Congruences in modular, Jacobi, Siegel, and mock modular forms with applications

We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have several applications. We describe the Ramanujan congruences in the counting functions for overparitions, overpartition pairs, crank differences, and Andrews' two-coloured generalized Frobenius partitions. We also study Ramanujan congruences in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of Jacobi forms and Siegel modular forms of degree two. Finally, the last chapter contains a completely unrelated result about harmonic weak Maass forms.

Effect of boiling on deposition of metallic colloids

In this study the relationship between heterogeneous nucleate boiling surfaces and deposition of suspended metallic colloidal particles, popularly known as crud or corrosion products in process industries, on those heterogeneous sites is investigated. Various researchers have reported that hematite is a major constituent of crud which makes it the primary material of interest; however the models developed in this work are irrespective of material choice. Qualitative hypotheses on the deposition process under boiling as proposed by previous researchers have been tested, which fail to provide explanations for several physical mechanisms observed and analyzed. In this study a quantitative model of deposition rate has been developed on the basis of bubble dynamics and colloid-surface interaction potential. Boiling from a heating surface aids in aggregation of the metallic particulates viz. nano-particles, crud particulate, etc. suspended in a liquid, which helps in transporting them to heating surfaces. Consequently, clusters of particles deposit onto the heating surfaces due to various interactive forces, resulting in formation of porous or impervious layers. The deposit layer grows or recedes depending upon variations in interparticle and surface forces, fluid shear, fluid chemistry, etc. This deposit layer in turn affects the rate of bubble generation, formation of porous chimneys, critical heat flux (CHF) of surfaces, activation and deactivation of nucleation sites on the heating surfaces. Several problems are posed due to the effect of boiling on colloidal deposition, which range from research initiatives involving nano-fluids as a heat transfer medium to industrial applications such as light water nuclear reactors. In this study, it is attempted to integrate colloid and surface science with vapor bubble dynamics, boiling heat transfer and evaporation rate. Pool boiling experiments with dilute metallic colloids have been conducted to investigate several parameters impacting the system. The experimental data available in the literature is obtained by flow experiments, which do not help in correlating boiling mechanism with the deposition amount or structure. With the help of experimental evidences and analysis, previously proposed hypothesis for particle transport to the contact line due to hydrophobicity has been challenged. The experimental observations suggest that deposition occurs around the bubble surface contact line and extends underneath area of the bubble microlayer as well. During the evaporation the concentration gradient of a non-volatile species is created, which induces osmotic pressure. The osmotic pressure developed inside the microlayer draws more particles inside the microlayer region or towards contact line. The colloidal escape time is slower than the evaporation time, which leads to the aggregation of particles in the evaporating micro-layer. These aggregated particles deposit onto or are removed from the heating surface, depending upon their total interaction potential. Interaction potential has been computed with the help of surface charge and van der Waals potential for the materials in aqueous solutions. Based upon the interaction-force boundary layer thickness, which is governed by debye radius (or ionic concentration and pH), a simplified quantitative model for the attachment kinetics is proposed. This attachment kinetics model gives reasonable results in predicting attachment rate against data reported by previous researchers. The attachment kinetics study has been done for different pH levels and particle sizes for hematite particles. Quantification of colloidal transport under boiling scenarios is done with the help of overall average evaporation rates because generally waiting times for bubbles at the same position is much larger than growth times. In other words, from a larger measurable scale perspective, frequency of bubbles dictates the rate of collection of particles rather than evaporation rate during micro-layer evaporation of one bubble. The combination of attachment kinetics and colloidal transport kinetics has been used to make a consolidated model for prediction of the amount of deposition and is validated with the help of high fidelity experimental data. In an attempt to understand and explain boiling characteristics, high speed visualization of bubble dynamics from a single artificial large cavity and multiple naturally occurring cavities is conducted. A bubble growth and departure dynamics model is developed for artificial active sites and is validated with the experimental data. The variation of bubble departure diameter with wall temperature is analyzed with experimental results and shows coherence with earlier studies. However, deposit traces after boiling experiments show that bubble contact diameter is essential to predict bubble departure dynamics, which has been ignored previously by various researchers. The relationship between porosity of colloid deposits and bubbles under the influence of Jakob number, sub-cooling and particle size has been developed. This also can be further utilized in variational wettability of the surface. Designing porous surfaces can having vast range of applications varying from high wettability, such as high critical heat flux boilers, to low wettability, such as efficient condensers.

A geometric study of liquid retention in open-cell metal foams

Open-cell metal foams show promise as an emerging novel material for heat exchanger applications. The high surface-area-to-volume ratio suggests increased compactness and decrease in weight of heat exchanger designs. However, the metal foam structure appears conducive to condensate retention, which would degenerate heat transfer performance. This research investigates the condensate retention behavior of aluminum open-cell metal foams through the use of static dip tests and geometrical classification via X-ray Micro-Computed Tomography. Aluminum open-cell metal foam samples of 5, 10, 20, and 40 pores per inch (PPI), all having a void fraction greater than 90%, were included in this investigation. In order to model the condensate retention behavior of metal foams, a clearer understanding of the geometry was required. After exploring the ideal geometries presented in the open literature, X-ray Micro-Computed Tomography was employed to classify the actual geometry of the metal foam samples. The images obtained were analyzed using specialized software from which geometric information including strut length and pore shapes were extracted. The results discerned a high variability in ligament length, as well as features supporting the ideal geometry known as the Weaire-Phelan unit cell. The static dip tests consisted of submerging the metal foam samples in a liquid, then allowing gravity-induced drainage until steady-state was reached and the liquid remaining in the metal foam sample was measured. Three different liquids, water, ethylene glycol, and 91% isopropyl alcohol, were employed. The behaviors of untreated samples were compared to samples subjected to a Beomite surface treatment process, and no significant differences in retention behavior were discovered. The dip test results revealed two distinct regions of condensate retention, each holding approximately half of the total liquid retained by the sample. As expected, condensate retention increased as the pores sizes decreased. A model based on surface tension was developed to predict the condensate retention in the metal foam samples and verified using a regular mesh. Applying the model to both the ideal and actual metal foam geometries showed good agreement with the dip test results in this study.