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# Continuum expression of the first law of thermodynamics

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:

${\displaystyle dE_{t}=dQ+dW}$

where

• ${\displaystyle E_{t}}$ is the total energy of a system.
• ${\displaystyle W}$ is the work on it.
• and ${\displaystyle Q}$ is the heat added to that system.

In fluid mechanics, the first law of thermodynamics takes the following form:[1][2]

${\displaystyle {\frac {DE_{t}}{Dt}}={\frac {DW}{Dt}}+{\frac {DQ}{Dt}}\to {\frac {DE_{t}}{Dt}}=\nabla \cdot ({\bf {\sigma \cdot v}})-\nabla \cdot {\bf {q}}}$

where

• ${\displaystyle {\bf {\sigma }}}$ is the Cauchy stress tensor.
• ${\displaystyle {\bf {v}}}$ is the flow velocity.
• and ${\displaystyle {\bf {q}}}$ 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 non-conservation form of the Navier–Stokes equations.[3]

## Note

${\displaystyle {\bf {\sigma }}=-p{\bf {I}}+{\bf {T}}}$

Where

• ${\displaystyle p}$ is the pressure
• ${\displaystyle {\bf {I}}}$ is the identity matrix
• ${\displaystyle {\bf {T}}}$ is the deviatoric stress tensor

That is, pulling is positive stress and pushing is negative stress.

## Compressible fluid

For a compressible fluid the left hand side of equation becomes:

${\displaystyle {\frac {DE_{t}}{Dt}}={\frac {\partial E}{\partial t}}+\nabla \cdot (E{\bf {v)}}}$

because in general

${\displaystyle \nabla \cdot {\bf {v\neq 0}}}$.

## Integral form

${\displaystyle \int _{V}{\frac {\partial E}{\partial t}}dV=-\oint _{\partial V}E{\bf {v}}\cdot d{\bf {A}}+\oint _{\partial V}({\bf {\sigma \cdot v}})\cdot d{\bf {A}}-\oint _{\partial V}{\bf {q}}\cdot d{\bf {A}}}$

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]