Fluid mechanics deals with the study of all fluids under static. and dynamic situations. Fluid mechanics is a branch of continuous

Transcription

1 Fluid Mechanics: Fluid mechanics deals with the study of all fluids under static and dynamic situations. Fluid mechanics is a branch of continuous mechanics which deals with a relationship between forces, motions, and statically conditions in a continuous material. This study area deals with many and diversified problems such as surface tension, fluid statics, flow in enclose bodies, or flow round bodies (solid or otherwise), flow stability, etc. In fact, almost any action a person is doing involves some kind of a fluid mechanics problem. Furthermore, the boundary between the solid mechanics and fluid mechanics is some kind of gray shed and not a sharp distinction. For example, glass appears as a solid material, but a closer look reveals that the glass is a liquid with a large viscosity. A proof of the glass liquidity is the change of the glass thickness in high windows in European Churches after hundred years. The bottom part of the glass is thicker than the top part. Materials like sand (some call it quick sand) and grains should be treated as liquids. It is known that these materials have the ability to drown people. Even material such as aluminum just below the mushy zone also behaves as a liquid similarly to butter. Furthermore, material particles that behaves as solid mixed with liquid creates a mixture that behaves as a complex liquid. After it was established that the boundaries of fluid mechanics aren t sharp, most of the discussion in this book is limited to simple and (mostly) Newtonian (sometimes power fluids) fluids which will be defined later. 1

2 Continuous Mechanics Solid Mechanics Fluid Mechanics Constant Fluid Statics Fluid Dynamics Multi Phase Boundaries Problems Stability flow Turbulent flow Laminar flow Problems Internal flow Figure. Diagram to explain part of relationships of fluid mechanics branches 2

3 The fluid mechanics study involves many fields that have no clear boundaries between them. Researchers distinguish between orderly flow and chaotic flow as the laminar flow and the turbulent flow. The fluid mechanics can also be distinguished between a single phase flow and multiphase flow (flow made more than one phase or single distinguishable material). The last boundary (as all the boundaries in fluid mechanics)isn t sharp because fluid can go through a phase change (condensation or evaporation)in the middle or during the flow and switch from a single phase flow to a multiphase flow. Moreover, flow with two phases (or materials) can be treated as a single phase(for example, air with dust particle).after it was made clear that the boundaries of fluid mechanics aren t sharp, the study must make arbitrary boundaries between fields. Then the dimensional analysis can be used explain why in certain cases one distinguish area/principle is more relevant than the other and some effects can be neglected. For example, engineers in Software Company analyzed a flow of a complete still liquid assuming a complex turbulent flow model. Such absurd analyses are common among engineers who do not know which model can be applied. Thus, one of the main goals of this book is to explain what model should be applied. Before dealing with the boundaries, the simplified private cases must be explained. There are two main approaches of presenting an introduction of fluid mechanics materials. The first approach introduces the fluid kinematic and then the basic governing equations, to be followed by stability, turbulence, boundary layer and internal and external flow. The second 3

4 approach deals with the Integral Analysis to be followed with Differential Analysis, and continue with Empirical Analysis. Fluid mechanics one of the oldest braches of physics and the foundation for the understanding of many other aspects of applied sciences and engineering concerns itself with the investigation of the motion and equilibrium of fluids. It is wide spread interest in all most all fields of engineering as well as in Astro Physics, biology, bio medicine, metrology, Physical Chemistry, Plasma Physics and Geo Physics. The frontier of fluid dynamic research has been extended into the exotic regimes of hyper velocity flight and flow of electrically conducting fluids. This has introduce new fields of interest such has hypersonic flow and magneto fluid dynamics. In this connection it has become necessary to combine knowledge of thermodynamic, mass transfer, heat transfer, electromagnetic theory and fluid mechanics to fully understand the physical phenomena involved. Anything that occupies space is a Matter. Matter is divided into three state Solid, Liquid and Gas. The matters have defined shape in given thermodynamic condition and in the absence of external force are called Solids. If the matter takes the shape of the container is called Liquids. The liquid and gases collectively called Fluids. In a fluid the volume does not change are called incompressible fluids (liquids), the volume changes significantly are called compressible fluids (gases). The properties of fluids are of general importance to study of fluid mechanics. mass per unit volume is called Density, weight for unit volume is called Specific Weight, volume per unit mass of a fluid 4

5 is called Specific Volume, the ratio of specific weight of the fluid to that of standard fluid is called Specific Gravity, the property of fluid that the flow of fluid is called the Viscosity, the capacity to do work is called Energy, the energy of a particle possessed by virtue of its position is called Kinematic Energy, sharing stress of fluid element is directly proportional to the rate shearing strain is called Newton s Law of Viscosity, the amount of heat required to raise the temperature of unit mass of the medium by one degree is called Specific Heat, an incompressible fluid having no viscosity is called Ideal Fluid, the fluid which possess viscosity is called Real Fluid, any fluid which obey Newton s law of viscosity is called Newtonian Fluid, any fluid which does not obey Newton s law of viscosity is called Non Newtonian Fluid. Magnetohydrodynamics (MHD): We can describe scientifically the interaction of electromagnetic fields and fluids by the proper application of the principles of the special theory of relativity. The practical applications of these principles, in Physical Engineering, Astro Physics, Geo Physics etc, have become an important in recent years. The study of three applications to continuum is known as Magnetohydrodynamics (MHD) or Magneto fluid dynamics. The study of Magnetohydrodynamics (MHD) plays an important role in agriculture, engineering and petroleum industries. MHD has won practical applications, for instance, it may be used to deal with problems such as cooling of nuclear reactors by liquid sodium and 5

