Eulerian Derivation of the Conservation Equation for Energy in a Non-Inertial Frame of Reference in Arbitrary Motion

The standard inertial Navier-Stokes equations consisting of the conservation equations for mass, momentum and energy are often used to investigate the motion of a compressible fluid around an object that is in arbitrary motion. The non-inertial form of the Navier-Stokes equations can be used to accurately capture the acceleration effects that arise from the unsteady motion. The acceleration source terms that arise in the conservation equation for momentum have been extensively documented. In this paper,
an Eulerian approach for deriving the apparent forces is presented to transform the governing conservation equation for energy into a non-inertial reference frame that is in arbitrary motion. The Eulerian approach is based on successive Galilean transformations between an inertial frame, an orientation-preserving non-inertial frame and a rotating non-inertial frame. The paper demonstrates that for an object in arbitrary motion, the rate of work done due to fictitious forces affects the rate of change of the total energy. The fictitious work arises in the kinetic energy equation while the internal energy and enthalpy equations remain invariant in the non-inertial frame. The present derivation is a step towards quantifying the contribution of the fictitious work terms to the heat transfer of a body that is accelerating/decelerating.