CHAPTER I INTRODUCTION

Transcription

1 CHAPTER I INTRODUCTION 1.1 Preface The classical theory of isotropic viscous incompressible fluids is based on Newtonian hypothesis, i.e., a linear relation between the stress tensor and first-order Rivlin- Ericksen tensor containing a parameter called co-efficient of viscosity which depends on temperature and pressure. A simple equation to describe Newtonian fluid (1687) behaviour is given as (1.1.1) where is the stress tensor, p is an undetermined hydrostatic pressure, is kronecker delta and 2,,, (1.1.2) Here is the co-efficient of viscosity and are velocity components. The classical linear theory of isotropic, viscous fluids based on the equation (1.1.1) provides a satisfactory approximation to that of ordinary mobile liquids like air, water, glycerine, honey and many thin oils. It is capable of explaining the phenomena of lift, skin friction, drag etc., however, it fails to explain a number of phenomena observed in a large number of liquids. 1.2 Beginning of the Study of non-newtonian Fluid Flows The study of fluid dynamics is a central theme in modern applied mathematics. The subject of fluid mechanics goes back at least to the days of ancient Greece, when Archimedes investigated fluid statics and buoyancy and formulated his famous law known now as the Archimedes' Principle. Rapid advancement in fluid mechanics began with Leonardo da Vinci (observation and experiment), Evangelista Torricelli (barometer), Isaac Newton (viscosity) and Blaise Pascal (hydrostatics), and was

2 2 continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics in Hydrodynamica (1738). Inviscid flow was further analyzed by various mathematicians (Leonhard Euler, D'Alembert, Lagrange, Laplace, Poisson) and viscous flow was explored by a multitude of engineers including Poiseuille and Gotthilf Heinrich Ludwig Hagen. Further mathematical justification was provided by Claude-Louis Navier and George Gabriel Stokes in the Navier Stokes equations, and boundary layers were investigated (Ludwig Prandtl, Theodore Von Kármán), while various scientists (Osborne Reynolds, Andrey Kolmogorov, Geoffrey Ingram Taylor) advanced the understanding of fluid viscosity and turbulence. Tremendous advances have been observed since Euler gave his famous equations of fluid flow for perfect (non-viscous incompressible and compressible) fluids in The perfect or ideal fluids are characterized by the assumption that for them the stress tensor and the stress vector on a plane surface in contact with a fluid is normal to the surface. Euler s partial differential equations were non-linear but solutions were obtained for flows past circular and elliptic cylinders and past spheres and ellipsoids. The ideal fluid theory gave excellent results for certain classes of problems such as wave formation and tidal motions. The ideal fluid theory however led to the conclusion that when a solid moves through a fluid at rest at infinity, it experiences no drag. This was in contradiction with observations (D'Alembert s paradox). To explain the drag, the concept of viscous Newtonian fluid (given by (1.1.1)) was introduced. Navier (1821) and Stokes (1845) obtained the equations of motion for these fluids independently and from different considerations. Exact solutions were obtained for viscous flows between plates, in pipes, annuli between translating and rotating coaxial cylinders etc., by using Navier- Stokes equations and the no-slip conditions at the surfaces. Although good approximate solutions were obtained for stagnation flows, for very slow motions, for lubrication flows, flows past suddenly accelerated plane walls near oscillating flat plates and near rotating discs, but their applications in engineering were not yet available. There had been a major breakthrough when Prandtl in 1904 proposed his assumptions that when viscous fluid flows past a surface, the viscous effect is dominant in a thin layer called the boundary layer near the surface and outside

