The virtual power principle in fluid mechanics

A conceptual framework on analytical mechanics for continuous fluid medium, which connects the fluid motion and all of the (internal and external) forces with mechanical power, is proposed by using the virtual power and the virtual velocity. Based on this framework, it is found that the internal virtual power is equal to the external virtual power in fluid dynamics, which is called the virtual power principle. This framework is also proved to be equivalent to the vector dynamics (Cauchy’s equation or Navier–Stokes equation). Furthermore, based on the virtual power principle, a theorem is introduced for continuous fluid medium, which indicates the relationship between the force (or torque) acting on a body immersed in a fluid and the specified virtual power. Subsequently, according to Galilean invariance, the detailed relationship for Newtonian fluids in incompressible flows is derived and used to illustrate the mechanisms on instantaneous forces: the added inertial effects, the boundary energy flux and dissipation effects, the vortex contribution, and the explicit body force contribution. As an application of the principle, the advantage of the V formation flight of geese is preliminarily discussed in the view of aerodynamics. Specifically, the total drag of the flock is reduced by contrast with the simple sum of the drag in solo fight and the optimal angle of V ranges from $60^{\circ }$ to $120^{\circ }$. The principle could be a useful approach to reveal the contributions of the flow structures and the moving or deforming boundaries to the force and torque acting on a body, especially in a multibody system.