The branch of mechanics that deals with bodies at rest as well as in motion. The

Transcription

1 Chapter 1 Preliminaries 1.1 Introduction to Fluid Mechanics The branch of mechanics that deals with bodies at rest as well as in motion. The subcategory of fluid mechanics is defined as the science that deals with the behavior of fluids at rest or in motion. Fluid mechanics itself is also divided into several categories. The study of the motion of fluids that can be approximated as incompressible such as liquids, especially water, and gases at low speed is usually referred to as hydrodynamics. A subcategory of hydrodynamics is hydraulics, which deals with liquid flows in pipes and open channels. Gas dynamics deals with the flow of fluids that undergo significant density changes, such as the flow of gases through nozzles at high speeds. The category aerodynamics deals with the flow of gases over bodies such as aircraft, rockets, and automobiles at high or low speeds. Some other specialized categories such as meteorology, oceanography and hydrology deal with naturally occurring flows. 1

2 Fluid mechanics is widely used both in everyday activities and in the design of modern engineering systems from vacuum cleaners to supersonic aircraft. For example, fluid mechanics plays a vital role in the human body. The heart is constantly pumping blood to all parts of the human body through the arteries and veins, and the lungs are the sites of airflow in alternating directions. All artificial hearts, breathing machines, and dialysis systems are designed using fluid dynamics. Fluid mechanics is also used in the design of the heating and air-conditioning system, the hydraulic brakes, the power steering, the automatic transmission, the lubrication systems, the cooling system of the engine block including the radiator and the water pump, and even the tires. The sleek streamlined shape of recent model cars is the result of efforts to minimize drag by using extensive analysis of flow over surfaces. 1.2 Types of Fluids Fluid: Fluid is a substance that continually deforms (flows) under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases and plasmas. Although the term fluid includes both the liquid and gas phases, in common usage, fluid is often used as a synonym for liquid. A portion of matter is called a liquid, if its expansion, when subjected to some external force or an 2

3 increase in temperature, is not much that is, the liquid have definite volume which changes slightly when subjected to external force or temperature difference. Thus a liquid may not occupy whole of the space of the container. In gases, the volume of a gas changes significantly when subjected to external force or temperature difference or a change in the pressure, and so, it does not have a definite volume. Thus a portion of matter is called gas if it has no definite volume, and occupies the whole of the space of the container. As there is no definite volume for gases and plasmas both are usually considered as gases. But plasma is characterized by an ionized state of the matter, it may be present even in liquids. Thus there is a distinction between plasmas and gases. Here again it is to be pointed out that the division between, gases and plasmas is not a sharp one. The fluids can be classified as; Ideal fluids Real fluids Ideal fluid or Inviscid fluid An Ideal fluid is one, which has no property other than density. No resistance is encountered when such a fluid flows or Ideal fluids or Inviscid fluids are those fluids in 3

4 which two contacting layers experience no tangential force (shearing stress) but act on each other with normal force (pressure) when the fluids are in motion. This is equivalent to stating that inviscid fluid offers no internal resistance to change in shape. The pressure at every point of an ideal fluid is equal in all directions, whether the fluid is at rest or in motion. Inviscid fluids are also known as effect fluids or frictionless fluids. In true sense, no such fluid exists in nature. The assumption of ideal fluids helps in simplifying the mathematical analysis. However fluids which have low viscosities such as water and air can be treated as ideal fluids under certain conditions Viscous fluid or Real fluid Viscous fluid or real fluid are those, which have viscosity, surface tension and compressibility in addition to the density or viscous fluid or real fluid are those when they are in motion the two contacting layers of those fluids experience tangential as well as normal stresses. The property of exerting tangential of shearing stress and normal stress in a real fluid when the fluid is in motion is known as viscosity of the fluid. In viscous fluid internal friction plays an important role during the motion of the fluid. One of the important characteristics of viscous fluid is that it offers internal resistance to motion of the fluid, viscosity, being the characteristic of the real fluids, exhibits a certain 4

5 resistance to alter the form also. Viscous or real fluids are classified into Newtonian fluids and non Newtonian fluids Newtonian fluid To understand the concept of Newtonian fluid, let us consider a thin layer of fluid between two parallel plates at distance dy. Here one plate is fixed and a shearing force F is applied to the other. When conditions are steady the force F is applied to the other and balanced by an internal force in the fluid due to its viscosity. Newton, while discussing the properties of fluid, remarked that in a simple rectilinear motion of a fluid two neighboring fluid layers, one moving over the 5

6 other with some relative velocity, will experience a tangential force proportional to the relative velocity between the two layers and inversely proportional to the distance between the layers, that is if the two neighboring fluid layers are moving with velocities u and u + δu are at a distance δy, then the shearing stress, is given by τ u y or τ = μ u y (1.2.1) This is called Newtonian hypothesis and a fluid satisfying this hypothesis is called a Newtonian fluid. It is clear from the Newton s law that If τ = 0 then μ = 0, equation (1.2.1) will represent an ideal fluid. If u = 0 then μ = 1, equation (1.2.1) will represent an elastic bodies. y A fluid for which the constant of proportionality μ does not change with rate of deformation (shear strain u u ) is said to be an Newtonian fluid and graph τ verses is a y y straight line. Where μ is known as Newtonian viscosity. It will be seen that μ is the tangential force per unit area exerted on layers of fluid a unit distance apart and having a unit velocity difference between them. The diagram relating shear stress and rate of shear for Newtonian fluids represents flow 6