6 induction flow water which depends on the potential differencing the fluid direction perpendicular to the motion and goes to the magnetic field. The study of Magnetohydrodynamics (MHD) of viscous conducting fluids playing a significant role, owing to its practical interest and abundant applications, in Astro physical and Geo physical phenomena. Astro Physicsts and Geo Physicsts realized the importance of MHD in stellar and planetary processes. The main impetus to the engineering approach to the electromagnetic fluid interaction studies has come from the concept of the magnetohydrodynamics, direct conversion generator, ion propulsion study of flow problems of electrically conducting fluid, particularly of ionized gasses is currently receiving considerable interest. Such studies have made for years in connection with Astro Physical and Geo Physical problems such as sun spot theory, motion of the instellar gas etc Recently, some engineering problems need the study of the flow of an electrically conducting fluid, in ionized gas is called plasma. Many names have been used in referring to the study of plasma phenomena. Hartmann called it mercury dynamics, as he worked with mercury. Astro Physics called it comical electro dynamics and some called it magnetohydrodynamics. Physics and electrical engines commonly use the term plasma physics or plasma dynamics. The aerodynamist has spoken of magnetohydrodynamics. 6

7 Applications of Magnetohydrodynamics (MHD): MHD has applications in many areas. A few brief details are given below. 1. The Earth: The Earth The outer core of the Earth is composed primarily of molten iron. It is here that it is believed that the Earth's magnetic field is generated. Studying and solving the equations of MHD should permit us to explain such phenomena as the gradual change of the field with time and the infrequent and irregular reversals of the field. This is an area of very active current research. MHD can also be used to describe the ionosphere. 2. The Sun: Much of the Sun is composed of ionized hydrogen. For MHD there are two areas of interest. First there is the convection zone. In this, or just below it, the solar magnetic field is generated. The basic mechanism (interaction of a moving electrically conducting fluid with a magnetic field) is similar to that operating in the Earth's core but results in a rather different magnetic field. The Solar field reverses regularly on a 22 year cycle. Second, the Solar atmosphere (chromospheres and corona) is much less dense than the convection zone. Here, features such as flares and prominences can be observed and studied. One of the major problems to be explained is the heating of the corona which reaches temperatures of up to 10 6 K while the photosphere (the narrow region separating the convection zone from the chromospheres) is only at a few thousand degrees K. 7

8 3. Industry: Industry Here there are many applications. For example electromagnetic forces can be used to pump liquid metals (Example. in cooling systems of nuclear power stations) without the need for any moving parts. They can shape the flow of a molten metal and so aid controlling its shape once solidified, and can even levitate and heat a sample of metal to prevent any contact with (and consequent contamination from) a container. 4. Fusion: The goal of copying the Sun; releasing huge quantities of energy from the fusing of hydrogen into helium, has so far eluded us. No material can withstand the huge temperatures required. One promising way around this problem is to contain the ionized hydrogen in a magnetic container, so that there is no contact between the hydrogen and any material container. Progress continues but so far the temperatures and containment times achieved have fallen short of break even where the energy put in to the system equals the energy given out from fusion. Porous Medium: A porous medium can be defined as a material consisting of solid matrix with an interconnected void. In recent years, the investigation of flow of fluids through porous media has become an important topic due to the recovery of crude oil from the pores of reservoir rocks. Also the flow through porous medium is of interest in Chemical Engineering (absorption, filtration), Petroleum Engineering, Hydrology, Soil Physics, 8

9 Bio Physics and Geo Physics. With the growing importance of Non Newtonian fluids in modern technology and industries, the thermal instability, thermal solution instability and Rayleigh Taylor instability, problems of Walters (model B) fluid and stress fluid are desirable. The definition of porosity of the porous medium can be given as the ratio of pore volume to the total volume of a given sample of material. A complete graduation exisists from large force easily, accessible fluids to very small openings in minerals that are caused by minor lattice imperfection. Moisture equivalent, effective porosity, specific rententation, drainage coefficient of storage such as degree of saturation, forces applied to the sample, length of test, degree of interconnection of pores and fluid chemistry. Permeability of the porous medium is a measure of ease with which fluids pass through a porous material. The intrinsic permeability is an important property of the solid material and it is independent of the density and viscosity of the fluid. The permeability K can be defined as 1 Q h K Ag S (1) Where Q is the total discharge of the fluid, A is the crossectional area, is the viscosity, is the density, g is the acceleration due to gravity, h S is the hydraulic gradient in the direction of the flow. The dimension of the permeability is 2 L. The unit of permeability is named as Darcy who is extensively used in petroleum industry. The value of 9

