The force on a body moving in an inviscid fluid

TL;DR

This paper develops a novel theoretical framework for understanding the forces on a body moving in an inviscid fluid, emphasizing vorticity generation and circulation effects during acceleration, with applications to biological propulsion.

Contribution

It introduces a new approach that accounts for vorticity generation independent of velocity and relaxes boundary conditions, expanding classical fluid dynamics theory.

Findings

Vorticity generation is linked to body acceleration, independent of velocity.

Circulation around the body can be non-zero and is communicated to infinity.

The generalized force calculation highlights the role of vorticity in propulsion.

Abstract

This paper presents some novel contributions to the theory of inviscid flow regarding the forces exerted on a body moving through such a fluid in two dimensions. It is argued that acceleration of the body corresponds to vorticity generation that is independent of the instantaneous velocity of the body and thus the boundary condition on the normal velocity. The strength of the vortex sheet representing the body retains a degree of freedom that represents the net effect of the tangential boundary condition associated with the viscous flow governed by the higher-order Navier-Stokes equations. This degree of freedom is the circulation of the vorticity generated by the acceleration of the body. Equivalently, it is the net circulation around a contour enclosing the body and any shed vorticity. In accordance with Kelvin's circulation theorem, a non-zero value of the circulation around this system is necessarily communicated to infinity. This contrasts with the usual acceptance of the theorem as requiring this circulation to be zero at all times; a condition that is incapable of capturing the effect of newly generated vorticity on the body surface when it accelerates. Additionally, the usual boundary condition of continuity of normal velocity is relaxed to allow for fluid entrainment into surfaces of discontinuity that represent the mass contained within the viscous layers of the physical problem. The generalized force calculation is presented in detail. The importance of the vorticity generation due to body acceleration is demonstrated on some modeled problems relevant to biological propulsion.