Does shear viscosity play a key role in the flow across a normal shock wave?

Abstract

Once there is a velocity gradient in a viscous fluid-flow, such as that across a shock wave, a viscous force and viscous energy loss exist inside the flow according to the Navier-Stokes equation, which may confuse the relative contribution of compressibility and viscosity. In this paper, a viscous shear vector is defined as the component of gradient vector of local velocity magnitude perpendicular to the velocity vector. Then, a local viscous energy flux vector is defined from the viscous shear vector after being multiplied by the viscosity and the velocity magnitude. The divergence of the viscous energy flux vector results in new expressions for viscous force and loss of viscous energy, in which all the square terms of derivative of velocity components correspond to irreversible energy loss. The rest part can be taken as a kind of mechanical energy transfer done by the viscous force, from which the viscous force components can be got based on the assumption that the viscous force vector is parallel to the velocity vector. The new equations are different from and more complex than those in the traditional Navier-Stokes equation. By the new theory, it is shown that there is no shear viscous force and shear viscous energy loss in the flow across a normal shock wave without velocity gradient perpendicular to the flow direction.