Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates With Suction and Oscillating Boundaries
Chapter One
Aim and Objectives of the Study
The main aim of this dissertation is to modify an existing work of Balasubramanyam et al. (2010), by introducing conditions that do change with time, coupled with and in the presence of modified Darcy term. With the following specific objectives of the study, to solve:
- Magnetohydrodynamic unsteady free convection flow past vertical porous plates with heat
- Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous plates with heat
- Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical porous plates with heat
CHAPTER TWO
LITERATURE REVIEW
Introduction
The flow of viscous incompressible liquid driven by a constant pressure gradient in the channel between two parallel plates or simply channel flow, is one of the most extensively studied models of hydrodynamic instabilities and transition to turbulence in shear flows. The problem of MHD flow through a porous medium has become very important in recent years because of its possible applications in many branches of Science and Technology, particularly in the field of Agricultural Engineering to study the underground water re- sources. In Chemical Engineering for filtration and purification process, and in Petroleum Technology to study the movement of natural gas, oil and water through the oil reservoirs and so on.
Some Related Literature Review
Elbashbeshy et al. (2010) investigated the effects of thermal radiation and magnetic field on unsteady boundary layer mixed convection flow and heat transfer problem from a vertical porous stretching surface. Messiha (1966) discussed the problem of laminar boundary layers in oscillatory flow along an infinite flat plate with variable suction. Ahmed (2007) studied influence of chemical reaction on transient MHD free Convective flow over a vertical plate. Asghar et al. (2004) studied the flow of a non-Newtonian fluid induced due to the oscillations of a porous plate. Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification have been presented by Kandasamy et al. (2005). The opposing buoyancy effects on simultaneous heat and mass transfer by natural convection in a fluid saturated porous medium was investigated by Angirasa et al. (1997). Singh et al. (1994) analyzed the unsteady MHD free convective flow through a porous medium between two infinite vertical parallel oscillatin porous plates with different amplitude. Ahmed (2010) studied the Magneto hydrodynamic and chemical reaction effects on unsteady flow, heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate in a slip-flow regime.
Further, Raptis and Perdikis (1985) studied the unsteady two-dimensional free convective flows through highly porous medium. Recently, Ahmed and Ahmed (2004) analyzed the effect of two dimensional MHD oscillatory flow along a uniformly moving infinite vertical porous plate bounded by porous medium. Later on, the effects of unsteady free convective MHD flow through a porous medium bounded by an infinite vertical porous plate were investigated by Ahmed (2007). Deka et al. (1997) showed the effect of free convection on MHD flow past an infinite vertical oscillating plate with constant heat flux. Jha et al. (2013) studied the problem of unsteady natural convection couette flow of a reactive viscous fluid in a vertical channel. Taneja and Jain (2002) solved the hydrodynamic flow in slip flow regime with time-dependent suction. Singh and Singh (2003) analyzed the heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity.
Pawan et al. (2007) investigated unsteady free convection oscillatory flow through a porous medium with periodic wall temperature. Effect of suction and injection on MHD three dimensional couette flow and heat transfer through a porous medium has been studied by Das (2009). Pathak et al. (2006) concentrated on effect of radiation on unsteady free convective flow bounded by an oscillating plate with variable wall temperature. Hayat et al. (2004) reported the flow of a visco-elastic fluid on an oscillating plate. Krishna et al. (2004) analyzed the hydromagnetic oscillatory flow of a second order Rivlin Erickson fluid in a channel. Sahim (2008) investigated transient three dimensional flows through a porous
CHAPTER THREE
METHODOLOGY
Introduction
There are many problems in physical science, applied mathematics and engineering where the governing equations are a set of coupled unsteady equations involving many variables. Almost invariably the equations are also nonlinear and difficult to make progress with analytically. However if there are large or small parameters present, the use of Perturbation methods, also known as linearisation or asymptotic techniques can allow for significant progress to be made in trying to understand the solution properties.
It may be possible to obtain solutions in analytical form, by reducing the equations to a much simpler set which can be tackled more easily. In this dissertation we will apply regular and multiparameter perturbation techniques to obtain semi analytical solution of the mathematical models with illustration taken largely from fluid flow problems.
The method hinges on the identification of a small parameter ε, 0 < ε 1. Typically there is an easily obtained solution when ε 1. One then uses this solution as a seed to construct a linear theory about it. The resulting sets of linear equations are then solved giving a solution which is valid in a regime near ε 1 Ali (1993).
CHAPTER FOUR
RESULTS AND DISCUSSION
Introduction
This chapter presents results and discussion to illustrate the flow characteristics for the velocity, temperature, skin friction coefficient and Nusselt number and then show how the flow fields are influenced by the material parameters of the flow problems with the aid of graphs and tables.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Introduction
This chapter presents summary, conclusion, recommendations, contributions and limita- tions of the study.
