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STUDY OF HYDROMAGNETIC FLOW PAST A VERTICAL SEMI INFINITE POROUS PLATE

CHEPKONGA DAVID

BACHELOR OF SCIENCE

(Mathematics and Computer Science)

A research proposal submitted in partial fulfillment for the Degree of Bachelor of Science in Mathematics and Computer Science in the Taita Taveta University College

2016

DECLARATION

This proposal is my original work and has not been presented for a degree in any other university.

Signature: ………………………………… Date:…………………………….

This proposal has been submitted for examination with our approval as university supervisors.

Signature: ………………………………… Date:…………………………….

Dr. Phineas Roy Kiogora

JKUAT, KENYA

Signature: ………………………………… Date:…………………………….

Mr. Nicholas Muthama

TTUC, KENYA

DECLARATION ii

NOMENCLATURE v

ABBREVIATIONS v

ABSTRACT v

1.0 INTRODUCTION AND LITERATURE REVIEW 1

1.1 BACKGROUND INFORMATION 1

1.2 LITERATURE REVIEW 1

1.3 STATEMENT OF THE PROBLEM 3

1.4 OBJECTIVES OF THE STUDY 3

1.To determine velocity profiles 3

2.To determine skin friction 3

3.To determine the rate of mass transfer 3

1.5 HYPOSTHESIS 3

1.6 GOVERNING EQUATIONS AND ASSUMPTIONS 4

1.6.1 Introduction 4

1.6.2 The equation of continuity 4

1.6.3 The equation of conservation of momentum 4

1.6.4 The equation of species concentration 4

1.7 METHODOLOGY 5

1.8 RESEARCH SCHEDULE 6

1.9 GANTT CHART 7

REFERENCES 9

NOMENCLATURE

Symbol                            Meaning

Magnetic field vector

Density

Body force

ABBREVIATIONS

Magneto hydrodynamic

ABSTRACT

In this research, the hydro magnetic flow past a vertical porous semi- infinite plate is to be considered, The effects of induced magnetic field arising due to the motion of the fluid is taken into account .The governing equations will be solved by adopting the finite difference method. The effects of various non-dimensional parameters on the velocity profile, the induced magnetic field and the temperature profile will be discussed. The results shall be tabulated and a computer program shall be written to help in the clear simulation of the solutions, where possible graphs shall also be used to present the results. It is expected that the variations of the parameters had no effect on the skin friction.

1.0 INTRODUCTION AND LITERATURE REVIEW

1.1 BACKGROUND INFORMATION

Matter is classified into fluids and solid. A solid can resist shear stress by static deformation but a fluid cannot .Fluid flow maybe one, two or three dimensional .Individual particles move in the direction of the flow to constitute fluid flow. Fluids are classified as incompressible or compressible .A fluid is said to be compressible if the fluid density varies with pressure whereas it’s not incompressible if the change in density with pressure is limited.

Hydro magnetic flow is the science which deals with the motion of electrically conducting fluid in the presence of magnetic fields. It is the synthesis of two classical sciences, Fluid Mechanics and Electromagnetic field theory. It is well known result in electromagnetic theory that when a conductor moves in a magnetic field, electric currents are induced in it. These current experience a mechanical force, called ‘Lorentz Force’, due to the presence of magnetic field. This force tends to modify the initial motion of the conductor. Moreover, induced currents generate their own magnetic field which is added on to the primitive magnetic field.

The science of fluid dynamics describes the motion of liquids and gases and their interaction with solid bodies. It is a broad, interdisciplinary field that touches almost every aspect of our daily lives, and it is central to much of science and engineering.  Fluid dynamics impacts defenses, homeland security, transportation, manufacturing, medicine, biology, energy and the environment. Predicting the flow of blood in the human body, the behavior of micro fluidic devices the aero-dynamics performance of airplanes, cars, and ships, the cooling of electronic components, or the hazard of weather and climate, all require a detailed understanding of fluid dynamics and therefore substantial research. Fluid dynamics is one of the most challenging and exciting fields of scientific activity simply because of the complexity of the subject and the breadth of the applications.

