Continuum expression of the first law of thermodynamics

Continuum mechanics
Laws Conservations Mass Momentum Energy Inequalities Clausius–Duhem (entropy)
Solid mechanics Deformation Elasticity (linear) Plasticity Hooke's law Stress Finite strain Infinitesimal strain Compatibility Bending Contact mechanics (frictional) Material failure theory Fracture mechanics
Fluid mechanics Fluids Statics⧼dot-separator⧽Dynamics Archimedes' principle⧼dot-separator⧽Bernoulli's principle Navier–Stokes equations Poiseuille equation⧼dot-separator⧽Pascal's law Viscosity (Newtonian⧼dot-separator⧽non-Newtonian) Buoyancy⧼dot-separator⧽Mixing⧼dot-separator⧽Pressure Liquids Surface tension Capillary action Gases Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law Plasma
Rheology Viscoelasticity Rheometry Rheometer Smart fluids Electrorheological Magnetorheological Ferrofluids
Scientists Bernoulli Boyle Cauchy Charles Euler Gay-Lussac Hooke Pascal Newton Navier Stokes

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:

$dE_{t}=dQ+dW$

where

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

In fluid mechanics, the first law of thermodynamics takes the following form:

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

where

• ${\bf {\sigma }}$ is the Cauchy stress tensor.
• ${\bf {v}}$ is the flow velocity.
• and ${\bf {q}}$ 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.

Note

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

Where

• $p$ is the pressure
• ${\bf {I}}$ is the identity matrix
• ${\bf {T}}$ 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:

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

because in general

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

Integral form

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

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.

See also

• Clausius-Duhem inequality
• Continuum mechanics
• First law of thermodynamics
• Material derivative
• Incompressible flow

References

This article "Continuum expression of the first law of thermodynamics" is from Wikipedia. The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Cameron–Fon-Der-Flaass IBIS Theorem
• Power of 6
• Modified hyperbolic tangent
• Math-U-See
• Schneider's sine approximation formula
• Approximate inequality
• Factoriangular number

---