3 3 this layer, the flow may be regarded as that of an ideal fluid. The boundary layer theory had enormous achievements into the discipline of engineering importance. For example, this theory gives correct predictions for resistance to motion of ships and for lift and drag on aeroplane wings. But the problems in the hypothesis of Newton, Reynolds, Chezy etc. could not explain the basis of viscous fluid theory. For example, the inadequacy of the Newtonian theory to account for normal stress effect, centripetal pump effect, Merrington s effect (1943), stress relaxation, memory effects etc., invoke diverse scientific thoughts to generalize the linear relationship given by Newton. In fact, the classical linear theory is incapable of explaining the experiments and phenomena observed by Garner and Nissan (1946), Weissenberg (1947), Rivlin and Ericksen (1955) and Reiner (1957). The linear theories describing the motion of various fluids did not work as fluids got more complicated. Since 1960, there had been many attempts to write all-encompassing new theories to be able to describe the flow conditions and separation of streamlines and nonlinearities to cover the gradually increasing difficulties of fluid list, which are branded as non-newtonian fluids. At a later stage, the postulate based on the flow behaviour of some materials which was explained by the linearity relationship between stress and rate of strain tensors immediately led to the development of the theory of non-newtonian fluids [Kapur et al. (1982), White, F.M. (2006), Wilkinson, W.L. (1960)]. 1.3 Classification of non-newtonian Fluids Non-Newtonian fluids are grouped property wise as follows: (1) Visco-inelastic fluids: Power-law fluid, Reiner-Rivlin fluid, Bingham plastic, Ellis fluid, Reiner-Philippoff fluid, Prandtl fluids, Eyring fluid, Powell-Eyring fluids, Williamson fluid, Rabinowlisch type fluids, Meter fluid, Peek-Mclean fluids, Casson fluids. (2) Visco-elastic fluids: Oldroyd fluid, Rivlin-Ericksen second-order fluid, Walters fluid, other visco-elastic fluids. (3) Polar fluids: Dipolar fluid, micropolar fluid, model of Condiff and Dahler, Stokes fluid with couple stresses.

4 4 (4) Anisotropic fluids. (5) Fluids with micro-structure. (6) Heat conducting nematic liquid crystals. Although this does not give an exhaustive list, the most important of these are Power-law fluids, Bingham plastics, Oldroyd fluids, Second-order fluids, Walters fluid, micropolar fluids and anisotropic fluids. The solution of each of the problems of such fluids is an intellectual challenge and it is human nature to accept the challenge of unsolved problems whether the solutions are immediately useful or not. But the important aspect of solving the problems of such fluids is to get insight into the flow behaviour of non-newtonian fluids as they occur more in industry, under conditions. In engineering practice, lubricants like oils, greases, diester lubricants, synthetic lubricants etc. are modelled with non-newtonian fluids. 1.4 Subsequent Analysis of Visco-elastic Fluids The theoretical calculations along with experimental observations can be used to estimate the non-newtonian parameter occurring in the constitutive equations. Out of this vast list of pending problems, we select the visco-elastic fluid characterized by the second-order fluid model to investigate the flow pattern and analyze the flow behaviour under different valid and suitable boundary conditions. These fluids are of increasing importance in modern technology due to their growing use in many activities such as molten plastic, paints, drilling and polymer solutions. The visco-elastic fluids possess a certain degree of elasticity in addition to viscosity. When a visco-elastic fluid is in motion, a certain amount of energy is stored up in the material as strain energy while some energy is lost due to viscous dissipation. In this class of fluids, strain is not neglected, as it is responsible for the recovery to the original state and for the possible reverse flow that follows the removal of the stress. During the flow, the natural state of the fluid changes constantly and tries to attain the instantaneous state of deformed state, but it does never succeed completely. Further improvement in this study became essential and that was done by Noll (1958) by giving more general definition of fluidity. Noll defined a simple fluid as a substance which has neither preferred intrinsic direction nor preferred rate of strain, i.e., a substance such that all local states of equal mass and density are intrinsically

