Ministry of Ecation) for the financial support through vote F19 & H7 for this research. Prof. S. Islam gratefully acknowledges the financial support of his visit provided by UTM. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * sharidan@utm.my Introction The subject of non-newtonian fluids is popular and is an area of active research specially in mathematics, instry and engineering problems. Examples of non-newtonian fluids include plastic manufacturing, performance of lubricants, food processing, movement of biological fluids, wire and fiber coating, paper proction, transpiration cooling, gaseous diffusion, drilling mud, heat pipes etc. These fluids are described by a non-linear relationship between stress and the rate of deformation tensors and therefore several models have been proposed. There are several subclasses of non-newtonian fluids. Third grade fluid is one of the important fluid in this category and its equation is based on strong theoretical foundations, where relation between stress and strain is not linear. Therefore, in this problem, we have considered third grade fluid. Considerable efforts have been made to study non-newtonian fluids for various geometrical configurations via analytical techniques. Some developments in this direction are discussed in [1 19]. On the other hand, the physical importance of thin film has been highlighted by scientists and engineers. Amongst them, Khalid and Vafai [] studied hydrodynomic squeezed flow and heat transfer over a sensor surface. Miladinova et al. [1] investigated thin film flow of a power law liquid falling from an inclined plate where it was observed that saturation of non-linear interaction occur in a permanent finite amplitude wave. Similarly, Taza Gul et al. [] investigated effects of slip condition on thin film flow of third grade fluids for lifting and drainage problem under the condition of constant viscosity. The effects of various parameters on the lift and drainage velocity profiles are also studied. It is crystal clear that the physical problems are frequently modeled, using non-linear differential equations. Recently, several analytical and numerical techniques were used for solution of such non-linear problems. In order to find analytical approximate solutions of non-linear differential equations, researchers usually use approximate techniques such as Homotopy Perturbation Method (HPM) [3], Homotopy Analysis Method (HAM) [] and Optimal Homotopy Asymptotic Method (OHAM) [5]. OHAM is a powerful mathematical technique and has already been applied to several non-linear problems. Marinca and Herisanu [6] used OHAM for solving non-linear equations arising in heat transfer problems. In another paper, Marinca [7] applied OHAM to study steady flow of a fourth grade fluid past a porous plate. Joneidi et al. [8] analyzed micropolar flow in a porous channel with high mass transfer. Siddiqui et al. [9] examined a thin film flow of non-newtonian fluid over a moving belt. In another study, Siddiqui et al. [3] discussed the thin film flow of a fourth grade fluid down a vertical cylinder. Costa and Macedonio [31] noticed that increase in velocity may proce additional growth of local temperature. Nadeem and Awais [3] investigated thin film unsteady flow with variable viscosity. They analyzed the effect of variable thermo capillarity on the flow and heat transfer. Ellahi and Riaz [33] discussed analytical solution for MHD flow in a third grade fluid with variable viscosity. Whereas Aksoy et al. [3] found an approximate analytical solution for flow of a third grade fluid through a parallel plate channel filled with a porous medium. The main objective of this research is to study thin film flow of MHD third grade fluid over a vertical belt under the influence of temperature with variable viscosity. More exactly, we are interested in showing the effects of MHD and variable viscosity PLOS ONE 1 June 1 Volume 9 Issue 6 e9755

