ABSTRACT
The problem of Magnetohydrodynamics rotational stagnation point flow over a shrinking and stretching sheet with heat generation and absorption was considered . The Partial differential equations were transformed using similarity variables to ordinary nonlinear coupled differential equations. The analytical solutions were presented using the Adomian decomposition method. The results presented were validated with the literature and a good agreement was observed. The effects of various dimensionless parameters like Rotational parameter , Thermal Grashof number , Concentration Grashof number , Magnetic parameter , Prandtl number , Heat generation , Schimidt number , shrinking/stretching parameter and Suction/injection parameter that appeared were graphically presented and the magnetic parameter was found to enhanced the fluid temperature.
CHAPTER ONE
1.0 INTRODUCTION
1.1 Background to the Study
Magnetohydrodynamic flows with or without the movement of heat in an electrically conducting fluids have attracted a large interest in the context of metallurgical fluid dynamics, aerothermodynamics, astronautics, geophysics, nuclear engineering and applied mathematics. Carrier and Greenspan (1960) considered unsteady hydromagnetic flows past a semi-infinite flat plate moving impulsively in its own plane. Gupta (1960) considered unsteady magneto- convection under buoyancy forces. Singer (1965) carried out further study rtyon unsteady free convection heat transfer with magnetohydrodynamic effects in a channel regime. Pop (1969) works on transient buoyancy-driven convective hydromagnetics from a vertical surface. Tokis (1986) implored the Laplace transforms to analyze the three dimensional free-convection hydromagnetic flows near an infinite vertical plate moving in a rotating fluid when the plate temperature undergoes a thermal transient. The influence of oscillatory pressure gradient on transiently rotating hydromagnetic flow was reported by Ghosh (1993).
Abd-El Aziz (2006) carried out a study on the thermal radiation flux effects on unsteady Magnetohydrodynamics micropolar fluid convection. Ogulu and Prakash (2006) in their work, presented an analytical solutions for variable suction and radiation effects on dissipative-free convective, optically-thin, Magnetohydrodynamic flow using a differential approximation to describe the radiative flux. Recent studies involving thermal radiation and transient hydromagnetic convection with specie transfer and viscous heating can be found in the analyses of Prasad et al. (2006) and Zueco (2007). In many geophysical and metallurgical flows, porous medium can arise also. Classically, the Darcian model is used to showcase the bulk effects of porous materials on flow dynamics and is valid for Reynolds numbers based on the pore radius. Chamkha (1996) works on the transient-free convection Magnetohydrodynamic boundary layer flow in a fluid-saturated porous medium channel, and later Chamkha (2001) extended the study to consider the influence of temperature-dependent properties and inertial effects on the convection regime. B´eg et al. (2005) presented perturbation solutions for the transient oscillatory hydromagnetic convection in a Darcian porous media with present of heat source. Chaudhary and Jain (2008) carried out the influence of oscillating temperature on Magnetohydrodynamic convection heat transfer past a vertical plane in a Darcian porous medium. Lately, Variational iteration method was applied for squeezing MHD Nano fluid flow in a rotating channel with the lower stretching porous surface, see Shahmohamadi and Rashidi (2016) for example. More extensive works as contained in the works of Mishra and Bhatti (2017), Rashidi et al. (2014), Sheikholeslami and Bhatti (2017), Abbas et al. (2017) and Bhatti and Rashidi (2016).
1.2 Statement of the Problem
Das and Ahmed (1992) employed the perturbation technique to analyze the buoyancy-driven magnetohydrodynamic (MHD) flow and heat transfer for a viscous incompressible fluid confined between a long vertical undulated plate and a parallel flat plate. They considered the effects of relative temperature of the channel walls on the velocity and temperature profiles without considering the wavy wall amplitude and inclination angle effects.
Recently, Bhatti et al. (2018) consider the problem of magnetohydrodynamic boundary layer flow with suction on a stretching and shrinking sheet. In their work, they used numerical approach to obtain the solution to the problem formulated. Natural convection, fluid rotation, fluid temperature and concentration were neglected. There by, no temperature and concentration dependent variables were considered.
1.3 Justification of the Study
A wide variety of industrial processes involve the transfer of heat energy. Throughout any industrial facility, heat must be added, removed, or moved from one process stream to another and it has become a major task for industrial necessity. These processes provide a source for energy recovery and process fluid heating/cooling.
Several works have been cited in this research work and we have decided to consider the areas that have been left out by the previous researchers, thus making the present study justifiable.
1.4 Aim and Objectives of the Study
1.4.1 Aim of the study
The aim of this study is to carry out a comparative analysis of magnetohydrodynamics boundary layer flow between stretching and shrinking sheet with heat generation and fluid rotation.
1.4.2 Objectives of the study
The objectives of this present study are :-
i. To transform the Partial differential equation (PDE) formulated to ordinary differential equations (ODE) using the similarity equations.
ii. To solve the set of transformed non linear, coupled, ordinary differential equations (ODE) using the Adomian Decomposition Method (ADM).
iii. To validate the results obtain with the work of Wang (2008) and Bhatti et al.(2018).
iv. To present and analyse the solutions with the help of the graphical representations.
v. To verify the effects of the various dimensionless parameters that appears in the solutions on both stretching and shrinking sheet.
1.5 Scope and Limitation
The Partial Differential Equation (PDE) formulated from the problem is presented in its rectangular coordinate system. The appropriate similarity transformations and stream functions are used to transform the partial differential equations to ordinary differential equations. Non linear coupled ordinary differential equations are derived, corresponding to momentum, energy equation and concentration equations. These equations are solved using improved Adomian Decomposition Method. The effect of various parameters that appears are analysed with the help of graphs. This work is limited to incompressible fluid dynamics.
1.6 Definition of Terms
Fluid: A substance which deforms continuously when shear stress is applied to it no matter how small, such as liquid or gas which can flow, has no fixed shape and offers little resistance to an external stress.
Grashof number: is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection. It is named after the German engineer Franz Grashof.
Prandtl number: the relationship between the thickness of two boundary layers at a given point along the plate depend on the dimensionless prandtl number which is the ratio of the momentum diffusivity 𝜈 or 𝜇 to the thermal diffusivity
Boundary Layers:- boundary layer is defined as that part of moving fluid in which the fluid motion is influenced by the presence of a solid boundary. As a specific example of boundary layer formation, consider the flow of fluid parallel with a thin plate, when a fluid flows at htigh Reynolds number past a body, the viscous effects may be neglected everywhere except in a thin region in the vicinity of the walls . This region is termed as the boundary layer.
Magnetohydrodynamics: is the study of magnetic properties and behavior of electrically conducting fluids.
Convection: is the conveying of heat from part of a liquid or gas to another by the movement of heated substances.
Stagnation Point: A point in the flow where the local velocity is zero.
𝐂�������𝐢���� ��𝐮𝐢�: change in density of fluid with time
Incompressible flow: fluid motion with negligible changes in density
This material content is developed to serve as a GUIDE for students to conduct academic research
COMPARATIVE DECOMPOSITION APPROACH OF MAGNETOHYDRODYNAMICS ROTATIONAL STAGNATION ON POINT FLOW BETWEEN A STRETCHING AND SHRINKING SHEET WITH HEAT GENERATION
Project 4Topics Support Team Are Always (24/7) Online To Help You With Your Project
Chat Us on WhatsApp » 09132600555
DO YOU NEED CLARIFICATION? CALL OUR HELP DESK:
09132600555 (Country Code: +234)
YOU CAN REACH OUR SUPPORT TEAM VIA MAIL: [email protected]
09132600555 (Country Code: +234)