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
Proper Orthogonal Decomposition Reduced Order Model for Tear Film Flows
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
- Award ID(s):
- 2012634
- NSF-PAR ID:
- 10283515
- Date Published:
- Journal Name:
- 2021 American Control Conference (ACC)
- Page Range / eLocation ID:
- 2763 to 2768
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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
-
more » « less
The combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto a subset of modes determined by proper orthogonal decomposition (POD). When applied to boundary layers that are horizontally homogeneous, POD and a horizontally averaged quasilinear approximation both assume flow features that are horizontally wavelike, making the pairing very efficient. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mean fields as well as certain energetically important second-order turbulent transport statistics and energies to within 30% error. Reduced-basis quasilinear statistics must approach the full-basis statistics as the basis size approaches completion; however, some quasilinear statistics resemble those found in the fully nonlinear simulations at smaller basis truncations. Thus, model reduction could possibly improve upon the accuracy of quasilinear dynamics.
-
Purpose: Little quantitative or mechanistic information about tear film breakup can be determined directly via current imaging techniques. In this paper, we present simplified mathematical models based on two proposed mechanisms of tear film breakup: evaporation of water from the tear film and tangential fluid flow within the tear film. We use our models to determine whether one or a combination of the two mechanisms causes tear film breakup in a variety of instances. In this study, we estimate related breakup parameters that cannot currently be measured in breakup during subject trials, such as tear film osmolarity and thinning rates. The present study validates our procedure against previous work.Methods: Five ordinary differential equation models for tear film thinning were designed that model evaporation, osmosis, and various types of tangential flow. Eight tear film breakup instances occurring within a time interval of 1–8 s postblink of five healthy subjects thatwere identified in fluorescence images in previous work were fit with these five models. The fitting procedure used a nonlinear least squares optimization that minimized the difference of the computed theoretical fluorescent intensity from the models and the experimental fluorescent intensity from the images. The optimization was conducted over the evaporation rate and up to three tangential flow rate parameters. The smallest norm of the difference was determined to correspond to the model that best explained the tear film dynamics.Results: All of the breakup instances were best fit by models with time-dependent tangential flow. Our optimal parameter values and thinning rate as well as tangential fluid flow profiles compare well with previous partial differential equation model results in most instances.Conclusion: Our fitting results suggest that a combination of tangential fluid flow and evaporation cause most of the breakup instances. Comparison with results from previous work suggests that the simplified models can capture the essential tear film dynamics in most cases, thereby validating this procedure for wider usage.more » « less
-
more » « less
Summary This article presents a nonlinear closed‐loop active flow control (AFC) method, which achieves asymptotic regulation of a fluid flow velocity field in the presence of actuator uncertainty and sensor measurement limitations. To achieve the result, a reduced‐order model of the flow dynamics is derived, which utilizes proper orthogonal decomposition (POD) to express the Navier‐Stokes equations as a set of nonlinear ordinary differential equations. The reduced‐order model formally incorporates the actuation effects of synthetic jet actuators (SJA). Challenges inherent in the resulting POD‐based reduced‐order model include (1) the states are not directly measurable, (2) the measurement equation is in a nonstandard mathematical form, and (3) the SJA model contains parametric uncertainty. To address these challenges, a sliding mode observer (SMO) is designed to estimate the unmeasurable states in the reduced‐order model of the actuated flow field dynamics. A salient feature of the proposed SMO is that it formally compensates for the parametric uncertainty inherent in the SJA model. The SMO is rigorously proven to achieve local finite‐time estimation of the unmeasurable state in the presence of the parametric uncertainty in the SJA. The state estimates are then utilized in a nonlinear control law, which regulates the flow field velocity to a desired state. A Lyapunov‐based stability analysis is provided to prove local asymptotic regulation of the flow field velocity. To illustrate the performance of the proposed estimation and AFC method, comparative numerical simulation results are provided, which demonstrate the improved performance that is achieved by incorporating the uncertainty compensator.
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC50511.2021.9483018
