Unsteady MHD Thin Film Flow of an Oldroyd-B Fluid over an Oscillating Inclined Belt

This paper studies the unsteady magnetohydrodynamics (MHD) thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt making a certain angle with the horizontal. The problem is modeled in terms of non-linear partial differential equations with some physical initial and boundary conditions. This problem is solved for the exact analytic solutions using two efficient techniques namely the Optimal Homotopy Asymptotic Method (OHAM) and Homotopy Perturbation Method (HPM). Both of these solutions are presented graphically and compared. This comparison is also shown in tabular form. An excellent agreement is observed. The effects of various physical parameters on velocity have also been studied graphically.


Introduction
In recent time, non-Newtonian fluids have become quite prevalent in industry and engineering. Some of their common examples are polymer solutions, paints, certain oils, exocitic lubricants, colloidal and suspension solutions, clay coatings and cosmetic products. As a consequence of diverse physical structures of these fluids, there is not even a single constitutive model which can predict all the salient features of non-Newtonian fluids. Generally there are three non-Newtonian fluids models. They are known as (i) the differential type, (ii) the rate type, and (iii) the integral type. But the most famous amongst them are the first two models. In this work, we will study the second model, the rate type fluid and consider its subclass known as Oldroyd-B fluid. The simplest subclass of rate type fluid is Maxwell fluid, however, this fluid model can only be described in terms of its relaxation time, while there are no information on its retardation time. The Oldroyd-B fluid model, on the other hand, has a measurable retardation time and can relate the viscoelastic manners of dilute polymeric solutions under general flow conditions. Fetecau et al. [1] obtained exact solutions in their study on constantly accelerating flow over a flat plate for Oldroyd-B fluid. In the following year, Fetecau et al. [2] studied the transient oscillating motion of an Oldroyd-B fluids in cylindrical domains and obtained the exact solutions. Haitao and Mingyu [3] studied the series solution for the plane Poiseuille flow and plane couette flow of an Oldroyd-B fluid using the sine and Laplace transformations. Hayat et al. [4] investigated the exact solution of Oldroyd-B fluid for five different problems. Liu et al. [5] discussed the MHD flow of an Oldroyd-B fluid between two oscillating cylinders. Khan et al. [6,7] investigated the solution for unsteady MHD flow of an Oldroyd-B fluid passing through a porous medium. They obtained the exact solutions for both of their problems by using the Laplace transform technique and discussed the physical behavior of relaxation and retardation times of fluid motion.
Burdujan [8] studied the unsteady flow of incompressible Oldroyd-B fluid between two cylinders. He obtained the exact solution by using Hankal and Laplace transformations. Asia et al. [9] investigated the oscillating motion of Oldroyd-B fluid between two sides wall. They obtained the starting solution of velocity field. Shahid et al. [10] examined the steady and unsteady flow of Oldroyd-B fluid. Steady state and transient solution have been obtained by using Laplace and Fourier series. Aksel et al. [11] discussed the flow of an Oldroyd-B fluid due to the oscillation of a plate. As a special case, they reduced their solutions to those of Maxwell and Newtonian fluids. Ghosh and Sana [12] analyzed hydromagnetic flow of an Oldroyd-B fluid near a pulsating plate. In subsequent papers, Ghosh and Sana [13] and Gosh et al. [14] discussed the unsteady flow of electrically conducting Oldroyd-B fluid induced by rectified sine pulses and half rectified sine pulses. Khan and Zeeshan [15] extended the work of Gosh and Sana [12] by taking the Oldroyd-B fluid into a porous medium. As discussed, most of Oldroyd-B fluid studies are confined to some specific geometries. Studies on Oldroyd-B fluids over an oscillating belt are scarce, especially when considering the thin film flow of an Oldroyd-B fluid over an inclined oscillating belt.
Having such motivation in mind, Gul et al. [16][17][18] studied the analytical solution of MHD thin film flow of non-Newtonian fluid on a vertical oscillating belt by using the ADM and OHAM methods. The result of lift and drainage velocity and temperature distributions are compared and presented graphically. The effects of various physical parameters are also discussed. Shah et al. [19] studied the solution of thin film flow of third grade fluid on moving inclined plane by using OHAM. Siddiqui et al. [20] investigated the thin film flow of a third grade fluid over an inclined plane. The non-linear equation of velocity field is solved by using OHAM and traditional perturbation method.
Based on the above motivation, the main goal of the present work is to venture further in the regime of Oldroyd-B fluid. More exactly, this article aims to study the unsteady MHD thin film flow of an Oldroyd-B fluid past an oscillating inclined belt using Optimal Homotopy Asymptotic Method (OHAM) and Homotopy Perturbation Method (HPM). These methods have been used successfully in the literature for the solutions of non-linear fluid problems. Marinca et al. [21][22][23][24] discussed the approximate solution of non-linear steady flow of fourth grade fluid by using OHAM. They noticed from the results that OHAM method is more effective and easy to use then other methods. Kashkari [25] studied the OHAM solution of nonlinear Kawahara equation. For comparison HPM, VHPM and VIM method is used but OHAM is more successful method. He [26][27][28][29][30] provided the fundamental introduction of HPM and solved the wave equation. Sanela et al. [31] solved nonlinear partial differential equations using HPM. Nofel [32] studied application of homotopy perturbation method for nonlinear differential equations. Ganji et al. [33] studied the solution of Blasius non differential equation using HPM. Anakira et al. [34] discussed the analytical solution of delay differential equation using OHAM. Mabood et al. [35,36] investigated the approximate solution of non-linear Riccati differential equation by using OHAM.

