4.1 Introduction to recent trends and Modification of OHAM, HAM and ADM

There have been many technological advancements, innovations, and inventions in the field of fluid economics resulting in notable progress in the field of thin film lows. Several scientific soft-wares such as Aquila Digital Signal and Mat-lab have made it easy to evaluate and gauge results of thin film flows. Most of these soft-wares keep being updated to enhance their performance and produce better results. These has ensured that the solutions obtained by using the aforementioned methods have a high and valid degree of convergence. Moreover, to enhance and produce better accurate authentic results regarding the analysis of thin film fluid flows, researchers now modify existing formulas.

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4.2 Introduction to Research Contributions

The current chapter analyses the contributions made by this research. Throughout the project, essential and vital decision making tools have been developed by applying the shooting method, HAM, ADM, and OHAM methods. These methods are used in solving problems on non-linear ODEs. Some of the advantages of these tools is that they deal with inherent uncertainties caused by high oscillation, problems of heat transfer, and porosity issues arising from thin-film fluid flows.

4.3 Summary of the Research Contribution

In the current research, there have been modifications of OHAM, ADM, and HAM methods. The results indicate that by using these techniques, it becomes possible to get durable results. However, among the three methods, the modified HAM method has more accurate results than the rest.

4.4 Engineering Applications (Formulation of lift problem)

There is a detailed analysis of the vibratory motion of fourth-order thin film. The flow problems of non-Newtonian fluids have been the subject of many investigations in the recent times. This is because of the occurrence of non-Newtonian fluids in nature and their industrial value. The significant number of investigations have been carried out on such flows (for example, see recent studies [13-15] and many references therein)

The distinction between Newtonian fluids and non-Newtonian fluids is based on the different relationship that exists in each other between the application of a tangential effort and the speed with which they deform. In other words: if a tangential force is applied to a Newtonian fluid, it will start moving no matter how small the tangential force and a certain velocity distribution will be generated in the fluid. In formulating the lift problem, an oscillating vertical plane is considered. The plate moves in an upward with a velocity. When the plate is moving upward, the belt comes along with a thin film of fourth-grade liquid. The liquid has a uniform thickness. The assumption made is that atmospheric pressure is not steady and is laminar. On the belt is applied a transverse magnetic field applied to the belt. Pressure considered atmospheric here as well as flow is supposed to be unsteady and laminar and the transverse magnetic-field applied to the belt. The relationship between the velocity and temperature fields is as presented below

(1)

(2)

Boundary conditions as mentioned in [8]

(3)

(4)

Here are the velocity and temperature components in the form of x-direction where is the frequency of oscillating belt.

4.4.1 ADM Solution of Lift Velocity and Heat Problem

The fluids in which the shear stress is not directly proportional to the deformation ratio are non-Newtonian. Strictly speaking, the definition of fluid is valid only for materials that have a strain of zero deformation. In general, non-Newtonian fluids are classified concerning their behavior over time, that is, they can be time-dependent or independent of it. To ensure that there is no repetition, the information on ADM method is used. The ADM technique is used in solving these governing problems. As such, we consider a nonlinear partial differential equation (PDE) in an operator form then the outcomes of equations (3) and (4) as follows:

(5)

The enormous calculations the analytical results of velocity and heat shape have been stated up to the first order although the graphical solutions are given up to the second order.

4.4.2 OHAM Solution for Lift Velocity Problem

In the study of hydrodynamics, Bernoulli's theorem, which deals with the law of conservation of energy, is of paramount importance, as it points out that the sum of the kinetic, potential and pressure energies of liquid moving in a determined point is equal to that of any other point. Hydrodynamics fundamentally investigates incompressible fluids, that is, liquids, because their density practically does not change when the pressure exerted on them changes. When a fluid is in motion, one layer resists the movement of another layer that is parallel and adjacent to it; this resistance is called viscosity. Just as is the case with the previous case, the details of the OHAM procedures are not provided in detail. Similar to the previous case we directly apply OHAM to equations. (4, 5) the zero and first order solutions of the velocity are given as,

(6)

The massive calculations the analytical results of velocity shape has been set up to first order although the graphical solutions are given up to the second order.

