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Finite amplitude planetary waves are studied on a homogeneous fluid on both the rotating sphere and on a mid-latitude β-plane. The integrated equations of motion are rederived both on the rotating sphere, in a spherical polar co-ordinate system whose axis is tilted relative to the rotation axis, and on a mid-latitude β-plane. The linear solutions are re-examined and the errors associated with the non-divergent and the β-plane approximations are each shown to be about 10 to 15% for waves of a few thousand kilometers wavelength. Using the integrated equations of motion both on the sphere and on the β-plane, the linear non-divergent Rossby wave solutions are shown to be exact finite amplitude solutions. An exact topographic wave solution is also given for the case of an exponential depth profile. Such behaviour is not found for the divergent waves. Using a Stokes-type expansion in terms of an amplitude parameter, the second order solution for divergent Rossby waves is obtained, and it is found that, as in surface gravity wave theory, the first order correction to the phase velocity is zero. It is also shown that the linear non-divergent Rossby wave solution on a uniformly sheared zonal current is not a finite amplitude solution, and the second order solution is then calculated. Once again, the phase speed is correct to the first order. A class of long waves of permanent form analogous to the solitary and cnoidal waves of surface wave theory is obtained for a β-plane channel of either constant or exponentially varying depth. Such waves are found to exist in the divergent case in the absence of any zonal current; however, if the divergence is weak, or if the non-divergent approximation is made , then it is found, as it was by Larsen (1965), that these waves will exist only in the presence of a weakly sheared zonal current. On the exponential depth profile, such waves exist in the absence of a sheared zonal current, even if the non-divergent approximation is made. It is suggested that such waves may also exist trapped along long ocean ridges or scarps. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The non-Newtonian incompressible power-law uid in jet ow models is investigated. An important feature of the model is the de nition of a suitable Reynolds number, and this is achieved using the standard de nition of a Reynolds number and ascertaining the magnitude of the e ective viscosity. The jets under examination are the two-dimensional free, liquid and wall jets. The two-dimensional free and wall jets satisfy a di erent partial di erential equation to the two-dimensional liquid jet. Further, the jets are reformulated in terms of a third order partial di erential equation for the stream function. The boundary conditions for each jet are unique, but more signi - cantly these boundary conditions are homogeneous. Due to this homogeneity the conserved quantities are critical in the solution process. The conserved quantities for the two-dimensional free and liquid jet are constructed by rst deriving the conservation laws using the multiplier approach. The conserved quantity for the two-dimensional free jet is also derived in terms of the stream function. For a Newtonian uid with n = 1 the twodimensional wall jet gives a conservation law. However, this is not the case for the two-dimensional wall jet for a non-Newtonian power-law uid. The various approaches that have been applied in an attempt to derive a conservation law for the two-dimensional wall jet for a power-law uid with n 6= 1 are discussed. In conjunction with the attempt at obtaining conservation laws for the two-dimensional wall jet we present tenable reasons for its failure, and a feasible way forward. Similarity solutions for the two-dimensional free jet have been derived for both the velocity components as well as for the stream function. The associated Lie point symmetry approach is also presented for the stream function. A parametric solution has been obtained for shear thinning uid free jets for 0 < n < 1 and shear thickening uid free jets for n > 1. It is observed that for values of n > 1 in the range 1=2 < n < 1, the velocity pro le extends over a nite range. For the two-dimensional liquid jet, along with a similarity solution the complete Lie point symmetries have been obtained. By associating the Lie point symmetry with the elementary conserved vector an invariant solution is found. A parametric solution for the two-dimensional liquid jet is derived for 1=2 < n < 1. The solution does not exist for n = 1=2 and the range 0 < n < 1=2 requires further investigation.

