Analysis of Unsteady MHD Thin Layer Flow of Fourth-Order Fluid Through a Vertical Belt

Objectives: To study the unsteady magneto-hydro-dynamic (MHD) thin layer flow of a fourth-order fluid through a moving and oscillating belt. Methods: The well-known analytical technique, namely Adomian decomposition method (ADM) is used to solve the non-linear partial differential equation for governing equations of velocity profile with subjects to initial and boundary conditions for both lift and drainage problems. Findings and novelty/ improvements: The solution is found in excellent agreement. The basic purpose to study the effect of MHD on velocity field and understand the behavior of this physical problem and the effects of different non-dimensional parameters the graphical results are provided.


Introduction
In researched articles, the various structures of non-Newtonian fluids have been advised to modify the research because of the heterogeneous physical nature of these fluids. These fluids are divided into three different representative types of non-Newtonian fluids (a) differential (b) the rate and (c) the integral types fluid models. But, the (a) and (b) types are the most generous and applicable. Therefore, we will study type (a) model in my research i.e. the differential type fluids model. Furthermore, these types are categorized into different sub-classes, it follows (a) the second-order fluid (b) the third-order fluid (c) the fourth-order fluid, etc. The easiest subtype is the second-order fluids also known as grade 2 fluid by which one can sensibly has an expectation to investigate the solution by analytical technique for the predictions of normal stress differences. But it drew back to consider the shearing thickness and shearing thinness cases that exclude the many fluids. On the contrary, the (b) subtype is also known as grade 3 fluid shows an un-idealistic descriptive behavior of these types of fluids. This fluid model can abduct the non-Newtonian behavior

Basic Equations
The basic equations which govern the problems are continuity and momentum equations with the interference of transversely magnetic field which is applied externally are as follows, .

DV T g J B Dt
Indian Journal of Science and Technology Vol 13(03), DOI: 10.17485/ijst/2020/v13i03/148554, January 2020 Here ρ is the density constant, g represents the gravity, V denotes the velocity vector of fluid, D Dt is material derivative, and T is a cauchy stress tensor, and the term of body force corresponds to MHD flow is the Lorentz force J × B, where B is applied magnetic field and J shows the density of current, according to ohm's law, density of current is given as, Here σ and E are electrical conductivity and electric field of the fluid, the imposed magnetic field is denoted by B = B 0 + b 1 and b 1 denotes the magnetic field which is induced. Here we also assume E = 0, b 1 = 0 and B = B 0 (0, B 0 , 0), where B 0 is the magnitude of total magnetic field, then (3) reduces to; The Cauchy stress tensor T which is define for fourth-order fluid as, Here p is pressure, I is an identity tensor while S r denotes the extra stress tensor define as, Where µ shows the viscosity's coefficient, (α 1 , α 2 ) (β 1 , β 2 , β 3 ) and (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 , γ 6 , γ 7 , γ 8 ) are the material constants of the second, third, and fourth-order of fluids. Furthermore, the Rivlin-Ericksen tensor A 1 , A 2 , A 3 , and A 4 are defined as;

Formulation of Lift Problem
We consider two different problems of a wide vertical belt with uniform thickness δ of a thin film layer of fourth-order fluid placed in a large bath. On the belt the transversely uniform magnetic field is applied. For analysis, we have chosen the Cartesian coordinate system and flow in assumed to be one dimensional in the direction of y where as x is normal to it, we suppose that the flow is unsteady and laminar with no pressure gradient the gravity is kept balanced by shearing forces of fluid and the thickness of the films does not change and by the above conditions the velocity is in y-direction [1]: Here we consider the flow at t = 0 + on a wide belt which is oscillating and moving upward with the velocity U. The belt is dipped into a bath as shown in Magnetic Field Where the belt which is oscillating whose frequency is ω.
Using the velocity profile given in equation (10), the continuity equation (1) is identically satisfied and the equation of momentum (2) becomes to the form Introducing the following non-dimensional variables is a stock number, and M is a non-dimensional magnetic variable.

