Flows with variable viscosity: an asymptotic model

Abstract

Introducing a variable dynamic viseosity coefficient in the Navier equations for an incompressible fluid of small viscosity (Re≫1, where Re is the classical Reynolds number), we exhibit a three-layer asymptotic model: ideal fluid layer, boundary layer and lower viscous layer. Surprisingly we find that the interaction between the boundary layer and the lower viscous layer is realized only starting with the second-order approximation. We give the full mathematical formulation of the corresponding boundary-layer problem, for the second approximation, with the new boundary conditions obtained by matching from the first-order lower viscous layer, of which the thickness is O(1/Re).

As an application of this three-layer asymptotic model we solve completely the classical Blasius problem. In this case the expression for the skin friction coefficient shows that the classical Blasius value is multiplied by a positive term, directly linked to the variability of the dynamic viscosity coefficient.