5 5 equivalent in response, with all observable differences in response being due to definite differences in past history. This definition implies that a fluid cannot support a shearing traction indefinitely, in any direction, without giving way to it. In this definition, the principle of material difference has been employed. This principle states that the constitutive equation must be of the same form for all observers, or equivalently, must be intensive to rigid motion of the body as a whole. It can exhibit the phenomenon of shear dependent viscosity, normal stress differences and gradual stress relaxation. Taking this definition of simple fluid and assuming that the stress is more sensitive to recent deformation than to which occurred in the distant past, Coleman and Noll (1960) suggested that the constitutive equation of an incompressible simple fluid can be written as (1.3.1) where is the extra stress and the functional obeys the following identity for all histories and orthogonal tensor Q: where is the transpose of Q and has its trace fixed by some convention such as. 0 Coleman and Noll used the postulate of gradually fading memory to the memory of functional occurring in the constitutive equation (1.3.1) by constructing a new theory, 0 a being the retardation factor given by 0 1. If all terms greater than 2 in a are neglected, then the constitutive equation of an incompressible simple fluid may be written as,, (1.3.2) where p is the hydrostatic pressure, is the shear viscosity, is the coefficient of viscosity, is the cross-viscosity. and are defined as,,,

6 6,, 2,, and being the components of velocity and acceleration respectively. The fluid governed by the constitutive equation (1.3.2) has been defined as second-order fluid for it gives complete second-order corrections to the incompressible perfect fluid. Using the principle of dimensional analysis, Truesdell deduced the same equation as in (1.3.2) for a second-order fluid. Markovitz and Coleman (1964) proved that is negative from thermodynamic consideration and experimentally also, it has been found so [Srivastava (1967)]. 1.5 Constitutive equations for an incompressible second-order fluid Let 1, 2, 3 be the rectangular co-ordinates of a point P. If three functions,, 1, 2, 3 in a neighbourhood of P possess a unique inverse,, 1, 2, 3, we say that P has curvilinear co-ordinates. For fixed,, the transformations give surfaces, called curvilinear surfaces. The intersection line of any curvilinear surfaces defines a curvilinear line so that through a point P, there pass three non-dimensional curvilinear lines at defining co-ordinates of point P. If tangents to the curvilinear lines at P are rectangular, the curvilinear coordinates are called orthogonal curvilinear co-ordinates. For orthogonal curvilinear coordinates, the square of the line element is (1.4.1) In orthogonal curvilinear co-ordinates, the dynamical equations can be written as follows: 0 (1.4.2) The equations of momentum take the form, (1.4.3) where is the density. The acceleration are given by

7 7 (1.4.4) and the physical components in orthogonal curvilinear coordinates of the stress tensor is given by,, For an incompressible second-order fluid they are as follows: (1.4.5)

8 The components 22 and 33can be obtained from (1.4.6) by changing the indices on h, x and v from 1 to 2, 2 to 3, 3 to 1 and from 1 to 3, 3 to 2, 2 to 1 respectively. Similarly, we may obtain 23 and 31 from (1.4.7). The above equations can be written in Cartesian coordinates x, y, z by putting 1 ; x, y, z for,, and u, v, w for 1, 2, 3 respectively. Their forms in cylindrical polar co-ordinates,, can be obtained by putting 1, ;,, and,, for 1, 2, 3 respectively. These equations can be written in spherical polar coordinates,, if we put 1,, ;,, and,, for 1, 2, 3 respectively. 1.6 Magnetodynamics and its Applications Magnetohydrodynamics or MHD is a branch of the science of the dynamics of matter moving in an electromagnetic field, especially where currents established in the matter by induction modify the field, so that the field and dynamics equations are coupled. It treats, in particular, conducting fluids, whether liquid or gaseous, in which certain simplifying postulates are accepted. These are, generally, that the Maxwell displacement current is neglected, and the fluid may be treated as a continuum, without mean-free-path effects. It is distinguished from the closely related plasma dynamics in which these postulates are relaxed, but there is still a large intermediate area in which similar treatment is possible. Solid matter is generally excluded from MHD, but it should be realized that the same principles apply there. Electrical conduction in metals and the Hall Effect are two

9 9 examples. In an electric motor, the magnetic field produced by the armature current affects the operation of the motor in an important way, so that the mechanical and electrical analyses are coupled, just as in MHD. Electromagnetic forces are an essential part of motors and generators, though they generally do not produce significant elastic deformations, and the motions occur with the help of rotating and sliding contacts. Homopolar generators (ones that produce DC currents) are, indeed, closely related to MHD analogues. MHD was originally applied to astrophysical and geophysical problems, where it is still very important, but more recently to the problem of fusion power, where the application is the creation and containment of hot plasmas by electromagnetic forces, since material walls would be destroyed. Astrophysical problems include solar structure, especially in the outer layers, the solar wind bathing the earth and other planets, and interstellar magnetic fields. The primary geophysical problem is planetary magnetism, produced by currents deep in the planet, a problem that has not been solved to any degree of satisfaction [Welty at al. (1976), White, F.M. (2003)]. 1.7 Heat Transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy and heat between physical systems. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system. Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature, as required by the second law of thermodynamics.