1.2 LITERATURE REVIEW

The phenomenon of hydro magnetic flow past a vertical semi-infinite porous plate has attracted attention of a good number of investigators because of its various applications in Engineering. The study of magneto hydrodynamics (MHD) was started with Faraday (1859), who carried out an experiment in which an electrically conducting fluid was passed between poles of a magnet in a vacuum glass.

Alam et al (2006) investigated the effects of mass transfer on steady two dimensional free convection flow past a continuously moving semi-infinite vertical porous plate in a porous medium. They observed that the temperature decreases with increase in suction parameter .They concluded from their analysis that the temperature and concentration fields are influenced by the dufour and soret effects.

Tamana et al (2009) analyzed heat transfer in a porous medium over a stretching surface with internal heat generation, suction or injection. They observed that velocity profiles decrease with increase in injection or suction.

Das et al (2011) investigated the mass transfer effect of unsteady hydromagnetic convective flow past a vertical porous plate in porous medium with heat source, where they observed that velocity of the flow field changes more or less with variation of flow parameters.

S.velmuragan (2014) conducted a research on hydro magnetic flow past a parabolic started vertical plate in the presence of homogenous chemical reaction of first order, he used Laplace transform solution of unsteady flow past a parabolic starting motion of an infinite vertical plate with variable temperature and uniform mass diffusion, in the presence of homogenous chemical reaction of first order. The plate temperature was raised linearly with time and the concentration fields were studied for the different physical parameter. He observed that the velocity increases with increasing values of the thermal Grashof number or mass of Grashof number .he noticed that the trend is just reversed with respect to the chemical reaction parameter as well as magnetic field parameter.

Ashok (1988) carried out research on a similarity solution for hydro magnetic flow of an incompressible viscous electrically conducting fluid past a continuously moving semi-infinite porous plate in the presence of a magnetic field in the case of small magnetic Reynolds number the perturbation technique was applied to solve the equations .he realized that the effect of the magnetic parameter was to increase the skin friction coefficient while it has no significant effect on the Nusset number.

Rapits et al (1987) studied the unsteady free convective flow through a porous medium adjacent to a semi-infinite vertical plate using finite difference scheme.

Ramana (1991) studied heat transfer in flow past a continuously moving semi-infinite flat plate in transverse magnetic field with heat flux. He obtained that a fall in the temperature of the thermal boundary layer due to increase in magnetic field parameter.

Emma Marigi (2012) et al studied hydromagnetic turbulent flow past a semi-infinite vertical plate subjected to heat flux, they noticed that an induced electric current known as Hall current exists due to the presence of both electric field and magnetic field. They used finite difference scheme to solve partial differential equations, they noted that the hall current, rotation parameter, Eckert number, injection and Schmidt number affect the velocity, temperature magnetic field and concentration profiles.

G.palani, U.srikanth (2009) carried out an analysis on MHD flow past a semi-infinite vertical plate with mass transfer, they carried out analysis of incompressible, viscous fluid past a semi-infinite vertical plate with mass transfer, under the action of transversely applied magnetic field, and the heat due to viscous dissipation and the induced magnetic field were assumed to be negligible. The dimensionless governing equations were unsteady, two dimensional, coupled and nonlinear partial differential equations. They used a fast converging implicit difference scheme to solve the non-dimensional governing equations.

A.S. Gupta (1974) carried out an investigation on hydro magnetic flow past a porous flat plate with Hall effects, he realized that asymptotic solution for the velocity and magnetic field exist both for suction or blowing at the plate. Also he concluded that when the magnetic Reynolds number is very small, the flow pattern is remarkably similar to that for a non-conducting flow past a flat plate in rotating frame.

Ashraf A,moniem(2013),researched on the model of mass transfer on free convective flow of a viscous incompressible electrically conducting fluid past vertically porous plate through a porous medium with time dependent permeability and oscillatory suction in presence of transverse magnetic field .he applied perturbation technique to obtain the solution for velocity field and concentration distribution analytically.