2 Figure 1. Geometry of the problem (a) Lift problem and (b) Drainage problem. doi:1.1371/journal.pone.9755.g1 with heat transfer in a thin film fluid flow such as silicate melts and polymers. In these fluids, viscous friction generates a local increase in temperature near the belt with decrease in resultant viscosity and frequently increases the flow velocity. The governing problem is solved using an analytical technique known as Adomian Decomposition Method (ADM). This technique was introced by Adomian [35,36] for finding the approximate solutions for linear and non-linear differential equations. Wazwaz [37,38] used ADM for reliable treatment of Bratu-type and Rmden-Fowler equations. For comparisons and accuracy of results, the governing problem has also been solved by using OHAM. Basic Equations The continuity, momentum and energy equations for incompressible, isothermal and electrically concting third grade fluid are; +:u~ Du Dt ~+:Tzrgzj B, r c p DH Dt ~k + Hztr(T:L), Here, r is the constant density, g denotes gravitational acceleration, u is the velocity vector of the fluid, H defines temperature, k is the thermal conctivity, C p is specific heat, L~+u, D Dt ~ L z(u:+) denotes material time derivative, j is the Lt ð1þ ðþ ð3þ Figure. Comparison of ADM and OHAM methods for lift velocity profile. St ~:1, M ~:,b~:6,l~:1, Br ~:, C1 ~:1619, C ~{:3971, C3 ~:1611, C ~{:1683. doi:1.1371/journal.pone.9755.g Figure 3. Comparison of ADM and OHAM methods for lift temperature distribution. St ~1, M ~:,b~1:,l~:1, Br ~, C1 ~{:989619, C ~{:57, C3 ~{13:673, C ~ {163: doi:1.1371/journal.pone.9755.g3 PLOS ONE June 1 Volume 9 Issue 6 e9755

3 Figure. Comparison of ADM and OHAM methods for drainage velocity profile. St ~:9, M ~:1,b~:6,L~:1, Br ~:, C1 ~{:989619, C ~{:57, C3 ~{13:673, C ~ {163: doi:1.1371/journal.pone.9755.g current density and T is the Cauchy stress tensor. Moreover, a uniform magnetic field B~ ð, B,Þ, is applied in a direction, perpendicular to the belt. The Lorentz force per unit volume is given by j B~,s B u(x),, ðþ The Cauchy stress tensor T, is given by Figure 6. Influence of the Brinkman number on the lift velocity profile when St ~1,L~:1, M ~:5,b~:. doi:1.1371/journal.pone.9755.g6 Here a ji~1, ð Þ,b j(j~1,,3) are the material constants and A, A 1, A, A 3 are the kinematical tensors given by A ~I A1 ~ ð+uþz ð+uþ T, A ~ D A1 Dt z A1 ð+uþz ð+uþ T A1, A3 ~ D A Dt z A ð+uþz ð+uþ T A, ð7þ T~{pIzt ð5þ where {pi denotes spherical stress, p is the hydrostatic pressure and shear stress tensor t, is defined as t~m A1 z a 1 A z a A 1 z b 1 A3 z b ða1 A z A A1Þz b 3 tr A A1, ð6þ Formulation of Lift Problem Consider, a wide flat belt moves vertically upward at a constant speed U through a large bath of third grade liquid. The belt carries a layer of liquid of constant thickness, d with itself. For analysis, coordinate system is chosen in which the y-axis is taken parallel to the surface of the belt and x-axis is perpendicular to the belt. Uniform magnetic field is applied transversely to the belt. It is assumed that the flow is steady and laminar after a small distance above the liquid surface layer and the external pressure is atmospheric everywhere. Figure 5. Comparison of ADM and OHAM methods for drainage temperature distribution. St ~:1, M ~:3,b~:6, L~:6, Br ~:3, C1 ~{1:39, C ~{:1138, C3 ~{:55, C ~:389. doi:1.1371/journal.pone.9755.g5 Figure 7. Influence of the Brinkman number on the lift temperature distribution when St ~1,L~:1, M ~:5,b~:. doi:1.1371/journal.pone.9755.g7 PLOS ONE 3 June 1 Volume 9 Issue 6 e9755