10 10 Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. If the motion is caused by the action of the body forces on the fluid, which arises as a result of density gradients due to changes in temperature, is called a free convection. The later process is sometimes called natural convection. All convective processes also move heat partly by diffusion, as well. Another form of convection is forced convection. In this case, the fluid is forced to flow by use of a pump, fan or other mechanical means. In free convection flow, the velocity and temperature fields are coupled. In problem of free convection or combined free and forced convection it is customary to express the body force term as a buoyancy term. In this case, density varies slightly from point to point because the variation in the temperature T and it can be expressed as 1 which is well known Boussinesq approximation where is the co-efficient of thermal expansion and is the density at some temperature. The variation of is taken into account only in the buoyancy term. The final major form of heat transfer is by radiation, which occurs in any transparent medium (solid or fluid) but may also even occur across vacuum (as when the Sun heats the Earth). Radiation is the transfer of energy through space by means of electromagnetic waves in much the same way as electromagnetic light waves transfer light. The same laws that govern the transfer of light govern the radiant transfer of heat [White, F.M. (2003), Batchelor, G.K. (1967)]. 1.8 Mass Transfer Mass transfer is the net movement of mass from one location, usually meaning a stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption, evaporation, adsorption, drying, precipitation, membrane filtration, and distillation. Mass transfer is used by different scientific disciplines for different processes and mechanisms. The phrase is commonly used in engineering for physical processes that involve diffusive and convective transport of chemical species within physical systems.

11 11 Some common examples of mass transfer processes are the evaporation of water from a pond to the atmosphere, the purification of blood in the kidneys and liver, and the distillation of alcohol. In industrial processes, mass transfer operations include separation of chemical components in distillation columns, absorbers such as scrubbers, absorbers such as activated carbon beds, and liquid-liquid extraction. Mass transfer is often coupled to additional transport processes, for instance in industrial cooling towers. These towers couple heat transfer to mass transfer by allowing hot water to flow in contact with hotter air and evaporate as it absorbs heat from the air. In astrophysics, mass transfer is the process by which matter gravitationally bound to a body, usually a star, fills its Roche lobe and becomes gravitationally bound to a second body, usually a compact object (white dwarf, neutron star or black hole), and is eventually accreted onto it. It is a common phenomenon in binary systems, and may play an important role in some types of supernovae and pulsars. Mass transfer finds extensive application in chemical engineering problems. It is used in reaction engineering, separations engineering, heat transfer engineering, and many other sub-disciplines of chemical engineering. The driving force for mass transfer is typically a difference in chemical potential, when it can be defined, though other thermodynamic gradients may couple to the flow of mass and drive it as well. A chemical species moves from areas of high chemical potential to areas of low chemical potential. Thus, the maximum theoretical extent of a given mass transfer is typically determined by the point at which the chemical potential is uniform. For single phase-systems, this usually translates to uniform concentration throughout the phase, while for multiphase systems chemical species will often prefer one phase over the others and reach a uniform chemical potential only when most of the chemical species has been absorbed into the preferred phase, as in liquid-liquid extraction. While thermodynamic equilibrium determines the theoretical extent of a given mass transfer operation, the actual rate of mass transfer will depend on additional factors including the flow patterns within the system and the diffusivities of the species in each phase. This rate can be quantified through the calculation and application of mass transfer coefficients for an overall process. These mass transfer coefficients are typically published in terms of dimensionless numbers, often including Péclet numbers,