D.latha Mathuri(2012) carried out study on an unsteady ,two dimensional hydromagnetic, laminar mixed convective boundary layer flow of an incompressible, Newton electrically conducting and radiating fluid along a semi-infinite vertical permeable moving plate with heat and mass transfer is analyzed ,by taking into account the effect of viscous dissipation .he solved the equations using finite difference method.

J.Anand Rao (2012), carried out study of hydromagnetic heat and mass transfer in MHD flow of an incompressible, electrically conducting viscous fluid past an infinite vertical porous medium of time dependent permeability under oscillatory suction velocity normal to the plate. He noted that the influence of the uniform magnetic field acts normal to the flow and the permeability of the porous medium fluctuate with time. He used Galerkin finite element method for velocity, temperature, concentration, field and expressions for skin friction.

Throughout literature review I noted that much is still not covered on the method of finite difference on hydromagnetic flow. Many researchers have dwelt on unsteady state of the flow but in my present work am going to consider a case where the flow field is steady. Another vital thing that we are bringing on board that remains untouched is that the semi-infinite plate is fixed it’s not movable.

1.3 STATEMENT OF THE PROBLEM

In this study we are going to use steady incompressible flow, where the equation of conservation of momentum will be analyzed under various conditions and also the concentration equation shall be considered, we will adopt the finite difference method to obtain solutions and compare with results from other researchers. I choose finite difference scheme because much have not been done on this method.

1.4 JUSTIFICATION

Flow of hydromagetic fluid through a porous media are very much prevalent in nature and therefore ,the study for such flows has become a principal of interest in many scientific and engineering in the study of movement of natural gas, oil and water trough oil reservoirs ;in chemical engineering for the filtration and water purification process .it’s also applicable in MHD generators ,plasma studies ,nuclear reactors ,oil exploration ,flows in oil ,control of pollutant in ground water ,coolers, fuel and gas filters ,geothermal energy extraction and in boundary layer control in the field of aerodynamics.

1.5 OBJECTIVES OF THE STUDY

The objectives of the study will be;

1. To determine  velocity profiles

2. To determine skin friction

3. To determine the rate of mass transfer

1.6 HYPOSTHESIS

The flow field variables and parameters on steady hydro magnetic flow past a vertical semi-infinite porous plate has no effect on the primary velocity profiles.

1.7 GOVERNING EQUATIONS AND ASSUMPTIONS

1.7.1 Introduction:

The equations governing hydro magnetic flow are as follows:

1.7.2 The Assumptions:

In order to reduce complexity and achieve the outlined objectives, the following assumptions are made:

i The flow is steady

ii The fluid is incompressible

iii There are no chemical reactions taking place

iv The fluid flow is laminar

1.7.3 The equation of continuity.

The principle of conservation of mass says that the mass of the fluid element remains the same as the mass moves through the fluid.in fluid dynamics the continuity equation is a mathematical statement that, in any steady state process ,the rate at which the mass enters a system is equal to the rate at which mass leaves the system .

The differential form of the equation is given by:

=

But since the fluid is incompressible

Hence it reduces to

For the case of incompressible flow,  is assumed to be a constant and the equation simplifies to

1.7.4 The equation of conservation of momentum

This is derived from the Newton’s second law of motion which states that the sum of the resultant force is equal to the rate of change of momentum of the flow. The law requires that the sum of all forces acting on a control volume must be equal to the rate of increase of the fluid momentum within the control volume. The equation in normal form can be expressed as:

The F term in this study represents the body force which will be taken as the magnetic force, while T represents the traction force.

From Maxwell’s electromagnetic equations ,the relation   which implies that   ,when magnetic Reynolds number is small ,induced magnetic field is negligible in comparison with applied magnetic field ,so that

(A constant), the equation of conservation of charge   gives  a constant when where current density   since the plate is none conducting, this constant is zero.

