Formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: evolution of a thin pollutant layer over water

Abstract

In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier-Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids, thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model is provided of a dissipative entropy inequality, up to a second order term. Moreover, we propose a correction of the model −by taking into account the second order extention for the pressure− that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. In the second test the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test the influence of the friction coefficient on the coastal area affected by the pollutant is studied.

Formal deduction of the Saint-Venant–Exner model including arbitrarily sloping sediment beds and associated energy

“…These processes are different for each layer. In [24] a similar development is performed coupling also shallow water and Reynolds equations in a multiscale framework but for a transport of pollutant problem. Thus, we consider this work as a reference in the development, which in turn is based on the original works [30] and [44].…”

On the Existence of Global Weak Solutions to 1D Pollutant Transport Model

“…This work follows the work done in (Roamba, Zabsonré & Zongo, 2017). In (Roamba, Zabsonré & Zongo, 2017) as in this present work, we use a model of transport of pollutant in 1D formally derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013). In (Roamba, Zabsonré & Zongo, 2017), the authors showed the existence of global weak solutions of similar model derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013).…”

“…In (Roamba, Zabsonré & Zongo, 2017) as in this present work, we use a model of transport of pollutant in 1D formally derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013). In (Roamba, Zabsonré & Zongo, 2017), the authors showed the existence of global weak solutions of similar model derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013). To lead well this result, the authors considered the condition according to which h 2 ≤ h 1 (the water layer is more important than the layer of the pollutant).…”

Another Proof of Existence of Global Weak Solutions to 1D Pollutant Transport Model

“…As a reminder, several studies are being done on pollutant transport models. The authors in ( Fernandez-Nieto, Narbona-Reina & Zabsonré, 2013) were the pawns in the formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations. From this derivation, the authors prove that our model verify a dissapative entropy inequality up to a second'order term.…”

“…The authors in (Roamba, Zabsonré & Zongo), have proven the existence of global weak solution of a similar model derived in ( Fernandez-Nieto, Narbona-Reina & Zabsonré, 2013). To achieve this, the authors have made a technical hypothesis on the height of water, namely the water layer is more important than the layer of the pollutant in the form…”