Steady state solutions for a lubrication two-fluid flow

Abstract : In this paper, we describe possible solutions for a stationary flow of two superposed fluids between two close surfaces in relative motion. Physically, this study is within the lubrication framework, in which it is of interest to predict the relative positions of the lubricant and the air in the device. Mathematically, we observe that this problem corresponds to finding the interface between the two fluids, and we prove that it is equivalent to solve some polynomial equation. We solve this equation using an original method of polynomial resolution. First, we check that our results are consistent with previous work. Next, we use this solution to answer some physically relevant questions related to the lubrication setting. For instance, we obtain theoretical and numerical results enabling to predict the apparition of a full film with respect to physical parameters (fluxes, shear velocity, . . . ). In particular, we present a figure giving the number of stationary solutions depending on the physical parameters. Moreover, in the last part, we give some indications for a better understanding of the multi-fluid case.
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European Journal of Applied Mathematics, Cambridge University Press (CUP), 2011, 22, pp.581-612. 〈10.1017/S095679251100026X〉
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Laurent Chupin, Bérénice Grec. Steady state solutions for a lubrication two-fluid flow. European Journal of Applied Mathematics, Cambridge University Press (CUP), 2011, 22, pp.581-612. 〈10.1017/S095679251100026X〉. 〈hal-00461635〉

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