SUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS. where most of body fluids like blood and mucus are non-newtonian ones.

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1 SUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS Non-Newtonian fluids abound in many aspects of life. They appear in nature, where most of body fluids like blood and mucus are non-newtonian ones. Also, many food products like, for example, mayonnaise, ketchup, egg white, honey, cream cheese, molten chocolate belong to such class of fluids. Paints, that must be easily spread under the action of stress, but should not flow spontaneously once applied to the surface, as well as printer inks, lipstick are further examples. Another huge area of appearance of non-newtonian fluids is plastic industry. The examples are molten plastic and other man-made materials formed to produce everyday wealth like textiles, plastic bags, plastic toys, through the processes like extrusion, molding, spinning, for example. Often non-newtonian materials are created by addition of various polymers. The detergent industry adds polymers to shampoos, gels, liquid cleaning to improve their rheological properties. Non-Newtonian fluids are also used in motor industry. Multi-grade oils have polymer additives that change the viscosity properties of viscoelastic fluids, mentioned above, can help to reduce the overall production cost of goods made of those fluids. One of the means to achieve this goal is to use simulation tools that involve mathematical (numerical) methods. The study of non-newtonian fluids involves the modeling of flow with dense molecular structure such as polymer solutions, slurries, pastes, blood and paints. There

2 2 materials exhibit both viscous properties like liquids and elastic properties like solids and the understanding of their complex behavior is crucial in many industrial applications. Due to increasing importance of non-newtonian fluids in modern technology and industries, the investigation of such fluids is desirable. The flows of non-newtonian fluids occur in a variety of applications, for example from oil and gas well drilling to well completion operations, from industrial processes involving waste fluids, synthetic fibers, foodstuffs, extrusion of molten plastic and as well as in some flows of polymer solutions. Some important studies dealing with the flows of non- Newtonian fluids are made by Abel-Malek et al. [1], Ariel et al. [6], Chen et al. [24], Fetecau and Fetecau [36, 37, 39], Hayat and Ali [64], Hayat and Kara [65], Hayat et al. [60, 62, 63, 66], Rajagopal and Gupta [99], Rajagopal and Na [98, 100] and Wafo- Soh [126]. Modeling viscoelastic flows is important for understanding and predicting the behavior of process and thus for designing optimal flow configurations and for selecting operating conditions. Because of the complex nature of these fluids there is not a single constitutive equation available in the literature which describes the flow properties of all non-newtonian fluids. For this reason various models have been suggested and among those models, power-law and differential type fluids have acquired a great deal of attention. Some relevant contributions dealing with this type of fluids are given in references Fetecau et al. [38, 40, 41], Hayat et al. [58, 65, 67], Khan et al. [76, 77], Tan and Masuoka [116, 117].

3 3 Power-law fluids, also referred to as fluids of grade one are the simplest models of non-newtonian fluids and it is well-known for accurately modeling the shear stress and shear rate of non-newtonian fluids, but it does not properly predict the normal stress differences that are observed in phenomena like die-swell and rodclimbing (Schowalter [107]) which are manifestation of the stresses that develop orthogonal to the plane of shear which can be well modeled by extending the study to the fluid of grade two. In turn this does not fit shear thinning and shear thickening fluids. The third grade fluid model represents a further, although inconclusive, attempt towards a more comprehensive description of the behavior of viscoelastic fluids. Accordingly certain effects may well be described by flow of fourth grade fluids. On the other hand, the governing equations resulting from non-newtonian fluid models are non-linear high order equations whose analysis presents a particular challenge to researchers. Hence progress was limited until recent times and closedform solutions are available to more problems of particular interest than before. Also the study of such flows in porous media are quite important in many engineering fields such as enhanced oil recovery, paper and textile, but little work seems to be available in the literature. Few recent studies Hayat at el. [61], Khan et al. [76, 78], Tan and Masuoka [116, 117] may be mentioned in this direction. Similarly the study of hydrodynamic flows with application of magnetic field (MHD flows) is of particular interest in chemical engineering, electromagnetic propulsions and the study of the flow of blood and yet again the literature is scarce. Mention maybe made here to the recent study of the topic by Hayat et al. [54, 55, 56, 58, 59].

4 4 During the past years, there has been a growing recognition of the fact that many substances of industrial significance, especially of multi-phase nature (foams, emulsions, dispersions and suspensions, slurries, for instance) and polymeric melts and solutions (both natural and manmade) do not conform to the Newtonian postulate of the linear relationship between shear stress (σ) and shear rate (γ), for instance. Accordingly, these fluids are variously known as non-newtonian, non-linear, complex or rheological complex fluids. In many fields, such as food industry, drilling operations, polymer chemical industry and bio-engineering, the fluids, either synthetic or natural, are mixtures of different stuffs such as water, particles, oils, red cells and other long chain molecules. Generally, the viscosity function varies non-linearly with the shear rate and the elasticity is felt through elongation effects and time-dependent effects. In these cases, the fluids have been treated as visco-elastic fluids. Because of the difficulty, to suggest a single model, which exhibits all properties of visco-elastic fluids, many models or constitutive equations have been proposed, most of them being empirical or semiemperical. Flows of a number of fluids such as polymeric liquids, food products, paints, slurries, foams, and so forth cannot be adequately described by means of the classical linearly viscous Newtonian model. Thus, there is need to have at hand an arsenal of non-newtonian fluid models and, over the past several decades, a variety of such models have been developed. Among these models, the model of fluids of differential type gained much attention. These fluids can explain, for instance, normal stress

