# 2024 Faculty and Student Summer Talks

The talks will take place every **Monday and Thursday from June 3, 2024, to August 1, 2024 at 2PM in Cullimore Hall Room 611**.

Date | Day | Speaker, Title, and Abstract |
---|---|---|

June 3 | M |
The motion of three point-vortices in a 2D inviscid, incompressible fluid has been widely studied. Gröbli (1877) derived a closed system for the evolution of the side lengths of the triangle formed by the vortices. This system has been the basis of most studies of this problem. |

June 6 | R |
The main focus of this talk is developing a fast and accurate boundary integral method to simulate the two-phase flow of a viscous drop immersed in a surrounding viscous fluid which contains dissolved soluble surfactant. The drop is stretched by an imposed extensional flow. Surfactant is a surface active agent important in microfluidic applications that lowers surface tension and introduces a Marangoni force that typically opposes the imposed outer flow. However, it is difficult to adapt boundary integral methods to problems with soluble surfactant as the equation for the bulk surfactant does not have a Green’s function formulation. In addition, at large Peclet numbers that are characteristic of real physical systems, the concentration of soluble surfactant develops a transition or boundary layer adjacent to the drop interface which is difficult to resolve using traditional numerical methods. To address these difficulties, the hybrid numerical method that was first introduced in Booty and Siegel (2010) and extended to two-phase flow with soluble surfactant in Xu, Booty, and Siegel (2013) is further developed. The hybrid method is based on an asymptotic reduction of the surfactant dynamics in the transition layer in the limit Pe→∞. This reduced advection-diffusion equation for the concentration of bulk soluble surfactant has a Green's function formulation. A fast method for computing a time-convolution integral that arises in the Green’s function formulation of the advection-diffusion equation is introduced. The fast algorithm is based on a method originally introduced for Abel integral equations by Johannes Tausch. Results illustrating the speed and accuracy of the fast method will be presented. The secondary focus of this talk is conducting a study on electroconvective flow. The model considered is a symmetric binary electrolyte bounded by an ion-selective surface on the bottom and a stationary reservoir on the top. The ion-selective surface is impermeable to anions and permeable to cations. An external electric field is applied to drive the flow and transport of ions. The model is based on the electrostatically forced incompressible Stokes equations coupled to the Poisson and Nernst-Planck equations. The streamfunction form of the Stokes equation is used, giving rise to the 2D force biharmonic equation. This fourth order boundary value problem results in high round-off error when using Chebyshev-Fourier representation, as the round-off error for the k-th Chebyshev derivative grows like O(M^(2k) |

Updated: June 29, 2024