Physics

Newtonian fluid

Mathematics

Mechanics

Boundary (topology)

Vortex

Vorticity equation

Mathematical analysis

Vorticity

Flow (mathematics)

Classical mechanics

Journal of Fluid Mechanics

This paper presents a general theory and physical interpretation of the interaction between a solid body and a Newtonian fluid flow in terms of the vorticity ω and the compression/expansion variable Π instead of primitive variables, i.e. velocity and pressure. Previous results are included as special simplified cases of the theory. The first part of this paper shows that the action of a solid wall on a fluid can be exclusively attributed to the creation of a vorticity-compressing ω–Π field directly from the wall, a process represented by respective boundary fluxes. The general formulae for these fluxes, applicable to any Newtonian flow over an arbitrarily curved surface, are derived from the force balance on the wall. This part of the study reconfirms and extends Lighthil's (1963) assertion on vorticity-creation physics, clarifies some currently controversial issues, and provides a sound basis for the formulation of initial boundary conditions for the ω-Π variables. The second part of this paper shows that the reaction of a Newtonian flow to a solid body can also be exclusively attributed to that of the ω-Π field created. In particular, the integrated force and moment formulae can be expressed solely in terms of the boundary vorticity flux. This implies an inherent unity of the action and reaction between a solid body and a ω-Π field. In both action and reaction phases the ω-Π coupling on the wall plays an essential role. Thus, once a solid wall is introduced into a flow, any theory that treats ω and Π separately will be physically incomplete.

Interactions between a solid surface and a viscous compressible flow field