7 curve of the type straight line. The example of Newtonian fluid are water, air, mercury, benzene, ethyl alcohol, glycerin and oil etc., Non-Newtonian fluid Non-Newtonian fluids are those fluids which do not obey Newtonian law. It can also be stated as the non-newtonian fluids are those for which flow curve is not linear, i.e., the viscosity of a non-newtonian fluid is not constant at a given temperature and pressure but depends on other factors such as the rate of shear in the fluid. The typical examples of these classes of fluids are paints, coal-tar, polymer solutions, condensed milk, paste, lubricants, honey, plastics, molasses, molten rubber, printer ink, collides, macro/molecular materials and so on. 7

8 1.2.5 Compressible and Incompressible fluid Fluids undergo density changes when temperature and pressure variations occur in them then they are consider compressible. For several flows situations, however density changes are negligible and the fluid may be treated as incompressible. Therefore, flow of liquids are treated as incompressible for small pressure and temperature variations. 8

9 1.2.5 Ferro fluids A magnetic colloid, also known as a ferrofluid, is a colloidal suspension of singledomain magnetic particles, with typical dimensions of about 10 nm, dispersed in a liquid carrier. The liquid carrier can be polar or nonpolar. Since the nineteen sixties, when these materials were initially synthesized, their technological applications did not stop to increase. Ferrofluids are different from the usual magnetorheological fluids (MRF) used for dampers, brakes and clutches, formed by micron sized particles dispersed in oil. In MRF the application of a magnetic field causes an enormous increase of the viscosity, so that, for strong enough fields, they may behave like a solid. On the other hand, a Ferro fluid keeps its fluidity even if subjected to strong magnetic fields (~ 10 kg). Ferrofluids are optically isotropic but, in the presence of an external magnetic field, exhibit induced birefringence. Wetting of particular substrates can also induce birefringence in thin Ferro fluid layers.in order to avoid agglomeration, the magnetic particles have to be coated with a shell of an appropriate material. According to the coating, the Ferro Fluid s are classified into two main groups: Surfacted ferro fluids Ionic ferro fluids 9

10 1.3 Some Important Types of Flow Steady and Unsteady Flow A flow in which properties and conditions of fluid motions do not change with change of time such that the flow pattern is not affected with time iscalled steady, i.e φ t = 0, where φ may be velocity, temperature, pressure, density etc. On the other hand, if the flow pattern depends upon time, it is called unsteady flow Laminar and Turbulent Flow A flow in which each fluid particle traces out a definite curve and curves traced out by any two different particle do not intersect, is said to be laminar. On the other hand, a flow, in which each fluid particle does not trace out a definite curve and the curves traced out by fluid particle intersect, is said to be turbulent flow. The most of the flows, which occur in practical applications are turbulent, and this term denotes a motion in which an irregular fluctuation (mixing, or eddying motion) is superimposed on the main stream 10

11 1.3.3 Rotational and Irrotational Flow The flow in which the fluid particles rotate about their own axis is called rotational and the flow in which the fluid particle does not rotate about their own axis is called irrotational. Mathematically, If q = 0 irrotational flow. If q 0 rotational flow Uniform and Non-uniform Flow A flow in which the velocity of fluid particles are equal at each section of the channel is called uniform flow and a flow in which the velocities of fluid particles are different at each section of the channel is called non-uniform flow One and Two-Dimensional Flow Considerable simplification in analysis may often be achieved, however, by selecting the coordinate directions so that appreciable variation of the parameters occurs only two directions, or even in only one. 11

12 One-dimensional flow is that in which all the flow parameters may be expressed as functions of time and one space coordinate only. Example: Flow in a pipe. In two-dimensional flow, the flow parameters are functions of time and two rectangular space coordinates (say x and y) only, there is no variation in z-direction. Examples: Flow between parallel planes, water flow over a weir of uniform cross section and infinite width Magnetohydrodynamic Flow (MHD): The flow of electrically-conducting fluid under influence of applied magnetic field is called MHD flow. In other words, Magnetohydrodynamics is the study of the motion of an electrically conducting fluid in the presence of external electromagnetic field. It is the combination of two branches viz., hydrodynamics and electromagnetism. The dictionary meaning of hydro is water but hydrodynamics includes study dynamical behaviour of electrically conducting medium, which may be a liquid or an ionized gas in presence of magnetic field. Both plasma and conducting fluids are related in common theory by assuming plasma as a continuous fluid for which the kinetic theory of gases still holds true. In MHD induced electric current produces mechanical force, which in turn modified 12

