MHD Thin Film Flows of a Third Grade Fluid on a Vertical Belt with Slip Boundary Conditions

Abstract

The problem of heat transfer analysis is considered in electrically conducting thin film flows with slip boundary conditions. The flow is assumed to be obeying the nonlinear rheological constitutive equation of a third grade fluid. We have solved the governing nonlinear equations of present problems using the traditional Adomian decomposition method (ADM). Particular attention is given to the combined effect of heat and MHD on the velocity field. The results include the profile of velocity, volume flux, skin friction, average velocity, and the temperature distribution across the film. The effects of model parameters on velocity, skin friction and temperature variation have been studied. Optimal homotopy asymptotic method (OHAM) is also used for comparison. The numerical results and absolute errors are derived in tables.

1. Introduction

The flow and heat transfer inside thin films are ubiquitous in civil, environmental sciences, mechanical engineering, biological sciences, geophysics, and elsewhere. This is due to their several applications at large scale such as wire and fiber coating, reactor fluidization, paper production, different food stuffs like ketchup, sauce, and honey, transpiration cooling, gaseous diffusion, drilling mud, oil wills, heat pipes, and fluid cells. The problem of chambers for chemical and biological detection systems like fluid of many chemicals was considered by Lavrik et al. [1]. Heat transfer inside thin film flow with variable pressure was discussed by Khaled and Vafai [2]. Thin film unsteady flow with variable viscosity was investigated by Nadeem and Awais [3]. Munson and Young [4] discussed the thin film flow of Newtonian fluids. Alam et al. investigated the thin film flow of Johnson-Segalman fluids for lifting and drainage problems [5]. The MHD thin film flow was discussed by Hameed and Ellahi for a non-Newtonian fluid on a vertical moving belt [6]. Due to complexity of non-Newtonian fluids, it becomes difficult to suggest a single model which exhibits all properties of non-Newtonian fluids; therefore various empirical and semiempirical models have been imposed. The use of heat transfer together with the MHD and non-Newtonian fluids under the influence of slip boundary conditions is of particular interest in chemical processing. Relevant and interesting work about present work may be found in [7, 8]. One of the established models amongst non-Newtonian fluids is class of third grade fluids which have their constitutive equations based on strong theoretical foundations, where the relation between the stress and strain is not linear. To solve real world problems, different approximate techniques have been used in mathematics, fluid mechanics, and engineering sciences [9]. Some of the common methods are VIM [10–14], DTM [15, 16], HPM [17–19], HAM, and OHAM [20–24]. These methods deal with the nonlinear problems effectively. The work under various configurations on the thin film flows was discussed by Siddiqui et al. [25]. In [26] they examined the thin film flows of Sisko and Oldroyd-6 constant fluids on a moving belt. The heat transfer analysis of thin film is also discussed by Chakraborty and Som [27]. The main aim of the present work is to study heat transfer into a thin film of a third grade fluid on a vertical belt under the influence of transverse magnetic field with slip boundary conditions using ADM. In 1992 Adomian [28, 29] introduced the ADM for the approximate solutions for linear and nonlinear problems. Wazwaz [30, 31] used ADM for the reliable treatment of Bratu-type and Emden-Fowler equations.

2. Basic Equations

The governing equations of an incompressible isothermal and electrically conducting third grade fluid are 𝜌𝛁⋅𝐔=𝟎,𝐷𝐔𝐷𝑡=𝛁⋅𝐓+𝜌𝐠+𝐉×𝐁,𝜌𝑐_𝑝𝐷Θ𝐷𝑡=𝜅∇²Θ+tr(𝝉⋅𝐋), (1)(2)(3) where 𝜌 is the constant density, 𝐠 is body force per unit mass, 𝐔 is velocity vector of the fluid, Θ is temperature, 𝜅 is thermal conductivity, 𝑐_𝑝 is specific heat, 𝐋=𝛁𝐔, 𝐷/𝐷𝑡=𝜕/𝜕𝑡+(𝐔⋅𝛁) denote material time derivative, 𝝉 is Cauchy stress tensor, and 𝐓 is the shear stress. A uniform magnetic field 𝐁=(0,𝐵₀,0) is imposed transversely on the belt. The Lorentz force per unit volume is given by 𝐉×𝐁=0,𝜎𝐵²₀𝑢(𝑥),0. (4)

Shear stress tensor 𝐓 is given by 𝐓=−𝑝𝐈+𝝉, (5) where −𝑝𝐈 denotes spherical stress and 𝝉 is defined as 𝝉=𝜇𝐀₁+𝛼₁𝐀_𝟐+𝛼₂𝐀^𝟐₁+𝛽₁𝐀₃+𝛽₂𝐀₁𝐀₂+𝐀_𝟐𝐀₁+𝛽₃tr𝐀^𝟐₁𝐀₁. (6)

Here 𝛼_𝑖 and 𝛽_𝑗 are the material constants, and 𝐀₁, 𝐀₂, and 𝐀₃ are the kinematical tensors given by 𝐀_𝑛=𝐷𝐀_𝑛−1𝐷𝑡+𝐀_𝑛−1(∇𝐮)+(∇𝐮)^𝑇𝐀_𝑛−1,𝑛≥1,𝜇≥0,𝛼₁≥0,𝛼₁+𝛼₂≤24𝜇𝛽₃,𝛽₃≥0. (7)(8)

3. Formulation of the Lift Problem

Consider a wide flat belt moving vertically upward at constant speed 𝑈₀, through a large bath of third grade liquid as shown in Figure 1. The belt carries with it a layer of liquid of constant thickness, 𝛿. For analysis coordinate system is chosen, in which the x-axis is taken parallel to the surface of the belt and y-axis is perpendicular to the belt. Uniform magnetic field is applied transversely to the belt. Assume the flow is steady and laminar after a small distance above the liquid surface layer. The external pressure is atmospheric everywhere.

Figure 1 

Geometry of the lifting problem.