The Lorentz force becomes   which is equal to:

1.7.5 The equation of species concentration

The equation of species concentration is based on the conservation of mass .it is applicable when

i The porous medium is saturated with fluid

ii Flow is steady

The equation for species concentration is given by;

1.8 METHODOLOGY

The partial differential equations that are obtained are non-linear.it is therefore not possible to obtain an exact analytical solution .the equations are solved numerically using the finite difference method, this a numerical method that make use of finite difference codes/solvers that take low computation memory and easy to program and modify, hence more advantageous to use in electrical problems.it is a second order method which is accurate and unconditionally stable and has less computation cost. According to Steven and Raymond  ,Crank Nicolson method provide an implicit scheme which is accurate in both space and time.to provide this accuracy ,difference approximations are developed at the midpoint of the time increment.

1.9 RESEARCH SCHEDULE

ACTIVITY DURATION(WEEKS) START DATE FINISH DATE DELIVERABLE

Preliminary Work 3 7th Jan 2016 28th Jan 2016 Problem Statement

Project Identification 1 29th Jan 2016 5th  Jan 2016 Research Definition

Draft Proposal 3 6th Feb 2016 27th Feb 2016 Draft Proposal

Proposal Presentation 1 28th Feb 2016 6th March 2016 Final Proposal

Proposal Defense 1 7th March 2016 14th March2016 Final Proposal

Literature search & Mathematical formulation 1 15stMarch 2016 22nd March 2016 Project report

Actual Mathematical Analysis 1 23rd March 2016 30th March 2016 Project report

Modeling of the solution 1 31st March 2016 7th April 2016 Project report

Drafting final report 3 8th April 2016 29th April 2016 Draft Project report

Report publishing & submission 5 30th April 2016 4th June 2016 Project report booklet

Final Presentation 1 5th June 2016 12th June 2016 Project presentation

2.0 GANTT CHART

Activity Duration(weeks)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Preliminary Work

Project Identification

Draft Proposal

Proposal Presentation

Proposal Defense

Literature search and Mathematical formulation

Actual Mathematical Analysis

Modeling of the solution

Drafting final report

Report publishing & submission

Final Presentation

Key

2.1BUDGET ESTIMATES

ITEM DESCRIPTION COST(KSH)

Flash disk For backing up research files and documents 1500.00

Travel (field study) To gather the necessary information as pertains to the research proposal 4000.00

Binding and Photocopying Proposal papers 3000.00

Internet bundles For internet access to facilitate efficient project research 1500.00

TOTAL 10000.00

2.2 REFERENCES

A.S.Gupta, hydro magnetic flow past a porous flat plate with Hall effects, Acta media 22, pp.281-287, 1975.

Asharaf A,moniem, Solution of MHD Flow past a Vertical Porous Plate through a Porous Medium under Oscillatory Suction,vol.4,no.4,April 2013.

Alam,M.S,(2006) ,Dufuor and soret effects on unsteady hydromagnetic convective flow past a vertical porous plate in a porous medium ,Internal journal of Applied Mechanics and Engineering ,11(30),535-545.

D.latha, Finite difference analysis on an unsteady mixed convection flow past a semi-                         infinite vertical permeable moving plate with heat and mass transfer with radiation and viscous    dissipaton,vol3(4),pp.2266-2279,2012.

Das ,Biswal ,(2011) ,mass transfer effects on unsteady hydromagnetic convective  flow past a vertical porous plate in a porous medium with heat source ,Journal of Applied Fluid Mechanics ,4(4),91-100.

G.palani, U, srikanth, MHD Flow past a semi-infinite vertical plate with mass transfer, vol.14, no.3, pp.345-356, 2009.

J.Anand Rao, Finite Element Solution of Heat and Mass Transfer in MHD Flow of a Viscous Fluid past a Vertical Plate under Oscillatory Suction Velocity, Journal of Applied Fluid Mechanics, Vol. 5, No. 3, pp. 1-10, 2012.

Tamana, (2009), heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection in the presence of radiation, Journal of Mechanical Engineering, 40(1), 22-28.

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