13 the motion in the fluid. Hence, study of electrically conducting fluid flow in the presence of transverse magnetic field assures significance. In 1937, Hartmann studied the motion of electrically conducting fluids in presence of magnetic field. Chapman and Ferraro developed theory of magnetic storms during Electromagnetic forces will be generated which may be of the same order of magnitude as the hydrodynamical and inertial forces in the case when the conductor is either a liquid or a gas. Thus, the equation of motion will have to take these electromagnetic forces into account as well as the other forces. The science that treats these phenomena is magnetohydrodynamics (MHD). Other variants of nomenclature are hydromagnetics, magneto-fluid dynamics, magneto-gas dynamics etc. As we know that MHD is relatively new but important branch of fluid dynamics. It is concerned with the interaction of electrically conducting fluid and electro magnetic fields, such interaction occurs both in nature and in new man-made device. The study of uniform magnetic field on the motion of a electrically conducting fluid over a stretching sheet find its application in various engineering disciplines such as polymer technology, where one deals with stretching plastic sheet and metallurgy, where hydro magnetic techniques have recently been used. The important application of MHD flow in metallurgy is the purification process of molten metal from non-metallic inclusion using magnetic field. 13

14 1.4 Boundary-layer theory The concept of boundary layer, which was introduced by a German scientist Ludwig Prandtl in the year 1904, provides a link between the ideal flow and the real flow. It is the great achievement of Ludwig Prandtl, at the beginning of this century, set forth the way in which these two diverging directions of fluid mechanics could be unified. In the case of real fluids, however small their viscosity may be the fluid particles adhere to the boundary and hence the condition of no-slip prevails. The fluid velocity at the stationary boundary vanishes, whereas the fluid adhering to the boundary will have the same velocity as boundary itself, if the boundary is moving. Further away from the boundary there exists a velocity gradient u y normal to the boundary and the fluid exerts a shear or tangential in the direction of motion. The force caused by this shear stress in the direction of motion is known as surface drag. The boundary in turn exists a shear resistance to the flow i.e., a force on the fluid which is equal to magnitude and opposite in direction to the surface drag. Due to no-slip condition, the fluid is always retardedat or near the boundary. The retardation due to the presence of viscosity is negligible and the velocity remains constant far away from the boundary. The transition from zero velocity at the boundary to the full magnitude at far away from the boundary takes place in a very thin layer. The fluid layer which has its velocity 14

15 affected by boundary shear is called the boundary layer. The boundary layer was developed mainly for the laminar flow of an incompressible fluid and it was extended to the practically important turbulent compressible boundary layer flows. The boundary layer theory can be used to describe the flow through the blade cascades in compressors and turbines as well as through diffusers and nozzles Viscous boundary layer The influence of boundary, due to no slip condition is confined to a very thin region in the immediate neighborhood of the solid surface, known as viscous or momentum boundary layer. In this thin layer there is rapid change in velocity of the fluid, from velocity of the surface to its value that corresponds to external frictionless flow. This is the viscosity of the fluid that gives rise to the boundary layer and for an inviscid fluid there exists no boundary layer. For flow over a surface of finite or semi infinite length the thickness of boundary layer increase in the down stream region. The thickness of the boundary layer decreases with decrease in viscosity of the medium, but even for small viscosity, the shear stress τ w = μ u is important due to large velocity gradient. y 15

16 The injection or suction across the porous surface also has a great effect on the size of the viscous boundary layer Thermal boundary layer A thin region in which the temperature of the fluid particles changes from its free stream value to body surface value is called Thermal Boundary Layer. The thermal boundary layer strongly depends upon the thermal conductivity of the medium i.e., higher the conductivity of the medium, thicker would be the thermal boundary layer. Like viscous boundary layer, injection or suction across the porous surface also has a great effect on the size of the thermal boundary layer. 1.5 Boundary conditions Slip condition: If the boundary is rough, impermeable and moving then normal velocity is zero but the tangential velocity is not zero (v = 0, u is nonzero) then slip condition exists No-Slip condition: When a viscous fluid flows over a solid surface, the fluid elements adjacent to the surface attain the velocity of the surface; in other words, the relative velocity between 16

17 the solid surface and the adjacent fluid particles is zero. This phenomenon is known as the no-slip condition. For example, we know that water in a river cannot flow through large rocks, and must go around them. That is, the water velocity normal to the rock surface must be zero, and water approaching the surface normally comes to a complete stop at the surface. What is not as obvious is that water approaching the rock at any angle also comes to a complete stop at the rock surface, and thus the tangential velocity of water at the surface is also zero. 1.6 Stretching Sheet Stretching sheet is the sheet being stretched with the stretching velocity Uw(x) along the x-axis, keeping the origin fixed in the fluid of ambient temperature T. Thus the sheet can be stretched linearly, non-linearly or exponentially. For illustrations, consider the flow of a fluid past a flat sheet coinciding with the plane y = 0, the flow being confined to y > 0. Two equal and opposite forces are applied along the x-axis, so that the wall is stretched keeping the origin fixed. Assume that the stretching sheet has a velocity Uw(x) as shown in figure 1.2. If the sheet stretching linearly, then the stretching velocity is Uw(x) = U 0 x. 17

