Helical flow of yield stress fluids.
Creators: name: Alharbi, Fahad
Publisher: name: RMIT University
Year: 2024

description: This thesis investigates helical flow for generalized Newtonian yield stress fluids in the region between two infinitely long concentric coaxial cylinders. Such a flow may arise either by superimposing an axial flow driven by applying an axial pressure gradient on an existing transverse rotational flow, generated by the rotation of the inner cylinder, or by superimposing such an annular flow on an existing pressure driven axial flow. Either may be viewed as the simplest truly three dimensional flow field, and each is relevant to studies of fluid flows involved in many physical and industrial applications. Further, since yield stress fluids havethe property that they behave as a solid below the yield stress value and flow as a fluid when the local stress exceeds this value, solid and fluid regions, separated by yield boundaries, are considered to occur within the intercylindrical gap.<br><br>Three yield stress fluid models are considered in this thesis, namely the Bingham, Casson and Robertson-Stiff fluid models, each of which displays differing characteristics and occurs in differing contexts. For each of these, the flow problem for each of the two types of the helical flow described above may not be solved analytically and numerical solution techniques must be employed. However, this thesis will show that in each case, a small parameter may be identified in the flow problem, so that a perturbation method based on that parameter can be used, leading to approximate analytical solutions to the flow problem. These will also be shown to display very good agreement with numerical solutions.<br><br>Chapter 1 presents an overview of fluids, including generalized Newtonian fluids and, yield stress fluids. Then follows a review of the literature relevant to generalized Newtonian fluids, with and without yield stress and their flows. This review focuses on four aspects of these fluids and flows - yield stress fluid in application, using helical flow in rheometers, unidirectional flows as well as helical ow of non-Newtonian fluid with and without yield stress properties.<br><br>Chapter 2 introduces the equations of motion for incompressible fluids in general form and the general constitutive equation for the generalized Newtonian fluids. Attention is then given to a discussion of the constitutive equations and the properties of Bingham, Casson and Robertson-Stiff fluid models, including comparison between them.<br><br>In Chapter 3 the equations of motion, presented in Chapter 2, are expressed in cylindrical polar coordinates. These are used to obtain equations of motion for the helical flow of a generalized Newtonian fluid between two concentric cylinders. For the case where the fluid is a yield stress fluid, two distinct helical flow scenarios are discussed, namely, helical flow with a core attached to the outer cylinder and helical flow with a floating core detached from both cylinders and the features of the two types of flow are discussed. For these two helical flow scenarios, two distinct nondimensional forms of the equations of motion are derived; and, in each case, a small parameter is identified. This sets up the flow problem in each case as a perturbation problem in terms of these small parameters.<br><br>Chapter 4 gives the analysis of the helical flow problem for a Newtonian fluid in both the nondimensional forms of Chapter 3. Note that this analysis has long been known in the literature and the helical flow problem can be solved exactly for both the scenarios above. It is included here in the notation of this thesis and provides a good cross-check on the perturbation solution to be obtained in the following Chapters, since the yield stress models considered there can be reduced to Newtonian model by selecting appropriate parameter values.<br><br>Chapter 5 solves the helical flow problem for Bingham fluid approximately using a perturbation method for both the cases above - the attached core and floating core scenarios. This introduces the pattern for the perturbation analysis of the later Chapters. Analytical approximate solutions are obtained for both problems using the perturbation approach and also numerical solution are constructed and compared with these analytical approximations. Expressions for the fluid flow properties are also obtained in general form using the perturbation expansions. The axial velocity profiles for the floating core case for the Newtonian and Bingham fluids are compared and contrasted. In particular, this shows that when the yield stress the Bingham fluid reduces, the velocity profiles tend to behave like those of a Newtonian fluid.<br><br>Chapter 6 follows the pattern of Chapter 5, with perturbation solutions of the two helical flow problems for the Casson fluid (with attached and floating cores) being obtained. The analytical approximations using the perturbation method and the numerical solutions are obtained and the agreement between the two methods is found to be very good. Furthermore, the velocity profiles for the Casson fluid based on these approximations are compared with those of the Bingham fluid.<br><br>Chapter 7 analyzes the two types of helical flow for a Robertson-Stiff model. The analytical and numerical approximations are obtained and compared. These solutions are able to describe the shear thinning, shear thickening and Bingham fluid via changing the flow index value, a characteristic parameter of the Robertson-Stiff model. General expressions for the fluid flow properties are obtained to give information about the fluid behaviour within the intercylindrical gap. In addition, the shear thinning and thickening properties of this fluid are shown to be consistent with the perturbation solutions obtained.<br><br>Chapter 8 closes with a brief Conclusion and considerations of possible future work in this area.