Transport theorem momentum equation

For a fluid flow, of course, one uses the Reynolds transport theorem to establish the relationship between a system (where the momentum balance applies directly) and a control volume (through which fluid flows). In terms of Eq. 3.2, the extensive variable N is the momentum vector P = mV and the intensive variable tj is the velocity vector V. Thus the fundamental approach yields the following vector equation... 

Beginning with appropriate forms of the Reynolds transport theorem, derive the continuity and momentum equations. Show that they can be written as... 

The fundamental quantities which are conserved in a collision are the mass of a particle and each component of the total momentum of the colliding particles. Thus, conservation equations can be obtained by the use of the transport theorem. Although the energy equation can also be derived from the transport theorem, the total kinetic energy of... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

Though the integral form of linear momentum balance written above as eqn (2.26) serves as the starting point of many theoretical developments, it is also convenient to cast the equation in local form as a set of partial differential equations. The idea is to exploit the Reynolds transport theorem in conjunction... 

Within the linear response approximation, the rate of transport (mass, momentum, or energy) through a system is proportional to the gradient (of concentration, velocity, and temperature), with the transport coefficient being the proportionality constant. This proportionality constant can be computed using equilibrium description of the system through the so-called fluctuation dissipation theorems. One such equation, relating equilibrium fluctuations to the diffusion constant, is Einstein s well-known equation ... 

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