12 12 Reynolds numbers, Sherwood numbers and Schmidt numbers, among others [White, F.M. (2003), Batchelor, G.K. (1967)]. 1.9 Porous Medium A porous medium or a porous material is a material containing pores (voids). 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 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. Even the concept of porosity is only straightforward for a poroelastic medium. Often both the solid matrix and the pore network (also known as the pore space) are continuous, so as to form two interpenetrating continua such as in a sponge. However, there is also a concept of closed porosity and effective porosity, i.e., the pore space accessible to flow. Many natural substances such as rocks and soil (e.g., aquifers, petroleum reservoirs), zeolites, biological tissues (e.g. bones, wood, cork), and manmade materials such as cements and ceramics can be considered as porous media. Many of their important properties can only be rationalized by considering them to be porous media. The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acoustics, 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 flow of a fluid through porous medium was theoretically studied by Darcy (1856). To study the flow of a fluid, he established an empirical law known as Darcy s law. According to this law, the flow is linearly dependent upon the pressure gradient and the gravitational force. For slowly, specially varying flow, Darcy s law is absolutely correct. If, however, the flow involves large shear, one expects further terms involving

13 13 velocity gradients to appear. Inadequacy of Darcy s law occurs in the formulation of boundary condition at the edge of the porous medium leads to the necessity of generalizing the linear relationship between the discharge velocity and pressure gradient. Numerous investigations gave generalization of Darcy s law and a number of hydrologist, geologist, geophysicist, petroleum geologists and chemical engineers have studied the flows in porous medium [White, F.M. (2003), Batchelor, G.K. (1967), Hughes, W.F. and Brighton, J.A. (1991)] A brief survey of the corresponding works In the analysis of second-order fluid, Coleman, Noll and Truesdell have derived the constitutive equation from the definition of simple fluid and discussed the viscometric flows for it in large number of papers. The solution of any viscometric flow which may be expressed in terms of three material functions known as shear viscosity and normal functions have discussed by Coleman and Noll (1959 1, ). Markovitz and Brown (1964) measured the normal stresses when a solution of poly-iso-butylene is confined between two circular plates and between a cone and a plate in steady torsional shearing. In another paper, they (1963) have studied the effect of concentration of the solution on the normal stress. Markovitz and Coleman ( ) have discussed non-steady helical flows. They ( ) have given an account of the viscometric flows, the steady extensions and non-steady helical flows and have proposed some practicable methods for experiments to determine the material constants and. The flow past a symmetric cylinder has been discussed by Srivastava and Maiti (1966). Srivastava and Sharma (1970) have studied the forced flow against a flat wall. Srivastava and Saroa (1971) have studied the phenomenon of separation in second-order fluid. Bruan (1977) has studied slow motion of a rigid particle in second-order fluid. Srivastava and Saroa (1978) have studied heat transfer of a second-order fluid past a circular cylinder. Choudhury (1981) has studied the flow of a second-order fluid through a contracting and expanding pipe. Saroa and Choudhury (1983) have discussed the flow between eccentric cylinders. Also, they have studied the flow past a sphere (1984), the induced flow of a fluid due to an oscillating disk (1985).

