# The Equations of Fluid Flow

## Abstract

The aim of this chapter is the presentation of the governing equations of fluid flow. The treatment of this topic requires the use of tensor and vector fields and the whole formalism will often be based on the simplified Einstein’s notation for the sum over repeated indices. The balance equation for an extensive property of the fluid, such as the mass or the energy, is deduced by employing Reynolds’ transport theorem. This theorem is the elementary tool for developing the set of local balance equations of mass, momentum and energy, namely the basis of fluid mechanics. The physical meaning of the angular momentum balance and of the entropy balance is also discussed. These partial differential equations are applied to the case of a Newtonian fluid. Then, the Oberbeck–Boussinesq approximation is introduced and discussed. This approximated scheme serves to model the phenomena of convection and buoyant flow that, in several cases, are the source of instability.

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