Formulation of Drainage Problem
Here we assume the belt is oscillating but not moving and the fluid falls downward direction in the presence of gravity, as shown in the geometry of drainage problem See Figure  Along with the boundary conditions ( ) All the remaining assumptions are similar to the lift problem but stock number is taken positive. Using xy T the governing equation of drainage problem.
The non-dimensional form of equation (22) is given by,

Solution Technique
Fundamental concept of ADM ADM is used to decompose the unknown function ( ) , u x t into a sum of an infinite number of components defined the decomposition series.
The decomposition method is used to find the components separately. The Determination of these components can be obtained through simple integrals. To give a clear overview of ADM, we consider the partial differential equation in an operator form as , Nu x t is analytical term which is non-linear and can be expanded in the n A that is A domian polynomials.
After applying the inverse operator 1 x L − to both sides of Equation (28), we write Here the function ( ) And so on.

The ADM Solution of Lift Problem
Rewrite Equation (17) Using the inverse operator 1 x L − , we get    x The components of velocity profile are obtained by comparing both side of equation (40) Zeroth component problem: Solution of zeroth component problem using boundary conditions given in equation (18) and (19) is: First component problem: The first-order analytical result has been computed while graphical solutions are provided up to second order due to massive calculation (Error! Reference source not found.).

The ADM Solution of Drainage Problem
The model for drainage problem is the same as for the lift problem. The only difference in this problem is that the belt is only oscillating and due to the draining of thin film, stock number is positively mentioned in Equation (23). After rewriting equation (23) in L operator form of ADM, then using boundary conditions Equations (24) (25) we get the components of the problem are Zeroth component problem: The zeroth component problem solution using boundary conditions which is given in Equations (24) and (25) is

Indian Journal of Science and Technology
Vol 13(03), DOI: 10.17485/ijst/2020/v13i03/148554, January 2020 First component problem: The first component problem solution by using boundary conditions which is given in Equations (24) and (25) Second component problem: The first-order analytical result has been computed while graphical solutions are provided up to second order due to massive calculation (Error! Reference source not found.).

Results and Discussion
The analytical solutions are examined for unsteady MHD thin layer flow of fourth-order fluid through a vertical belt which is oscillating and also moving. The arrived non-linear partial differential equations of both drainage and lift problems are solved by using the ADM and results are also shown by graphically.
Figure show the geometry of lift and drainage problems, respectively. Figure 1 and Figure 2 show the graphical results for the velocity profile of both lift and drainage problems respectively at altered values of the inserted parameters. Whereas the effects of other physical parameters t, s t , α, β, γ and M on velocity field for both lift and drainage problems are examined in Figures 3-10. All the results are computed in the x − coordinate only for a certain domain 0,1 x ∈     . In Figure 3 and Figure 4 showing the effect of β on lift and drainage velocities respectively in both cases we noticed that the velocity profile increases as the value of β increases. In general, the Newtonian fluid shows much thinner boundary film than non-Newtonian fluid due to the reduced apparent viscosity. The Indian Journal of Science and Technology Vol 13(03), DOI: 10.17485/ijst/2020/v13i03/148554, January 2020 outward viscosity of the fluid become more bulky (domination of viscous forces) due to the increment in fourth-order parameters then the flow will adjust simultaneously to the driving force which is present and closely oscillate with the congruent phase in the whole domain of flow. Therefore, increment in non-Newtonian parameters α, β, γ, β 1 , and γ 1 , of second, third, and fourth-order fluids causes more thickening of the boundary layer. So increment in these parameters increases the field of velocity for both lift and drainage problems shown in Figure 3, Figure 4, Figure 5, Figure 6. The effects of t S are shown in Figure 7 and   increasing in the value of t S drainage velocity increases. This relates to the frictional force that causing the effect of gravity and it appears to be smaller near to the belt. While towards the free surface it gradually increases. In Figure 9 and Figure 10 showing the variation of the magnetic parameter M on the lift and drainage velocity profiles Increase in magnetic

Conclusion
The analytical solutions for unsteady MHD thin layer flow of fourth-order fluid through an oscillating and moving vertical belt is obtained. The belt is translating and oscillating for lift velocity distribution while the belt is only oscillating for drainage velocity profile. The