Some Remarks on the Navier-Stokes Equations with a Pressure-Dependent Viscosity.
Abstract
In most mathematical treatments of the Navier-Stokes equations, it is assumed that the viscosity is a constant. Viscosities of real fluids, however, depend not only on the temperature, but may also change significantly with pressure, in particular at high pressures. In this paper, the mathematical consequences of such a prssure dependence are investigated. It is found that, in contrast to the ordinary Navier-Stokes equations, ellipticity can be lost, and the equations are not necessarily well-posed. The complementing condition for traction boundary conditions is investigated, and an existence theorem for the initial-boundary value problem with prescribed velocities on the wall is proved. One of the main differences to ordinary Navier-Stokes theory lies in the elimination of the pressure, which now leads to a nonlinear elliptic partial differential equation instead of Laplace's equation. Keywords: Navier-Stokes equations; Pressure dependent viscosity; Nonlinear Neumann problems; Complementing conditions; Initial value problems; Change of type.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1985
- Accession Number
- ADA163726
Entities
People
- Michael Renardy
Organizations
- University of Wisconsin–Madison