10 one Darcy is C m. Permeability is very high with air and other non polar fluids. Boundary Conditions: 1. No slip condition: When the flow takes place over a rigid plate, the velocity component vanishes at the boundaries. This is called no slip condition. 2. Free surface boundary condition: Vertical component of velocity vanishes at a horizontal free surface. Further, if there is no surface tension, the free surface will be free form shear stress. 3. Beavers and Joseph slip condition: When a fluid flows, an imperable surface, the no slip condition is valid on the boundary. But when a fluid flows over a permeable surface, it is necessary to specify some condition on the tangential component of the velocity of the free fluid at the boundary within the permeable surface at the permeable interface. In this case, there will be a migration of the fluid tangential drag due to the transfer of forward momentum across the permeable interface. The velocity inside the permeable bed will be different from the velocity of the fluid past over the permeable bed. These two velocities are to be matched at the normal boundary (surface) of the permeable bed. The nominal boundary of a permeable bed is defined as a smooth geometric surface with the assumption that the outermost perimeters of all surface pores of the permeable material are in this 10

11 surface. Thus, if the surface is filled with solid material to the level of their respective perimeters a smooth rigid boundary of the assumed shape results. The Newtonian fluid flows past a permeable bed the no slip condition is not valid there. Firstly, Beavers and Joseph proved that there exisists a slip on the velocity at the surface of the porous bed. Thermal Radiation: The third mode of heat transmission due to electromagnetic wave propagation, which can occur in a total vacuum as well as in a medium. Thermal radiation is an important factor in the thermo dynamic analysis of many high temperature systems like solar connectors, boilers and furnaces. The simultaneous effects of heat and mass transfer in the presence of thermal radiation play an important role in manufacturing industries. For the design of fins, steel rolling, nuclear power plants, cooling of towers, gas turbines and various propulsion devices for aircraft, combustion and furnace design, materials processing, energy utilization, temperature measurements, remote sensing for astronomy and space exploration, food processing and cryogenic engineering, as well as numerous agricultural, health and military applications. Experimental evidence indicates that radiant heat transfer is proportional to the fourth power of the absolute temperature, where as conduction and convection are proportional to a linear temperature difference. The fundamental Stefan Boltzmann law is 4 ' q AT (2) 11

12 When thermal radiation strikes a body, it can be absorbed by the body, reflected from the body, or transmitted through the body. The fraction of the incident radiation which is absorbed by the body is called Absorptivity (symbol ). Other fractions of incident radiation which are reflected and transmitted are called reflectivity (symbol 1 ) and Transmissivity (symbol * )respectively. The sum of these fractions should be unity i.e. * 1. Free or Natural convection: Free or Natural convection is a mechanism, or type of heat transport, in which the fluid motion is not generated by any external source (like a pump, fan, suction device, etc.) but only by density differences in the fluid occurring due to temperature gradients. In natural convection, fluid surrounding a heat source receives heat, becomes less dense and rises. The surrounding, cooler fluid then moves to replace it. This cooler fluid is then heated and the process continues, forming convection current; this process transfers heat energy from the bottom of the convection cell to top. The driving force for natural convection is buoyancy, a result of differences in fluid 12

13 density. Because of this, the presence of a proper acceleration such as arises from resistance to gravity, or an equivalent force (arising from acceleration, centrifugal force or Coriolis effect), is essential for natural convection. For example, natural convection essentially does not operate in free fall (inertial) environments, such as that of the orbiting International Space Station, where other heat transfer mechanisms are required to prevent electronic components from overheating. Natural convection has attracted a great deal of attention from researchers because of its presence both in nature and engineering applications. In nature, convection cells formed from air raising above sunlight-warmed land or water are a major feature of all weather systems. Convection is also seen in the rising plume of hot air from fire, oceanic currents, and sea-wind formation (where upward convection is also modified by Coriolis forces). In engineering applications, convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat dissipation fins, and solar ponds. A very common industrial application of natural convection is free air cooling without the aid of fans: this can happen on small scales (computer chips) to large scale process equipment. Couette flow: In fluid dynamics, Couette flow refers to the free convection flow of a viscous fluid in the space between two parallel plates, one of which moving relative to the other. The flow is driven by virtue of viscous 13

14 drag force acting on the fluid and the applied pressure gradient parallel to the plates. This type of flow is named in honor of Maurice Marie Alfred Couette, a professor of physics at the French university of Angers in the late 19 th century. Couette flow is frequently used in undergraduate physics and engineering courses to illustrate shear driven fluid motion. Couette flows find widespread applications in geophysics, planetary sciences and also many areas of industrial engineering. For many decades engineers have studied such flows with and without rotation and also for both the steady case and unsteady case. Newtonian and non Newtonian flows with for example magnetic field effects and heat transfer have also been examined. Such studies have entailed many configurations including the flow between rotating plates, rotating concentric cylinders, etc. In rotating Couette flows a viscous layer at the boundary is instantaneously set into motion. 14