Summary
In this dissertation, the problems of Magnetohydrodynamic unsteady free convection flow past vertical porous plates with suction and oscillating boundaries are studied. The dissertation has been presented in five (5) chapters. Chapter one serves as the background to the research report. It consists of statement of the problem, aim and objectives of the study as well as definitions of the basic concept. Chapter two examines the related literature review, while Chapter three discusses the methodology adapted for the research report, which includes problem formation and solutions to the mathematical models under study, and the perturbation techniques applied in getting the semi analytical solutions for the velocity, temperature, the skin friction and Nusselt number. Chapter four presents the result and discussion and shows how the flow fields are influenced by the material parameters of the flow problems with the aid of graphs and tables. Finally Chapter five draws together the research project, that means it showcases summary, conclusion and recommendations, as well as contributions to knowledge. The chapter ends with limitations of the study.
Conclusion
The intent of the work reported in this dissertation was to demonstrate and convey the idea of using perturbation technique in the area of hydrodynamics. Both graphical and tabular results are presented to show the effects of flow characteristics for velocity, temperature, skin friction and Nusselt number. The following conclusions are drawn:
Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Heat Deposition
- Thevelocity profile within the boundary layer increases with increasing values of Gr, k and Q, while it deceases with increasing values of Pr, M, A and λ .
- Thetemperature profile within the boundary layer increases with increasing value of Q, while it deceases with increasing values of Pr and λ .
- Thelocal Skin friction coefficient increases with increasing values of Gr, Pr, A, Q, k and λ , while it decreases with increasing
- Anincrease in Pr, Q and λ increases the Nusselt
Magnetohydrodynamic Unsteady Free Convection Flow Past an Infinite Verti- cal Porous Plates with Heat Deposition
- Itis observed that the flow fluid has constant suction or
- Thevelocity profile within the boundary layer increases with increasing values of S, Gr, Ec and ε.
- Thetemperature profile within the boundary layer increases with increasing values of S, Gr, Ec and ε.
- Thelocal Skin friction coefficient increases with increasing values of S, Gr, Ec and
- Anincrease in S, Gr, Ec and ε increases the Nusselt
Darcy Forchcheimer Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Heat Deposition
- Thevelocity profile within the boundary layer increases with increasing value of Gr, Ec, b, λ and Q, while it deceases with increasing values of A and
- Thetemperature profile within the boundary layer increases with increasing value of k, while it deceases with increasing values of Gr, Ec and b.
- Thelocal Skin friction coefficient increases with increasing values of Ec, b and Q, while it decreases with increasing values of Gr, A, k, and λ .
- Anincrease in Q, A, Ec and Gr increases the Nusselt number, while it decreases with increasing .
Recommendations
Based on the research reported in this dissertation, several methods are recommended for comparison with perturbation techniques for further research and these methods are as follows:
FiniteDifference
- FiniteElement
- FiniteVolume
- HomotopyAnalysis
- LaplaceTransform
Contributions
This research, contributes to the body of knowledge based on the following facts, which are not taking care in the work of Balasubramanyam et al. (2010):
- Unsteadystate of the modified adapted work was
- Thepresence of coupled between momentum and energy equations of problem (3.3.2) and (3.3.3)
- Inclusionof Darcy Forcheimer term in the problem
- Solutionof problem (3.3.1) e. Magnetohydrodynamic unsteady free convection flow past vertical porous plates with heat deposition was obtained.
- Solutionof problem (3.3.2) e. Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous plates with heat deposition was obtained.
- Solutionof problem (3.3.3) e. Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical porous plates with heat deposition was obtained.
Limitations of the Study
This report was limited to vertical porous plates with suction and oscillating boundaries. This is due to many reasons, among which includes:
- Thatonly perturbation techniques which required the use of small parameter in analysis
- Theunsteady state solutions were obtained for the model governing
- Thetime constraint and nature of the
REFERENCES
- Abdel-Naby, M. A. (2003). Finite difference solution of radiation effects on mhd unsteady free convection flow over vertical plate with variable surface temperature. Journal of Applied Mathematics, 2:65 – 86.
- Ahmed, N., Sarma, D., and Deka, H. (2012). Mixed convection and mass transfer from an infinite vertical porous plate with chemical reaction in presence of heat source. Applied Mathematical Sciences, 6(21):1011 – 1020.
- Ahmed, S. (2007). Effects of unsteady free convective mhd flow through a porous medium bounded by an infinite vertical porous plate. Bull. Cal. Math. Soc., 99(5):511 – 522.
- Ahmed, S. (2010). Influence of chemical reaction on transient mhd free convective flow over a vertical plate in slip flow regime. Emirates Journal for Engineering Research, 15(1):25 – 34.
- Ahmed, S. and Ahmed, N. (2004). Two dimensional mhd oscillatory flow along a uniformly moving infinite vertical porous plate bounded by porous medium. International Journal of Pure and Applied Maths, 35:1309 – 1319.
- Ahmed, S. and Batin, A. (2013). Convective laminar radiating flow over an accelerated ver- tical plate embedded in a porous medium with an external magnetic field. International Journal of Engineering and Technology, 3(1):66 – 72.
- Ali, H. N. (1993). Introduction to Perturbation Techniques. John Wiley and Sons, INC., wiley classics library edition.
- Anand, R. J. and Shivaiah, S. (2011). Chemical reaction effects on an unsteady mhd flow past a semi infinite vertical porous plate with viscous dissipation. Journal of Applied Mathematics and Mechanics, 32(8):1065 – 1078.