18 If the sheet stretching non-linearly or exponentially, then the stretching velocity is Uw(x) = U o e x/l. Figure Figure - 1.2: Schematic diagram of the Stretching sheet 1.7 Heat Transfer Temperature The word temperature indicates a physical property on which depends the sense impression of hotness or coldness. Temperature has been defined as the the state of a substance or body with regard to sensible warmth referred to some standard of 18

19 comparison. Senseimpressions can give only a crude estimate and temperature is usually measured by means of a Thermometer Heat The conception of heat which passes from the hotter to the colder body and is thought of as bringing about the change of temperatures. According to Max Planck, the conception of heat, like all other physical concepts originates in the sense-perception, but it acquires its physical significance of the events which excite the sensation. So heat regarded physically, has no more to do with the sense of hotness than color in the physical sense and has to do with the perception of color. The terms Heat and Temperature in older philosophy drew little or no distinction between them and we still use words like blood-heat and summer heat, which introduce the term heat in connection with the idea of temperature. Joseph Black was the first to perceive clearly the necessity of removing this confusion and he pointed out that we must distinguish between quantity and intensity of heat, quantity corresponding to the amount of heat and intensity to temperature. As we know that the knowledge of heat transfer is very important for construction and designing of power plan, which will perform in the prescribed fashion, is the objective of the engineer. This clearly requires detailed knowledge of the principles governing heat 19

20 transfer in the various components, which may be involved i.e., boilers, turbines, condenser, pumps and compressors. Some of the other industrial fields of heat transfer play an important role like heating and air conditioning, chemical reactions and process. A detailed heat transfer analysis is essential, since the dimensions of boilers, heaters, refrigerators and heat exchangers not only depend on the amount of heat to be transmitted but also on the rate at which heat is to be transferred under given conditions Types of heat transfer Heat transfer is a science that predicts the transfer of heat energy from one body to another by virtue of temperature difference. Heat transfer occurs as a result of three mechanisms. Conduction Convection Radiation Conduction: In conduction heat flows due to molecular interaction, molecules not being displace or due to the motion of free electrons. Heat conduction may be stated as the transfer of 20

21 internal energy between the molecules. Heat flows from a region of higher temperature to a region of lower temperature by kinetic motion or direct impact of molecules whether the body is at rest or in motion. Convection: Heat transfer due to convection involves the energy exchange between a solid surface and an adjacent fluid. Convection is a mechanism in which heat flows or transferred between a fluid and a solid surface as a consequence of motion of fluid particles relative to the solid surface when there exists a temperature gradient. Convection heat transfer may be classified as Forced Convection and Free or Natural Convection. Forced convection: If heat transfer between a fluid and a solid surface occurs by the fluid motion induced by by external agencies or forces then the mode of heat transfer is termed as forced convection. Heat transfer in all types of heat exchangers, nuclear reactor and air conditioning apparatus is by forced convection. 21

22 Natural or Free convection: If heat transfer between a fluid and a solid surface occurs by the fluid motion due to the density differences caused by the temperature differences between the surface and the fluid, then the mode of heat transfer is termed as Free Convection or Natural Convection. Heat flows from a heated metal plate to the atmosphere, heat flows from hot water to the container are certain examples of free convection. Radiation: The phenomenon or the mode of heat transfer in the form of electromagnetic waves without the presence of any intervening medium is called Radiation. The transfer of heat energy from the sun to the earth is an example of Radiation. Heat Flux: The heat transfer per unit area is called heat flux. If q is the amount of heat transfer and A is area normal to the direction of the heat flow, then the heat flux is Q = q/a 22

23 Heat Dissipation: The heat generated by internal friction within the volume element of the fluid per unit time is called heat dissipation. Thermal Conductivity: The concept of thermal conductivity is that The quantity of heat passing in unit time through each unit of area when there is a difference of temperature of one degree between the inside and outside face of a wall of unit thickness. To be more specific about discussion of thermal conductivity we consider two parallel layers of a fluid, at distance d apart are kept at different temperatures T1 and T2 (One of the layers may be a solid surface). Fourier noticed that a flow of heat is set up through the layer such that the quantity of heat q transferred through unit area in unit time is directly proportional to the difference of the temperature between the layers and inversely proportional to the distance d. Thus he found q = k T 1 T 2 d, where k is the constant of proportionality and is known as the coefficient of thermal conductivity. 23

24 If the distance d between the two layers of fluid is infinitesimal the above law can be written in the differential form as = k dt, where the negative sign indicated that the heat dy flows in the direction of decreasing temperature. The dimensions of the coefficient of thermal conductivity can be determined as follows k = Heat flux temperature gradient Thermal Diffusivity: The effect of conductivity on the temperature field is determined by the ratio of k to the product of density ρ and specific heat Cp rather than k alone. This ration is known as the thermal conductivity and it is usually denoted by α = k ρc p 1.8. Porous Media A porous medium is a material containing pores (voids). Examples like sponges, clothes wicks, paper sand gravel, filters, concrete brickes, plaster walls, many naturally occurring rocks, packed beds used for distillation, absorption etc. The skeletal portion of the material is often called the matrix or frame. The pores are typically filled with a fluid (liquid or gas). The skeletal material is usually a solid, but structures like foams are often also usefully analyzed using concept of porous media. A porous medium is most 24

