Expressions of force and moment exerted on a body in a viscous flow via the flux of vorticity generated on its surface

Nonlinear and viscous contributions in the equation of a vorticity evolution in three-dimensional flow are represented in the divergent form. The concept ”vorticity transfer tensor” which describes transference of the vorticity in a flow is introduced. The vortex flux for all the flow points and from the body surface are defined (the vortex flux from the body surface in general case is not the same as the boundary vorticity flux used in the literature). Exact expressions of force and moment via the vortex flux from the body surface for non stationary viscous incompressible flow under the no-slip boundary condition are derived directly from Navier-Stokes equations without applying the conservation law in the whole flow region. The expressions contain only surface integrals, and are valid for calculating force and moment for each body of the system in infinite or bounded space. They are most useful when meshless vortex methods are applied for the flow simulation. That is demonstrated in the numerical example of the flow around the sphere. The formulas obtained can also be useful for checking the accuracy of the computations which are performed in natural variables.