4 Figure 8. Effect of Stock number on the lift velocity profile. when M ~:,b~1:,l~:1, Br ~:. doi:1.1371/journal.pone.9755.g8 Figure 1. Effect of viscosity parameter on the lift velocity profile. when M ~:, St ~:,b~:6, Br ~:. doi:1.1371/journal.pone.9755.g1 Velocity and temperature fields are u~ ð,uðxþ,þ and H~HðxÞ ð8þ Tzz ~{p, ð1þ Using the velocity field given in Eq. (8) the continuity Eq. (1) satisfies identically and Eq. (5) gives the following components of stress tensor: Txz ~ Tyz ~, ð13þ Txx ~{pzð a 1 z a Þ, ð9þ Txy ~m z b 3 ð z b 3 Þ, ð1þ Incorporating Eqs. (9 13) into the momentum and energy equations (, 3), we get ~ m d u d x z dm z6 b d u ð1þ ð z b 3 Þ {rg{s d x B u, Tyy ~{pz a, ð11þ ~k d H d x zm zðb z b 3 Þ, ð15þ Figure 9. Effect of Stock number on the lift temperature distribution when M ~:,b~1:,l~:1, Br ~:. doi:1.1371/journal.pone.9755.g9 Figure 11. Effect of viscosity parameter on the lift temperature distribution when M ~:, St ~1,b~1:, Br ~5. doi:1.1371/journal.pone.9755.g11 PLOS ONE June 1 Volume 9 Issue 6 e9755

5 Figure 1. The effect of magnetic force on lift velocity profile when Br ~:,L~:1,b~:, St ~:. doi:1.1371/journal.pone.9755.g1 Figure 1. The effect of Stock number on velocity for drainage problem when M ~:,b~1:,l~:1, Br ~:. doi:1.1371/journal.pone.9755.g1 The corresponding boundary conditions are: For Reynold s model, the dimensionless viscosity u~u,h~ H at x~, ð16þ m~ e {LH, ð19þ ~,H~ H 1 at x~d, Introcing the following non-dimensional variables ð17þ Using Taylor series expansion, one may represent viscosity and its derivative as follows: m%1{lh, dm dh %{L, ðþ u~ u U,x~ x d,h~ H{ H,m~ m, Br ~ m U H 1 { H m kðh 1 { H Þ, St ~ d rg m U, M ~ s B d ð,b~ b z b 3 ÞU m m d, ð18þ where B r is the Brinkman number, M is the magnetic parameter, b is the non-newtonian parameter and S t is the Stock s number. Using the above dimensionless variables into Eqs. (1 17) and dropping out the bar notations, we obtain. Eq. (1) has been rectified for Eq. (1, ) d u d x ~{L dh zh d u d x { St { M u~, z6b d u d x ð1þ Figure 13. The effect of magnetic force on lift temperature distribution when Br ~:,L~:1,b~:, St ~:. doi:1.1371/journal.pone.9755.g13 Figure 15. The effect of Stock number on temperature for drainage problem when M ~:,b~1:,l~:1, Br ~:. doi:1.1371/journal.pone.9755.g15 PLOS ONE 5 June 1 Volume 9 Issue 6 e9755

6 Figure 16. Drain velocity for various values of viscosity parameter when M ~:,m~:b~1:, Br ~:. doi:1.1371/journal.pone.9755.g16 " d H d x z Br {LH zb # ~, ðþ u~1,h~ atx~, ð3þ Figure 18. The influence of non-newtonian parameter b on velocity for drainage problem when M ~:5, St ~1,L~:1, Br ~5:. doi:1.1371/journal.pone.9755.g18 L (x)~ d H d x, N ðxþ " ~ Br {LH zb # ð6þ, GðxÞ~, ~,H~1 atx~, ðþ Now, OHAM is applied to non-linear coupled ordinary differential Eqs. (1, ) and Eqs. (5, 6) as follows: Solution of Lifting Problem The OHAM solution. In order to solve the system of equations (1 ), we define the linear, non-linear functions and source terms respectively as follows: L1 (x)~ d u d x, N1 ðxþ~{l dh zh d u d x ð5þ d u z6b d x { M u, G1ðxÞ~{ St, ½1{pŠ½L1ðxÞz G1 (x) Š{ H1ðpÞ½L1ðxÞz N1ðxÞz G1 (x) Š~, ½1{pŠ½LðxÞz G (x) Š{ HðpÞ½LðxÞz NðxÞz G (x) Š~, ð7þ We consider ux ð Þ,HðxÞ, H1ðpÞ, HðpÞ as the following. H1 ðpþ~p C1 z p C, HðpÞ~p C3 z p C,uðxÞ ~ u ðxþzp u 1 ðxþz p u ðxþ, HðxÞ~ H ðxþzp H 1 ðxþz p H ðxþ, ð8þ Substituting ux ð Þ,HðxÞ, H1ðpÞ, HðpÞ, L1ðxÞ, LðxÞ, N1ðxÞ, NðxÞ, G1ðxÞ, GðxÞ from Eq. (8) into Eq. (7) and after some simplifications based on power of p-terms, we get the following. Zero components: p : { St z d u d x ~, d H ~, ð9þ d x Figure 17. Drain temperature distribution for various values of viscosity parameter when M ~:,m~1:b~:6, Br ~5:. doi:1.1371/journal.pone.9755.g17 PLOS ONE 6 June 1 Volume 9 Issue 6 e9755

7 First components: p 1 d u : St z C1 St zm C1 zl d u d H C1 { d u d x { d u C1 d x d u z C1 LH d x {1b d u d u C1 d x z d u 1 d x ~ d u { Br C3 z H Br C3 L d u d u {b Br C3 " x # ux ð Þ~AxzBz St z L{1 M u {6b L {1 d u d x zl dh zh d u d x, ð35þ { d H d x { C3 d H d x z d H 1 d x ~: ð3þ Solving Eqs. (9, 3) along with boundary conditions (3, ), we get the term solutions as fallows. Zero term solution: HðxÞ~ExzF{ Br L {1 { Br b L {1, The series solutions of Eqs. (35, 36): z Br L L {1 H ð36þ First term solution: u (x)~ 1 ({x St z x St ), H (x)~x, ð31þ M {8 M St {96b S 3 3 t x u 1 (x)~ C1 { 1 M {1L St {1b St 3 x 6 z M St {8L St {96b St 3 x { M St {b St 3 x, 3L St {15 St {b St xz 3 St z6b S 3 t x H 1 (x)~ Br C 3 { St z1l St z8b St x 3, 6 6 z 5 St z1l St z6b St x { 3 St zb St x z b St x 6 ð3þ The second term solution for velocity and temperature are too bulky, therefore, only graphical representations up to second order are given. The series solutions of velocity profile and temperature distribution are n~ n~ x u n ~AxzBz St z M L {1 " # X? {1 zl L Bn n~ X? {1 H n ~ExzF{ Br L " # X? {1 { Br b L Fn, n~! " # X? {1 u n {6b L An n~ n~ " # ð37þ, X? {1 zl L Cn n~ " # Dn n~ " # X? {1 z Br L L En n~ ð38þ The Adomian polynomials An, Bn, Cn, Dn,E n and Fn, for Eqs. (37, 38) are defined as n~ An~ d u d x, ð39þ u(x, Ci )~ u ðxþz u 1 ðxþz u ðxþ, and Hðx, CiÞ~ H (x)z H 1 (x)z H (x) ð33þ n~ Bn~ dh, ðþ The arbitrary constants Ci,i~1,,3, are found out by using the resial R~L½ux, ð c i ÞŠzN½ux, ð c i ÞŠzG½ux, ð c i ÞŠ: ð3þ n~ Cn~H d u d x, ð1þ For velocity profile and temperature distribution the arbitrary constants are mentioned in graphs. The constants C1, C C3 C can also be obtained from Collocation and Ritz methods. The ADM solution. The inverse operator L {1 ~ ÐÐ, of the ADM on the second order coupled Eqs. (1,) is used: n~ Dn~, ðþ PLOS ONE 7 June 1 Volume 9 Issue 6 e9755

8 n~ En~H, ð3þ H ðxþ~exzf, ð5þ n~ Fn~, ðþ u 1 ðxþ~m L {1 ½u Š{6b L {1 ½AŠzL L {1 ½BŠzL L {1 ½CŠ, ð51þ The components of Adomian polynomials are derived from Eqs. (39 ) as: H 1 ðxþ~{ Br L {1 ½DŠz Br L L {1 ½EŠ{ Br b L {1 ½FŠ, ð5þ A ~ d u d u d x, B ~ d u d H, d u C ~ H d x, D ~ d u d u, E ~ H, E ~ d u, ð5þ u ðxþ~ M L {1 ½u 1 Š{6b L {1 ½A1ŠzL L {1 ½B1ŠzL L {1 ½C1Š,ð53Þ H ðxþ~{ Br L {1 ½D1Šz Br L L {1 ½E1Š{ Br b L {1 ½F1Š, ð5þ A1 ~ d u d u 1 d x z d u d u 1, B1 ~ d u 1 d H z d u d H 1, subject to the boundary conditions d u C1 ~ H 1 d x z H d u 1 d x, D1 ~ d u F1 ~ d u 3 d u 1, d u 1, d u E1 ~ H 1 d u 1 z H d u, The series solutions of Eqs. (37, 38) are derived as: ð6þ u ðþ~1, d u ð1þ~, u nðþ~, d u n ð1þ~,n 1, H ðþ~, H ð1þ~1, H n ðþ~, H n ð1þ~,n 1, ð55þ ð56þ Using boundary conditions from Eqs. (55, 56) into Eqs. (9 5), we obtain x u z u 1 z u z:::~axzbz St z M L {1 ðu zu 1 zu z::: Þ{6b L {1 ð7þ ½Az A1 z A z::: Šz L L {1 ½B z B1 z B z::: ŠzL L {1 ½C z C1 z C z::: Š, u ðxþ~ 1 { St xz St x, ð57þ H ðxþ~x, ð58þ H z H 1 z H z:::~exzf{ Br L {1 ½D z D1 z D z::: Š z Br L L {1 ½E z E1 z E z::: Š { Br b L {1 ½F z F1 z F z::: Š, ð8þ The velocity and temperature components are obtained by comparing both sides of Eqs. (7, 8): Components of the lift problem up to second order are: x u ðxþ~axzbz St, ð9þ u 1 ðxþ~ {M z M St zb St 3 x 3 z M z L St {6b S3 t x z { M St z L St 6 3 zb S3 t x 3 z M St {b S3 t x, ð59þ PLOS ONE 8 June 1 Volume 9 Issue 6 e9755

