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<div>Free surface flows where the shape of the interface separating two fluids is unknown <i>apriori</i> are an important area of interest in fluid dynamics. The study of free surface flows such as the breakup and coalescence of drops, and thinning and rupture of films lends itself to a diverse range of industrial applications, such as inkjet printing, crop spraying, foam and emulsion stability, and nanolithography, and helps develop an understanding of natural phenomena such as sea spray generation in oceans, or the dynamics of tear films in our eyes. In free surface flows, singularities are commonly observed in nite time, such as when the radius of a thread goes to zero upon pinchoff or when the thickness of a film becomes zero upon rupture. Dynamics in the vicinity of singularities usually lack a length scale and exhibit self-similarity. In such cases, universal scaling laws that govern the temporal behavior of measurable physical quantities such as the thickness of a lm can be determined from asymptotic analysis and veried by high-resolution experiments and numerical simulations. These scaling laws provide deep insight into the underlying physics, and help delineate the regions of parameter space in which certain forces are dominant, while others are negligible. While the majority of previous works on singularities in free-surface flows deal with Newtonian fluids, many fluids in daily use and industry exhibit non-Newtonian rheology, such as polymer-laden, emulsion, foam, and suspension flows.</div><div><br></div><div><div>The primary goal of this thesis is to investigate the thinning and rupture of thin films of non-Newtonian fluids exhibiting deformation-rate-thinning (power-law) rheology due to attractive intermolecular van der Waals forces. This is accomplished by means of intermediate asymptotic analysis and numerical simulations which utilize a robust Arbitrary Eulerian-Lagrangian (ALE) method that employs the Galerkin/Finite-Element Method for spatial discretization. For thinning of sheets of power-law fluids, a signicant finding is the discovery of a previously undiscovered scaling regime where capillary, viscous and van der Waals forces due to attraction between the surfaces of the sheet, are in balance. For thinning of supported thin films, the breakdown of the lubrication approximation used almost exclusively in the past to study such systems, is shown to occur for films of power-law fluids through theory and conrmed by two dimensional simulations. The universality of scaling laws determined for rupture of supported films is shown by studying the impact of a bubble immersed in a power-law fluid with a solid wall.</div></div><div><br></div><div><div>Emulsions, which are ne dispersions of drops of one liquid in another immiscible liquid, are commonly encountered in a variety of industries such as food, oil and gas, pharmaceuticals, and chemicals. Stability over a specied time frame is desirable in some applications, such as the shelf life of food products, while rapid separation into its constituent phases is required in others, such as when separating out brine from crude oil. The timescale over which coalescence of two drops of the dispersed phase occurs is crucial in determining emulsion stability. The drainage of a thin film of the outer liquid that forms between the two drops is often the rate limiting step in this process. In this thesis, numerical simulations are used to decode the role played by fluid inertia in causing drop rebound, and the subsequent increase in drainage times, when two drops immersed in a second liquid are brought together due to a compressional flow imposed on the outer liquid. Additionally, the influence of the presence of insoluble surfactants at the drop interface is studied. It is shown that insoluble surfactants cause a dramatic increase in drainage times by two means, by causing drop rebound for small surfactant concentrations, and by partially immobilizing the interface for large surfactant concentrations.</div></div>

Multiphase transport phenomena primarily encompass the fundamental principles and applications concerning the systems where overall dynamics are precept by phase change evolution. On the other hand, multiscale transport phenomena essentially corroborate to a domain where the transport characteristics often contain components at disparate scales. Relevant examples as appropriate to multiphase and multiscale thermofluidic transport phenomena comprise solid-liquid phase change during conventional solidification process and hydrodynamics through narrow confinements. The additional effect of superimposed magnetic field over such multiphase and multiscale systems may give rise to intriguing transport characteristics, significantly unique in nature as compared to flows without it. The present investigation focuses on multiphase, multi-scale transport phenomena in physical systems subjected to the superimposed magnetic field, considering four important and inter-linked aspects. To begin with, for a multiphase system concerning binary alloy solidification, a normal mode linear stability analysis has been carried out to investigate stationary and oscillatory convective stability in the mushy layer in the presence of external magnetic field. The stability results indicate that the critical Rayleigh number for stationary convection shows a linear relationship with increasing Ham (mush Hartmann number). Magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of  (a function of the Stefan number and concentration ratio), and vice versa for lower  values. Analogous to the behaviour for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer. In multiscale paradigm, the combined electroosmotic and pressure-driven transport through narrow confinements have been firstly analyzed with an effect of spatially varying non–uniform magnetic field. It has been found that a confluence of the steric interactions with the degree of wall charging (zeta potential) may result in heat transfer enhancement, and overall reduction in entropy generation of the system under appropriate conditions. In particular, it is revealed that a judicious selection of spatially varying magnetic field strength may lead to an augmentation in the heat transfer rate. It is also inferred that incorporating non–uniformity in distribution of the applied magnetic field translates the system to be dominated by the heat transfer irreversibility. Proceeding further, a semi-analytical investigation has been carried out considering implications of magnetohydrodynamic forces and interfacial slip on the heat transfer characteristics of streaming potential mediated flow in narrow fluidic confinements. An augmentation in the streaming potential field as attributable to the wall slip activated enhanced electromagnetohydrodynamic transport of the ionic species within the EDL has been found. Furthermore, the implications of Stern layer conductivity and magnetohydrodynamic influence on system irreversibility have been shown through analysis of entropy generation due to fluid friction and heat transfer. The results being obtained in this analysis have significant scientific and technological consequences in the context of novel design of future generation energy efficient devices, and can be useful in the further advancement of theory, simulation, and experimental work. Finally, the combined consequences of interfacial electrokinetics, rheology, and superimposed magnetic field subjected to a non-Newtonian (power-law obeying) fluid in a narrow confinement are studied in this work. The theoretical results demonstrate that the applied magnetic field imparts a retarding influence in the induced streaming potential development, whereas, triggers the heat transfer magnitude. Moreover, additional influences of power law index show reduction in heat transfer as well as the streaming potential magnitude. It is unveiled that the optimal combinations of power law index and the magnetic field lead to the minimization of the global total entropy generation in the system.