Galilean-Invariant Expression for Bernoulli’s Equation

Abstract

The Bernoulli principle is one of the basic and most famous concepts in fluid mechanics, and the related equation is an extremely useful and effective tool in the solution of a wide range of practical engineering problems. However, the Bernoulli equation is not Galilean-invariant (i.e., it does not remain the same as a result of a change in the inertial frame of reference when the new frame moves with constant velocity relative to the original one). In this paper, a frame-independent expression of the Bernoulli equation is obtained from the Euler equation of inviscid motion for the case of steady flow, for the two classic versions valid along a streamline, and for irrotational flow. Compared to the conventional formulation of the equation, an additional term is present in the definition of the Bernoulli constant. An interpretation of the new equation is provided on the basis of the first law of thermodynamics. Some classic applications are presented, and the results found are in accordance with the consolidated knowledge of fluid mechanics.