More Like this

1. The influence of a lipid reservoir on the tear film formation

   https://doi.org/10.1093/imammb/dqz018  

   Maki, Kara L; Braun, Richard J; Barron, Gregory A (January 2020, Mathematical Medicine and Biology: A Journal of the IMA)

   null (Ed.)

   Abstract We present a mathematical model to study the influence of a lipid reservoir, seen experimentally, at the lid margin on the formation and relaxation of the tear film during a partial blink. Applying the lubrication limit, we derive two coupled non-linear partial differential equations characterizing the evolution of the aqueous tear fluid and the covering insoluble lipid concentration. Departing from prior works, we explore a new set of boundary conditions (BCs) enforcing hypothesized lipid concentration dynamics at the lid margins. Using both numerical and analytical approaches, we find that the lipid-focused BCs strongly impact tear film formation and thinning rates. Specifically, during the upstroke of the eyelid, we find specifying the lipid concentration at the lid margin accelerates thinning. Parameter regimes that cause tear film formation success or failure are identified. More importantly, this work expands our understanding of the consequences of lipid dynamics near the lid margins for tear film formation. 

   more » « less
2. Proper Orthogonal Decomposition Reduced Order Model for Tear Film Flows

   https://doi.org/10.23919/ACC50511.2021.9483018  

   Sahyoun, Samir; Wilson, Dan; Djouadi, Seddik M.; Wise, Steven M.; Abderrahmane, Hamid Ait (May 2021, 2021 American Control Conference (ACC))

   null (Ed.)

   Tear film plays a key role in protecting the cornea surface against contaminations and dry eye syndrome which can lead to symptoms of discomfort, visual trouble, and tear film instability with the potential to damage the ocular surface. In this paper, coupled nonlinear partial differential equations of the fourth order proposed by Aydemir et al. to describe the evolution of tear film dynamics are considered. These equations are of Benney type and known to suffer from unbounded behavior and lack of a global attractor. The objective here is to identify a reduced order modeling framework with the potential to be used as a basis for control in future work using smart tears with a surfactant that can modify the surface tension to prevent tear film breakup. Since the dynamics are infinite dimensional and nonlinear, a reduced order model based on the proper orthogonal decomposition (POD) is developed, analyzed, and compared to the full order model. Numerical simulations illustrate that only a small number of POD modes are required to accurately capture the tear film dynamics allowing for the full partial differential model to be represented as a low-dimensional set of coupled ordinary differential equations. 

   more » « less
3. Extensional flow of a free film of nematic liquid crystal with moderate elasticity

   https://doi.org/10.1063/5.0151809  

   Taranchuk, M. J.; Cummings, L. J.; Driscoll, T. A.; Braun, R. J. (June 2023, Physics of Fluids)

   The human tear film is a multilayer structure in which the dynamics are often strongly affected by a floating lipid layer. That layer has liquid crystalline characteristics and plays important roles in the health of the tear film. Previous models have treated the lipid layer as a Newtonian fluid in extensional flow. Motivated to develop a more realistic treatment, we present a model for the extensional flow of thin sheets of nematic liquid crystal. The rod-like molecules of these substances impart an elastic contribution to the rheology. We rescale a weakly elastic model due to Cummings et al. [“Extensional flow of nematic liquid crystal with an applied electric field,” Eur. J. Appl. Math. 25, 397–423 (2014).] to describe a lipid layer of moderate elasticity. The resulting system of two nonlinear partial differential equations for sheet thickness and axial velocity is fourth order in space, but still represents a significant reduction of the full system. We analyze solutions arising from several different boundary conditions, motivated by the underlying application, with particular focus on dynamics and underlying mechanisms under stretching. We solve the system numerically, via collocation with either finite difference or Chebyshev spectral discretization in space, together with implicit time stepping. At early times, depending on the initial film shape, pressure either aids or opposes extensional flow, which changes the free surface dynamics of the sheet and can lead to patterns reminiscent of those observed in tear films. We contrast this finding with the cases of weak elasticity and Newtonian flow, where the sheet retains the same qualitative shape throughout time. 

   more » « less
4. Optimal boundary control of a model thin-film fiber coating model

   https://doi.org/10.1016/j.physd.2023.133942  

   Biswal, Shiba; Ji, Hangjie; Elamvazhuthi, Karthik; Bertozzi, Andrea L (January 2024, Physica D: Nonlinear Phenomena)

   This paper considers the control of fluid on a solid vertical fiber, where the fiber radius is larger than the film thickness. The fluid dynamics is governed by a fourth-order partial differential equation (PDE) that models this flow regime. Fiber coating is affected by the Rayleigh–Plateau instability that leads to breakup into moving droplets. In this work, we show that control of the film profile can be achieved by dynamically altering the input flux to the fluid system that appears as a boundary condition of the PDE. We use the optimal control methodology to compute the control function. This method entails solving a minimization of a given cost function over a time horizon. We formally derive the optimal control conditions, and numerically verify that subject to the domain length constraint, the thin film equation can be controlled to generate a desired film profile with a single point of actuation. Specifically, we show that the system can be driven to both constant film profiles and traveling waves of certain speeds. 

   more » « less
5. Evaporation and breakup effects in the shock-driven multiphase instability

   https://doi.org/https://doi.org/10.1017/jfm.2020.871  

   Duke-Walker, Vasco; Maxon, W. Curtis; Almuhna, Sahir R.; McFarland, Jacob A. (February 2021, Journal of fluid mechanics)

   null (Ed.)

   Evaporation and breakup of liquid droplets are common in many applications of the shock-driven multiphase instability (SDMI), such as in liquid-fuelled detonation engines, multiphase ejector pumps and turbines and explosive dispersal of liquid particles (i.e. chemical or biological agents). In this paper, the effects of evaporation and breakup of droplets on the mixing induced by the SDMI are considered through simulations and compared with experimental results. The evaporation model is validated against previous experimental data. The capabilities of the simulations and particle models are then demonstrated through a qualitative comparison with experimental results where breakup effects are negligible (i.e. small droplets). The simulation results are explored further to quantify the effects of evaporation (i.e. mixing enhancement) in the SDMI, providing further insight into the experimental results. A new breakup model, derived from previous works, is then presented for low Reynolds number (below 500), low Weber number (below 100) droplets in a shock-driven multiphase instability. The breakup model capabilities are then demonstrated through a comparison with experimental results where breakup effects are significant (larger droplet sizes). Finally, the simulation results are used to highlight the importance of breakup parameters on the evaporation rate and large-scale mixing in the SDMI. Overall, it is shown that evaporation is enhanced by the large-scale hydrodynamics instability, the SDMI, and that breakup of the droplets significantly increases the strength of the instability, and rate of droplet evaporation. 

   more » « less