This is a 'sandwich thesis' consisting of four publications I contributed to during my M.Sc. work. These papers are the results of three types of experiments. Paper 1 studies the formation of non-uniform spin-cast polymer films. Spincoating is widely used to prepare thin polymer films of reproducible thickness. Typically spincoating produces highly uniform films, however in certain circumstances the process results in films with non-uniform surface topographies. The origin of such topographies is not fully understood and the formation of non-uniform films represents a practical problem in both research laboratories and industrial settings. In Paper 1 we find that the formation of non-uniform films is dependent on temperature. Furthermore, our results indicate that surface instabilities form as a result of the Marangoni effect. Finally, we demonstrate that non-uniformities in spin-cast films can be avoided simply by spincoating at lower temperatures. In Papers 2 and 3 we study the capillary driven levelling of polymer films with non-uniform surface geometries and compare our results to the theoretical predictions of the two-dimensional capillary-driven thin film equation. In Paper 2 we prepare polymer films with small surface perturbations and track their evolution above $T_g$ as the surface flattens. We find that all perturbations approach a universal self-similar attractor at long times, as predicted by theory. Our results also show that the time taken for the perturbations to convergence to the attractor depends on the initial volume of the perturbation. In Paper 3 we prepare samples with a rectangular trench geometry and follow their evolution above $T_g$ as surface forces cause the trench to fill in. At long times we observe a change in the levelling dynamics that is associated with a change in the boundary conditions governing the flow. In Paper 4 we use crazing experiments to probe two types of non-equilibrium entanglement networks. First, we study spincast polymer films and find that chains are stretched compared to equilibrium Gaussian chains. Furthermore, we find that the entanglement network relaxes on timescale on the order of one reptation time. Next, we stack two films in the glassy state to create a bilayer. Chains on either side of the mid-plane of the bilayer suffer a loss of entropy because of their restricted conformations. In the melt, the interface heals. We find it takes less than one reptation time for the bilayer film to become indistinguishable from a single film. / Thesis / Master of Science (MSc)

This thesis consists of two related studies presented as three separate manuscripts (all three have been published in peer-reviewed journals) and a study protocol that has been submitted for peer-reviewed funding. The over-arching theme of this thesis was to better characterize the efficacy of different intravenous fluids used for the resuscitation of intensive care unit (ICU) patients with severe sepsis or septic shock. We performed an extensive search including multiple databases which found 20 randomized controlled trials (RCTs) that examined the effects of different intravenous fluids used in septic patients and met our a priori inclusion and exclusion criteria. In the first manuscript, we described in detail the composition of the 19 unique fluid products that were used in the various studies. This description included the fluid type, trade name, osmolality, tonicity, electrolyte content, molecular composition, pH, and manufacturer. We reviewed manufacturer’s websites, product monographs, and emailed industry representatives or study authors for more information regarding the fluids as required. The results of this study and systematic review led us to the second and third manuscripts which reported on a Bayesian network meta-analysis (NMA) of all fluid type comparisons. Despite multiple well-done RCTs, comparative data regarding the clinical effect of different resuscitative fluids when used for sepsis was incomplete. Most RCTs used 0.9% saline (normal saline) as control fluid and very few studies compared colloids directly. The advantage of using an NMA model in this setting was the ability to include indirect data into the overall point estimates. Data was abstracted from the 14 studies which focused on adult ICU patients and analyzed examining the outcomes of mortality (manuscript #2) and the use of renal replacement therapy (RRT) (manuscript #3). Certainty of evidence was evaluated for both outcomes using the GRADE approach. Results of the analysis clearly document the harm of starch-based fluids when used in septic patients. Albumin containing fluids and crystalloids (such as normal saline and Ringer’s Lactate) are better options. Lower chloride solutions, such as Ringer’s Lactate, showed a signal towards decreased mortality and a decreased use of renal replacement therapy when compared to higher chloride fluids, such as normal saline, however this was based on indirect data, not statistically significant, and warrants direct comparison trials. The final component of this thesis is a pilot study protocol for a study assessing the feasibility of a larger RCT examining the effect of low chloride versus high chloride fluids for resuscitation in patients with sepsis and septic shock. This protocol has been submitted as part of a peer-reviewed grant with the hopes of addressing this clinically important and timely question. / Thesis / Master of Science (MSc) / This thesis examines the ideal intravenous fluid to be given to patients with severe infection causing low blood pressure. A review of the current literature is presented with a protocol for future work.