14 14 Srivastava and Sharma (1991) have discussed the flow of a second-order fluid through a circular pipe and its surrounding porous medium. Saroa and Choudhury (1992, 1993) have studied the flow and heat transfer through a circular cylinder and the flow between rotating eccentric discs. Bhattacharjee and Gupta (1995) have discussed the flow to a stretching surface rotating in a non-newtonian fluid. Tanner (1996) has studied plane creeping flows of incompressible second-order fluids. The flow and heat transfer of a visco-elastic fluid over a flat plate with a magnetic field and pressure gradient have been studied by Kumari et al. (1997). Choudhury and Sharma (1999) have studied the torsional oscillations of disk in elastico-viscous fluid. Ardrienko et al. (2000) have studied the resonance behaviour of visco-elastic fluids in Poisseuille flow and application to flow enhancement. Fatecau (2001) has studied the unsteady Coutte flows of second-order fluids in heated cylindrical domains. The effect of visco-elastic heating on linear stability of visco-elastic cone and plate flow problem in axisymmetric case has been studied by Olaguju et al. (2002). Nigen and Walters (2002) have studied the visco-elastic contraction flows. Eldabe et al. (2003) have investigated the magnetohydrodynamic flow of a visco-elastic fluid through a porous medium near an accelerated plate. Attia (2004) has considered the unsteady Hartmann flow with heat transfer of a visco-elastic fluid considering the Hall Effect. Khayat and Pan (2004) have discussed the transient free-surface flow of visco-elastic fluid in a narrow channel. Eldabe and Sallam (2005) have studied the non-darcy Coutte flow through a porous medium of magnetohydrodynamic visco-elastic fluid with heat and mass transfer. Fatecau and Fatecau (2005) have discussed starting solutions for some unsteady unidirectional flows of second-order fluid. Shivakumara et al. (2006) have discussed the onset of convection in a visco-elastic fluid. Ardekani, Rangel and Joseph (2007) have studied the motion of a sphere normal to a wall in a second-order fluid. The unsteady free convective MHD flow and heat transfer of a second-order fluid between two heated vertical plates through a porous medium has been studied by Das et al. (2007). Sreehari Reddy et al. (2008) have studied the hydromagnetic elastic free convection of a conducting elastico-viscous liquid between heated vertical plates. Erdogan and Imrak (2008) have studied the steady flow of a second-grade fluid in an annulus with porous walls. Hayat et al. (2008) have discussed MHD flows of a secondgrade fluid between two side walls perpendicular to a plate through a porous medium.

15 15 Massoudi and Vaidya (2008) have studied some generalizations of the second-grade fluid model. Xi and Graham (2009) have studied a mechanism for oscillatory instability in visco-elastic cross-slot flow. The particle-wall collision in a visco-elastic fluid has been investigated by Ardekani et al. (2009). Choudhury and Das ( ) have discussed the oscillatory visco-elastic flow through porous medium in a rotating porous channel with injection. Flow of second-order fluid in a curved duct with square cross-section has been studied by Norouzi et al. (2010). Mukherjee et al. (2010) have studied the effects of visco-elasticity on the retraction of a shared drop. Chen et al. (2010) have studied unsteady unidirectional flow of a second-order fluid through a microtube with a wallslip and different given volume flow rate. Nandeppanavar et al. (2010) have investigated flow and heat transfer characteristics of a visco-elastic fluid in a porous medium over an impermeable stretching sheet with viscous-dissipation. The nonsimilarity boundary layer flows of a second-order fluid on a stretching sheet have been studied by You et al. (2010). The two-dimensional boundary layer flow past a wedge with slip velocity of a visco-elastic, electrically conducting fluid in presence of a strong transverse magnetic field has been studied analytically by Choudhury and Deb (2010). Hussain et al. (2010) have studied the oscillatory flows of second-grade fluid in porous plate. Choudhury and Deb ( ) have discussed the flow of an elastico-viscous fluid due to a rotating disk in presence of an infinite saturated porous medium. Snijkers et al. (2011) have studied the effects of viscoelasticity on the rotation of a sphere in shear flow. Choudhury et al. (2011) have discussed the effects of MHD flow and heat transfer of a visco-elastic fluid in a vertical channel with slip velocity. Kang et al. (2011) have studied the thermal instability of visco-elastic fluids in a rotating porous layer heated from below. Choudhury and Deb ( ) have studied the MHD flow of visco-elastic fluid between a stretching sheet and an oscillatory porous parallel plate with constant suction. Kavitha et al. (2011) have studied the convective MHD flow of a second-order fluid in an inclined porous channel considering Darcy s effect. Choudhury and Das (2011) have studied the boundary layers in oscillatory MHD visco-elastic flow past a vertical plate with periodic suction. Moreover, a number of flow problems have been discussed by Choudhury and Deb (2007, ), Choudhury and Das (2008, , 2012) to study the flow patterns of second-order fluids and analyze their behaviours

