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This study explores how electroosmosis and buoyancy forces affect flow regimes in electrokinetic systems within rectangular capillaries and how these regimes shape concentration profiles under varying temperature and non-symmetric conditions caused by uneven wall convection. The advective impact of Joule heating and convection on solute migration is investigated, with emphasis on determining which force dominates and how non-symmetric environments influence flow regimes and dispersion—key considerations for designing efficient electrokinetic devices and effective soil-remediation protocols. Using generalized (Robin-type) boundary conditions, the study introduces a skewness parameter 𝑅2 to help predict flow reversal behavior and mixing issues based on system parameters. The analysis applies heat-transfer modeling, solves the Navier–Stokes equation for buoyancy-driven cases limited by 𝑅2, and solves the molar species continuity equation to obtain concentration profiles across scenarios of 𝑅2 values and Joule heating. The area-averaging method is used for the advective case and limiting scenarios (including insulation and uneven environments) are reported, along with reverse-flow conditions and their mixing impact on concentration profiles. 

I joined the 3 Pasos Program at California Baptist University as an undergraduate teaching assistant, first by working on a research project about the geometry of beehive cells and later by helping with the Primer Paso summer bridge program for high school students. The 3 Pasos program is built around a Familia-Cohort mentorship model that emphasizes community, belonging, and hands-on STEM experiences for underrepresented students. Our project asked a simple but fascinating question: why do bees build hexagonal hives? Exploring this led me to study ideas such as surface-area-to-volume ratios and isoperimetric properties and then share those ideas with younger students. During the summer bridge, I helped design and run activities where students used indirect measurement to estimate the height of buildings, applied dimensional analysis to physical problems, and explored how natural designs like hives connect to mathematical efficiency. My role was to guide students through the problem-solving process, encourage them when they were stuck, and help them present their findings at the end of the program. Pre- and post-program surveys were administered to measure changes in non-cognitive factors such as academic self-efficacy, sense of belonging, motivation, academic hope, and knowledge of campus resources, and learning assessments were conducted to measure gains in academic knowledge and skills taught during the bridge curriculum. The results were encouraging: participants reported increased confidence in tackling STEM problems and greater excitement about seeing math in everyday contexts. This work illustrates how combining research, teaching, and near-peer mentoring can support student learning and persistence in STEM through a cohort-based model. 

The use of mathematical models for predicting the concentration gradient of a molar species under electrophoretic and electroosmotic forces has been exploited in a variety of applications, including water desalination, electrokinetic remediation of soil pollutants, and modeling drug delivery methods. Many existing models are only suited for narrow applications and are rooted primarily in data analysis rather than the governing equations, limiting flexibility under parameter changes. This contribution provides a generalized mathematical model for the concentration gradient of a molar species undergoing electrophoretic and electroosmotic forces in a rectangular channel. The model couples electrostatic potential and velocity profile formulations to produce an accurate concentration profile, relying on solutions to fluid mechanics equations including the Poisson–Boltzmann equation, the Navier–Stokes equation, and the molar species continuity equation. The model treats diffusivity, susceptibility, and electrostatic potential as variable parameters rather than fixed constants, and it leverages area-averaging techniques to solve the molar species continuity equation in this context. The work includes analysis of the resulting model and describes useful parameter configurations, including behavior in highly convective systems. 

This study investigates electrochemotherapy by introducing a modified mathematical formulation that captures the effects of an applied DC electrical field on the penetration of chemotherapeutic drugs into tumorous cells. The model uses differential equations (second order) based on a cylindrical depiction of a blood vessel as the drug source. Drug concentration is treated as radially distributed at steady state, with the influence of the electrical field incorporated through an additional evaluation, and electroporation-driven tumor uptake modeled as a first-order chemical reaction. Nondimensionalization is applied to broaden applicability across scenarios, and a unique solution is proposed using the modified Bessel function. The resulting equations yield simulated predictions for drug penetration depth under varying applied electrical fields and the fraction of tumorous cells killed, with noted needs for improved linkage to tumor microenvironments and realistic electrical-field distributions. 

Due to rising concerns with environmental issues, electroosmosis and electrokinetic techniques are currently being explored in various applications within the field of environmental engineering. Current techniques include desalination and the remediation of soil due to the necessity of non-polluted soil and safe water. Mathematical modeling approaches to electroosmosis and electrokinetic techniques have been proven to be effective; unfortunately, existing models are often specialized to a specific set of circumstances and lack flexibility for other situations (e.g., drug delivery and biomedical engineering). This work introduces a generalized mathematical model for describing the concentration gradient of a molar species within a cylindrical channel undergoing electroosmotic and electrokinetic influence. The model couples electrostatic potential in a cylindrical channel with the velocity profile to obtain concentration predictions. Core to the formulation are established fluid mechanics equations, including the Poisson–Boltzmann equation, the Navier–Stokes equation, and the molar species continuity equation, with parameters such as diffusivity, susceptibility, and electrostatic potential treated as variables. A distinct aspect of this study is its use of area-averaging techniques to resolve the molar continuity equation. The study provides analysis of the cylindrical model and highlights patterns under standard parameters (e.g., higher susceptibility yields a more uniform concentration gradient from the wall to the channel center), with further research directions discussed. 

Electrostatic potential profiles are vital for understanding and controlling electroosmosis as well as particle mixing. The shape and magnitude of this profile—particularly a key parameter called the zeta potential (ζ)—directly dictate the speed and direction of fluid flow when an electric field is applied. The role of temperature in modifying this important parameter has not been analyzed from a mathematical approach, and it is relevant for improving the design of microfluidic devices and technologies involving capillary electrophoresis for non-isothermal systems. In this study, two electrophoretic cell geometries are investigated under non-isothermal conditions. A heat-transport model (with heat generation and Dirichlet boundary conditions) is coupled into the Poisson–Boltzmann equation to obtain zeta potential profiles for different temperature distributions. In addition, the Navier–Stokes equation for the electroosmotic case is solved to obtain velocity profiles, including examples of flow reversal under temperature development. Numerical analysis for rectangular and cylindrical geometries indicates that large temperature gradients produce significant zeta-potential changes and can induce multiple flow reversals; effects are more pronounced at high Joule-heating values, while small temperature differences yield approximately linear electrostatic potential behavior and typical laminar profiles.