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A generalization of the classical Euler and Korteweg fluids. (English) Zbl 07729508

Summary: The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by D. J. Korteweg [Arch. Néerl. (2) 6, 1–24 (1901; JFM 32.0756.02)] that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set of observations. (The implicit generalization developed in this paper is not a sub-class of the implicit generalization of the Navier-Stokes fluid developed by K. R. Rajagopal [Appl. Math., Praha 48, No. 4, 279–319 (2003; Zbl 1099.74009); J. Fluid Mech. 550, 243–249 (2006; Zbl 1097.76009)] or the generalization due to V. Průša and K. R. Rajagopal [J. Non-Newton. Fluid Mech. 181–182, 22–29 (2012; doi:10.1016/j.jnnfm.2012.06.004)], as spatial gradients of the density appear in the constitutive relation developed by Korteweg [loc. cit.].) Third, we introduce a challenging set of partial differential equations that would lead to new techniques in both analysis and numerical analysis to study such equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q31 Euler equations
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