25 often characterised by its porosity. Other properties of the medium (e.g., permeability, tensile strength, electrical conductivity) can sometimes be derived from the respective properties of its constituents (solid matrix and fluid) and the media porosity and pores structure, but such a derivation is usually complex. Most of the studies of flow in porous media assume the Darcy s law is valid. However this law is known to be valid only for relatively slow flows through porous media. In general we must consider the effect of fluid inertia as well as of viscous diffusion at boundaries which may become significant for material with high porosities such as fibrous and foams. The concept of porous media is of great interest in many areas of applied science and engineering due to their important applications in the field of agricultural engineering to study the under-ground water resource, seepage of water in river beds, in petroleum technology to study the movement of natural gas, oil and water through oil reservoirs, in chemical engineering for filtration and purification processes. The petroleum industry has been showing a lot of interest in these problems in connection with the crude oil production from the underground reservoirs. These problems are also of much interest in geophysics and in the study of the interaction of the geomagnetic field with the fluid in the geothermal region. The textile technologist is interested in fluid flow through fibers, whereas biologists are interested in water movement through plant roots of the cells of living systems. On the other hand, it has also encountered in the field of mechanics 25

26 (acoustics, geomechanics, soil mechanics, rock mechanics), engineering (petroleum engineering, bio-remediation, construction engineering), geosciences (hydrogeology, petroleum geology, geophysics), biology and biophysics, material science, etc. Fluid flow through porous media is a subject of most common interest and has emerged a separate field of study. The study of more general behaviour of porous media involving deformation of the solid frame is called poromechanics Darcy s Law Based on the experimental research of Darcy in flow through porous medium, Navier- Stokes equation are replaced by linear partial differential equations. Suitable approximations are to be made to get the solution, as the governing equations of porous media are partial differential equations. In 1856, Henri Darcy formulated the law which governs the flow through a porous medium. Darcy s law is given by, q = constant( p + ρg). (1.8.1) where p is the pressure, ρ is the density and g is the acceleration due to gravity. Equation (1.8.1) express that Darcy s velocity q is proportional to the sum of pressure gradient and the gravitational force. Moreover, q is inversely proportional to viscosity. 26

27 This Darcy s law is macroscopic equation of motion for Newtonian fluid in porous median at small Reynolds numbers. Many researchers verified this law experimentally. The constant in the equation is replaced by the permeability k by Musket. Now equation (1.8.1) becomes, q = k μ p (1.8.2) This law is valid for the flow through isotropic porous media. By using Darcy s law various flows through porous media have been invistigated by Musket, De Wiest, Bear and many other researchers. The most general form of Darcy s law is given by, ρ = du i = p + ρx E dt x i μ u k i (1.8.3) x i = The i th component of body force per unit mass, u i = The i th component of velocity, E = The Porosity, d dt = Substantial derivative. The dimension of permeability is L 2. The unit of permeability is Darcy which is used in petroleum industry. The value of one darcy is cm 2. The hydraulic 27

28 conductivity of the porous medium is measured in meinzers. If the porous medium has a permeability of one Darcy, then it has the hydraulic conductivity 18.2 meinzers. Darcy s law is valid when the flow takes place at low speeds. But for high speed flows, Darcy s law is not valid. Also Darcy s law fails to describe the flows with high speeds or the flow near surfaces which are either permeable or rigid. In such cases, Brinkman equation will be useful Brinkman Model The following equation is proposed by Brinkman for the flow through porous media where u is the velocity vector This equation is valid when the permeability k is very high. In general, the particles of the porous media are loosely packed so that k is small. Hence there exist a two boundary layer very near to the surface. In 1966, Tam supplemented a theoretical proof for this equation. Katto and Masuoka experimentally found that Brinkman equation is valid up to the magnitude of k h 2 of order 28

29 10 or so. If the porous medium is made up of spherical particles then k h2 corresponds to considerably high values of d h where d is the diameter of the fillings and h is the recital thickness of the porous media. Yamamoto and Yoshida made improvements on Darcy s law by adding corrective terms. Saffman gave the equations of motion for the flow through porous medium by incorporating viscous stresses Non-Darcy Law In many practical problems, the flow through porous media is curvilinear and the curvature of the path yields the inertia effect, so that the streamlines become more distorted and the drag increase more rapidly. Lapwood was the first person who suggested the inclusion of convective inertial term (q. )q in the momentum equation. Subsequently many research articles have appeared on the non-darcy model. Now the equation can be written as, (1.8.5) However, equation (1.8.5) does not take care of possible unsteady nature of velocity. The flow pattern in a certain region may be unsteady and one has to consider the local acceleration term 1 δ 2 q t also. Adding this term equation (1.8.5) it becomes, 29

30 (1.8.6) This equation is known as Darcy-Lapwood-Brinkman equation. For an isotropic porous medium equation (1.8.5) takes the form. (1.8.7) 1.9 Dimensionless 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 terms 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 significance of a particular phenomenon associated with the problem. There are usually two general methods for obtaining dimensionless parameters. 1. The inspection analysis 2. The dimensionless analysis 30

