1. The living forces theorem and the work of internal surface forces

One of the most important general consequences of the dynamic equations of motion of a continuous medium is the living forces theorem.

Let V – be an arbitrary finite volume moving together with particles of the material medium, \Sigma – the surface bounding it. Suppose that inside the volume V the components of the stress tensor P = p^{ij}\vec{э}_{i}\vec{э}_{j} and components the speed vector \vec{v} = v^i\vec{э}_i = v_i\vec{э}^i – are continuous differentiable functions of spatial coordinates and time.

Take the vector d\vec{r} = \vec{v}dt – the displacement vector of an infinitely small volume of a continuous medium d\tau during the time dt; scalarly multiply the equation of impulses by d\vec{r} and integrate by volume V. We get

\int\limits_V \rho\vec{a} \cdot \vec{v} dt d\tau = \int\limits_V \rho\vec{F}\cdot d\vec{r}d\tau + \int\limits_V\left( \nabla_i p^{ij} \right) v_i dt d\tau(1.1)

Lets transform the integrals involved in this relation.

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From the “Basic concepts and equations of thermodynamics” by Sedov

  1. The living forces theorem and the work of internal surface forces
  2. The first law of thermodynamics (the law of conservation of energy) and the equation of heat influx
  3. Thermodynamic equilibrium, reversible and irreversible processes
  4. Two-parameter environments. Perfect gas. Carnot cycle
  5. The second law of thermodynamics and the concept of entropy
  6. Thermodynamic potentials of two-parameter media
  7. Examples of ideal and viscous media and their thermodynamic properties. Thermal conductivity
  8. The first and second laws of thermodynamics for finite volumes of a continuous medium. Entropy production in some irreversible processes
  9. Introduction to the theory of models of a mixture of liquids or gases, taking into account chemical reactions and diffusion of components
  10. Modeling mixtures in reversible processes
  11. Models of mixtures taking into account irreversible processes