5 5 differences in a simple shear flow, shear thinning, shear thickening, and nonlinear creep characteristics exhibited by some non-newtonian fluids but they cannot describe the stress relaxation exhibited by certain liquids. Another class of fluids is that of the rate-type fluids, whose models can describe stress relaxation, nonlinear creep, shear thinning, shear thickening and normal stress differences in simple shear flows. Maxwell [84] developed the first rate type one-dimensional model that could describe stress relaxation. Oldroyd [90] was the first to develop systematically threedimensional rate-type models that met the requirements of frame indifference. Although, the Oldroyd-B model can describe stress relaxation and normal stress differences in a simple shear flow, it is incapable of describing shear thinning and shear thickening. It can, however, be easily generalized to describe shear thinning and shear thickening The primary objective of the present study was to develop a new methodology to investigate flow of the visco-elastic non-newtonian fluid. In this thesis we determined some exact solutions of generalized second grade and Oldroyd-B fluids in cylindrical domains and to underline some energetic aspects corresponding to some unsteady motions of Maxwell fluids. To find exact solutions, we used Laplace and Hankel transforms. The solutions that have been obtained are presented as a sum between the Newtonian solutions and the corresponding non-newtonian contributions. They satisfy both the governing equations and all imposed initial and boundary conditions.

6 6 Chapter I contains the basic preliminaries regarding fluids of rate type, the fundamental flow equations, the constitutive equations, energetic balance and Laplace and Hankel transforms. In chapter II, we have determined the exact solutions of velocity field and the associated shear stress corresponding to flow of a generalized second grade fluid between two infinite concentric circular cylinders. The motion is produced by the two cylinders which at time t = 0 + begin to rotate around their common axis with angular velocities t and t 1 2. The solutions, obtained by means of Laplace and Hankel transforms, are presented under integral and series forms in terms of the generalized a, b, c, G function, and satisfy all imposed initial and boundary conditions. For 1or 1and 0, the similar solutions for the ordinary second grade fluids, respectively, Newtonian fluids are recovered. The velocity field and the adequate shear stress corresponding to the flow between two cylinders, one of them being at rest, are obtained as particular cases of our general solutions. Making 1 0 and 2, we obtain the velocity field corresponding to the flow between cylinders, the inner cylinder being at rest. In chapter III, the flow of a Bingham fluid over a rotating disk was considered. The flow is characterized by the dimensionless yield stress Bingham number, by which is the ratio of the yield and local viscous stress. Using von Karman s similarity transformation, and introducing the rheological behavior law of the fluid into the conservation equations, the corresponding non-linear two-point boundary value

7 7 problem is formulated. A solution to the problem under investigation of the set of ordinary differential equations, using a multiple shooting method, which employs a fourth order Runge-Kutta method to implement the numerical integration of the equations, and Newton iteration to determine the unknowns F 0 and 0 G. In chapter IV, the exact and approximate expressions for the power due to the shear stress at the wall L, the dissipation and the boundary layer thickness corresponding to the unsteady motion induced by a constantly accelerating plate in a Maxwell fluid have been established. As a consequence, the changing of the kinetic energy with time is obtained from the balance energy. In the special case when the retardation time tends to zero, the results for Newtonian fluids are recovered. Further, our result regarding the boundary layer thickness is different of that obtained in Zierep and Fetecau for the flow due to the impulsive motion of the plate where decreases in comparison with Newtonian fluids. Finally, in contrast with the result obtained by Teipel, it is shown that the corresponding series solution is completely determined by means of the appropriate boundary conditions. Furthermore, this series solution as well as the associated series solution for the shear stress is identical to the solution obtained by means of the asymptotic approximations from the general solution and respectively. In chapter V, the velocity field and the adequate shear stress corresponding to the rotational flow induced by an infinite circular cylinder in an incompressible generalized Maxwell fluid have been determined using Laplace and Hankel transforms. The motion is produced by the circular cylinder that at the initial moment

8 8 begins to rotate around its axis with an angular velocity of constant acceleration. The solutions that have been obtained, written under integral and series forms in terms of the generalized G functions satisfy all imposed initial and boundary conditions. Furthermore, they are presented as a sum between the Newtonian solutions and the adequate non-newtonian contributions. In the special case when 1, the similar solutions for ordinary Maxwell fluids, performing same motion, are obtained. Maxwell model was very helpful in the development of linear viscoelasticity but in some cases it was found inadequate linear to describe linear viscoelastic data. During the time, many other models were suggested. Time derivative of rate of strain tensor was introduced in Jeffreys model which contained relaxation and retardation times. The chapter VI is to provide the velocity field for the unsteady flow of an incompressible generalized Oldroyd-B fluid induced by a suddenly moved plate between two side walls perpendicular to the plate and to spotlight the influence of the fractional parameters on the fluid motion. The exact solutions, obtained using Fourier sine and Laplace transforms, are presented under integral and series form in terms of the generalized G,, (, ) functions. Oldroyd-B fluids are also obtained as limiting a b c cases of our general solutions. Furthermore, the present solutions are presented as a sum of the Newtonian solution and the corresponding non-newtonian contributions. In the absence of side walls, that is, h, all solutions that have been obtained reduce to the solutions corresponding to the motion over an infinite suddenly moved plate.