ABSTRACT
The study discusses the effect of suction/injection on MHD transient free convection flow of Newtonian viscous, incompressible, electrically conducting fluid in a vertical plate, is analysed to obtain a partial differential equation, in which the partial differential equations with relevant initial and boundary conditions are transform using Laplace transform. It was solved to obtained velocity, temperature, skin fraction and mass flow rate. To verify the accuracy of these solutions, Riemann Sum Approximation is applied to obtained the solution in time domain. Numerical values of different parameters were run in MATLAB, in order to generate graphs for better understanding of the relationships between them. Then the effect of suction/injection as well as various non-dimension parameters controlling the physical situation are extensively discussed with the aid of graphs.
TABLE OF CONTENTS
DECLARATION ii
CERTIFICATION iii
DEDICATION iv
ACKNOWLEGDEMENTS v
ABSTRACT vi
NOMENCLATURE vii
TABLE OF CONTENTS viii
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study 1
1.2 Aim and Objectives 4
1.3 Methodology 5
1.4 Significance of the Study 5
1.5 Scopes and Limitations 6
1.6 Definition of Some Terms 6
1.7 Organization of the Project 7
CHAPTER TWO
LITERATURE REVIEW
2.1 Empirical Literature 9
CHAPTER THREE
METHODOLOGY AND TOOLS
3.1 Mathematical formulation 15
3.2 Solutions to the problem 16
CHAPTER FOUR
RESULTS AND DISCUSSIONS
4.1 Introduction 22
CHAPTER FIVE
SUMMARY, RECOMMENDATIONS, AND CONCLUSIONS
5.1 Summary 31
5.2 Conclusions 31
5.3 Recommendations 32
REFERENCES 33
APPENDIX A 36
APPENDIX B 38
LIST OF FIGURES
Figure 4.1. Temperature profile θ showing the effect of t for s=2 23
Figure 4.2. Temperature profile θ showing the effect of t for s=-2 24
Figure 4.3. Temperature profile θ showing the effect of pr . 25
Figure 4.4. Velocity profile U showing the effect of t 26
Figure 4.5. Velocity profile U showing the effect of t 27
Figure 4.6. Velocity profile U showing the effect of s 28
Figure 4.7. Skin friction at y=1 (τ1) showing the effect of G and s . 29
Figure 4.8. Skin friction at y=0 (τ0) showing the effect of G and s . 29
NOMENCLATURE
B_0: Uniform magnetic field (Tesla).
G: Grashof number (dimensionless).
M: Magnetic Field (dimensionless).
S: Suction/injection.
Pr : Prandtl number (dimensionless).
s ̅: Laplace parameter.
t: time (dimensionless).
GREEK LETTERS
τ: Skin fraction (dimensionless).
μ : Mass flow rate (dimensionless).
θ : Temperature (dimensionless).
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
In our daily life, we come across three states of matter: solid, liquid, and gas. Even though they are dissimilar in many respects, liquids, and gases differ from solids in their characteristics and they are called fluids, the term fluid is generally used to describe either a liquid or a gas since both have a common feature that makes it possible to construct a unified dynamical theory for liquids and gases simultaneously. This common property is referred to as fluidity, which is broadly defined as a tendency of either medium to flow under the action of any external force, no matter how small; in other words, a fluid moves and deforms continuously as long as an external force is applied, since the fluid motion continues under the action of a shear stress (Kim et al., 2019).
Fluid mechanics are divided into two parts: dynamics and kinematic. Kinematics describes the motion of the fluid without any consideration of forces that cause fluid motion. Fluid motion where the forces are considered is called fluid dynamics. Governing equations are formulated by considering the balance of these. However, fluid dynamics deals with fluid flow that offers a systematic structure, which embraces empirical and semi empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time. It has several sub disciplines itself, including aerodynamics and hydrodynamic.
During Ancient Greece, Archimedes investigated fluid statics and buoyancy and formulated his famous law known as the Archimedes' principle, which was published in his work on floating bodies generally considered to be the first major work on fluid mechanics.