9 H 1 ðxþ~ 1 Br S t { 1 L Br S t z 3 b Br S t x z { 1 Br S t zb Br S t x z 1 3 Br S t z 1 6 L Br S t z 8 3 b Br S t x 3 ð6þ z { 1 1 Br S t { 1 6 L Br S t zb Br S t x z 1 L Br S t z 5 b Br S t x 5 z { 15 b Br S t x 6, Due to lengthy calculations, the analytical results have been given up to first order but they have been shown graphically up to second order. Formulation of Drainage Problem Under the same assumptions as in the previous problem, we consider a film of non-newtonian liquid draining down the vertical belt. The belt is stationary and the fluid drains down the belt e to gravity. The gravity in this case is opposite to the previous case. The coordinate system is selected same as in the previous case. Assuming that the flow is steady and laminar, external pressure is neglected whereas the fluid shear forces keep gravity balanced and the thickness of the film remains constant. Boundary conditions for the drainage problem are u~, at x~, ~, at x~d, ð61þ Using non-dimensional variables, the boundary conditions for drainage problem become OHAM is applied to non-linear coupled ordinary differential Eqs. (63, 6) as ½1{pŠ½L1ðxÞz G1ðxÞŠ{ H1ðpÞ½L1ðxÞz N1ðxÞz G1ðxÞŠ~, ½1{pŠ½LðxÞzGðxÞŠ{ HðpÞ½LðxÞz NðxÞz GðxÞŠ~, ð65þ We consider ux ð Þ,HðxÞ, H1ðpÞ, HðpÞ as the following H1 H ðpþ~p C 1 z p C, ðpþ~p C 1 z p C, ux ð Þ~ u ðxþzp u 1 ðxþz p u ðxþ, HðxÞ~ H ðxþzp H 1 ðxþz p H ðxþ, ð66þ Substituting ux ð Þ,HðxÞ, H1ðpÞ, HðpÞ, L1ðxÞ, N1ðxÞ, G1ðpÞ, L ðxþ, NðxÞ and GðxÞ from Eq. (66) into Eq. (65) we have the following components of velocity and temperature. Zero components: p : St z d u d x ~, d H ~, ð67þ d x First components: d u d H u {d d x {C d u 1 d x d x {1b C d u d u 1 d x z d u 1 d x ~, d u d u zl Br C nolimits 3 H ðxþ {b Br C 3 p 1 : { St{ St C 1 zm C 1 d u zl C 1 zl C 1 H ðxþ d u d u { Br C 3 u n ðþ~, and dn ð 1 Þ~, n, ð6þ { d H d x { C d H 3 d x z d H 1 d x ~, ð68þ For temperature distribution, the boundary conditions are same as given in Eq. (56). Solution of Drainage Problem The OHAM solution. From Eqs. (1, ), the linear, nonlinear functions and source term (in drainage case), it is opposite e to gravity) are respectively defined as L1 ~ d u d x, N1 ~{L { M u, G1 ~ St, L1 ~ d u d x, N1 ~ Br dh zh d u d x {LH zb z6b d u! d u d x ð63þ, GðxÞ~ ð6þ Solving Eqs. (67, 68) with boundary conditions (61, 6), we get u (x)~ 1 x St { x St, H (x)~x, ð69þ u 1 ðxþ 3 8 M St z96b S 3 ~ C t xz 1b S 3 t {1L St x 1 6 z 8L St { M St z96b St 3 x 3 7 5, z M St {b St 3 x H 1 ðxþ ~ Br C 3 6 ð7þ 3 St L{15 S t {b 3 S t xz 3 S t z1b St x { St z16b S t z1 S t L x 3 z 5 St z1b S t z1 S t L, 6 x 7 5 { 8b St z3 S t L x 5 z 8b St x 6 PLOS ONE 9 June 1 Volume 9 Issue 6 e9755

10 Like previous problem, results up to first order terms haven been obtained. ADM solution. Using ADM on Eqs. (1, ), the Adomian polynomials in equations (5, 6) for both problems are same whereas the different velocity components are obtained as: Components of the Problem The boundary conditions of first and second components for drainage velocity profile are same as given in Eq. (56). Also, the boundary conditions for temperature distribution are same as given in Eq. (57) but solution of these components is different, depends on the different velocity profile of drainage and lift problems. Due to