Velocity and temperature fields are 𝐮=(0,𝑢(𝑥),0),Θ=Θ(𝑥). (9)

Boundary conditions are 𝑢=𝑈₀−𝛾𝑇_𝑥𝑦at𝑥=0,𝑑𝑢𝑑𝑥=0,at𝑥=𝛿,Θ=Θ₀,at𝑥=0,Θ=Θ₁,at𝑥=𝛿. (10)

Inserting the velocity field given in (9) in (1) and (5)–(7), the continuity equation (1) satisfies identically and (5) gives the following components of stress tensor 𝑇_𝑥𝑥=−𝑃+2𝛼₁+𝛼₂𝑑𝑢𝑑𝑥²,𝑇_𝑥𝑦=𝜇𝑑𝑢𝛽𝑑𝑥+2₂+𝛽₃𝑑𝑢𝑑𝑥³,𝑇_𝑦𝑦=−𝑃+𝛼₂𝑑𝑢,𝑇𝑑𝑥_𝑧𝑧𝑇=−𝑃,_𝑥𝑧=𝑇_𝑦𝑧=0. (11)

Using (11), the momentum and energy equations reduce to 𝑑0=𝜇²𝑢𝑑𝑥²+6𝛽₂+𝛽₃𝑑𝑢𝑑𝑥²𝑑²𝑢𝑑𝑥²−𝜌𝑔−𝜎𝐵²₀𝑢(𝑥),𝑑0=𝜅²Θ𝑑𝑥²+𝜇𝑑𝑢𝑑𝑥²+2𝛽₂+𝛽₃𝑑𝑢𝑑𝑥⁴. (12)

Introducing the following nondimensional variables: 𝛿𝑢=𝜈𝑈,𝑥𝑥=𝛿,Θ=Θ−Θ₀Θ₁−Θ_𝟎,𝛾=𝜇𝛾𝛿,Λ=𝛾𝜈𝛿,𝜆=𝜇𝜈²𝑘Θ₁−Θ₀𝛿²𝛿,𝑚=³𝑔𝜈²,𝑀=𝜎𝐵²₀𝛿²𝜇𝛽,𝛽=²+𝛽₃𝜈²𝜇𝛿⁴,𝛼=𝛿𝑈₀𝜈,𝑅_𝑒=𝑈𝛿𝜈𝜇,𝜈=𝜌, (13) where 𝑚 is the gravitational parameter, 𝑀 is magnetic parameter, 𝛽 is non-Newtonian effect, Λ is slip parameter, 𝜆 is heat dimensionless number, 𝑅_𝑒 is the local Reynolds number, and 𝛼 is the nondimensional variable using the above dimensionless variables in (10) and in (12) and dropping bars we obtain 𝑑²𝑢𝑑𝑥²+6𝛽𝑑𝑢𝑑𝑥²𝑑²𝑢𝑑𝑥²(𝑥)𝑑−𝑚−𝑀𝑢=0,₂Θ𝑑𝑥²+𝜆𝑑𝑢𝑑𝑥²+2𝛽𝑑𝑢𝑑𝑥⁴𝑢=0,_𝑛(0)=𝛼−Λ𝑑𝑢_𝑛𝑑𝑥+2𝛽𝑑𝑢_𝑛𝑑𝑥³,𝑑𝑢_𝑛(1)𝑢𝑑𝑥=0,𝑛=0,_𝑛(0)=−Λ𝑑𝑢_𝑛𝑑𝑥+2𝛽𝑑𝑢_𝑛𝑑𝑥³,𝑑𝑢_𝑛(1)Θ(0)(1)𝑑𝑥=0,𝑛>0,=0,Θ=1. (14)(15)(16)(17)(18)

4. Adomian Decomposition Method (ADM)

A general description of the method is as follows. Begin with an equation 𝐹(𝑢)=𝑔(𝑥), where 𝐹 represents a general nonlinear ordinary differential operator involving both linear and nonlinear terms and 𝑔 is a source term. The linear term is decomposed into 𝐿+𝑅, where 𝐿 is easily invertible and 𝑅 is the remainder of the linear operator. For convenience, 𝐿 may be taken as the highest order derivative which avoids difficult integrations which result when complicated Green’s functions are involved. Thus the equation may be written as follows: 𝐿𝑢+𝑅𝑢+𝑁𝑢=𝑔. (19) Solving for 𝐿𝑢, 𝐿𝑢=𝑔−𝑅𝑢−𝑁𝑢, (20) where 𝑁 is non-linear operator and 𝑔 is a source term, since 𝐿 is easily invertible.

Equation (20) can be written as follows: 𝐿⁻¹𝐿𝑢=𝐿⁻¹𝑔−𝐿⁻¹𝑅𝑢−𝐿⁻¹𝑁𝑢. (21) We use 𝐿⁻¹ depending on the order of the differential equation. If differential equation is an initial-value problem, if it is desired for boundary value problem as well, integrations are used. The constants of integration are evaluated from the given initial and boundary conditions. 𝐿⁻¹ can also be treated as definite integral from (𝑡₀̃ to 𝑡̃). Solving (21) for 𝑢 we obtained 𝑢=𝐶₁+𝐶₂𝑡+𝐿⁻¹𝑔−𝐿⁻¹𝑅𝑢−𝐿⁻¹𝑁𝑢. (22)

The non-linear term 𝑁𝑢, is defined as follows: 𝑁𝑢=^∞_𝑘=0𝐴_𝑛. (23) Here 𝐴_𝑛 are special polynomials called Adomian polynomials and 𝑢 will be equated to ∑^∞_𝑘=0𝑢_𝑛. The initial velocity is identified as follows: 𝑢₀=𝐶₁+𝐶₂𝑡+𝐿⁻¹𝑔,_∞𝑘=0𝑢_𝑛=𝑢₀−𝐿⁻¹𝑅^∞_𝑘=0𝑢_𝑛−𝐿^∞−1_𝑘=0𝐴_𝑛. (24) Comparison for different components of velocity profile is as follows: 𝑢₁=−𝐿⁻¹𝑅𝑢₀−𝐿⁻¹𝐴₀,𝑢₂=−𝐿⁻¹𝑅𝑢₀−𝐿⁻¹𝐴₁,⋮𝑢_𝑛+1=−𝐿⁻¹𝑅𝑢_𝑛−𝐿⁻¹𝐴_𝑛. (25) The Adomian polynomials 𝐴_𝑛 depend on the velocity components 𝑢₀,𝑢₁,𝑢₂,…,𝑢_𝑛, which play a flourishing role in the rapid convergence of the series. In the above series 𝐴₀ depends only on 𝑢₀, 𝐴₁ depends on 𝑢₀ and 𝑢₁, 𝐴₂ depends on 𝑢₀, 𝑢₁, and 𝑢₂, and so forth. Relevant discussion about ADM can be seen in [16, 17].