15 Hall Effect: The Hall Effect was discovered in 1879 by Edwin Herbert Hall while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland. His measurements of the tiny effect produced in the apparatus he was used an experimental tour de force, accomplished 18 years before the electron was discovered. When the strength of applied magnetic field is very strong, one cannot neglect the effect of hall currents. Due to the gyration and drift of charged particles, the conductivity parallel to the electric field is reduced and the current is induced in the direction normal to both electric and magnetic fields. This phenomenon is known as the Hall Effect. The Hall Effect is the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and a magnetic 15

16 field perpendicular to the current. The Hall Effect comes about due to the nature of the current in a conductor. Current consists of the movement of many small charge carriers, typically electrons, holes, ions (see Electromigration) or all three. Moving charges experience a force, called the Lorentz force, when a magnetic field is present that is perpendicular to their motion. When such a magnetic field is absent, the charges follow approximately straight, 'line of sight' paths between collisions with impurities, phonons, etc. However, when a perpendicular magnetic field is applied, their paths between collisions are curved so that moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element that is perpendicular to both the 'line of sight' path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electrical potential builds up for as long as the charge is flowing. It shall be noted that in the classical view, there are only electrons moving in the same average direction both in the case of electron or hole conductivity. This cannot explain the opposite sign of the Hall Effect observed. The difference is that electrons in the upper bound of the valence band have opposite group velocity and wave vector direction when moving, which can be effectively treated as if positively charged particles (holes) are moved in opposite direction than the electrons do. 16

17 For a simple metal where there is only one type of charge carrier (electrons) the Hall voltage V H is given by V H IB ned (3) The Hall coefficient is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made, since its value depends on the type, number and properties of the charge carriers that constitute the current. The Hall coefficient is defined as R H Ey J B (4) x In SI units, this becomes R H Ey J B dv IB 1 ne H (5) x 17

18 As a result, the Hall Effect is very useful as a means to measure either the carrier density or the magnetic field. One very important feature of the Hall Effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. The Hall Effect offered the first real proof that electric currents in metals are carried by moving electrons, not by protons. The Hall Effect also showed that in some substances (especially p type semiconductors), it is more appropriate to think of the current as positive "holes" moving rather than negative electrons. A common source of confusion with the Hall Effect is that holes moving to the left are really electrons moving to the right, so one expects the same sign of the Hall coefficient for both electrons and holes. This confusion, however, can only be resolved by modern quantum mechanical theory of transport in solids. It must be noted though that the sample in homogeneity might result in spurious sign of the Hall Effect, even in ideal vander Pauw configuration of electrodes. For example, positive Hall Effect was observed in evidently n type semiconductors. The Hall Effect is a conduction phenomenon which is different for different charge carriers. In most common electrical applications, the conventional current is used partly because it makes no difference whether you consider positive or negative charge to be moving. But the Hall voltage has a different polarity for positive and negative charge carriers and it has been used to study the details of conduction in semiconductors and other materials which show a combination of negative and positive charge carriers. 18

19 The Hall Effect can be used to measure the average drift velocity of the charge carriers by mechanically moving the Hall probe at different speeds until the Hall voltage disappears, showing that the charge carriers are now not moving with respect to the magnetic field. Other types of investigations of carrier behaviour are studied in the quantum Hall Effect. The effect of hall currents on the fluid with variable concentration has a lot of applications in MHD power generators, several astrophysical and meteorological studies as well as in flow of plasma through MHD power generators. From the point of applications, this effect can be taken into account within the range of magnetohydrodynamical approximation. Hall probes are often used as magnetometers, i.e. to measure magnetic fields, or inspect materials (such as tubing or pipelines) using the principles of magnetic flux leakage. Hall Effect devices produce a very low signal level and thus require amplification. While suitable for laboratory instruments, the vacuum tube amplifiers available in the first half of the 20 th century were too expensive, power consuming and unreliable for everyday applications. It was only with the development of the low cost integrated circuit that the Hall Effect sensor became suitable for mass application. Many devices now sold as Hall Effect sensors in fact contain both the sensor as described above plus a high gain integrated circuit ( IC ) amplifier in a single package. Recent advances have further added into one package an analogue to digital converter and I 2 C (Inter integrated circuit communication protocol) IC for direct connection to a microcontroller's I / O port. 19

20 Basic equations in vector form: The basic equations in vector form of an unsteady incompressible viscous, electrically conducting fluid are given as follows. 1. Continuity Equation: The continuity equation is. ( q) 0 t (6) Where is the fluid density, q is the fluid velocity vector. is the rate of increase of the density in control volume. t. ( q) is the rate of mass flux passing out of the control surface (which surrounds control volume) per unit volume. Where q ui vj wk is the velocity of the fluid. 2. Momentum Equation: The momentum equation is q p ( q. ) q g t 2 q q K J B (7) Where q t ( q. ) q is the inertia forces, p is the pressure gradient, J B is the Lorentz force per unit volume, B is the magnetic induction vector, g is the acceleration due to gravity, 2 q is the viscous flow, 20

