||The general area of
investigation is extremal combinatorics. The work will
involve exploring problems of forbidden families of
configurations. The goal is to expand on the M.Sc. work of
Christina Koch and also continue the work of Lu and myself
on bounds for a special family of configurations which made
use of Ramsey Theory. The following is a typical problem
given in matrix language. Let m be given and let F be a
given kxt (0,1)-matrix. Let A be an mxn (0,1)-matrix with no
repeated columns and no submatrix that is a row and column
permutation of F. We seek bounds on n in terms of m,F (so
called problem of Forbidden Configurations). It is perhaps
surprising that n < cm^k, for some c, but we can do even
better for many F.
Interested students could contact my previous USRA students: Ron Estrin and Foster Tom.
| Project 1: Mixing and displacement in pipe
We seek a motivated individual to help in modifying an existing experimental apparatus in order to be able to conduct experiments involving two-fluid displacement flows in an inclined pipe. The applicant will need to understand the flow experiments to be run, help in design of new components and other modifications, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. Once modified, the person will assist in various operations associated with the flow loop: mixing and fluid preparation, operating the flow loop, running careful experiment, image processing of the data. Machining and instrumentation experience, data acquisition, etc. are considered as advantages. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.
Displacement of one fluid by another is a common process in industrial applications, where the fluids are not always Newtonian and where a range of fluid properties and densities are used. Here we focus on pipe flow displacements in inclined pipes, where there is also a significant density difference. Depending on the fluid properties and flow rates the fluids either mix, or displace with a clean interface, or stratify during the displacement. We seek to understand these flows mostly experimentally also partly computationally though numerical simulations.
Project 2: Gas migration in viscoplastic fluids
We seek a motivated individual to help run experiments in a small-scale apparatus. The apparatus is made of an acrylic container and involves viscoplastic fluids, ultra-sensitive pneumatic components and high speed flow imaging. The applicant will need to understand the flow experiments to be run and may need to help in design of new components, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. The person will assist in various operations associated with the experiment: mixing and fluid preparation, running careful experiment, image processing of the data and rheometry measurements of the non-Newtonian fluids. Experience with pneumatic, machining and instrumentation, data acquisition etc. is an advantage. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.
Through fundamentally studying the topic of gas bubble movement in a viscoplastic bed with intend to resolve the gas-migration problem in cemented oil & gas wells. After drilling oil & gas wells, the annulus section between the production casing and rock formation is cemented using cement slurry. The slurry is then left to set and solidify. In this stage of the process gas may enter the cemented annulus creating channels that provide an undesirable flow path of the reservoir fluids including hydrocarbons into the wellbore and near-surface environment. Our aim is to study this problem mostly experimentally (also partly computationally and analytically) in order to finally design the cement slurry fluid such that it minimizes the gas intrusion from formation into the wellbore. This will in return, decreases the environmental impacts and increases the well productivity.
The positions are likely to appeal to applied mathematics or engineering physics students. The mathematical content of the positions lies in understanding physical fundamentals, in data analysis and some computation.
||Novel Approximation Schemes to model
Propagation Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. Examples of HF occur in nature as well as in industrial applications. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy. They have recently received considerable attention in the media due to the intense hydraulic fracturing of horizontal wells in order to release the natural gas embedded in shale-like rocks – a procedure referred to as “fracking.”
We plan to investigate the numerical solution of an integral equation that governs the propagation of a fracture in an elastic medium under conditions of plane strain. We will consider a collocation scheme to solve this integral equation. Of particular interest is the robustness of the solution to mesh refinement. The objective of this project is to devise autonomous mesh refinement strategies that will be able to achieve uniformly convergent schemes that are much more efficient than can be achieved using a uniform mesh. The new adaptive scheme scheme will then be used to solve the dynamic model for a hydraulic fracture propagating in an elastic medium. It is proposed that a computer code will be developed in MATLAB and the numerical solutions will be checked against existing asymptotic solutions.
For more information please check my web site:
||Project 1. Computation of eigenfunctions on
Abstract: Motivated by the Polymath7 project and the collocation method we will investigate a finite-element method for computing approximate eigenfunctions on plane domains. Strong programming background required.
Project 2. Topology of modular links
Abstract: We will use computer calculations to investigate topological invariants of the spaces obtained by removing closed geodesics from unit cotangent bundle of the modular surface. Programming experience will be required (familiarity with python an advantage); background in group theory, algebra and topology will be useful.