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The polymers are integral part of vast number of products used in day to day life due to their anomalous viscoelastic behaviour. The remarkable flow behaviour exhibited by the polymeric fluids including rod climbing, extrudate swell, tube-less siphon, viscoelastic jet, elastic recoil and sharkskin instability is attributed to the complex microstructures in the polymeric liquids that arise due to the interactions of long chain polymer molecules with each other and with the surrounding fluid particles. The significance of polymer in transportation, packaging, pharmaceutical, chemical, biomedical, textiles, food and polymer processing industries highlights the requirement to comprehend the complex rheology of polymeric fluids. First, we investigate the flow features exhibited by different shear thinning vis-coelastic fluids in rectangular cavities over a wide range of depth to width ratio. We have developed a viscoelastic flow solver in order to perform numerical simulations of highly elastic flow of viscoelastic fluids. In particular, we discuss the simulations of flows of constant viscosity Boger and shear thinning viscoelastic fluids in the complex flow problems using different constitutive equations. The effects of elasticity and inertia on the flow behaviour of two shear thinning vis-coelastic fluids modeled using Giesekus and linear PTT constitutive equations in rectangular cavities is studied. The size of the primary eddies and critical aspect ratio over which the corner eddies merge to yield a second primary eddy in deep cavities is discussed. We demonstrate that the flow in the shallow and deep cavities can be characterized using Weissenberg number, defined based on the shear rate, and Deborah number, specified based on the convective time scale, respectively. The study of flow in driven cavities is important in understanding of the mixing process during synthesis of blends and composites. Next, we study two phase polymeric flow in confined geometries. Nowadays, polymer processing industries prefer to develop newer polymer with the desired material properties mechanically by mixing and blending of different polymer components instead of chemically synthesizing fresh polymer. The microstructure of blends and emulsions following drop deformation, breakup and coalescence during mixing determines its macroscopic interfacial rheology. We developed a two phase viscoelastic flow solver using volume conserving sharp interface volume-of-fluid (VOF) method for studying the dynamics of single droplet subjected to the complex flow fields. We investigated the effects of drop and matrix viscoelasticity on the motion and deformation of a droplet suspended in a fully developed channel flow. The flow behaviour exhibited by Newtonian-Newtonian, viscoelastic-Newtonian, Newtonian-viscoelastic and viscoelastic-viscoelastic drop-matrix systems is presented. The difference in the drop dynamics due to presence of constant viscosity Boger fluid and shear thinning viscoelastic fluid is represented using FENE-CR and linear PTT constitutive equations, respectively. The presence of shear thinning viscoelastic fluid either in the drop or the matrix phase suppresses the drop deformation due to stronger influence of matrix viscoelasticity as compared to the drop elasticity. The shear thinning viscoelastic drop-matrix system further restricts the drop deformation and it displays non-monotonic de-formation. The constant viscosity Boger fluid droplet curbs the drop deformation and exhibits flow dynamics identical to the shear thinning viscoelastic droplet, thus indicating that the nature of the drop viscoelasticity has little influence on the flow behaviour. The matrix viscoelasticity due to Boger fluid increases drop deformation and displays non-monotonic deformation. The drop deformation is further enhanced in the case of Boger fluid in viscoelastic drop-matrix system. Interestingly, the pressure drop due to the presence of viscoelastic drop in a Newtonian matrix is lower than the single phase flow of Newtonian fluid. We also discuss the effects of inertia, surface tension, drop to matrix viscosity ratio and the drop size on these drop-matrix systems. Finally, we investigate the emulsion rheology by studying the motion of a droplet in the square lid driven cavity flow. The viscoelastic effects due to constant viscosity Boger fluid and shear thinning viscoelastic fluid are illustrated using FENECR and Giesekus rheological relations, respectively. The presence of viscoelasticity either in drop or matrix phase boosts the drop deformation with the drop viscoelasticity displaying intense deformation. The drop dynamics due to the droplet viscoelasticity is observed to be independent of the nature of vis-coelastic fluid. The shear thinning viscoelastic matrix has a stronger influence on the drop deformation and orientation compared to the Boger fluid matrix. The different blood components, cells and many materials of industrial importance are viscoelastic in nature. Thus, the present study has significant applications in medical diagnostics, drug delivery, manufacturing and processing industries, study of biological flows, pharmaceutical research and development of lab-on-chip devices.

I begin this study by presenting an overview of the theory of magnetohydrodynamics and the necessary conditions to justify the fluid treatment of a plasma. Upon establishing the fluid description of a plasma we move on to a discussion of magnetohydrodynamics in both the ideal and Hall regimes. This framework is then extended to include multiple plasmas in order to consider two problems of interest in the field of theoretical space physics. The first is a study on the evolution of a partially ionized plasma, a topic with many applications in space physics. A multi-fluid approach is necessary in this case to account for the motions of an ion fluid, electron fluid and neutral atom fluid; all of which are coupled to one another by collisions and/or electromagnetic forces. The results of this study have direct application towards an open question concerning the cascade of Kolmogorov-like turbulence in the interstellar plasma which we will discuss below. The second application of multi-fluid magnetohydrodynamics that we consider in this thesis concerns the amplification of magnetic field upstream of a collisionless, parallel shock. The relevant fluids here are the ions and electrons comprising the interstellar plasma and the galactic cosmic ray ions. Previous works predict that the streaming of cosmic rays lead to an instability resulting in significant amplification of the interstellar magnetic field at supernova blastwaves. This prediction is routinely invoked to explain the acceleration of galactic cosmic rays up to energies of 10¹⁵ eV. I will examine this phenomenon in detail using the multi-fluid framework outlined below. The purpose of this work is to first confirm the existence of an instability using a purely fluid approach with no additional approximations. If confirmed, I will determine the necessary conditions for it to operate.

published_or_final_version / abstract / Biological Sciences / Doctoral / Doctor of Philosophy