31 In this thesis the latter method has been used. The basic equations are made dimensionless using certain dependent and independent characteristics values. In this process certain dimensionless numbers appear as the co-efficient of various terms in these equations.the some of the dimensionless parameters used in this thesis are explained below Ekman Number It is the ratio of viscous forces in a fluid to the fictitious forces arising from planetary rotation Reynolds Number The Reynolds number is the ratio of inertial forces to viscous forces and consequently it quantifies the relative importance of these two types of forces for given flow conditions. Thus, it is used to identify different flow regimes, such as laminar or turbulent flow. It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. It is named after Osborne Reynolds ( ), who first introduced this number while discussing boundary layer theory in Typically it is given as 31

32 follows: R e = ρu2 /h = ρuh μu/h 2 μ = Uh v, where U - some characteristic velocity, h - some characteristic length, v = μ - kinematic fluid viscosity, ρ ρ- fluid density Hartmann Number Hartmann number is the ratio of electromagnetic force to the viscous force. It was first introduced by Hartmann and is defined as: H a = BL σ μ where B - the magnetic field, L - the characteristic length scale, σ- the electrical conductivity, μ- the viscosity 32

33 1.8.4 Hall parameter It is defined the product of cyclotron frequency of electrons and electron collision time and is given by m = w e T e where w e is cyclotron frequency of electrons and _e is electron collision time Prandtl Number It is an important dimension parameter dealing with the properties of a fluid. It is defined as the ratio of viscous force to thermal force of a fluid. Prandtl number physically means or signifies the relative speed with which the momentum and heat energy are transmitted through a fluid. It thus associates the velocity and temperature fields of a fluid. For gases Prandtl number is of unit order and varies over a wide range in case of liquids. P r = Viscous force = μc p Thermal force k where μ - Coefficifent of viscosity, C p - Specific heat at constant pressure, 33

34 k - Coefficient of thermal conductivity Eckert Number It is equal to the square of the fluid velocity far from the body divided by the product of the specific heat of the fluid at constant temperature and the difference between temperatures of the fluid and the body. E C = V 0 2 C p (T w T ), where Tw - Temperature near the plate, T - Temperature far away from the plate, Cp - Specific heat at constant pressure, V0 - Characteristic value of velocity Number Density Number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space, or Number density is the number of specified objects per volume i.e., n = N / V. 34

35 1.9.8 Grashof Number It plays a significant role in free convection heat transfer. The ratio of the product of the inertial force and the buoyant force to the square of the viscous force in the convection flow system is known as Grashof number. Grashof number in free convection is analogues to Reynolds number in forced convection. G r = gβ (T W T )l 2 v 2 where g - acceleration due to gravity, β- volumetric coefficient of thermal expansion, l - characteristic length, Tw - temperature of the wall, T - constant temperature far away from the sheet Non-uniform heat source/sink parameter It is defined as q = ( ku w(x) ) [A (T xv w T )f (η) + B (T T )] 35

36 where A and B are the parameters of the space and temperature dependent internal heat generation/absorption. It is to be noted that A and B are positive to internal heat source and negative to internal heat sink, v is the kinematic viscosity, Tw and T denote the temperature at the wall and at large distance from the wall respectively Radiation parameter It is defined as N r = 16 σ T 3 3kk where σ - Stefan-Boltzman constant k - mean absorption co-efficient Melting parameter M = C p(t T m ) γ+c s (T m T 0 ), where M is the dimensionless melting parameter, where Cp is the heat capacity of the 36

37 fluid at constant pressure. The melting parameter is a combination of the Stefan numbers C f(t T 0 ) γ and C s(t T 0 ) γ for the liquid and solid phases, respectively. We Take Tm is the temperature of the melting surface, while the temperature in the free-stream condition is T, where Tm > T Curvature parameter γ = lv ba 2 is the curvature parameter, γ = 0, corresponds to flat plate Shear stress/skin Friction Any real fluids moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary is zero, but atsome height from the boundary the flow speed must equal that of the fluid. The region between these two points is aptly named the boundary layer. For all Newtonian fluids in laminar flow the shear stress is proportional to the strain rate in the fluid, where the viscosity is the constant of proportionality. However for Non-Newtonian fluids, this is no longer the case as for these fluids the viscosity is not constant. The shear stress is 37

38 imparted onto the boundary as a result of this loss of velocity. The shear stress, for a Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by: τ(y) = μ u y where μ is the dynamic viscosity of the fluid, u is the velocity of the fluid along the boundary, and y is the height of the boundary. Specifically, the wall shear stress is defined as: τ w = τ(y = 0) = μ u y y = 0 In case of wind, the shear stress at the boundary is called wind stress Nusselt Number The convective heat transfer from the surface will depend upon the magnitude of Ch(Tw T), where, Ch is the heat transfer coefficient and Tw and T are the temperatures of wall and fluid respectively. Also, if there was no flow, the heat transfer was purely due to 38

