A first integral form of the energy–momentum equations for viscous flow, with comparisons drawn to classical fluid flow theory.

An elegant four-dimensional Lorentz-invariant first-integral of the energy–momentum equations for viscous flow, comprised of a single tensor equation, is derived assuming a flat space–time and that the energy–momentum tensor is symmetric. It represents a generalisation of corresponding Galilei-invariant theory associated with the classical incompressible Navier–Stokes equations, with the key features that the first-integral: (i) while taking the same form, overcomes the incompressibility constraint associated with its two- and three-dimensional incompressible Navier–Stokes counterparts; (ii) does not depend at outset on the constitutive fluid relationship forming the energy–momentum tensor. Starting from the resulting first integral: (iii) a rigorous asymptotic analysis shows that it reduces to one representing unsteady compressible viscous flow, from which the corresponding classical Galilei-invariant field equations are recovered; (iv) its use as a solid platform from which to solve viscous flow problems is demonstrated by applying the new general theory to the case of propagating acoustic waves, with and without viscous damping, and is shown to recover the well-known classical expressions for sound speed and damping rate consistent with those available in the open literature, derived previously as solutions of the linearised Navier–Stokes equations. [ABSTRACT FROM AUTHOR]