lengthy analytical calculation, solutions up to first order terms are included whereas the graphical representations up to second order terms are given. Using boundary conditions (6) and (56) into Eqs. (9 5), the components, solution are obtained as: u (x)~ 1 x St { x St, ð71þ H (x)~x, u 1 ðxþ~ { 1 3 M St {b St 3 xz L St z M St { L St 6 3 {b S3 t x 3 z6b S3 t x 3 z { M St zb S3 t ð7þ ð73þ x, H 1 ðxþ~ 1 Br S t { 1 L Br S t z 3 bb r St x z { 1 Br S t { bb r St x z 1 3 Br S t z 1 6 L Br S t z 8 3 bb r St x 3 ð7þ z { 1 1 Br S t { 1 6 L Br S t { bb r St x z 1 1 L Br S t { 5 bb r St x 5 z { 15 bb r St x 6, Figure 19. The influence of non-newtonian parameter b on temperature for drainage problem when M ~:5, St ~1,L~:1, Br ~5:. doi:1.1371/journal.pone.9755.g19 problems. A comparison of the ADM and OHAM solutions is shown in Figs. 5 for various values of physical parameters. It is found from these figures that ADM and OHAM solutions are in good agreement. Figs. 6 and 7 provide variation of velocity and temperature distribution for different values of Brinkman number. It has been found that velocity decreases whereas temperature inside the fluid increases by increasing Br while keeping the other parameters fixed. In Fig. 8, we observed that velocity decreases with an increase in the Stock number St. Physically, it is true as increasing Stock number causes the fluid s thickness and reces its flow. The effect of Stock number St on temperature distribution has been illustrated in Fig. 9. It is observed that temperature H increases monotonically for large values of Stock number St. The effect of viscosity parameter L on lift velocity u is shown in Fig. 1. It is observed that the speed of flow decreases by increasing L. The speed of flow is actually caused by shear s thickening and thinning effects e to increase and decrease in viscosity parameter. A similar situation is observed in Fig. 11 where an increase in viscosity parameter L decreases temperature distribution. Here, the velocity profiles are parabolic in nature and their amplitudes depend on the magnitude of the viscosity parameter L. Variations of the magnetic parameter M on lift velocity have been studied in Fig. 1. Here, it is clear that the boundary layer thickness is reciprocal to the transverse magnetic field and velocity decreases Results and Discussion The effect of Stock number St magnetic parameter M, Brinkman number Br non-newtonian parameter b and viscosity parameter L in lifting and drainage problems together with the physical interpretation of the problem have been discussed in Figs. 1. Fig. 1 shows the geometry of lift and drainage Figure. Comparison of the present results with published work [] when St ~:5, M ~:1,b~:6,L~,a~1:. doi:1.1371/journal.pone.9755.g PLOS ONE 1 June 1 Volume 9 Issue 6 e9755

11 Table 1. Numerical results show the comparison of present results with published work [] when St ~:5, M ~:1,b~:6,L~,a~1:. x Present work Published work[] Absolute error doi:1.1371/journal.pone.9755.t1 as flow progresses towards the surface of the fluid. On the other hand, temperature profile as shown in Fig. 13 indicates that fluid temperature increases with magnetic parameter. Fig. 1 shows that velocity increases in drainage flow when Stock number St increases. Physically, it is e to friction which seems smaller near the belt and higher at the surface of the fluid. Further, it is found from Fig. 15 that temperature profile also increases when St is increased. Fig. 16 illustrates the effect of variable viscosity parameter L on the drain flow. It is observed that at higher