4.1. The ADM Solution of Lifting Problem

Using the inverse operator 𝐿⁻¹=∬𝑑𝑢^′ of the Adomian decomposition method on (14), we obtained 𝑢=𝐶₁𝑥+𝐶₂𝑥+𝑚²2+𝐿⁻¹[𝑀𝑢]−6𝛽𝐿⁻¹𝑑𝑢𝑑𝑥²𝑑²𝑢𝑑²𝑥. (26) For series solution we use summation on (26) ^∞_𝑛=0𝑢_𝑛=𝐶₁𝑥+𝐶₂𝑥+𝑚²2+𝑀𝐿^−1∞_𝑛=0𝑢_𝑛−6𝛽𝐿⁻¹×𝑑𝑑𝑥^∞_𝑛=0𝑢_𝑛²𝑑²𝑑𝑥^2∞_𝑛=0𝑢_𝑛. (27) Adomian polynomials are defined from (27) as follows: 𝐴_𝑛=𝑑𝑑𝑥^∞_𝑛=0𝑢_𝑛²𝑑²𝑑𝑥^2∞_𝑛=0𝑢_𝑛,𝑛≥0. (28)

From (28) when 𝑛≥0, the Adomian polynomials in components form are 𝐴₀=𝑑𝑢₀𝑑𝑥²𝑑²𝑢₀𝑑𝑥²,𝐴₁=𝑑𝑢₀𝑑𝑥²𝑑²𝑢₁𝑑𝑥²+2𝑑𝑢₀𝑑𝑥𝑑𝑢₁,𝐴𝑑𝑥₂=𝑑𝑢₀𝑑𝑥²𝑑²𝑢₂𝑑𝑥²+𝑑𝑢₁𝑑𝑥²𝑑²𝑢₀𝑑𝑥²+2𝑑𝑢₀𝑑𝑥𝑑𝑢₂𝑑𝑑𝑥²𝑢₀𝑑𝑥². (29) The series solution becomes 𝑢₀+𝑢₁+𝑢₂+⋯=𝐶₁𝑥+𝐶₂𝑥+𝑚²2+𝑀𝐿⁻¹𝑢₀+𝑢₁+𝑢₂+⋯−6𝛽𝐿⁻¹𝐴₀+𝐴₁+𝐴₂+⋯. (30)

The velocity components are obtained by comparing both sides of (33).

4.1.1. Zero Component Problem

Consider 𝑢₀(𝑥)=𝐶₁𝑥+𝐶₂𝑥+𝑚²2. (31) From (16) the boundary condition, for 𝑛=0, is 𝑢₀(0)=𝛼−Λ𝑑𝑢₀𝑑𝑥+2𝛽𝑑𝑢₀𝑑𝑥³,𝑑𝑢₀(1)𝑑𝑥=0. (32) By making use of (32) in (31), after simplification we obtain 𝑢₀(𝑥)=𝛼+𝑚𝑥²2−𝑚𝑥+Λ𝑚+2𝛽𝑚³. (33)

4.1.2. First Component Problem

Consider 𝑢₁(𝑥)=𝑀𝐿⁻¹𝑢₀−6𝛽𝐿⁻¹𝐴₀. (34) From (17) the boundary condition, for 𝑛=1, is 𝑢₁(0)=−Λ𝑑𝑢₁𝑑𝑥+6𝛽𝑑𝑢₀𝑑𝑥²𝑑𝑢₁,𝑑𝑥𝑑𝑢₁(1)𝑑𝑥=0. (35)

The solution is 𝑢₁(𝑥)=1+6𝑚²𝛽Λ×𝑚𝑀3−𝑀𝛼+2𝑚³𝛽−𝑚𝑀Λ−2𝑚³+𝑀𝛽Λ𝑚𝑀3−𝑀𝛼+2𝑚³𝛽−𝑚𝑀Λ−2𝑚³𝑥+𝑀𝛽Λ𝑀𝛼2−3𝑚³𝛽+𝑚𝑀Λ2+𝑚³𝑀𝛽Λ𝑥²+2𝑚³𝛽−𝑚𝑀6𝑥³+𝑚𝑀−𝑚24³𝛽2𝑥⁴. (36)

4.1.3. Second Component Problem

Consider 𝑢₂(𝑥)=𝑀𝐿⁻¹𝑢₁−6𝛽𝐿⁻¹𝐴₁. (37) From (17) the boundary condition, for 𝑛=2, is 𝑢₂(0)=−Λ𝑑𝑢₂𝑑𝑥1+6𝛽𝑑𝑢₀𝑑𝑥²+6𝛽𝑑𝑢₁𝑑𝑥²𝑑𝑢₀,𝑑𝑥𝑑𝑢₂(1)𝑑𝑥=0. (38)

The solution is 𝑢₂(𝑥)=Ψ₀−Ψ₁𝑥+Ψ₂𝑥²−Ψ₃𝑥³+Ψ₄𝑥⁴−Ψ₅𝑥⁵+Ψ₆𝑥⁶. (39)

The series solution up to the second component is 𝑢(𝑥)=𝑢₀(𝑥)+𝑢₁(𝑥)+𝑢₂(𝑥). (40) Using (33), (36), and (39) in (40), we have 𝑢(𝑥)=Φ₀−Φ₁𝑥+Φ₂𝑥²−Φ₃𝑥³+Φ₄𝑥⁴−Φ₅𝑥⁵+Φ₆𝑥⁶. (41) The constants (Ψ₀--Ψ₆) and (Φ₀--Φ₆) are given in the appendix.

