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In this thesis we examine some of the interesting aspects of stability for some convection problems. Specifically, the first part of the thesis deals with the Bénard problem for various Non-Newtonian fluids, whereas the second part develops a stability analysis for convection in a porous medium. The work on stability for viscoelastic fluids includes nonlinear stability analyses for the second grade fluid, the generalised second grade fluid, the fluid of dipolar type and the fluid of third grade. It is worth remarking that throughout the work the viscosity is supposed to be any given function of temperature, with the first derivative bounded above by a positive constant. The connection between the two parts of the thesis is made through the method used to approach the nonlinear stability analysis, namely the energy method. It is shown in the introductory chapter how this method works and what are its advantages over the linear analysis. Nonlinear stability results established in both Part I and Part II are the best one can get for the considered physical situations. Different choices of energy have been considered in order to achieve conditional or unconditional nonlinear stability results.

In this work, we develop a method for solving viscoelastic fluid flows using the Navier-Stokes equations coupled with the Oldroyd-B model. We solve the Navier-Stokes equations in skew-symmetric form using the mixed finite element method, and we solve the Oldroyd-B model using the discontinuous Galerkin method. The Crank-Nicolson scheme is used for the temporal discretization of the Navier-Stokes equations in order to achieve a second-order accuracy in time, while the optimal third-order total-variation diminishing Runge-Kutta scheme is used for the temporal discretization of the Oldroyd-B equation. The overall accuracy in time is therefore limited to second-order due to the Crank-Nicolson scheme; however, a third-order Runge-Kutta scheme is implemented for greater stability over lower order Runge-Kutta schemes. We test our numerical method using the 2D cavity flow benchmark problem and compare results generated with those found in literature while discussing the influence of mesh refinement on suppressing oscillations in the polymer stress.

In this thesis we describe a technique for animating the behavior of viscoelastic fluids such as mucus, liquid soap, pudding, toothpaste, or clay, that exhibit a combination of both fluid and solid characteristics. The technique builds upon prior Eulerian methods for animating incompressible fluids. Our method computes viscoelastic fluid behavior by supplementing the basic Navier--Stokes equations with additional terms for elastic body forces. These terms can be readily computed on rectilinear grids using a staggered discretization scheme, and the use of an Eulerian formulation easily accommodates modeling flows that undergo large deformations with topological changes. These elastic terms require computing the material strain throughout the fluid. Because the fluid simulations do not make use of an explicit reference configuration, strain is computed by integrating strain rate and advecting the results. The transition from elastic resistance to viscous flow is controlled by von Mises' yield condition, and subsequent behavior is governed by a quasi-linear plasticity model.

We develop a viscoelastic version of the volume of fluid algorithm for tracking deformable interfaces. The code uses the piecewise linear interface calculation method to reconstruct the interface, the continuous surface force formulation to model interfacial tension forces and utilizes the semi-implicit Stokes solver (enabling computations at low Reynolds numbers). The algorithm is primarily designed to simulate the flow of superposed fluids and the drop in a flow problem in 2D shear flows of viscoelastic and/or Newtonian fluids. The code is validated against linear stability theory for the two-layer flow case and against experimental and other documented numerical investigations for the droplet-matrix case.

The rheological behavior of selected polymer solutions used for mobility control in waterflooding, micellar fluids used in low-tension chemical flooding, and micellar/polymer mixtures has been examined. Poly-mers used were Kelco's Xanflood (a xanthan gum) and Tiorro's Watercut 110 and 160. The micellar solutions contained Witco's TRS 10-80 or TRS 10-410 (both petroleum sulfonates), isobutyl alcohol, sodium chloride, and one of several hydrocarbons (iso-octane, n-decane, or Nectone 37 oil). A variety of porous media were utilized, ranging from packs made with glass beads to sandpacks and consolidated porous media (Berea rock) . Use was made of the modified Blake-Kozeny model of the porous medium to correlate the steady flow viscometric behavior of the fluids with that of the apparent viscosity as the fluid is displaced through a porous medium. Measurements were made to extend the shear rate range of investigation to values beyond those used in earlier experiments. Static measurements were made of phase volumes and viscosities of various mixtures of surfactant, water, alcohol, sodium chloride, polymer, and oil to obtain a more complete characteriza-tion of the phase behavior and viscosities of the fluids used in the flow experiments. Three different rheological measurements were made for each fluid. A couette viscometer was used for steady state measurements at low shear rates, a capillary viscometer measured the steady flow viscosity at medium to high rates of shear, and the-apparent viscosity was calculated from pressure drop-flow rate data as the fluid was flowing through the porous medium. For the micellar solutions studied, use of the modified Blake-Kozeny relation proved to be adequate to characterize their rheological behavior in porous media. Micellar fluids containing Witco's TRS 10-410 or TRS 10-80 petroleumsulfonates,isobutyl alcohol, and brine were mixed with various hydrocarbons. The apparent viscosity for these fluids as they flowed in glass bead packs was correlated with steady viscosity and good agreement between the values was achieved. A single experiment was done with a micellar fluid in Berea rock and a good correlation between apparent viscosity and the steady flow values was also realized, despite the existence of a residual water phase. The maximum viscosity at low shear rates and the onset of pseudoplastic behavior for these fluids was a strong function of oil type, oil concentration, and salinity. No shear degradation was observed for any of these fluids up to several thousand -1 sec The apparent viscosities of two Xanflood polymer solutions in Berea rock did not show very good agreement with the measured steady flow viscosity values. The apparent viscosity values were lower than the measured bulk viscosities over the middle shear rate range. At very high shear rates, the apparent viscosity of the fluids became slightly greater than the viscosity as measured in the capillary viscometer. A dispersion experiment done for these same Xanflood polymer solutions in Berea rock has indicated some polymer transport problems characterized by a polymer breakthrough lag behind that of tritiated water. v An experiment to characterize the rheological properties of a micellar/polymer mixture in a glass bead pack showed that the modified Blake-Kozeny relation was adequate under the conditions used. A polymer cross-linking experiment utilizing the cationic Watercut 160 and the anionic Watercut 110 was conducted with the cation Al3+. Alternating slugs of the anionic Watercut 110 and the aluminum ion solution resulted in no significant changes in the resistance factor in a high permeability sandpack. Alternating slugs of the cationic Watercut 160 and the anionic Watercut 110 produced a slight increase in resistance factor; however, further injections of Al3+ and Watercut 110 resulted in no change in resistance factor.