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Mathematical problems for miscible, incompressible fluids with Korteweg stresses. (English) Zbl 0741.76001

Summary: It is shown that the equations governing the motion and diffusion of miscible liquids can be reduced to a form like the Navier-Stokes equations when the equation of state is for the density of a simple mixture. In particular, in this case, \(\mathbf W=C\mathbf u+D\nabla \varphi\), where \(C\) and \(D\) are constants, is solenoidal. This allows one to introduce a generalized stream and diffusion function which may be useful in the study of two-dimensional problems. Problems of unidirectional shearing flows in the presence of gradients of composition are briefly considered. Korteweg terms do not enter these problems. We consider the problem of the stability of a vertically stratified incompressible motionless Korteweg fluid of variable concentration analogous to the classical BĂ©nard problem. In general the stability problem is not self- adjoint and it may be possible to have complex eigenvalues at criticality. One and only one Korteweg constant enters into this calculation.

MSC:

76A05 Non-Newtonian fluids
76V05 Reaction effects in flows
76E99 Hydrodynamic stability
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