The dimensionless shear stress is 𝜏_𝑥𝑦=𝜇𝜈𝛿²𝑑𝑢𝑑𝑥+2𝛽𝑑𝑢𝑑𝑥³. (42) The shear rate becomes 𝜏_{𝑥𝑦𝑥=0}=𝜇𝜈𝛿²2590𝑚²𝑀𝛼𝛽+90𝑚²𝑀²𝛼𝛽Λ−15𝑀𝛼+5𝑀²𝛼+15𝑀²𝛼Λ−180𝑚⁵𝛽²+180𝑚²𝛽²𝑀²Λ²−15𝑚+5𝑚𝑀−2𝑚𝑀²−15Λ𝑚𝑀Λ+15𝑚𝑀²Λ²+30𝑚³𝛽−36𝑀𝑚³𝛽+30Λ𝑀𝑚³𝛽−20Λ𝑀²𝑚³𝛽+20Λ²𝑀²𝑚³𝛽+90𝑚²𝑀𝛼𝛽+90𝑚²𝑀²𝛼𝛽Λ−15𝑀𝛼+5𝑀²𝛼+15𝑀²𝛼Λ−180𝑚⁵𝛽²+180𝑚²𝛽²𝑀²Λ²−15𝑚+5𝑚𝑀−2𝑚𝑀²−15Λ𝑚𝑀Λ+15𝑚𝑀²Λ²+30𝑚³𝛽−36𝑀𝑚³𝛽+30Λ𝑀𝑚³𝛽−20Λ𝑀²𝑚³𝛽+20Λ²𝑀²𝑚³𝛽³. (43)

The coefficient of skin friction is defined as follows: 𝐶_𝑓(0)=𝜏_{𝑥𝑦𝑥=0}(1/2)𝜌𝑈². (44)

By use of (43), we have 𝐶_𝑓(0)=2𝑅²_𝑒2590𝑚²𝑀𝛼𝛽+90𝑚²𝑀²𝛼𝛽Λ−15𝑀𝛼+5𝑀²𝛼+15𝑀²𝛼Λ−180𝑚⁵𝛽²+180𝑚²𝛽²𝑀²Λ²−15𝑚+5𝑚𝑀−2𝑚𝑀²−15Λ𝑚𝑀Λ+15𝑚𝑀²Λ²+30𝑚³𝛽−36𝑀𝑚³𝛽+30Λ𝑀𝑚³𝛽−20Λ𝑀²𝑚³𝛽+20Λ²𝑀²𝑚³𝛽+90𝑚²𝑀𝛼𝛽+90𝑚²𝑀²𝛼𝛽Λ−15𝑀𝛼+5𝑀²𝛼+15𝑀²𝛼Λ−180𝑚⁵𝛽²+180𝑚²𝛽²𝑀²Λ²−15𝑚+5𝑚𝑀−2𝑚𝑀²−15Λ𝑚𝑀Λ+15𝑚𝑀²Λ²+30𝑚³𝛽−36𝑀𝑚³𝛽+30Λ𝑀𝑚³𝛽−20Λ𝑀²𝑚³𝛽+20Λ²𝑀²𝑚³𝛽³. (45)

5. Volume Flow Rate and Average Velocity

Volume flow rate in nondimensional form is as follows: 𝑄=¹₀𝑢(𝑥)𝑑𝑥. (46)

Using velocity field from (41), volume flow rate is obtained as follows.

Average velocity 𝑢 in dimensionless form is given by 𝑚𝑢=𝑄,𝑄=−3+2𝑚𝑀−1517𝑚𝑀²315+𝛼−𝑀𝛼3+2𝑀²𝛼+152𝑚³𝛽5−64𝑚105³+6𝑀𝛽5𝑚²𝑀𝛼𝛽−12𝑚⁵𝛽²7−1+𝑚Λ9𝑚𝑀²2Λ−𝑀𝛼Λ+3𝑀²𝛼Λ+4𝑚³8𝛽Λ−𝑚15³𝑀𝛽Λ−33𝑚20³𝑀²+𝛽Λ152𝑚²𝑀²𝛼𝛽Λ−6𝑚𝑀²𝛼²−𝛽Λ1435𝑚⁵𝑀𝛽²Λ+78𝑚⁴𝑀𝛼𝛽²Λ−144𝑚⁷𝛽³Λ−𝑚𝑀Λ²+13𝑚𝑀²Λ²+𝑀²𝛼Λ²−4𝑚³𝑀𝛽Λ²+296𝑚³𝑀²𝛽Λ²+54𝑚⁵𝑀𝛽²Λ²+12𝑚⁵𝑀²𝛽²Λ²+12𝑚⁴𝑀²𝛼𝛽²Λ²+84𝑚⁷𝑀𝛽³Λ²+𝑚𝑀²Λ³+8𝑚³𝑀²𝛽Λ³+36𝑚⁵𝑀²𝛽²Λ³+48𝑚⁷𝑀²𝛽³Λ³. (47)(48)

6. Temperature Distribution in Case of Lift Problem

On substituting the series solution for velocity field given in (41) in (15) and solving corresponding to the boundary condition given in (18) for fixed values of 𝛼=0.1; 𝑚=0.5; 𝑀=1; Λ=0.4; 𝛽=1; 𝜆=100, we obtained Θ(𝑥)=7.1923𝑥−12.7305𝑥²+9.0310𝑥³−2.4798𝑥⁴−0.3026𝑥⁵+0.4717𝑥⁶−0.2248𝑥⁷+0.03702𝑥⁸+0.01515𝑥⁹−0.0133𝑥¹⁰+0.0044𝑥¹¹−0.00032𝑥¹²−0.00037𝑥¹³+0.000202𝑥¹⁴−0.000048𝑥¹⁵+0.00000156𝑥¹⁶+0.000003𝑥¹⁷−0.0000014𝑥¹⁸+3.3326×10⁻⁷𝑥¹⁹−5.1691×10⁻⁸𝑥²⁰+5.0040×10⁻⁹𝑥²¹−2.2745×10⁻¹⁰𝑥²². (49)

7. Formulation of Drainage Problem

The geometry and assumptions of the problem are the same as those in the previous problem. Consider a film of non-Newtonian liquid draining at volume flow rate 𝑄 down the vertical belt, as shown in Figure 2. The belt is stationary and the fluid drain down the belt due to gravity. The coordinate system is selected in the same way as that in the previous case. Assuming the flow is steady, laminar and external pressure is neglected. Consider that fluid shear forces balance gravity and the film thickness remain constant.

Figure 2 

Geometry of the drainage problem.