16 16 under different valid and suitable boundary conditions, some of which are included here Purpose, Extension and Scope of this work Although the visco-elastic behaviour of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Advancements in computational techniques are making possible much more detailed analyses of complex flows and more sophisticated simulations of the structural and molecular behaviour that gives rise to visco-elastic behaviour. Engineers, chemists, physicists, and mathematicians are actively pursuing research in rheology, particularly as more technologically important materials are found to display visco-elastic behaviour. In this thesis, we have studied a number of steady as well as unsteady flow problems with different geometries. Out of different fluid models mentioned above, we have chosen second-order fluid governed by the constitutive equation (1.3.2). The results reveal various aspects of the additional terms in the constitutive equations as compared to the Newtonian fluid. In chapter II, the unsteady free convective magnetohydrodynamic flow of a visco-elastic, electrically conducting fluid past an infinite vertical porous plate with constant suction and heat absorbing sinks has been investigated. Approximate solutions of the equations governing the flow have been derived for the velocity and temperature fields, mean skin friction and mean rate of heat transfer by using multi-parameter perturbation technique. The profiles of mean velocity, the mean temperature, the temperature and the skin-friction have been presented graphically for different values of the visco-elastic parameter. The mean skin-friction and the mean rate of heat transfer with corresponding amplitudes and phases have been numerically worked out for different values of the parameters involved in the solution to observe the effects of visco-elastic parameter encountered in the problem under investigation. In chapter III, the unsteady hydromagnetic flow of an electrically conducting visco-elastic fluid past an infinite vertical porous plate in a porous medium of time dependent permeability under oscillatory suction velocity normal to the plate has been

17 17 investigated. It is considered that the uniform magnetic field acts normal to the flow and the permeability of the porous medium fluctuates with time. The perturbation technique has been used to solve the problem. The profiles of velocity and skin friction have been presented graphically for different values of parameters involved in the solution to observe the effects of the visco-elastic parameter. In chapter IV, the unsteady MHD periodic flow of visco-elastic fluid through a planar channel has been investigated. The fluid flow and heat transfer is considered with saturated porous medium in the presence of a transverse magnetic field. Solutions for the velocity, temperature and skin friction have been obtained. The profiles of velocity and skin friction have been presented graphically for different values of parameters involved in the solution to observe the effects of the visco-elastic parameter. In chapter V, a theoretical analysis of mixed convective unsteady flow of a visco-elastic incompressible fluid past an accelerated infinite vertical porous plate subjected to a uniform suction has been investigated under the influence of a uniform transverse magnetic field. Approximate solutions for fluid velocity, temperature, concentration field and skin friction have been obtained. The effects of the various parameters involved in the solution have been studied. The profiles of fluid velocity and the skin friction are presented graphically to observe effects of the visco-elastic parameter. In chapter VI, the unsteady MHD oscillatory slip flow of a viscous incompressible electrically conducting optically thin fluid with variable temperature and concentration has been investigated. The flow through a planar channel filled with saturated porous medium is considered. The governing equations of the flow field are solved and the expressions for velocity, temperature, concentration, skin friction, rate of heat transfer and rate of mass transfer are obtained. The velocity profile and the skin friction are analyzed graphically to observe the effect of the visco-elastic parameter involved in the solution. In chapter VII, a theoretical study of two-dimensional free convective flow of a visco-elastic fluid through a porous medium bounded by a uniformly moving long vertical wavy wall and a parallel flat wall in presence of heat and mass transfer has been discussed. The solutions consist of two parts: a mean part and a perturbed part. To solve the perturbed part, long wave approximation has been applied and for the mean part, the

18 18 well known approximation used by Ostrach has been utilized. Solutions for the velocity, temperature, concentration, skin friction, rate of heat transfer and rate of mass transfer have been obtained. The results are discussed for the positive values of Grashof number for heat transfer (i.e. flow on cooled plate). The visco-elastic effects on velocity profile, temperature field, concentration field, skin friction, are analyzed graphically with the combination of other flow parameters involved in the solution. The above problems can also be studied with different rheological models apart from second-order fluid model. The differential equations involved in these flow problems are solved by analytical method. Different numerical methods may also be applied to solve the flow problems and can be verified with the above results.