(213h) A Unified Framework for Total Variation Regularized Optimization in Fluid Dynamics and Related Physical Systems

An optimization framework is presented for minimizing the energy functional developed around a generalized equation governing physical systems such as fluid dynamics, particle transport, phase transition, and other related systems. The convexity of the energy functional is investigated to derive the necessary conditions for a smooth and global optimum solution. Furthermore, the Total Variation (TV) regularization term is introduced to gain insights into the solution space and convergence analysis of convection- dominated problems. The framework optimizes energy functionals associated with fluid systems, offering improved convergence and smoothness in solutions where non-linear interactions—such as those seen in turbulent flow or agitated systems—lead to complex behavior. By applying TV regularization, the optimization reduces noise and sharp gradients in the solution space, critical in accurately modeling and optimizing mixing efficiency in industrial settings, such as stirred tank reactors and turbulent flow mixers. Through applications to the Navier-Stokes equations, we demonstrate how the framework can be used to model and optimize mixing patterns, helping to understand better fluid motion in scenarios where convection plays key roles.