Boundary condition for electrically conducting drainage problem is as follows: 𝑢=−𝛾𝑇_𝑥𝑦at𝑥=0,𝑑𝑢𝑑𝑥=0,at𝑥=𝛿. (50) Using nondimensional variables the slip boundary conditions for drainage problem become 𝑢_𝑛(0)=−Λ𝑑𝑢_𝑛𝑑𝑥+2𝛽𝑑𝑢_𝑛𝑑𝑥³,𝑑𝑢_𝑛(1)𝑑𝑥=0,𝑛≥0. (51)

7.1. Solution of the Drainage Problem by ADM

Using ADM on (14), the Adomian polynomials in (29) for both problems are the same. The different velocity components are obtained as follows.

7.1.1. Zero Component Problem

Consider 𝑢₀(𝑥)=𝐶₁+𝐶₂𝑥𝑥−𝑚²2,𝑢₀(0)=−Λ𝑑𝑢₀𝑑𝑥+2𝛽𝑑𝑢₀𝑑𝑥³,𝑑𝑢₀(1)𝑑𝑥=0. (52) The solution is 𝑢₀(𝑥)=𝑚𝑥−𝑚𝑥²2−Λ𝑚+2𝛽𝑚³. (53)

7.1.2. First Component Problem

Consider 𝑢₁(𝑥)=𝑀𝐿⁻¹𝑢₀−6𝛽𝐿⁻¹𝐴₀,𝑢₁(0)=−Λ𝑑𝑢₁𝑑𝑥+6𝛽𝑑𝑢₀𝑑𝑥²𝑑𝑢₁,𝑑𝑥𝑑𝑢₁(1)𝑑𝑥=0. (54) For different velocity components, the Adomian polynomials mentioned in (29) are used: 𝑢₁(𝑥)=1+6𝑚²𝛽Λ×−𝑚𝑀3−2𝑚³𝛽+𝑚𝑀Λ+2𝑚³+−𝑀𝛽Λ𝑚𝑀3−2𝑚³𝛽+𝑚𝑀Λ+2𝑚³𝑥+𝑀𝛽Λ3𝑚³𝛽−𝑚𝑀Λ2−𝑚³𝑀𝛽Λ𝑥²+𝑚𝑀6−2𝑚³𝛽𝑥³+−𝑚𝑀+𝑚24³𝛽2𝑥⁴. (55)

7.1.3. Second Component Problem

Consider 𝑢₂(𝑥)=𝑀𝐿⁻¹𝑢₁−6𝛽𝐿⁻¹𝐴₁,𝑢₂(0)=−Λ𝑑𝑢₂𝑑𝑥1+6𝛽𝑑𝑢₀𝑑𝑥²+6𝛽𝑑𝑢₁𝑑𝑥²𝑑𝑢₀,𝑑𝑥𝑑𝑢₂(1)𝑑𝑥=0. (56) The solution is 𝑢₂(𝑥)=𝜉₀−𝜉₁𝑥+𝜉₂𝑥²−𝜉₃𝑥³+𝜉₄𝑥⁴−𝜉₅𝑥⁵+𝜉₆𝑥⁶. (57) The series solution of the velocity field is 𝑢(𝑥)=𝑢₀(𝑥)+𝑢₁(𝑥)+𝑢₂(𝑥). (58) Substituting (53), (55), and (57) in (58), 𝑢(𝑥)=𝜔₀−𝜔₁𝑥+𝜔₂𝑥²−𝜔₃𝑥³+𝜔₄𝑥⁴−𝜔₅𝑥⁵+𝜔₆𝑥⁶. (59) The constants (𝜉₀--𝜉₆) and (𝜔₀--𝜔₆) are defined and listed in the appendix.

Substituting (59) the shear rate is obtained as 𝜏_{𝑥𝑦𝑥=0}=𝜇𝜈𝛿²12𝑚⁵𝛽²−12𝑚⁵𝛽²𝑀²Λ²+𝑚−𝑚𝑀3+𝑚Λ𝑀+2𝑀²𝑚15−𝑀²Λ²𝑚−2𝑚³𝛽+36𝑚15³𝑀𝛽−2𝑚³4𝑀𝛽Λ+3𝑚³𝑀²𝛽Λ−8𝑚³𝑀²𝛽Λ²×+2𝛽12𝑚⁵𝛽²−12𝑚⁵𝛽²𝑀²Λ²+𝑚−𝑚𝑀3+𝑚Λ𝑀+2𝑀²𝑚15−𝑀²Λ²𝑚−2𝑚³𝛽+36𝑚15³𝑀𝛽−2𝑚³+4𝑀𝛽Λ3𝑚³𝑀²𝛽Λ−8𝑚³𝑀²𝛽Λ²³. (60)

8. Volume Flow Rate and Average Velocity of Thin Film Flow

Using (59) in (47) we obtain the following.

The average velocity is defined as 𝑢/𝛿𝑚𝑢=𝑄,𝑄=3−2𝑚𝑀+1517𝑚𝑀²−3152𝑚³𝛽5+64𝑚105³𝑀𝛽+12𝑚⁵𝛽²7+1−𝑚Λ9𝑚𝑀²Λ−4𝑚³8𝛽Λ+𝑚15³+𝑀𝛽Λ33𝑚20³𝑀²𝛽Λ+1435𝑚⁵𝑀𝛽²Λ+144𝑚⁷𝛽³Λ+𝑚𝑀Λ²−13𝑚𝑀²Λ²+4𝑚³𝑀𝛽Λ²−296𝑚³𝑀²𝛽Λ²−54𝑚⁵𝑀𝛽²Λ²−3𝑚⁵𝑀²𝛽²Λ²−84𝑚⁷𝑀𝛽³Λ²−𝑚𝑀²Λ³−8𝑚³𝑀²𝛽Λ³−36𝑚⁵𝑀²𝛽²Λ³−48𝑚⁷𝑀²𝛽³Λ³. (61)

9. Heat Distribution for Drainage Problem

Solving (15) with boundary conditions given in (18), after making use of (59) for fixed values of 𝑚=0.5, 𝑀=1, Λ=0.4, 𝛽=1, and 𝜆=100, we obtain Θ(𝑥)=7.7356𝑥−14.3046𝑥²+11.1370𝑥³−4.2154𝑥⁴+0.57907𝑥⁵+0.241003𝑥⁶−0.2503𝑥⁷+0.0947𝑥⁸−0.01404𝑥⁹−0.0068𝑥¹⁰+0.0053𝑥¹¹−0.0016𝑥¹²+0.000103𝑥¹³+0.000127𝑥¹⁴−0.0000609𝑥¹⁵+0.0000126𝑥¹⁶−8.11926×10⁻⁸𝑥¹⁷−7.8766446×10⁻⁷𝑥¹⁸+2.651929×10⁻⁷𝑥¹⁹−4.828779×10⁻⁸𝑥²⁰+5.004021×10⁻⁹𝑥²¹−2.274555×10⁻¹⁰𝑥²². (62)

