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In a recent paper, Liu, Zhu and Wu (2015, {\it J. Fluid Mech.} {\bf 784}:304) present a force theory for a body in a two-dimensional, viscous,compressible and steady flow. In this companion paper we do the same forthree-dimensional flow. Using the fundamental solution of the linearizedNavier-Stokes equations, we improve the force formula for incompressible floworiginally derived by Goldstein in 1931 and summarized by Milne-Thomson in1968, both being far from complete, to its perfect final form, which is furtherproved to be universally true from subsonic to supersonic flows. We call thisresult the \textit{unified force theorem}, which states that the forces arealways determined by the vector circulation $\pGamma_\phi$ of longitudinalvelocity and the scalar inflow $Q_\psi$ of transverse velocity. Since thistheorem is not directly observable either experimentally or computationally, atestable version is also derived, which, however, holds only in the linear farfield. We name this version the \textit{testable unified force formula}. Afterthat, a general principle to increase the lift-drag ratio is proposed.

Lift and drag in three-dimensional steady viscous and compressible flow