Extended Navier–Stokes equations, ultrasonic absorption and shock structure

Abstract

The Navier–Stokes (N–S) equations for a steady or unsteady, compressible, continuum flow are modified. The extension is based on a Stokesian fluid with a single nonlinear term in an isotropic stress, rate-of-deformation relation. This is the simplest possible nonlinear extension that also satisfies the second law of thermodynamics. The transport coefficient of this term is referred to as the third viscosity coefficient. In the extended version, the momentum and energy equations each contain a nonlinear term that is proportional to this new coefficient. These terms are significant only when the velocity gradient is extremely large. They are inconsequential, e.g., in a laminar boundary layer. Nevertheless, there are flows where the extended version of the N–S equations is relevant. The first of these is an ultrasonic, unsteady, one-dimensional flow, which is used for evaluating the bulk viscosity. In this case, the linearized N–S equations become singular as the ultrasonic frequency increases toward infinity. When the frequency is sufficiently large, nonlinear terms in the extended N–S equations need to be retained. The terms that are proportional to the third viscosity coefficient increase in importance, relative to linear terms, as the fourth power of the frequency. A second example is shock wave structure. A model is established and numerically solved for the normalized density derivative. Results are compared with corresponding measurements for argon when the upstream Mach number is 1.058 and 1.23. Good agreement between the extended N–S predictions and measurements is obtained for both Mach numbers with a single, but extremely small, value for the third viscosity coefficient. An important difference between conventional and extended N–S shock structure solutions is that the extended-model solution depends on the upstream pressure, whereas the conventional solution does not.