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.

假设流动时间平坦且能量动量张量是对称的，则可以得出一个优雅的四维粘滞流的能量动量方程式的洛伦兹不变式第一积分，它由一个张量方程式组成。它代表了与经典不可压缩Navier–Stokes方程相关的相应伽利略不变理论的推广，其主要特征是第一积分：（i）采用相同形式，克服了与其二，三相关的不可压缩性约束维不可压缩的Navier-Stokes对应物；（ii）一开始并不依赖于构成能量-动量张量的本构关系。从产生的第一个积分开始：（iii）严格的渐近分析表明，它减小到代表不稳定的可压缩粘性流的值，从中恢复相应的经典伽利略不变场方程；（iv）通过将新的一般理论应用于传播声波的情况（具有和不具有粘滞阻尼），证明了其作为解决粘滞流动问题的坚实平台，并且可以恢复已知的经典表达式。声速和阻尼率与公开文献中提供的一致，以前是线性Navier–Stokes方程的解。