#### Abstract

This thesis focuses on the fluid phenomena observed within what is known as the constrained vapor bubble system. The constrained vapor bubble system is a closed system consisting of a quartz cuvette partially filled with liquid and used as a coolant device. Heat is applied to the heater end which causes the liquid to evaporate and condense on the cooled end of the cuvette. Liquid travels back to the heated end via capillary flow in the corners. A pure vapor bubble is formed in the center of the cuvette giving rise to the name of the experiment. The constrained vapor bubble system is important due to its potential use for cooling devices in microgravity since it does not require a metal wick or gravity used by most micro heat pipes. Experiments done onboard the International Space Station showed fluid phenomena inconsistent with mathematical models and experiments performed in earth's environment. Most notably a flooded heater end and droplets forming on the four walls of the cuvette. 1-dimensional and 2-dimensional heat transfer models are presented. Novel mathematical models of heat transfer and fluid flow in the constrained vapor bubble system are developed in the thesis. Fitting the experimental data from \cite{chatterjee2013} to the 1D heat transfer model in the region near the hot end leads to an estimate of the internal heat transfer coefficient of 400 W/(m$^2$ K) there. However, the heat transfer coefficient is found to increase in the condensation zone near the middle of the cuvette, an observation explained by increased liquid-vapor interface area. Finally, near the cold end the heat transfer is dominated by axial conduction in the liquid phase that fills most of the cross-section and the heat transfer coefficient drops to zero. In the 2D cross-sectional model for temperature the evaporative flux is calculated by taking into account heat transfer in the liquid phase in the corners of the cuvette and introducing a localized cooling parameter into the boundary conditions at the cuvette walls. Heat flux at the liquid-vapor interface is determined and used to estimate the evaporative loss and thus the axial velocity of the flow, with typical average axial flow velocity found to be of the order of 1 mm/s. An analytical estimate of flow velocity is obtained and is shown to be consistent with the numerical results. Effects of 3D heat conduction in the cuvette and the Marangoni stresses are also studied. Further investigation is needed to fully understand the mechanisms of the flow slow-down in the evaporation region. A mathematical model is developed of an evaporating droplet observed in the constrained vapor bubble experimental set-up. The motion of receding contact line is described using two-component disjoining pressure coupled with the effects of phase change and capillarity. The results include dynamics of interface shapes during droplet evaporation, including the radius of curvature at the top of the droplet expressed in dimensional form. The evaporative flux is found to increase toward the contact line, but not as sharply as in the case of evaporating meniscus due to the addition of localized cooling of the substrate. Detailed studies of the the effect of evaporative cooling parameter $K_s$ on the solutions are conducted. Increase in $K_s$ leads to lower local heat flux near the contact line and thus slower evaporation. Radius of curvature at the top of the droplet is found to decrease in a linear fashion with the slope consistent with experimentally measured values. A second term in the disjoining pressure gives excellent control over contact angle for matching with experiments or observing different regimes.

#### Degree Date

Fall 12-19-2020

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### Advisor

Dr. Vladimir Ajaev

#### Subject Area

Mathematics, Applied

#### Number of Pages

74

#### Format

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

#### Recommended Citation

Barrett, James, "Modeling Fluid Phenomena in the Context of the Constrained Vapor Bubble System" (2020). *Mathematics Theses and Dissertations*. 11.

https://scholar.smu.edu/hum_sci_mathematics_etds/11