10. Results and Discussion

The effects of magnetic parameter 𝑀, non-Newtonian parameter 𝛽, gravitational parameter 𝑚, slip parameter Λ, and the dimensionless number 𝜆, for both lift and drainage velocity profiles, are discussed in Figures 5–14. Figures 1 and 2 show the geometry of lift and drainage velocity profiles. Figures 3 and 4 show the comparison of OHAM and ADM for lift and drainage velocity profiles. Numerical results and absolute error for both problems are shown in Tables 1 and 2, respectively. Figures 5 and 10 show that the rise in the non-Newtonian parameter 𝛽 increases the speed of flow. For small values of 𝛽, the velocity profile differs little from the Newtonian one; however when 𝛽 is increased, these profiles become more flattened showing the shear-thinning effect. Behavior of the velocity field 𝑢, for different values of 𝑀, by fixing other physical parameters, is shown in Figures 6 and 11 for lifting and drainage of fluid, respectively. Here, it can be seen that the boundary layer thickness is reciprocal to the transverse magnetic field and the velocity decreases as one progresses towards the surface of the fluid. We note that the velocity of fluid is maximum at the surface of the belt and minimum at the surface. Moreover, it is to be noted that for large values of 𝑀, the velocity increases rapidly as compared to small values. When the gravitational parameter 𝑚 increases, the velocity decreases in lifting flow and increases in case of drainage. This can be seen from Figures 7 and 12, respectively. Due to friction force the gravitational effect seems to be smaller near the belt. It can be seen that there is a point in the domain, where the velocity of the fluid becomes approximately the same for different values of gravitational parameter. The reason is that the friction of the belt becomes negligible at this point. On increasing 𝑚 after this point in lifting flow, the velocity decreases due to negligible friction but in drainage flow after this point, the velocity of the fluid increases. The effect of slip parameter can be observed from Figures 8 and 13. It is noticed that the speed of the fluid near the belt is greater than the speed at the surface. When we increase the slip parameter, the velocity of the fluid increases and comparatively this increase can be seen more clearly between Λ=1.0 and Λ=1.5 because the friction goes on decreasing. Figures 9 and 14 indicate the dimensionless temperature distribution for different values of 𝜆. It can be seen that temperature distribution increases as the dimensionless parameter 𝜆 increases and becomes more flattened for large values of 𝜆. Figure 15 shows the effect of local Reynolds number versus skin friction. This figure shows that the Reynolds number decreases the skin friction. For large values of Reynolds number, the skin friction vanishes. Figure 16 shows the effect of non-Newtonian parameter 𝛽 versus skin friction.

Table 1 

Comparison of (OHAM) and (ADM) for lift velocity profile.

Table 2 

Comparison of (OHAM) and (ADM) for drainage velocity profile.

Figure 3 

Comparison of ADM and OHAM methods for lift velocity profile is shown for the given parameters and auxiliary constants. 𝑚=0.5, 𝑀=0.1, 𝛽=0.3, Λ=0.01, 𝛼=0.1, 𝐶1=−0.6221311197, and 𝐶2=−0.0208205048.

Figure 4 

Comparison of ADM and OHAM methods for drainage velocity profile is shown for the given parameters and auxiliary constants. 𝑚=0.1, 𝑀=0.1, 𝛽=0.3, Λ=0.01, 𝛼=0.1, 𝐶1=−0.93798569, and 𝐶2=−0.000741729.

Figure 5 

The influence of non-Newtonian 𝛽 on velocity profile for lifting problem keeping 𝛼=0.1, 𝑚=0.5, 𝑀=1, and Λ=0.4.

Figure 6 

The effect of magnetic force “𝑀” on the lift velocity profile keeping other parameters fixed. 𝛼=0.1, 𝑚=0.2, 𝛽=1.2, and Λ=0.4.

Figure 7 

The figure shows this gravity effect of “𝑚” on velocity profile for lift problem, where 𝛼=0.1, 𝛽=1, 𝑀=1, and Λ=0.2.

Figure 8 

Variation of velocity for various values of slip parameter Λ, by fixing 𝛼=0.1; 𝑚=0.2; 𝛽=1.2; 𝑀=0.6.

Figure 9 

The influence of the dimensionless “𝜆” on the temperature distribution for lift problem keeping, 𝛼=0.5, 𝛽=0.2, 𝑀=2, and 𝑚=1.

Figure 10 

The influence of the non-Newtonian effect “𝛽” on the velocity profile 𝑢(𝑥) of drainage problem keeping 𝑚=0.5, 𝑀=1, and Λ=0.2.

Figure 11 

The influence of the magnetic force “𝑀” on velocity profile shown in this figure for the drainage problem, where 𝑚=0.5, 𝛽=1, and Λ=0.2.

Figure 12 

This figure shows the influence of the gravitational parameter 𝑚 on the velocity profile “𝑢(𝑥)” for drainage problem keeping 𝛽=1, 𝑀=1, and Λ=0.2.

Figure 13 

Variation of velocity for various values of slip parameter Λ, by fixing 𝑀=0.6, 𝑚=0.2, and 𝛽=1.2.

Figure 14 

The influence of dimensionless number “𝜆” on the temperature distribution 𝑢(𝑥), for drainage problem keeping 𝑚=0.5, 𝑀=1, Λ=0.4, and 𝛽=1.

Figure 15 

Variation of skin friction versus Reynolds number.

Figure 16 

Variation of skin friction versus non-Newtonian parameter 𝛽.

