Reduced Models for Thick Liquid Layers with Inertia on Highly Curved Substrates

Wray, Alexander; Papageorgiou, Demetrios T.; Matar, Omar

doi:10.1137/16m1060686

Cited by 10 publications

(4 citation statements)

References 52 publications

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“…The convergence criterion was that L had not varied by more than 10 −6 across 10 iterations. This was implemented in C++, based on a well-tested centred-finite-difference code [9,35].…”

Section: A Optimal Control Of the Rommentioning

confidence: 99%

Electrostatic control of the Navier-Stokes equations for thin films

Wray¹,

Cimpeanu²,

Gomes³

2022

Preprint

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A robust control scheme is derived and tested for the Navier-Stokes equations for two-dimensional multiphase flow of a thin film underneath an inclined solid surface. Control is exerted via the use of an electrode parallel to the substrate, which induces an electric field in the gas phase, and a resultant Maxwell stress at the liquid-gas interface. The imposed potential at the second electrode is derived using a Model Predictive Control loop, together with optimal control of a high-fidelity reduced-order model. In this implementation the interfacial shape of the fluid is successfully controlled, however the algorithm is sufficiently general to control any other quantity of interest.

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“…The convergence criterion was that L had not varied by more than 10 −6 across 10 iterations. This was implemented in C++, based on a well-tested centred-finite-difference code [9,35].…”

Section: A Optimal Control Of the Rommentioning

confidence: 99%

Electrostatic control of the Navier-Stokes equations for thin films

Wray¹,

Cimpeanu²,

Gomes³

2022

Preprint

Self Cite

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“…The derivation of the lubrication equation for generic curved substrates can be found in Roy et al. (2002), Thiffeault & Kamhawi (2006) and Wray, Papageorgiou & Matar (2017). The geometry is sketched in figure 2.…”

Section: The Coating Problem Of a Generic Substratementioning

confidence: 99%

“…For a complete description of differential geometry and general coordinates, we refer to Deserno (2004). The derivation of the lubrication equation for generic curved substrates can be found in Roy et al (2002), Thiffeault & Kamhawi (2006) and Wray, Papageorgiou & Matar (2017). The geometry is sketched in figure 2.…”

Section: Problem Definition and Metric Terms In General Coordinatesmentioning

confidence: 99%

Gravity-driven coatings on curved substrates: a differential geometry approach

et al. 2022

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Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.

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“…determined from (14), were solved using a fully-implicit centred-finite-difference C++ code that has been extensively tested in other film problems [9,42]. We used 200 points in space and a fixed time step to facilitate the OC code, resulting in 1000 in time.…”

Section: B Framework For Control Of Dns Using Mpcmentioning

confidence: 99%