Continuum expression of the first law of thermodynamics
Continuum mechanics  

Laws




In physics, the first law of thermodynamics is an expression of the conservation of total energy of a system. The increase of the energy of a system is equal to the sum of work done on the system and the heat added to that system:
where
 is the total energy of a system.
 is the work on it.
 and is the heat added to that system.
In fluid mechanics, the first law of thermodynamics takes the following form:^{[1]}^{[2]}
where
 is the Cauchy stress tensor.
 is the flow velocity.
 and is the heat flux vector.
Because it expresses conservation of total energy, this is sometimes referred to as the energy balance equation of continuous media. The first law is used to derive the nonconservation form of the Navier–Stokes equations.^{[3]}
Note[edit]
Where
 is the pressure
 is the identity matrix
 is the deviatoric stress tensor
That is, pulling is positive stress and pushing is negative stress.
Compressible fluid[edit]
For a compressible fluid the left hand side of equation becomes:
because in general
 .
Integral form[edit]
That is, the change in the internal energy of the substance within a volume is the negative of the amount carried out of the volume by the flow of material across the boundary plus the work done compressing the material on the boundary minus the flow of heat out through the boundary. More generally, it is possible to incorporate source terms. ^{[2]}
See also[edit]
 ClausiusDuhem inequality
 Continuum mechanics
 First law of thermodynamics
 Material derivative
 Incompressible flow
References[edit]
 ↑ F.M. White (2006). Viscous fluid flow (3rd ed.). McGraw Hill. pp. 69&endash, 72. Search this book on
 ↑ ^{2.0} ^{2.1} Truesdell; Toupin (1960). "The classical field theories". In Flügge. Encyclopedia of physics: Principles of classical mechanics and field theory. III. p. 609. Search this book on
 ↑ Chung (2002). Computational fluid dynamics. Cambridge University Press. pp. 33–34. Search this book on
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