11. Conclusion

The constitutive equation governing the flow of a third grade fluid for lifting and drainage of fluid with slip boundary conditions is solved analytically by using Adomian decomposition method. Expression for velocity field, volume flow rate, skin friction, and temperature distribution is derived and sketched. It is concluded that velocity increases as the gravitational parameter 𝑚 decreases in lifting case while velocity increases as this parameter increases in drainage. For small values of 𝛽, the velocity profile differs little from the Newtonian one; however when 𝛽 is increased, these profiles become more flattened showing the shear-thinning effect. It can be seen that the boundary layer thickness is reciprocal to the transverse magnetic effect and the velocity decreases as it progresses towards the surface of the fluid. On increasing the slip parameter, the velocity of the fluid increases and comparatively this increase can be seen more clearly between Λ=1.0 and Λ=1.5 because the friction goes on decreasing. According to the best of our knowledge there is no previous literature about discussed problem; this is our first attempt to handle this problem with slip boundary condition. Also this problem is more general when compared to linear viscous model and second grade fluid model.

Appendix

Ψ₀2=−15Λ𝑚𝑀²+Λ³𝑚𝑀²+13Λ𝑀²𝛼+Λ²𝑀²𝛼−125Λ𝑚³𝑀𝛽+4Λ²𝑚³−5𝑀𝛽4Λ𝑚³𝑀²𝛽+136Λ²𝑚³𝑀²𝛽+8Λ³𝑚³𝑀²𝛽+6Λ𝑚²+𝑀𝛼𝛽112Λ𝑚²𝑀²𝛼𝛽−6Λ𝑚𝑀²𝛼²𝛽−12Λ𝑚⁵𝛽²−27Λ𝑚⁵𝑀𝛽²+66Λ²𝑚⁵𝑀𝛽²−Λ²𝑚⁵𝑀²𝛽²+36Λ³𝑚⁵𝑀²𝛽²+78Λ𝑚⁴𝑀𝛼𝛽²+12Λ²𝑚⁴𝑀²𝛼𝛽²−144Λ𝑚⁷𝛽³+42Λ²𝑚⁷𝑀𝛽³+48Λ³𝑚⁷𝑀²𝛽³,Ψ₁=2𝑚𝑀²15−Λ²𝑚𝑀²−𝑀²𝛼3−Λ𝑀²𝛼+125𝑚³𝑀𝛽−4Λ𝑚³+4𝑀𝛽3Λ𝑚³𝑀²𝛽−8Λ²𝑚³𝑀²𝛽−6𝑚²𝑀𝛼𝛽−6Λ𝑚²𝑀²𝛼𝛽+12𝑚⁵𝛽²−12Λ²𝑚⁵𝑀²𝛽²,Ψ₂=16Λ𝑚𝑀²−12Λ²𝑚𝑀²−12Λ𝑀²𝛼+2𝑚³𝑀𝛽−8Λ𝑚³𝑀𝛽+Λ𝑚³𝑀²𝛽−4Λ²𝑚³𝑀²𝛽−9𝑚²𝑀𝛼𝛽−3Λ𝑚²𝑀²𝛼𝛽+30𝑚⁵𝛽²−12Λ𝑚⁵𝑀𝛽²−6Λ²𝑚⁵𝑀²𝛽²,Ψ₃=−𝑚𝑀²+1186Λ𝑚𝑀²+𝑀²𝛼6−23𝑚³𝑀𝛽−6Λ𝑚³1𝑀𝛽+3Λ𝑚³