THE NAVIER-STOKES FLOWS CHANGING TO NON-NEWTONIAN FLOWS

The motion of viscous incompressible fluids is described by the Navier-Stokes equations in a bounded domain in R³. A modification of the Navier-Stokes equations is investigated. In the physical fluid dynamics, the Navier-Stokes equations have been formulated under the assumption that the rate of deformation of fluids is sufficiently small and therefore the relation between the viscous stress and the rate of deformation is linear. The rate of deformation depends on the velocity gradient in the fluids. As a result the Navier-Stokes equations are found to be well representing the motion of fluids for small velocity gradients. From such a consideration as this, we have found out a modified equation by taking the motion of fluids for large velocity gradients also into account. The flow based on the modified Navier-Stokes equations varies from the Navier-Stokes flow into a non-Newtonian flow, namely the solution of the modified Navier-Stokes equations is switched so as to satisfy a non-Newtonian equation at all the times when the velocity gradient gets larger.