39 conduction, the Fouriers law states that the quantity k(t w T) l would be the measure of the heat transfer rate, where k is the thermal conductivity and l is the length. Now Nusselt number can be written as N u = C h(t w T) k(t w T)/l = C h l k i.e., Nusselt Number is the measure of the ratio of magnitude of the convective heat transfer rate to the magnitude of heat transfer rate that would exist when there was pure conduction Laplace Transforms Laplace Transform is an essential mathematical tool which can be used to solve several problems in science and engineering. This technique becomes popular when Heaviside function applied to the solution of an ODE representing a problem in electrical engineering. Transforms are used to accomplish the solution of certain problems with less effort and in a simple routine way. The Laplace transform method reduces the solution of an ODE to the solution of an algebraic equation. Also, when the Laplace transform technique is applied to a PDE, it reduces the number of independent variables by one 39

40 Definition Let f(t) be a continuous and single-valued function of a real variable t defined for all t, 0 < t <, and is of exponential order. Then the Laplace transform of f(t) is defined as a function F(s) denoted by the integral L[f(t)] = 0 e st f(t)dt (1.10.1) Definition Error Function: The error function is defined as, erf(x) = 2 π π 0 e t2 dt (1.10.2) and its compliment is erf c(x) = 1 erf(x) = 2 π x e t2 dt (1.10.3) The Laplace transform of the error function is L(erf(x)) = 1 s es2 4 erfc( s 2 ) 40

41 Complex Inversion Formula/Mellin-Fourier integral In solving partial differential equations using Laplace transform method, complex variable theory may come in handy for finding inverse transform. Inverse Laplace transform can be expressed as an integral which is known as inverse integral and this integral can be evaluated by using contour integration methods. The inverse Laplace Transforms of U, V are u, v respectively and are given by the Integrals 41

42 u = 1 2iπ r i r+i e st U dt and v = 1 2iπ r i r+i e st V dt Which can be evaluated by means of contour integration. Since there is no branch point, the contour chosen is the closed curve ABC formed by the line x = r and a semi circle C with origin as center and radius R (See figure 1.1) so that r+i r i B R A e st U dt = lim e st U dt == lim [ e st U dt e st U dt] R ABC C Using Cauchy s theorem of residues and Jordans lemma, we have u = 1 Similarly, v = 1 r+i 2iπ r i r+i 2iπ r i e st U dt e st V dt = sum of residues of {e st U} at its poles. = sum of residues of {e st V} at its poles. Table 1.1: Laplace transform of some important functions 42

43 1.11 Similarity Transformation Birkhoff (1950) first recognized that Boltzmann s method of solving the diffusion equation with a concentration-dependant diffusion co-efficient is based on the algebraic symmetry of the equation and special solutions of this equation can be obtained by solving a related ordinary differential equation. Such solutions are called similarity solutions because they are geometrically similar. He also suggested that the algebraic symmetry of the partial differential equations can be used to find similarity solutions of other partial differential equations by solving associated ordinary differential equations. 43

44 Thus, the method of similarity solutions has become a very successful dealing with the determination of a group of transformation under which a given partial differential equation is invariant. The simplifying feature of this method is that a similarity transformation of the form u(x, t) = t p v(η), η = x t q can be found which can, then, we used effectively to reduces the partial differential equations to an ordinary differential equations with η as the independent variable. The resulting ordinary differential equations is relatively easy tosolve. In practice this method is simple and useful in finding solutions of both linear andnonlinear partial differential equations Numerical Methods Numerical methods are the way to do higher mathematics problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage for numerical analysis is that a numerical answer can be obtained even when a problem has no analytical solution. It is important to realize that a numerical analysis solution is always numerical. Analytical methods usually give a result in terms of mathematical functions that can then be evaluated for specific instances. There is thus advantage to the analytical results, in that the behavior and properties of the function are often apparent. However, numerical results can be plotted to show some of the behavior 44

45 of the solution. Another important distinction is that the result from numerical method is an approximation, but results can be made as accurate as desired. To achieve high accuracy, many separate operations must be carried out. Here are some of the operations that numerical methods can do: Solve for the roots of a nonlinear equation. Solve large systems of linear equations. Get the solutions of a set of nonlinear equations. Interpolate to find intermediate values within a table of data. Solve ODE when given initial values for the variables. Solve boundary-value problems and determine eigenvalues and eigenvectors. Obtain numerical solutions to all types of partial differential equations and so on. In connection with numerical analysis many symbolic algebraic programmes are available,namely Mathematica, DERIVE, Maple, MathCad, MATLAB, and MacSyma. In this thesis the numerical solutions of the problem are solved by RKF -45 method with the help of algebraic software MAPLE Runge Kutta Fehlberg Method 45

46 Runge-Kutta-Fehlberg is adaptive; that is, the method adapts the number and position of the grid points during the course of the iteration in attempt to keep the local error within some specified bound. Sketch of the ideas: Begin with two RK approximation algorithms, one with order p and with order p + 1. Apply the algorithms to get two approximations at a given grid point tk. These approximations are used to approximate the local discretization error at the grid point. This error approximations is then used to make several decisions. If the error approximation exceeds some prescribed maximum bound on accuracy, then a smaller step size is assigned, a new grid point tk is assigned, and the preceding steps are repeated. If the error approximation falls below some present minimum bound on accuracy, then the step size is increased and the next step in the iteration is performed. If the error approximation falls in between some user-specified minimum and maximum values, then we may choose to leave the step size alone or we may compute an optimal step size for the next step. The term optimal is used loosely because there are some assumptions made and some approximations involved in getting this value. Typically, the approximation given to the user is reported as the more accurate p + 1 order approximation, even through, in the analysis, that approximation is used to approximate the error in the pth order approximation 46

