On physics of boundary vorticity creation in incompressible viscous flow

Abstract

The present paper provides some arguments surrounding the controversies of boundary vorticity creation for incompressible viscous flow. Our discussion shows that boundary vorticity creation must be a viscous physical process. Importantly, it is emphasized that not only viscosity is responsible for spreading the generated vorticity out of the boundary, but also must be involved in the process of boundary vorticity creation to realize the no-slip boundary condition. Lyman flux is a part of the boundary vorticity flux under the Lighthill-Panton-Wu’s definition, which provides an alternative interpretation of boundary vorticity dynamics. Different from the existing inviscid interpretation, we insist that viscosity is fully indispensable for generating the Lyman flux through the tangential boundary acceleration and surface pressure gradient where the acceleration adherence is shown to be derived from the velocity adherence directly. Through a detailed discussion on interfacial vortex sheet and slip velocity, it is revealed that the velocity jump across the material interfacial vortex sheet (a thin viscous shear layer as the fluid viscosity approaches to zero) is physically different from that across the interface. In addition, it is shown that the formation of surface pressure distribution is an inviscid process while the subsequent boundary vorticity generation by the tangential pressure gradient must be a viscid process (contributed by the non-equilibrium particle relaxation effect). These two processes are separated by a non-zero time increment with the same order as the particle relaxation time. Then, the hydrodynamic limit of the Boltzmann equation is revisited to elaborate the crucial roles of viscosity for both the continuum and slip regimes. For continuum flows with a no-slip boundary, the physical carrier of the slip velocity in the inviscid Euler theory originates from the produced vorticity concentrated in the thin material vortex sheet. Interestingly, we find that Lyman flux must be a viscous boundary flux even for a slip boundary where the implicit viscid mechanism is attributed to the non-continuum effect hidden in the Knudsen layer. The present exploration suggests that a complete physical picture including the boundary vorticity creation and the formation of airfoil circulation should be built upon the viscous flow theory.