values of viscosity parameter L, velocity of the fluid increases graally towards the surface of the fluid. However, it is found from Fig.17, that an increase in viscosity parameter L causes graal decrease in temperature field. The effects of non- Newtonian parameter b on drain velocity have been studied in Fig. 18. We observed that an increase in b raises drain velocity profile and decreases temperature profile as shown in Fig. 19. Finally for the accuracy purpose the present results are compared with published work in [] in Fig. and in table 1. An excellent agreement is found. Future Work We intend to carry out researches in future on third grade fluid on vertical belt regarding the following discussions: 1) Time dependent third grade fluid on vertical belt. ) Vogel Model third grade fluid on vertical belt with slip boundary conditions. References 1. Fetecau C, Fetecau C (5) Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int J Eng Sci 3: Fetecau C, Fetecau C (6) Starting solutions for the motion of a second grade fluid e to longitudinal and torsional oscillations of a circular cylinder. Int J Eng Sci : Tan WC, Masuoka T (7) Stability analysis of a Maxwell fluid in a porous medium heated from below. Phys Lett A 36: Fakhar K, Zenli Xu, Cheng Yi (8) Exact solution of a third grade fluid on a porous plate. Appl Math Comput : Yao Y, Liu Y (1) Some unsteady flows of a second grade fluid over a plane wall. Non-linear Anal Real World Appl 11: Nazar M, Fetecau C, Vieru D (1) New exact solutions corresponding to the second problem of stokes for second grade fluids. Non-linear Anal Real World Appl 11: Ali F, Norzieha M, Sharidan S, Khan I, Hayat T (1) New exact solution of Stokes second for an MHD second fluid in a porous space. Int J Non-linear Mech 7: ) Third grade fluid on vertical belt wit surface topography. ) Third grade fluid on vertical rotating disc with surface topography. Conclusion In this work, we have investigated the thin film flow non- Newtonian third grade fluid e to vertical belt and the fluid was subjected to lifting and drainage. Analytical solutions of the lifting and drainage problems have been obtained using ADM and OHAM. It has been shown graphically that these solutions are identical. The results for velocity and temperature have been plotted graphically and discussed in detail. It has been observed that these solutions are valid not only for small but also for large values of the emerging parameters. It has been observed that in both cases of lift problem velocity decreases while temperature increases with increasing Brinkman number Br. However, in drainage problem both velocity and temperature increases. Author Contributions Conceived and designed the experiments: TG IK SI RAS. Performed the experiments: TG SI. Analyzed the data: TG IK SS. Contributed reagents/ materials/analysis tools: IK SS. Wrote the paper: TG SI IK SS. Revised the manuscript: TG IK SI. 8. Qasim M (13) Heat and mass transfer in a Jeffrey fluid over a stretching sheet with heat source. Alex Eng J 5: Qasim M, Hayat T, Obaidat S (1) Radiation effect on the mixed convection flow of a viscoelastic fluid along an inclined stretching sheet. Z Naturforsch 67: Hayat T, Shehzad SA, Qasim M, Asghar S, Alsaedi A (1) Thermally stratified radiative flow of third grade fluid over a stretching surface. J Thermophysics and Heat Transfer 8: Qasim M, Noreen S (1) Heat transfer in the boundary layer flow of a Casson fluid over a permeable shrinking sheet with viscous dissipation. The Europ Physical J Plus 19: Turkyilmazoglu M (11) Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int J of Thermal Sc 5: Turkyilmazoglu M (1) A note on the homotopy analysis method. Appl Math Lett 3: PLOS ONE 11 June 1 Volume 9 Issue 6 e9755