21 q K is the porous media. 3. Energy Equation: The energy equation is T t 2 k 2 J ( q. ) T T Q T T (8) C p Where T is the temperature, T is the temperature in the free stream, C is the specific heat at constant pressure ( J. Kg 1. K), p is the density, is the viscous dissipation per unit volume, 2 J is the ohmic dissipation per unit volume, J is the current conduction, is the electrical conductivity, Q is the heat source. 4. Species Diffusion Equation: The species diffusion or species concentration equation is C t 2 C D C 2 ( q. ) C D C K C T (9) Where D T is the chemical diffusivity, C is the species concentration, C is the species concentration in free stream, 2 1 D is the chemical molecular diffusivity ( m. S ), K is chemical reaction. 21

22 5. Maxwell Equations: The Maxwell s equations in RMKS are. D l (Coulomb s law) (10) D is the displacement.. B 0 (Absence of free magnetic poles) (11) B is the local magnetic field B (Faraday s law) (12) t D H J (Ampere s law) (13) t J E q B q (Ohm s law) (14) l The current conservation equation is. J 0 (15) t J is the conduction current. Non Dimensional Parameters: Every physical problem involved some physical quantities, which can be measured in different units. But the physical problem itself should not depend on the unit used for measuring these quantities. In dimensional analysis of any problem we write down the dimensions of each physical quantity in term of fundamental units. Then by dividing and rearranging the different units, we get some non dimensional numbers. Dimensional analysis of any problem provides information on qualitative behaviors of the physical problem. The dimensionless parameter helps us to understand the physical significance of a 22

23 particular phenomenon associated with the problem. There are usually two general methods for obtains dimensionless parameters. 1. The inspection analysis 2. The dimensionless analysis In this thesis the latter method has been used. In this method the basic equations are made dimensionless using certain dependent and independent characteristic values. In this processes certain dimensionless numbers appears as the some of the dimensionless parameters used in this thesis are explained below. Grashof number for heat transfer (Gr) : The Grashof number is usually occurring in free convection heat transfer problems. This gives the relative importance of buoyancy force to the viscous forces. This number is defined as: Gr(Grashof number) = g T T V w 3 o Modified Grashof number for mass transfer (Gc) : The Modified Grashof number is usually occurring in natural convection mass transfer problems. It is defined as Gc (Modified Grashof number) = * g C V w 3 o C Prandtl number (Pr): Prandtl number is the ratio of viscous forces to the thermal forces. It is a measure of the relative importance of heat conduction and viscosity of the fluid. The Prandtl number, like the viscosity and thermal conductivity, is a material property and it thus varies from fluid to 23

24 fluid. Usually Prandtl number is large when thermal conductivity is small and viscosity is large, and small when viscosity is small and thermal conductivity is large. It is defined as Pr (Prandtl number) = c p c p Thus it gives the relative importance of viscous dissipation to the thermal dissipation. Usually for gases Prandtl number is of the order of unity and for the liquids the Prandtl number is large. Schmidt number (Sc) : Schmidt number is a dimensionless number defined as the ratio of momentum diffusivity (viscosity) and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. It physically relates the relative thickness of the hydrodynamic layer and mass transfer boundary layer. It is defined as Sc (Schmidt number) 24 V o D Hartmann number (or) Magnetic parameter (M ) : The dimensionless quantity denoted by M is known as the Hartmann number. It was first introduced by Hartmann in 1930, in the study of the plane Poiseuille flow of an electrically conducting fluid in the presence of transverse magnetic field, where the important forces are magnetic and viscous force. Therefore, M = Magnetic force/viscous force and mathematically 2 B o v defined as M 2 v o

25 Hall parameter (m) : The Hall parameter (m) in plasma is the ratio between the electron gyro frequency ( e ) and the electron heavy particles collision frequency ( ). Mathematically, it is defined as e e B m me Eckert number (Ec) : It is equal to the square of the fluid far from the body divided by the product of the specific heat of the fluid at constant temperature and the difference between the temperatures of the fluid and the body. It is denoted Ec by and mathematically defined as Ec C p 2 v o T T w Reynold s number (Re) : In fluid mechanics, the Reynold s number (Re) concept was introduced by George Gabriel Stokes in 1851,but the Reynolds number is named after Osborne Reynolds ( ), who popularized its use in The Reynold s number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions (or) The ratio between total momentum transfer and molecular momentum transfer is Reynolds number. It is defined as U x Re o Darcy number () : In fluid mechanics, Darcy number () is a non dimensional number used in the study of the flow of fluids in 25