𝑀²𝛽−6𝑚²𝑀𝛼𝛽+40𝑚⁵𝛽²−12Λ𝑚⁵𝑀𝛽²,Ψ₄=124Λ𝑚𝑀²+𝑀²𝛼24−2𝑚³−3𝑀𝛽2Λ𝑚³1𝑀𝛽+12Λ𝑚³𝑀²𝛽−32𝑚²𝑀𝛼𝛽+30𝑚⁵𝛽²−3Λ𝑚⁵𝑀𝛽²,Ψ₅=𝑚𝑀²−12011𝑚10³𝑀𝛽+12𝑚⁵𝛽²,Ψ₆=𝑚𝑀²−72011𝑚60³𝑀𝛽+2𝑚⁵𝛽²,Φ₀=Λ𝑚+Λ𝑚𝑀3−Λ²2𝑚𝑀−15Λ𝑚𝑀²+Λ³𝑚𝑀²1+𝛼−Λ𝑀𝛼+3Λ𝑀²𝛼+Λ²𝑀²𝛼+4Λ𝑚³2𝛽−5Λ𝑚³𝑀𝛽−4Λ²𝑚³5𝑀𝛽−4Λ𝑚³𝑀²𝛽+136Λ²𝑚³𝑀²𝛽+8Λ³𝑚³𝑀²𝛽+32Λ𝑚²𝑀²𝛼𝛽+12Λ²𝑚²𝑀²𝛼𝛽−27Λ𝑚⁵𝑀𝛽²+54Λ²𝑚⁵𝑀𝛽²−Λ²𝑚⁵𝑀²𝛽²+36Λ³𝑚⁵𝑀²𝛽²+8Λ𝑚²𝑀𝛼𝛽²−24Λ²𝑚²𝑀𝛼𝛽²+54Λ𝑚⁴𝑀𝛼𝛽²+36Λ²𝑚⁴𝑀²𝛼𝛽²−144Λ𝑚⁷𝛽³+84Λ²𝑚⁷𝑀𝛽³+48Λ³𝑚⁷𝑀²𝛽³+48Λ𝑚⁴𝛼𝛽³−48Λ²𝑚⁴𝑀𝛼𝛽³−24Λ𝑚𝛼²𝛽³,Φ₁=𝑚−𝑚𝑀3+Λ𝑚𝑀+2𝑚𝑀²15−Λ²𝑚𝑀²𝑀+𝑀𝛼−²𝛼3−Λ𝑀²𝛼−2𝑚³𝛽+125𝑚³𝑀𝛽−2Λ𝑚³4𝑀𝛽+3Λ𝑚³𝑀²𝛽−8Λ²𝑚³𝑀²𝛽−6𝑚²𝑀𝛼𝛽−6Λ𝑚²𝑀²𝛼𝛽+12𝑚⁵𝛽²−12Λ²𝑚⁵𝑀²𝛽²,Φ₂=𝑚2+Λ𝑚𝑀2+16Λ𝑚𝑀²−12Λ²𝑚𝑀²+𝑀𝛼2−12Λ𝑀²𝛼−3𝑚³𝛽+2𝑚³𝑀𝛽−7Λ𝑚³𝑀𝛽+Λ𝑚³𝑀²𝛽−4Λ²𝑚³𝑀²𝛽−9𝑚²𝑀𝛼𝛽−3Λ𝑚²𝑀²𝛼𝛽+30𝑚⁵𝛽²−12Λ𝑚⁵𝑀𝛽²−3Λ²𝑚⁵𝑀²𝛽²,Φ₃=𝑚𝑀6−𝑚𝑀²+1186Λ𝑚𝑀²+𝑀²𝛼6−2𝑚³2𝛽−3𝑚³𝑀𝛽−6Λ𝑚³1𝑀𝛽+3Λ𝑚³𝑀²𝛽−6𝑚²𝑀𝛼𝛽+40𝑚⁵𝛽²−12Λ𝑚⁵𝑀𝛽²,Φ₄=𝑚𝑀+12424Λ𝑚𝑀²+𝑀²𝛼−𝑚24₃𝛽2−2𝑚³3𝑀𝛽−2Λ𝑚³+1𝑀𝛽12Λ𝑚³𝑀²3𝛽−2𝑚²𝑀𝛼𝛽+30𝑚⁵𝛽²−3Λ𝑚⁵𝑀𝛽²,Φ₅=𝑚𝑀²−12011𝑚10³𝑀𝛽+12𝑚⁵𝛽²,Φ₆=𝑚𝑀²−72011𝑚60³𝑀𝛽+2𝑚⁵𝛽²,𝜉₀=215𝑚𝑀²Λ+125𝑚³+5𝑀𝛽Λ4𝑚³𝑀²𝛽Λ+12𝑚⁵𝛽²Λ+27𝑚⁵𝑀𝛽²Λ+144𝑚⁷𝛽³Λ−4𝑚³𝑀𝛽Λ²−136𝑚³𝑀²𝛽Λ²−66𝑚⁵𝑀𝛽²Λ²+𝑚⁵𝑀²𝛽²Λ²−84𝑚⁷𝑀𝛽³Λ²−𝑚𝑀²Λ³−8𝑚³𝑀²𝛽Λ³−36𝑚⁵𝑀²𝛽²Λ³−48𝑚⁷𝑀²𝛽³Λ³,𝜉₁=𝑚𝑀²Λ²−2𝑚𝑀²−15125𝑚³𝑀𝛽−12𝑚⁵𝛽²+4𝑚³4𝑀𝛽Λ−3𝑚³𝑀²𝛽Λ+8𝑚³𝑀²𝛽Λ²+12𝑚⁵𝑀²𝛽²Λ²,𝜉₂=6𝑚⁵𝑀²𝛽²Λ²−2𝑚³𝑀𝛽−30𝑚⁵𝛽²−16𝑚𝑀²Λ+8𝑚³𝑀𝛽Λ−𝑚³𝑀²𝛽Λ+12𝑚⁵𝑀𝛽²1Λ+2𝑚𝑀²Λ²+4𝑚³𝑀²𝛽Λ²,𝜉₃=𝑚𝑀²+2183𝑚³𝑀𝛽−40𝑚⁵𝛽²−16𝑚𝑀²Λ+6𝑚³−1𝑀𝛽Λ3𝑚³𝑀²𝛽Λ+12𝑚⁵𝑀𝛽²𝜉Λ,₄=𝑚³𝑀𝛽−30𝑚⁵𝛽²−124𝑚𝑀²3Λ+2𝑚³−1𝑀𝛽Λ𝑚12³𝑀²𝛽Λ+3𝑚⁵𝑀𝛽²𝜉Λ,₅=11𝑚10³𝑀𝛽−12𝑚⁵𝛽²−𝑚𝑀²,𝜉120₆=−𝑚𝑀²+72011𝑚120³𝑀𝛽−2𝑚⁵𝛽²,𝜔₀=−𝑚Λ−𝑚𝑀Λ3+215𝑚𝑀²Λ−4𝑚³2𝛽Λ+5𝑚³5𝑀𝛽Λ+4𝑚³𝑀²𝛽Λ+27𝑚⁵𝑀𝛽²Λ+144𝑚⁷𝛽³Λ+𝑚𝑀Λ²+4𝑚³𝑀𝛽Λ²−136𝑚³𝑀²𝛽Λ²−54𝑚⁵𝑀𝛽²Λ²+𝑚⁵𝑀²𝛽²Λ²−84𝑚⁷𝑀𝛽³Λ²−𝑚𝑀²Λ³−8𝑚³𝑀²𝛽Λ³−36𝑚⁵𝑀²𝛽²Λ³−48𝑚⁷𝑀²𝛽³Λ³,𝜔₁=𝑚𝑀3−𝑚−2𝑚𝑀²15+2𝑚³𝛽−125𝑚³𝑀𝛽−12𝑚⁵𝛽²−𝑚𝑀Λ+2𝑚³4𝑀𝛽Λ−3𝑚³𝑀²𝛽Λ+𝑚𝑀²Λ²+8𝑚³𝑀²𝛽Λ²+12𝑚⁵𝑀²𝛽²Λ²,𝜔₂𝑚=−2+3𝑚³𝛽−2𝑚³𝑀𝛽−30𝑚⁵𝛽²−𝑚𝑀Λ2−16𝑚𝑀²Λ+7𝑚³𝑀𝛽Λ−𝑚³𝑀²𝛽Λ+12𝑚⁵𝑀𝛽²1Λ+2𝑚𝑀²Λ²+4𝑚³𝑀²𝛽Λ²+6𝑚⁵𝑀²𝛽²Λ²,𝜔₃=𝑚𝑀²−18𝑚𝑀6+2𝑚³𝛽+23𝑚³𝑀𝛽−40𝑚⁵𝛽²−16𝑚𝑀²Λ+6𝑚³−1𝑀𝛽Λ3𝑚³𝑀²𝛽Λ+12𝑚⁵𝑀𝛽²𝜔Λ,₄=−𝑚𝑀+𝑚24³𝛽2+2𝑚³𝑀𝛽−30𝑚⁵𝛽²−124𝑚𝑀²Λ+32𝑚³1𝑀𝛽Λ−𝑚12³𝑀²𝛽Λ+3𝑚⁵𝑀𝛽²𝜔Λ,₅=11𝑚10³𝑀𝛽−𝑚𝑀²120−12𝑚⁵𝛽²,𝜔₆=11𝑚60³𝑀𝛽−𝑚𝑀²720−2𝑚⁵𝛽². (A.1)