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GENERALIZED DEFORMATION-RATES in SECONDARY FLOWS of VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES Redacted for Privacy Kbstract Approved: M.N.L

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GENERALIZED DEFORMATION-RATES in SECONDARY FLOWS of VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES Redacted for Privacy Kbstract Approved: M.N.L AN ABSTRACT OF THE THESIS OF RONALD NORMAN KNOSHAUG for the MASTER OF ARTS (Name) (Degree) in MATHEMATICS presented on October 10, 1968 (Major) (Date) Title: GENERALIZED DEFORMATION-RATES IN SECONDARY FLOWS OF VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES Redacted for Privacy kbstract approved: M.N.L. Narasimhan By various experiments, it has been found that the re- sponse of real materials to external forces is, in general, nonlinear in character. In classical continuum mechanics, the use of ordinary measures of strain have forced the constitutive equations to take complex forms and since the orders of these measures are not fixed, many unknown re- sponse coefficients have to be introduced into the consti- tutive equations. In general, there is no basis of choos- ing these coefficients. Seth attempted to resolve this difficulty by introducing generalized measures in continuum mechanics and Narasimhan and Sra extended these measures in such a way as to adequately explain some rheological behav- ior of materials. The constitutive equation of Narasimhan and Sra essentially contains two terms and four rheological constants and, unlike some previous theories, it does not contain any unknown functions of the invariants of kine- matic matrices while at the same time explains many visco- elastic phenomena. In the present investigation, a theorem has been proved establishing certain criteria for fixing the orders of generalized measures suitably so as to predict different types of viscoelastic phenomena, such as dilatancy. We have found during the course of this investigation that the constitutive equation of Narasimhan and Sra does not ade- quately explain such physical phenomena as pseudoplasticity. However, in order to donstruct a constitutive equation so that it does explain such phenomena, we have found it nec- essary to construct combinations of sets of generalized measures. The resulting constitutive equation is found to be quite general and is able to explain a vast range of physical behavior of fluids. To illustrate the use of this constitutive equation based upon combined generalized measures, we have investi- gated the important problem of secondary flows for fluids in the presence of moving boundaries. This problem is very important, since the investigation of secondary flows al- lows us to obtain a clearer picture of the actual motion of the fluid. For the problem of flow of a fluid in thean- nulus of two rotating spheres, we have obtained the solu- tion for the velocity and pressure fields. In order to in- vestigate the secondary flow pattern more thoroughly, we have obtained the streamline function of the flow. The streamlines in meridian planes containing the axis of ro- tation are found to be closed loops and the nature of the closed loops is found to be strongly dependent upon the viscoelastic parameter S. For S less than critical value, the flow is found to be very much like that of a Newtonian fluid, with the fluid advancing toward the inner sphere along the pole and outward along the equation. At the critical value the flow region is found to split into two subregions, each containing closed loops of stream- lines. As S is increased further, another critical value is reached whereby the streamlines again become one set of closed loops, with the sense of rotation reversed from that of Newtonian fluid. Generalized Deformation-Rates in Secondary Flows of Viscoelastic Fluids Between Rotating Spheres by RONALD NORMAN KNOSHAUG A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Arts June 1969 APPROVED: Redacted for Privacy Professor of Mathematics in charge of major Redacted for Privacy Acting Chairman of Department cif Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented October 10, 1968 Typed by Charlene Laski for Ronald Norman Knoshaug Acknowledgements The author wishes to express his sincere appreciation to his major professor, Professor Narasimhan, for his sug- gestion of the problem, criticisms and suggestions during the course of the research, and especially his giving of unlimited time during the writing of this thesis. The author wishes to express his sincere appreciation to his wife Judy for the moral support she gave him during the course of research. The author thanks Charlene Laski for the excellent typing of the thesis The author is indebted to the Atomic Energy Commission for their financial support, (AT(45-1)-1947). Table of Contents Chapter Page I. Introduction 1.1 Classical Strain Measures 1 1.2 Present Constitutive Theory of Materials and Their Limitations 3 1.3 Purpose and Need for the Present Investigation 9 1.4 Plan of the Present Investigation 10 II. Measure Concept in Continuum Mechanics 2.1 Introduction 11 2.2 Generalized Measures of Deformation in Elasticity 12 2.3 Generalized Measures of