47 The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to resolve problem. It has a procedure to determine, if the proper step size h is being used. At each step, twomdifferent approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each step requires the use of the following six values: Now the approximation solution to the given I.V.P. is made using a Runge-Kutta method of order 4: 47

48 where the four function values f1, f3, f4 and f5 are used. Notice that f2 is not used in the above formula. A better value for the solution is determined using a Runge-Kutta method of order 5: The optimal step size sh can be determined by multiplying the scalar s times the current step size h. The scalar s is 48

49 Magnetite and Mn-Zn Ferrite-Water BasedNanofluidsFlow atstagnation Point over a Stretching Surface Inspired By Nonlinear Thermal Radiation 2.1 INTRODUTION The thermal properties of the ordinary fluids are not sufficient to meet today s cooling rate requirements in industrial applications. Therefore, the nanofluids are introduced to enhance the thermal performance of ordinary fluids. Basically, the nanofluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles are typically made up of metals (Al, Cu), oxides (Al 2 O 3, TiO 2 and CuO), carbides (SiC), nitrides (AlN, SiN) or nonmetals (Graphite, carbon nanotubes) and the base fluid is usually a conductive fluid, such as water, ethylene glycol and etc., The term of nanofluid was first introduced by Choi [1]. Buongiorno [2] proposed seven slip mechanisms to write down conservation equations based on the Brownian diffusion and thermophoresis effects. Recently, Gorla and Chamkha [3] have analyzed the flow of nanofluid with natural convective boundary layer over a horizontal plate along with porous medium. Rashidi et al. [4] have investigated entropy generation in steady MHD flow of nanofluid due to a rotating porous disk. Gireesha et al. [5] have analyzed the effect of suspended particles on nanofluid flow and heat transfer over a stretching sheet 49

50 saturated by a porous medium. The heat and mass transfer of water based nanofluid flow over a stationary/moving vertical plate wereexplored by Mahanthesh et al. [6]. Besides, theferrofluid comprises of iron-based nanoparticles such as magnetite, hematite, cobalt ferrite, etc.,researchers and scientists have focused considerably on the surface driven ferrofluid flows owing to their numerous industrial and biomedical demands. For instance, iron-based nanoparticles can be used for efficient drug delivery by guiding the particles via external magnets;magnetic nanoparticles are prominent in hyperthermia. Several researchers investigated diversified characteristics of such ferrofluid problems. For instance,tangthieng et al. [7] addressed heat transfer enhancement in ferrofluids subjected to steady magnetic fields.jue [8] used semiimplicit finite element method in orderto simulate magnetic gradient and thermal buoyancy induced cavity ferrofluid flow.nanjundappa et al. [9] analyzed the influenceof magnetic field dependent viscosity on the horizontal layer of ferrofluid.sheikholeslami and Ganji [10] have invstigatedthe MHD flow and heattransfer of ferro nanofluid with the effet of convective heat transfer. The stagnation point flow and heat transfer of ferrofluid towards a stretching sheet in the presence of viscous dissipationwere investigated by Zafar et al. [11]. They have consideredthree types of ferroparticles magnetite (Fe 3 O 4 ), cobalt ferrite (CoFe 2 O 4 ) and Mn-Zn ferrite (Mn ZnFe 2 O 4 ) with water and kerosene as 50

51 conventional base fluids.recently Sheikholeslami, Rashidi andganji [12, 13] have studied the effect of magnetic field with suspended ferroparticles. The fluid flow at astagnationpoint over a stretching sheet iscrucial in theoretical and application point of view in fluid dynamics. Chaim [14] was the first toinvestigate the stagnation-point flow towards a stretching sheet. Mahapatra and Gupta[15] have studied the similar problem by considering the strain-rate and the stretching rate to be different. They obtained boundary layer adjacent to thesheet which completely depends on the ratio between the strain-rate of stagnation-point flow and the stretching rate of thesheet. The steady and unsteady stagnation-point flow of an incompressible viscous fluid over a stretching surface wasstudied by Paullet and Weidman [16]. The stagnation point flow of an electrically conducting fluid over a stretching surface under theinfluenceof magnetic field wasinvestigated by Mahapatra et al. [17]. Pal et al. [18] have analyzed the nanofluid flow and non-isothermal heat transfer at the stagnationpointover a stretching/shrinking sheet embedded in a porous medium.gireesha et al. [19] have presented the numerical solution for boundary layer stagnation-point flow past a stretched surface with melting effect and aligned magnetic field. They have incorporated the nanofluid model by considering Brownian motion with thermophoresis mechanisms.furthermore, the stagnation-point flow over stretching sheet under 51