26 porous media, equal to the fluid velocity times the flow path divided by 2 2 the permeability of the medium. It is defined as 2 kw Skin friction ( ) : The dimensionless shearing stress on the surface of a body, due to a fluid motion, is known as skin friction and is u defined by the Newton s law of viscosity is given by x. y We can calculate the shearing stress component in dimensionless form 0 as x V x 2 o u y y0 Rate of heat transfer (or) Nusselt number (Nu) : The heat transfer co efficient is generally known as Nusselt number (Nu) is the ratio of the heat flow by convection process under a unit temperature gradient to the heat flow by conduction under a unit temperature gradient through a stationary thickness of meter. We can calculate the dimensionless coefficient of heat transfer as follows T y Nu x T T w y0 Nu Re 1 x y y0 Rate of mass transfer (or) Sherwood number (Sh) : The mass transfer coefficient is generally known as Sherwood number (Sh) is a diffusion rate constant that relates the mass transfer rate, mass transfer area and concentration gradient as driving force. It is defined as C y Sh x C C w y0 Sh Re 26 1 x C y y0

27 The objective of chapter 1 is to find the numerical solution of unsteady magnetohydrodynamic free convective Couette flow of viscous incompressible fluid confined between two vertical permeable parallel plates in the presence of thermal radiation is performed. A uniform magnetic field which acts in a direction orthogonal to the permeable plates, and uniform suction and injection through the plates are applied. The magnetic field lines are assumed to be fixed relative to the moving plate. The momentum equation considers buoyancy forces while the energy equation incorporates the effects of thermal radiation. The fluid is considered to be a gray absorbing emitting but non scattering medium in the optically thick limit. The Rosseland and approximation is used to describe the radiative heat flux in the energy equation. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. The non linear coupled pair of partial differential equations are solved by an efficient Crank Nicholson method. With the help of graphs, the effects of the various important flow parameters entering into the problem on the velocity, temperature and concentration fields within the boundary layer are discussed. Also the effects of these flow parameters on skin friction coefficient and rates of heat and mass transfer in terms of the Nusselt and Sherwood numbers are presented numerically in tabular form. Jha and Apere [7] extended the work of Jha [4] by considering the unsteady MHD free convection Couette flow between two vertical parallel porous plates with uniform suction and injection. The cases 27

28 where the magnetic field is considered fixed relative to the fluid and fixed relative to the moving plate were considered. The velocity and temperature distributions were obtained using the Laplace transform technique. The results revealed that both temperature and velocity decrease with increasing Prandtl number and with increasing suction/injection parameter. The effect of magnetic field strength on the velocity is consistent with the results obtained in [3] and [5]. The velocity has also been found to increase with increasing Grashof number. An early study of unsteady Couette flow was reported by Vidyanidhi and Nigam [8] who studied the viscous flow between rotating parallel plates under constant pressure gradient. Verma and Sehgal [9] used the micropolar flow model to obtain analytical solutions for the Couette flow of fluids which can support couple stresses and distributed body couples. Liu and Chen [10] investigated computationally the transient rotating Couette flow problem. Jana and Datta [11] studied the steady Couette flow of a viscous incompressible fluid between two infinite parallel plates, one stationary and the other moving with uniform velocity in a rotating frame of reference. Heat transfer rates were shown to decrease with an increase in rotation parameter. Mandal and Mandal [13] obtained analytical solutions for the effects of magnetic field and Hall currents on rotating parallel plate Couette flow. They are also studied the cases, where the plates have arbitrary conductivity and thickness. The transient dusty suspension Couette flow problem was studied by Kythe and Puri [14]. Singh et al. [15] obtained closed form solutions for velocity and skin friction for 28

29 rotating hydromagnetic Couette flow, showing that the Ekman number decreases primary velocities but boosts the secondary velocity values. The converse effect was reported for the magnetic parameter (Hartmann number). Other studies of rotating Couette flows include those by Ghosh [18] who considered magnetic field effects, Hayat et al. [19] who studied non Newtonian visco elastic hydromagnetic fluids, Choi et al. [20] who reported on free convection effects who considered the transient Couette flow in a rotating infinitely long parallel plate system. These studies were all confined to purely fluid regimes. Chauhan and Rastogi [30] analyzed the effects of thermal radiation, porosity and suction on unsteady convective hydromagnetic vertical rotating channel. Ibrahim and Makinde [31] investigated radiation effect on chemically reaction MHD boundary layer flow of heat and mass transfer past a porous vertical flat plate. Pal and Mondal [32] studied the effects of thermal radiation on MHD Darcy Forchheimer convective flow pasta stretching sheet in a porous medium. Palani and Kim [33] analyzed the effect of thermal radiation on convection flow past a vertical cone with surface heat flux. Recently, Mahmoud and Waheed [34] examined thermal radiation on flow over an infinite flat plate with slip velocity. The effects of thermal radiation and heat source/sink on the natural convection in unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid confined between two vertical parallel plates with constant heat flux at one boundary are analyzed by Rajput and Sahu [35]. The magnetic lines of force are assumed to be fixed relative to the 29

