Student Number : 9701675D - MSc dissertation - School of Mathematics - Faculty of Science / The role of invariants in obtaining exact solutions of di#11;erential equations is reviewed. The examples considered are nonlinear evolution type equations like the Fisher and Fitzhugh-Nagumo equations. Finally, we look directly at an equation (formulated) governing some non-Newtonian uid in a rotating system.

The objective of the thesis is to characterize the linear and nonlinear aspects of inertial and pressure effects in turbulent flows. In the first part of the study, computations of Navier-Stokes and 3D Burgers equations are performed in the rapid distortion (RD) limit to analyze the inviscid linear processes in homogeneous turbulence. By contrasting the results of Navier- Stokes RD equations and Burgers RD equations, the effect of pressure can be isolated. The evolution of turbulent kinetic energy and anisotropy components and invariants are examined. In the second part of the thesis, the velocity gradient dynamics in turbulent flows are studied with the help of inviscid 3D Burgers equations and restricted Euler equations. The analytical asymptotic solutions of velocity gradient tensor are obtained for both Burgers and restricted Euler equations. Numerical computations are also performed to identify the stable solutions. The results are compared and contrasted to identify the effect of pressure on nonlinear velocity gradient dynamics. Of particular interest are the sign of the intermediate principle strain-rate and tendency of vorticity to align with the intermediate principle strain-rate. These aspects of velocity gradients provide valuable insight into the role of pressure in the energy cascade process.

abstract: Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ordinary and partial differential equation (ODE, PDE) models for two problems: Vicodin abuse and impact cratering. The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable. Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes. Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous. / Dissertation/Thesis / Psyche asteroid impact simulation initialization / Psyche asteroid impact simulation video / Doctoral Dissertation Applied Mathematics 2019