Iranian scholar Abu Rayhan Biruni and later Al-Khazini applied experimental scientific methods to fluid mechanics. In 1739, Rapid advancement in fluid mechanics began with Leonardo da Vinci (observations and experiments), Evangelista Torricelli (invented the barometer), Isaac Newton (investigated viscosity), and Blaise Pascal (researched hydrostatics, formulated Pascal's law), and continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics i.e. hydrodynamic. Considering the work of Newton i.e. concept of lack of "slipperiness" which is an important quantity that we call viscosity all fluids has this natural resistance to flow. In a fluid, the molecules feel an “attraction” toward other molecules. We call this attraction “cohesive” force and leads to surface tension in liquids (White, 2017).
When placed in a container, the molecules also experience an attractive force toward the interior of the container. This is called “adhesive” force. When fluid flows, viscosity results in a frictional force, both against the surface it is flowing on and within the fluid itself (White, 2017). Viscosity makes the flow interesting and of course challenging to understand and calculate. It is viscosity that causes many of the physical features. Based on viscosity we have viscous and in viscid flow. Based compressibility we have compressible and incompressible fluid flow. A fluid flowing in a pipe, it is behaviour is governed mainly by the effects of viscosity and gravity relative to the inertial forces of the flow. Based on this we have laminar, turbulent and transient flow.
Laminar flow generally happens when dealing with small pipes and low flow velocities. Laminar flow can be regarded as a series of liquid cylinders in the pipe, where the innermost parts flows the fastest, and the cylinder touching the pipe isn't moving at all. Shear stress in a laminar flow depends almost only on viscosity (v) and is independent of density (ρ).
In turbulent flow vortices, eddies and wakes make the flow unpredictable. Turbulent flow happens in general at high flow rates and with larger pipes. Shear stress in a turbulent flow is a function of density (ρ). Transitional flow is a mixture of laminar and turbulent flow, with turbulence in the centre of the pipe, and laminar flow near the edges. Each of these flows behaves in different manners in terms of their frictional energy loss while flowing and have different equations that predict their behaviour. Turbulent or laminar flow is determined by the dimensionless Reynolds number. The Reynolds number is important in analysing any type of flow when there is substantial velocity gradient (i.e. shear.) It indicates the relative significance of the viscous effect compared to the inertia effect. The Reynolds number is proportional to inertial force divided by viscous force. The flow is laminar when Re < 2300, transient when 2300 < Re < 4000 and turbulent when 4000 < Re (White, 2017).
In 1970, the field of Magnetohydrodynamics (MHD) was initiated by Hannes Alfven for which he received the Noble price. The word Magnetohydrodynamics is derived three words derived from magneto- meaning-magnetic-field, hydro-meaning-water, and dynamic-meaning-movement. It is defined as model of electrically conducting fluids that treats all interpenetrating particle species together as a single continuous medium. It is primarily concerned with the low-frequency, large-scale, magnetic behaviour in plasmas and liquid metals and has applications in numerous fields includes geophysics, astrophysics, and engineering.
Heat transfer (or heat) is thermal energy in transit due to a spatial temperature difference. Whenever there exists a temperature difference in a medium, heat transfer must occur. Likewise, if we have a multicomponent system with a concentration gradient, one constituent of the mixture gets transported from the region of higher concentration to the region of lower concentration till the concentration gradient reduces to zero.
There are three types of heat transfer: conduction, convection and radiation. Conduction is heat transfer by means of molecular agitation within a material without any motion of the material as a whole.
This occurs in a solid or stationary fluid due the random motion of its constituent atoms, molecules or electrons. Radiation is heat transfer by the emission of electromagnetic waves which carry energy away from the emitting object. Net radiation heat exchanges between two surfaces. Convection is the transfer of thermal energy from one place to another by the movement of fluids or gases. Furthermore, in convection, if the fluid motion is induced by some external resources such as fluid machinery or vehicle motion, the process is generally called forced convection flow. While if the motion in the fluid is induced by body forces such as gravitational or centrifugal forces, this kind of flow is said to be free or natural convection.
Free convection flows occur not only due to temperature difference, but also due to concentration difference or the combination of these two.
The study of free convection flow with heat and mass transfer plays an important role in the design of chemical processing equipment, formation and dispersion of fog, crop damage due to freezing, nuclear reactors and environmental pollution. In the advancement of space technology and in processes involving high temperatures thermal radiation effects play an important role.
