Modelling of Miscible Liquids with the Korteweg Stress

Abstract

Abstract. When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an "effective interfacial tension". To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a twodimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.Mathematics Subject Classification. 35K50, 76D05.

Convergence and Error Estimates of Two Iterative Methods for the Strong Solution of the Incompressible Korteweg Model

“…However, the mathematical study of (KM ) becomes more difficult than that of the parabolic model considered in Ref. [5], because in (KM ) there is no diffusion for the density (in our case, a parabolic and hyperbolic coupled system is considered). The existence of local in time regular solutions for (KM ) is obtained in Ref.…”

“…For instance, in order to model two miscible liquids brought in contact, the same Korteweg tensor is considered in Ref. [5] together with a convection-diffusion equation for the density. In fact, the tensor ∇ρ∆ρ acts like an effective interfacial tension that relaxes with the time due to the mass diffusion.…”

Local strong solution for the incompressible Korteweg model⁎⁎The second, third and fifth authors have been partially supported by D.G.E.S. and M.C. y T. (Spain), Projet BFM2003-06446-C02-01.

“…He proposed in 1901 that a nonuniform density (concentration or temperature) distribution could induce stresses and convection in a fluid. In [4], the authors study the case of two miscible liquids brought in contact, this tensor acts like an effective interfacial tension that relaxes with time due to the mass diffusion. We take a similar model that does not consider diffusion for the concentration ρ.…”

“…Dans [4], ilétudient le cas de deux liquides miscibles, dans lequel le tenseur est une tension interfaciale effective qui se relaxe avec le temps dûà l'effet de la diffusion. Ici, on considère un modèle similaire mais sans diffusion.…”

Chemistry and Biochemistry

“…This treatment used a dispersive flux formulation as shown in [22]. We found that this formulation depended nonlinearly on the sign of the advective flux term, leading in the naive implementation to the formation of soliton/compacton type behavior; solutions which are well-known in the 'formally' equivalent formulation of Korteweg fluids (see [3,11,20,21]) -up to turbulence effects etc., as explained in §3 -which model diffuse fluid interfaces as well as having phenomenological interpretation in the context of the nonlinear Schrödinger equation (see [35]) and the Gross-Pitaevskii equation (see [1,19]) given nearly identical initial conditions to the ones we use in §5. However, in the context of chemical dynamics it is not clear that these types of solutions carry physical significance, and so we have isolated our analysis to the MDG method formulation presented in (2.14).…”