12 1. Turkyilmazoglu M (1) Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection. Int J Mechanical Sc 5: Turkyilmazoglu M (1) Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous rotating disk. Computers and Fluids 39: Fetecau C, Vieru D, Corina Fetecau, Akhter S (13) General solutions for magnetohydrodynamic natural convection flow with radiative heat transfer and slip condition over a moving plate, Z. Naturforsch. 68a, (13). 17. Fetecau C, Corina Fetecau, Mehwish Rana (13) General solutions for the unsteady flow of second-grade fluids over an infinite plate that applies arbitrary shear to the fluid, Z. Naturforsch. 66a, Rubbab Q, Vieru D, Fetecau C, Fetecau C (13) Natural convection flow near a vertical plate that applies a shear stress to a viscous fluid, PLoSONE 8(11): e7835. doi:1.1371/journal.pone Khan A, Khan I, Ali F, ulhaq S, Shafie S (1) Effects of wall shear stress on unsteady MHD conjugate flow in a porous medium with ramped wall temperature. PLoS ONE 9(3): e98. doi:1.1371/journal.pone.98.. Khaled ARA, Vafai K () Hydrodynomic squeezed flow and heat transfer over a sensor surface, Int J Eng Sci : Miladinova S, Lebon G, Toshev E () Thin film flow of a power law liquid falling down an inclined plate. J Non-Newtonian fluid Mech 1: Gul T, Shah RA, Islam S, Arif M (13) MHD thin film flows of a third grade fluid on a vertical belt with slip boundary conditions. J Appl Math: Article ID Siddiqui AM, Mahmood R, Ghori QK (8) Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos Sol Fract 35: Liao S J (3) Beyond Perturbation: Introction to Homotopy Analysis Method. Chapman & Hall/CRC Press Boca Raton. 5. Marinca V, Herisanu N, Nemes I (8) Optimal homotopy asymptotic method with application to thin film flow. Cent. Eur J Phys 6: Marinca V, Herisanu N (8) Application of Optimal homotopy asymptotic method for solving non-linear equations arising in heat transfer. Int Commun Heat Mass Transfer 35: Marinca V, Herisanu V, Bota C, Marinca B (9) An Optimal homotopy asymptotic method applied to the steady flow of a fourth grade fluid past a porous plate. Appl Math Lett : Joneidi AA, Ganji DD, Babaelahi (9) Micro polar flow in a porous channel with high mass transfer. Int Commun Heat Mass Transfer 36: Siddiqui AM, Ahmed M, Ghori QK (7) Thin film flow of non-newtonian fluid on a moving belt. Chaos Sol Fract 33: Siddiqui AM, Mahmood R, Ghori QK (6) Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys Lett A 35: Costa A, Macedonio G (3) Viscous heating in fluids with temperaturedependent viscosity implications for magma flows. Nonlinear Proc Geophys 1: Nadeem S, Awais M (8) Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity. Phy Let A 37: Ellahi R, Riaz A (1) Analytical solution for MHD flow in a third grade fluid with variable viscosity. Math Comput Mod 5: Aksoy Y, Pakdemirly M (1) Approximate analytical solution for flow of a third grade fluid through a parallel-plate channel filled with a porous medium. Springer 83: Adomian G (199) Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers. 36. Adomian G (199) A review of the decomposition method and some recent results for non-linear equations. Math Comput Model 13: Wazwaz AM (5) Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl Math Comput 166: Wazwaz AM (5) Adomian decomposition method for a reliable treatment of the Emden Fowler equation. Appl Math Comput 161: PLOS ONE 1 June 1 Volume 9 Issue 6 e9755