Algorithms for Bohemian Matrices

This thesis develops several algorithms for working with matrices whose entries are multivariate polynomials in a set of parameters. Such parametric linear systems often appear in biology and engineering applications where the parameters represent physical properties of the system. Some computations on parametric matrices, such as the rank and Jordan canonical form, are discontinuous in the parameter values. Understanding where these discontinuities occur provides a greater understanding of the underlying system. Algorithms for computing a complete case discussion of the rank, Zigzag form, and the Jordan canonical form of parametric matrices are presented. These algorithms use the theory of regular chains to provide a unified framework allowing for algebraic or semi-algebraic constraints on the parameters. Corresponding implementations for each algorithm in the Maple computer algebra system are provided. In some applications, all entries may be parameters whose values are limited to finite sets of integers. Such matrices appear in applications such as graph theory where matrix entries are limited to the sets {0, 1}, or {-1, 0, 1}. These types of parametric matrices can be explored using different techniques and exhibit many interesting properties. A family of Bohemian matrices is a set of low to moderate dimension matrices where the entries are independently sampled from a finite set of integers of bounded height. Properties of Bohemian matrices are studied including the distributions of their eigenvalues, symmetries, and integer sequences arising from properties of the families. These sequences provide connections to other areas of mathematics and have been archived in the Characteristic Polynomial Database. A study of two families of structured matrices: upper Hessenberg and upper Hessenberg Toeplitz, and properties of their characteristic polynomials are presented

Flows in Grooved Channels

This dissertation presents the analysis of effects of two-dimensional grooves on flow responses in laminar channel flows. Straight grooves have been considered which may have an arbitrary cross-section and an arbitrary orientation with respect to the flow direction. It has been shown that the grooves effects can be split into two parts; one due to the change in the mean positions of the walls and the other due to the flow modulations created by the groove geometry. The former effect can be determined analytically, while the latter effect requires numerical modelling. Projection of groove shape onto a Fourier space creates a basis for a reduced-order geometry model which has been used to capture the modulation effects. A spectral algorithm based on Fourier and Chebyshev expansions has been developed for numerical simulation which provides solutions with high levels of accuracy. The difficulties associated with the enforcement of the boundary conditions on the irregular geometries have been overcome either by using the immersed boundary conditions (IBC) or the domain transformation (DT) methods. Three types of flow have been considered; (i) pressure-driven flow, (ii) kinematically-driven flow, and (iii) flow driven by a combination of these two driving mechanisms. The effect of grooves on flow losses have been assessed based on either the additional pressure gradient required to maintain the same mass flow rate as in the case of reference smooth channel or the change in the mass flow rate induced by the grooves for flows driven with the same pressure gradient as in the case of reference flow. Detailed analyses of the extreme cases, i.e. grooves that are orthogonal to the flow direction (transverse grooves) and those that are parallel to the flow direction (longitudinal grooves or riblets) have been carried out. Mechanisms of drag generation for each case have been identified. Analytical solutions have been determined in the limit of long wavelength grooves in order to simplify identification of these mechanisms. It has been shown that longitudinal grooves with wavelengths larger than a critical value are able to reduce drag to values lower than the smooth channel value despite increase of the wetted surface area. For sufficiently short wavelength grooves, shear is eliminated over a majority of the wetted area but there is a rapid rise of local shear and pressure forces around the tips of grooves which counteracts the elimination of shear and results in an overall increase of drag. Potential for drag-reducing surfaces for this case exists if a method for reduction of undesired pressure and shear forces around groove tips can be found through proper shaping of the wall. Optimization method has been used in order to find forms of longitudinal grooves which minimize the flow losses in grooved channel and optimal shapes for different flow conditions have been identified