Therefore, the values of for lift velocity components are given below

C1 = -0.12299772501253805, C2 = 0.16377089853857172

4.4.3 ADM Solution of Drainage Problem

As a first postulate to obtain the equation we are looking for, we demand that said equation be consistent with the De Broglie formula. As a second postulate that is somewhat related to the first one, we will ask that the function that describes the mathematical form of the wave associated with a particle whose de Broglie wavelength is l is a sinusoidal trigonometric function, for example, a sine function, the justification of this requirement being the fact that a sinusoidal function is the simplest oscillatory function for which it is possible to define a constant wavelength. In analogy to the wave functions that we find in classical physics (acoustic, electromagnetic, etc.), for the case of matter waves we postulate a wave function, usually symbolized nowadays as either ps or as Ps. In this section, we applying the ADM method on the equation (6), the adomian polynomials are same in previous problem and after the simplification of zeroth and first components of the velocity and heat problems are,

(7)

(8)

The massive calculations the analytical results of velocity as well as heat shape have been stated up to the first order while the graphical solutions are given up to the second order.

4.4.4 OHAM Solution of Drainage Problem

Note that l is the wavelength of this sine wave, because if the value of x is increased by an amount l (a x + l), then the argument of the sine function is increased by 2p so that the sine will go through a complete cycle of the wave. Note also that this function is valid only for a particle moving at a constant velocity for all positions along the x coordinate where it is defined, that is, for a particle with a constant wavelength l, since it is not consistent to speak of a wavelength that varies significantly with the position when this concept is not defined (if there is any doubt in this, make a sketch in a role of an oscillatory function in which the oscillations are getting closer and closer in some positive or negative direction along the horizontal axis, and try to define a value for l on this oscillatory function).

The standard form of OHAM is expressed in equation (19) by comparing the powers of then we set up zero and first components of the velocity problems as given below,

(9)

(10)

Solutions of zerorth and first order component problem by using the boundary conditions given in the equations (20, 21) are,

(11)

It is noted that due to the massive calculations and analytical result of velocity as well as heat shape have been set up to the first order while the graphical solutions are given up to the second order.

Therefore the value of for the drainage velocity is given below

C1 = -0.8871443225163201, C2 = 1.3110786901741657

4.5 Engineering Investigations of Dufour and Soret effect on MHD (Definition of the Problem.

Affected by the examinations and applications indicated over, the central subject of the present work is to research the direct of an erratic thick liquid impacted by Dufour and Soret impacts with warmth and mass exchange. Execution of HAM prompts focused arrangement (Hayat et al., 2016). The making fluctuating parameters are plotted for particular respects to inquire about their effect on the speed and warm fields. The estimations of surface drag power and warmth transformation scale are numerically cleared up. The directing conditions are poor around HAM, and for realness, the outcomes are separated and numerical BVP4C bundle. In the running with go, the issue is portrayed, bankrupt down and reviewed.

(4.1)

, (4.2)

The issue is displayed by the properties of fluid portrayed by Dufour and Soret impacts; the crucial issue is that of the nonstraight or volatile stream of liquids. Dufour impacts show how particle improvement impacts temperature changes, while on the other hand Soret affect reacts these. Soret affect outlines how temperature changes influence particle improvement. These effects influence fluids directed in rotational development. To fathom this test, MHD fluids are used. These are electrically driving liquids, whose convention and rate of trade of mass is gooey and incompressible inside sight of warm radiation and warm dispersion.

4.5.1 Control

A terminal plate is used, to modify the customs of hydrodynamics. This is the social parts of fluid, on a very basic level laminar and turbulent stream displayed and managed by speed and course of the stream. To control the structure of the stream in thermodynamics, an alluring movement is used to affect stream, speed, and heading of the surge of MHD fluids. These fluids twist keep up their properties of the room, under the appealing movement and in this manner kept up in their method for the stream. How much these is possible is managed by numerical figuring's, which have been thought up since the issue was first perceived.

4.5.2 Homotopy Analysis Method (HAM) solution

The Homotopy Analysis Method (HAM) provides a perfect arrangement of obscure functions. However, the efficiency of the HAM method depends on the assistant capacities and the starting theories.

(4.21)

, (4.22)

4.5.3 Determination

One of the properties of MHD liquids is that they maintain the properties and characteristics of the room. As such, it becomes imperative to connect and use differential equations in discerning the different points of confinement conditions for the effects of Dufour and Soret passable for a fluid to continuing...