30 moving plate. In deriving the governing equations, a temperature dependent heat source/sink term is employed and the Rosseland approximation for the thermal radiation term is assumed to be valid. The non dimensional governing equations involved in the present analysis are solved analytically, to the best possible extent. Main purpose of this chapter is to find the numerical solution of unsteady magnetohydrodynamic free convective Couette flow of viscous incompressible fluid confined between two vertical permeable parallel plates in the presence of thermal radiation is performed. A uniform magnetic field which acts in a direction orthogonal to the permeable plates and uniform suction and injection through the plates are applied. The magnetic field lines are assumed to be fixed relative to the moving plate. The momentum equation considers buoyancy forces while the energy equation incorporates the effects of thermal radiation. The fluid is considered to be a gray absorbing emitting but non scattering medium in the optically thick limit. The Roseland s approximation is used to describe the radiative heat flux in the energy equation. The non linear coupled pair of partial differential equations are solved by an efficient Crank Nicholson method which is more economical from computational point of view. The resulting system of equations are solved to obtain the velocity and temperature distributions. These solutions are useful to gain a deeper knowledge of the underlying physical processes and it provides the possibility to get a benchmark for numerical solvers with reference to basic flow configurations. 30

31 Chapter 2 investigates the effect of thermal radiation on an unsteady magnetohydrodynamic free convective oscillatory Couette flow of an optically, viscous thin fluid bounded by two horizontal porous parallel walls under the influence of an external imposed transverse magnetic field embedded in a porous medium. The fluid is considered to be a gray, absorbing emitting but non scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The non dimensional governing coupled equations involved in the present analysis are solved by an efficient, accurate, and extensively validated and unconditionally stable finite difference scheme of the Crank Nicholson method and the expressions for velocity, temperature, Skin friction and rate of heat transfer has been obtained. Numerical results for velocity and temperature are presented graphically and the numerical values of Skin friction and Nusselt number have been tabulated. The effect of different parameters like thermal Grashof number, Magnetic field (Hartmann number), Prandtl number, Porosity parameter and Thermal radiation parameter on the velocity, temperature, Skin friction and Nusselt number are discussed. Sharma and Pareek [40] explained the behaviour of steady free convective MHD flow past a vertical porous moving surface. Singh and his co workers [41] have analyzed the effect of heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Makinde et al. [42] discussed the unsteady free convective flow with suction on an accelerating porous plate. 31

32 Sarangi and Jose [43] studied the unsteady free convective MHD flow and mass transfer past a vertical porous plate with variable temperature. Das and his associates [44] estimated the mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction employing finite difference analysis. Das et al. [45] investigated numerically the unsteady free convective MHD flow past an accelerated vertical plate with suction and heat flux. Das and Mitra [46] discussed the unsteady mixed convective MHD flow and mass transfer past an accelerated infinite vertical plate with suction. Bestman and Adjepong [52] studied the magnetohydrodynamic free convection flow, with radiative heat transfer, past an infinite moving plate in rotating incompressible, viscous and optically transparent medium. Das et al. [53] have analyzed radiation effects on flow past an impulsively started infinite isothermal vertical plate. Raptis and Perdikis [54] considered the effects of thermal radiation and free convection flow past a moving vertical plate. The governing equations were solved analytically. Ghaly and Elbarbary [58] have investigated the radiation effect on MHD free convection flow of a gas at a stretching surface with a uniform free stream. In all the above studies, only steady state flows over a semi infinite vertical plate have been considered. The unsteady free convection flows over a vertical plate has been studied by Gokhale [59] and Muthucumaraswamy and Ganesan [60]. Bejan and Khair [64] have investigated the vertical free convective boundary layer flow embedded in a porous medium resulting from the combined heat and mass transfer. Lin and Wu [65] were analyzed the problem of 32

33 simultaneous heat and mass transfer with the entire range of buoyancy ratio for most practical and chemical species in dilute and aqueous solutions. Rushi Kumar and Nagarajan [66] studied the mass transfer effects of MHD free convection flow of an incompressible viscous dissipative fluid past an infinite vertical plate. Mass transfer effects on free convection flow of an incompressible viscous dissipative fluid have been studied by Manohar and Nagarajan [67]. Choi et al. [68] studied the buoyancy effects in plane Couette flow heated uniformly from below with constant heat flux. Attia and Sayed Ahmed [69] investigated the problem of the effect Hall currents on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection. The effectiveness of variation in the physical variables on the generalized Couette flow with heat transfer in presence of porous medium studied by Attia [70]. Makinde and Osalusi [71] considered the problem of MHD steady flow in a channel filled with porous material with slip at the boundaries, while, Narahari [72] studied the effects of thermal radiation and free convection currents on the unsteady Couette flow between two vertical parallel plates with constant heat flux at one boundary. Israel Cookey et al. [73] discussed oscillatory magnetohydrodynamic Couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature. The object of this chapter is to analyze the effect of thermal radiation on an unsteady magnetohydrodynamic free convective oscillatory Couette flow of an optically, viscous thin fluid bounded by two horizontal porous parallel walls under the influence of an external 33