Deformation-Rates in Fluid Dynamics 16 2.4 Generalized Measures of Deformation-Rates in Viscoelasticity 19 III. Extension of Generalized MeaSure Concept 3.1 Introduction 22 3.2 A Theorem on the Classification of Visco- elastic Fluid based on Orders of Measures 23 3.3 Combined Generalized Measures in Visco- elasticity 34 3.4 Cpnstitutive Equation based on Combined Generalized Measures 36 IV. Primary and Secondary Flows in the Presence of Moving Boundaries 4.1 Introduction 39 4.2 Secondary Flows in Classical Fluids 40 (Theoretical Results) 4.3 Secondary Flows in Non-Newtonian Fluids (Theoretical Results) 42 4.4 Experimental Results on Secondary Flow 50 V. Application of Generalized Measures of Deformation- Rates to Secondary Flows of Viscoelastic Fluids between Rotating Spheres 5.1 Formulation of the Problem 54 5.2 Method of Solution of the Equations of Motion 65 5.3 Primary Flows in Pseudoplastic Fluids 66 Chapter Page 5.4 Secondary Flows in Pseudoplastic Fluids 68 5.5 Discussion of the Results 78 VI. Summary, Discussion, and Scope of Further Work 6.1 Summary and Discussion 82 6.2 Scope of Further Work 85 Bibliography 86 Appendix 89 List of Figures Figure Page 3.1 Simple shear model 25 3.2 Graph of apparent coefficient of viscosity for a pseudoplastic fluid 38 4.1 Typical Streamlines of steady radial-axial flow. One disk oscillating 41 4.2 Typical Streamlines of steady radial-axial flow. Two disks oscillating 41 4.3 Streamlines of secondary flow about a sphere rotating in a fixed spherical container 42 4.4 Secondary flow in a tube with elliptical cross- section 43 4.5 The secondary flow pattern in one quadrant of annulus of two spheres for a fluid whose para- meters have passed the critical values 46 4.6 The flow pattern of a viscoelastic fluid between two rotating cones 48 4.7 The projections of the particle paths on any plane containing the axis of rotation when m = 1/6. 49 4.8 The projections of the particle path on any plane containing the axis of rotation when m = 1/4. 50 4.9 Drawing showing flow characteristics 51 4.10 Vortices generated by a rotating cone immersed in 7.5% solution of Polyisobutylene in Cetane 52 5.1 Notation 56 5.2 Typical streamlines of the secondary flow for the case S = 0[01 = 0.2, Q2 = 0.1, and A = 2]. 80 List of Figures (con't) Figure Page 5.3 Typical streamlines of the secondary flow for the caseS = 4.5[Q1 = 0.2, Q2 = 0.1, and A = 2]. 80 5.4 Typical streamlines of the secondary flow for the case S = 5.5[Q = 0.2, Q = 0.1, 1 2 and A = 2]. 81 Notation We will be using the notation of tensor analysis and hence shall assume some knowledge of tensors. We shall ex- plain the meaning of the various symbols used as and when they arise. However, we shall list the more frequently used symbols which may be referred to whenever needed: Symbol Meaning ai ithcomponent of acceleration. a,b,c physical components of acceleration in spherical-polar coordinates along the r-, 8-, and cp- directions res- pectively. A,B,C non-dimensional components of accel- eration in spherical-polar coordin- ates along the r-, 8-, and (1)- directions respectively. dimension correcting constants. a2 b.. second deformation-rate tensor. 13 Symbol Meaning b*.. generalized second deformation-rate 1J tensor. B second deformation-rate matrix. B* generalized second deformation-rate matrix B** combined generalized second deforma- tion-rate matrix radius of inner sphere d.. first deformation-rate tensor 13 d*.. generalized first deformation-rate 13 tensor D first deformation-rate matrix D* generalized first deformation-rate matrix D** combined generalized first deforma- tion-rate matrix a13.. kronecker delta A Laplacian in spherical-polar coordin- ates A* non-dimensional Laplacian in spheri- cal-polar coordinates D material derivative Dt a intrinsic derivative e strain tensor e.. strain tensor 1J Symbol Meaning e*.. generalized strain tensor 13 E strain matrix E* generalized strain matrix dimension correcting constant y dimension correcting constant h non-dimensional rheological parameter H coefficient determinant I unit matrix I II III first, second, and third invariants B' B' B of the second deformation-rate matrix II III first, second, and third invariants D ' D' D of the first deformation-rate matrix II , III first, second, and third invariants e e e of the strain tensor J, set of positive integers, positive j0' je odd integers, positive even integers respectively k, 1(1, k2, k' dimension correcting constants K non-dimensional rheological parameter deformed and undeformed lengths res- 510 pectively L length A ratio of outer sphere to inner sphere m, m, m, m', m dimension correcting constant 1 2 M mass 1.1 dimension correcting constant Symbol Meaning n, n1,, n2,n' measure indices v apparent viscosity a 0 0 angular velocities of the inner and 1' 2 out sphere respectively hydrostatic pressure P non-dimensional hydrostatic pressure irreversibility indices q, q2, q' r, e, spherical-polar coordinate system density of fluid S non-dimensional rheological parameter t time stress tensor deviatoric part of stress tensor t( , t(yz) ,t , physical components of the stress xy) (xz) tensor in the rectangular coordinate etc.
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