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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Notes
- 1.
As opposed to the Lagrangian description, the Eulerian description is based on a fixed volume through which the fluid flows. Thus, the Eulerian description is focussed on an open thermodynamic system.
- 2.
The term solenoidal comes from electromagnetism as the magnetic induction field in a solenoid is a vector field with zero divergence.
References
Apostol TM (1967) Calculus. vol. 2: multi-variable calculus and linear algebra with applications to differential equations and probability. Wiley, New York
Baehr H, Park N, Stephan K (2011) Heat and mass transfer, 3rd edn. Springer, Berlin
Barletta A (2009) Local energy balance, specific heats and the Oberbeck-Boussinesq approximation. Int J Heat Mass Transf 52:5266–5270
Callen HB (1985) Thermodynamics and an introduction to thermostatistics. Wiley, New York
Davidson PA (2001) An introduction to magnetohydrodynamics. Cambridge University Press, Cambridge
Gyftopoulos EP, Beretta GP (2005) Thermodynamics: foundations and applications. Dover, New York
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Barletta, A. (2019). The Equations of Fluid Flow. In: Routes to Absolute Instability in Porous Media. Springer, Cham. https://doi.org/10.1007/978-3-030-06194-4_5
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DOI: https://doi.org/10.1007/978-3-030-06194-4_5
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