34 imposed transverse magnetic field embedded in a porous medium. The fluid is considered to be a gray, absorbing emitting but non scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The non dimensional governing coupled equations involved in the present analysis are solved by an efficient, accurate, and extensively validated and unconditionally stable finite difference scheme of the Crank Nicholson method which is more economical from computational view point. The effects of various governing parameters on the velocity, temperature, skin friction coefficient and Nusselt number are shown in figures and tables and discussed in detail. From computational point of view it is identified and proved beyond all doubts that the Crank Nicholson method is more economical in arriving at the solution and the results obtained are good agreement with the results of Israel Cookey et al. [73] in some special cases. Chapter 3 is an investigation on the non linear problem of the effect of Hall current on the unsteady magnetohydrodynamic free convective Couette flow of incompressible, electrically conducting fluid between two permeable plates is carried out, when a uniform magnetic field is applied transverse to the plate, while the thermal radiation, viscous and Joule s dissipations are taken into account. The fluid is considered to be a gray, absorbing emitting but non scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The dimensionless governing coupled, non linear boundary layer partial differential equations are 34

35 solved by an efficient, accurate, and extensively validated and unconditionally stable finite difference scheme of the Crank Nicholson method. The effects of thermal radiation and Hall current on primary and secondary velocity, skin friction and rate of heat transfer are analyzed in detail for heating and cooling of the plate by convection currents. Physical interpretations and justifications are rendered for various results obtained. Hellums and Churchill [76], using an explicit finite difference method. Because the explicit finite difference scheme has its own deficiencies, a more efficient implicit finite difference scheme has been used by Soundalgekar and Ganesan [77]. A numerical solution of transient free convection flow with mass transfer on a vertical plate by employing an implicit method was obtained by Soundalgekar and Ganesan [78]. Hossain et al. [88] analyzed the influence of thermal radiation on convective flows over a porous vertical plate. Seddeek [89] explained the importance of thermal radiation and variable viscosity on unsteady forced convection with an align magnetic field. Muthucumaraswamy and Senthil [90] studied the effects of thermal radiation on heat and mass transfer over a moving vertical plate. Pal [91] investigated convective heat and mass transfer in stagnation point flow towards a stretching sheet with thermal radiation. Aydin and Kaya [92] justified the effects of thermal radiation on mixed convection flow over a permeable vertical plate with magnetic field. Mohamed [93] studied unsteady MHD flow over a vertical moving porous plate with heat generation and Soret effect. Chauhan and 35

36 Rastogi [94] analyzed the effects of thermal radiation, porosity, and suction on unsteady convective hydromagnetic vertical rotating channel. Ibrahim and Makinde [95] investigated radiation effect on chemically reaction MHD boundary layer flow of heat and mass transfer past a porous vertical flat plate. Pal and Mondal [96] studied the effects of thermal radiation on MHD Darcy Forchheimer convective flow pasta stretching sheet in a porous medium. Gebhart [99] has shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate. Gebhart and Mollendorf [100] have considered the effects of viscous dissipation for external natural convection flow over a surface. Soundalgekar [101] has analyzed viscous dissipative heat on the two dimensional unsteady free convective flow past an infinite vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Maharajan and Gebhart [102] have reported the influence of viscous dissipation effects in natural convective flows, showing that the heat transfer rates are reduced by an increase in the dissipation parameter. Israel Cookey et al. [103] have investigated the influence of viscous dissipation and radiation on an unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Suneetha et al. [104] have analyzed the effects of viscous dissipation and thermal radiation on hydromagnetic free convective flow past an impulsively started vertical plate. 36

37 Katagiri [112] has studied the effect of Hall currents on the magnetohydrodynamic boundary layer flow past a semi infinite flat plate. Hall effects on hydromagnetic free convection flow along a porous flat plate with mass transfer have been analyzed by Hossain and Rashid [113]. Hossain and Mohammad [114] have discussed the effect of Hall currents on hydromagnetic free convection flow near an accelerated porous plate. Pop and Watanabe [115] have studied the Hall effects on the magnetohydrodynamic free convection about a semi infinite vertical flat plate. Hall effects on magnetohydrodynamic boundary layer flow over a continuous moving flat plate have been investigated by Pop and Watanabe [116]. Sharma et al. [117] have analyzed the Hall effects on an MHD mixed convective flow of a viscous incompressible fluid past a vertical porous plate immersed in a porous medium with heat source/sink. Effects of Hall current and heat transfer on the flow in a porous medium with slip condition have been described by Hayat and Abbas [118]. Guria et al. [119] have investigated the combined effects of Hall current and slip condition on unsteady flow of a viscous fluid due to non coaxial rotation of a porous disk and a fluid at infinity. Hall currents in MHD Couette flow and heat transfer effects have been investigated in parallel plate channels with or without ion slip effects by Soundalgekar et al. [129], Soundalgekar and Uplekar [130] and Attia [131]. Hall effects on MHD Couette flow between arbitrarily conducting parallel plates have been investigated in a rotating system by Mandal and Mandal [132]. The same problem of MHD Couette flow rotating flow in a rotating system 37