Multiscale analysis of emulsions and suspensions with surface effects

• Main Author: Nika, Grigor
• Other Authors: Bogdan M. Vernescu, Advisor
• Format: Others
• Published: Digital WPI 2016
• Subjects: Γ; -convergence; Magnetorheological fluids; Mosco convergence; suspensions; emulsions; Homogenization
• Online Access: https://digitalcommons.wpi.edu/etd-dissertations/146; https://digitalcommons.wpi.edu/cgi/viewcontent.cgi?article=1145&context=etd-dissertations

The better understanding of the behavior of emulsions and suspensions is important in many applications. In general, emulsions allow the delivery of insoluble agents to be uniformly distributed in a more efficient way. At the same time suspensions of rigid particles are used as “smart materialsâ€� as their properties can be changed by the interaction with a magnetic or electric field. In the first part of the talk we consider a periodic emulsion formed by two Newtonian fluids in which one fluid is dispersed under the form of droplets of arbitrary shape, in the presence of surface tension. We assume the droplets have fixed centers of mass and are only allowed to rotate. We are interested in the time-dependent, dilute case when the characteristic size of the droplets aε, of arbitrary shape, is much smaller than the period length ε. We obtain a Brinkman type of fluid flow for the critical size aε = O(ε3) as a replacement of the Stokes flow of the emulsion. Additionally, using Mosco convergence and semigroup theory we extend the convergence to the parabolic case. For the case when the droplets convect with the flow, it can be shown again using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase and no “strangeâ€� term appears. Moreover, we determine the rate of convergence of the velocity field for the emulsion to that of the velocity for the one fluid problem in both the H1 and L2 norms. Additionally, a second order approximation is determined in terms of the bulk and surface polarization tensors for the cases of uniform and non-uniform surface tension. The second part of the talk is devoted to the study of MR fluids. We consider a suspension of rigid magnetizable particles in a non-conducting, viscous fluid with an applied external magnetic field. Thus, we use the quasi-static Maxwell equations coupled with the Stokes equations to capture the magnetorheological effect. We upscale using two scale asymptotic expansions to obtain the effective equations consisting of a coupled nonlinear system in a connected phase domain as well as the new constitutive laws. The proposed model generalizes the model of Rosenweig by coupling the velocity of the fluid and the magnetic field intensity. Using the finite element method we compute the effective coefficients for the MR fluid. We analyze the resulting MR model for Poiseuille and Couette flows and compare with experimental data for validation.