Suction refers to the process of creating a partial vacuum or low-pressure are to draw a fluid from a system. It often used in pumps devices to pull a liquid or gas into the system. While injection is opposites of suction where the fluid is pushed or forced to into a system or container. It involves applying pressure to introduce a fluid into a specific location.
Both suction and injection are essential techniques in various applications, including fluid transport, hydraulic systems and medical devices.
1.2 Aim and Objectives
The aim of this work is to study the role of suction/injection on MHD free convection flow in a vertical plate.
The objectives of the study are;
i. to determine the velocity and temperature of the model.
ii. to analyse the effect of suction/injection and magnetic field on flow and temperature field.
iii. to find the skin friction and mass flow rate of the model.
to present and discuss the effect of role of suction/injection graphically.
1.3 Methodology
In this work, the relevant physical problem will be modelled in form of couple linear partial differential equation prescribed with a set of linear initial and boundary conditions. Then a set of non-dimensional variables will be introduced from the transform of couple partial differential equation along with imposed initial and boundary conditions.
However, the non-dimensional equations are transform using Laplace transform to get velocity and temperature in Laplace domain and concentration field with help of numerical values were obtained and graphical display using MATLAB. The graphs obtained are used to interpret the flow behaviours using different flow condition. Moreover, the corresponding skin friction and mass flow rate.
1.4 Significance of the Study
It is equally important to via in mind that there are several motivations for studying the role of suction/injection on MHD transient natural convection flow in a vertical plate:
First, it is a simple model for more complex flow in engineering application, such as astrophysics and geophysics.
Secondly, the effect suction/injection and magnetic field on natural convection flow are still not well understood, so, further research is needed.
Finally, clear understanding of flow in a vertical plate will motivate in developing more efficient designs in heat exchangers and other engineering applications.
1.5 Scopes and Limitations
This study will focus on the unsteady, transient free convection flow of an incompressible viscous fluid with either heat or mass transfer together.
The Laplace inverse of temperature and velocity was not possible using closed form Laplace transform, but using Riemann Sum approximation (RSA.)
1.6 Definition of Some Terms
Viscous Flow: In viscous flow, frictional effects are significant. for example, boundary layer flows.
In viscid (ideal) flow: In viscid flow is nothing but the viscous terms neglected in the governing equations.
Compressible Flow: When a fluid moves at a speed equivalent to 0.3 times the speed of sound, density variation becomes predominant and the flow is compressible. Such flows do not occur easily in liquids, Water hammer and cavitations are examples of the significance of compressibility in liquid flows.
Incompressible Flow: If the effect of pressure on the density of the fluid is negligible, the flow is called an incompressible flow; the fluid volume fraction remains constant along the flow path. For an incompressible flow, the equation of continuity simplifies to ∇.v=0
Unsteady flow: If it is time-dependent in this case the flow parameters such as velocity, acceleration, etc. dependent on the time as well as the space variable. Example of an unsteady flow is a short pipe (circulation).
Steady flow: A flow is said to be steady if the velocity vector and other flow quantity are independent of the time variable. flow of fluid in long pipes is the common example.
Laplace transform is an integral transform that converts a function of a real variable (usually t, in the time domain) to a function of complex variable s (in the complex frequency domain, also known as s-domain, or s-plane).
For suitable functions f, the Laplace transform is the integral
Where, s is the transform variable and e^(-st) is the kernel of the transform, for time t > 0.
Riemann Sum approximation: It is certain kind of approximation of an integral by a finite sum. It applied for approximating the length of curves and other approximation.
1.7 Organization of the Project
This project is divided into five chapters as follows: Chapter one discusses the background of the study, aim and objectives, significance, scope and limitation, research methodology, definition of some terms and organisation of the project. Chapter two provides the required literature of the previous work related to problems outlined in the objectives. Chapter three gives physical and mathematical formulation for the problem.
Then, solved the mathematical formation using Laplace transform to obtained velocity, temperature, skin friction and mass flow rate. Chapter four constitute the analysis of the fluid flow. The numerical computation is carried out for various values of the major parameters such as Suction/injection parameters (S), Magnetic field (M), Grashof number (G), Prandtl number (Pr) and dimensionless time (t). And chapter five consists of summary, conclusions and recommendations.
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