Basic Equation
Let us consider the unsteady MHD incompressible flow over an inclined belt defined by the following equations Where V is the velocity vector of the fluid, ρ is the fluid density, D Dt is the material time derivative, and g is the external body force. Thus,the Lorentz force perunit volume is where B = (0,B 0 ,0) is the uniform magnatic filed, B 0 is theapplied magnetic field and σ is the electrical conductivity. The current density J is Here, μ 0 is the magnetic permeability, E is an electric field which is not considered in this study, and The above model can be reduced to different types of fluid depend on λ 1 (relaxation time) and λ 2 (retardation time). In Eq (5), if λ 1 = λ 2 the fluid becomes viscous. When λ 2 = 0, it becomes a Maxwell fluid and reduced to Oldroyd-B fluid when 0<λ 2 <λ 1 <1.
The cauchy stress tensor, T is Where S is the extra tress tensor, pI is the isotropic stress, A 1 is the Rivlin Ericksen stress tensor and μ is the viscosity cofficient.

Formulation of the Problem
Let us consider a thin film flow of a non-Newtonian Oldroyd-B fluid on an oscillating inclined belt. The force of gravity will initiate the motion of a layer of liquid in the downward direction. The thickness, δ, of the of liquid layer is considered to be uniform. A uniform magnetic field is applied to the belt in the direction perpendicular to the fluid motion. The external electric field is not considered and the magnetic Reynolds number is negligible, which implies that the current is totally dependent on the induced electric field and the electric current flowing in the fluid does not affect the magnetic field. The induced magnetic field created by the fluid motion is very small compared to the applied magnetic field. Therefore, the Lorentz force term in Eq (2) is reduced to ÀsB 0 2 v; assuming that the flow is unsteady, laminar, incompressible, and pressure gradient is zero.
The velocity field is of the form v ¼ ðvðy; tÞ; 0; 0Þ and S ¼ ðy; tÞ; ð9Þ subject to the boundary conditions vð0; tÞ ¼ Vcos ot ; @vðd; tÞ where ω is the frequency of the oscillating belt. The momentum Eq (2) is reduced to @p @y ¼ @S yy @y ; ð12Þ It follows from (7) and (9) that S xy þ l 1 @S xy @t À S yy @v @y ¼ m @v @y þ l 2 m @ 2 v @t@y ; ð15Þ then Eq (16) reduces to Here B(y) is used as an arbitrary function. When t<0, then S yy is reduced to zero, which demonstrates that B(y) must also be zero. Therefore, from the Eqs (11) and (15) and in the presence of zero pressure gradient, we obtain where, ω is the oscillating parameter, k 1 is the relaxation paramter, k 2 is the retardation parameter, m is the gravitational parameter and M is the magnetic parameter.
To find the approximate solution, we expand the unknown function ψ(y,t,p)as By inserting Eq (37) into Eq (31) and equating the identical power of p, we obtain the zero, first and second order problem, v 0 (y,t), v 1 (y,t) and v 2 (y,t), so the governing equation is: The general governing equations for u k (y,t) are given by Here N m (v 0 (y,t),v 1 (y,t). . ..v m-1 (y,t)) is the coefficient of p m , in the expansion of Nψ(y,t,p).

HPM Solution
By applying HPM method to Eq (20) with boundary condition (21), we obtain zero, first and second component problems of the velocity profile.

Results and Discussion
Unsteady MHD thin film flow of an Oldroyd-B fluid over an oscillating inclined belt has been examined. The governing partial differential equations for velocity are analytically solved by using OHAM and HPM methods. Both of these results are compared. It is found that these results are in excellent agreement. In tables 1 and 2, we calculated the numerical comparisons of OHAM and HPM. Absolute errors of both methods are also calculated.   In case of oscillation, it can be observed in Fig (9) that the boundary layer thickness is reciprocal to the perpendicular magnetic field and the fluid motion decreases one step    Thin Film Flow of an Oldroyd-B Fluid oscillation case. In the presence of friction force, the gravitational effect seems to be smaller near the belt and greater at the fluid surface. By increasing m, the speed of fluid layer increases. The effects of non-Newtonian parameters k 1 and k 2 on velocity profiles are shown in Figs (11 and 12). It is observed that in the presence of magnetic field, the structure of the  thin layer becomes similar with those of Ekman and classical Stokes layers. It is also observed that for all frequencies, the thickness of the hydromagnetic thin layers remain bounded. The reason is that the magnetic field controls the growth of the thin layer thickness at the resonant frequency.

Conclusion
In this paper, the approximate solutions of unsteady MHD thin film flow of an Oldroyd-B fluid through oscillating inclined belt has been obtained using OHAM and HPM methods for  velocity field. Both of these solutions are compared numerically and graphically. It is found that the solution obtained by OHAM and HPM are in excellent agreement. The concluded remarks have been precised as follows: • It is found that for a specific region y[0,1], thin film near the belt oscillates together with the belt in the same period and the velocity amplitude of the fluid layer increases gradually towards the free surface of the belt.
• Due to no-slip condition, the force of friction reduces the gravitational effect near the belt and this effect seems to be greater at the fluid surface.
• Since magnetic field controls the growth of thin film thickness, therefore, the thickness of thin film remains the same for different frequencies.