Generalized transport theorem

All conservation equations in continuum mechanics can be derived from the general transport theorem. Define a variable F(t) as a volume integral over an arbitrary volume v(t) in an r-space... 

The equation describing the general transport theorem can be obtained by substituting Eq. (5.7) into Eq. (5.6) as... 

The averaging theorem for the time derivative can be derived directly from the general transport theorem. Consider the phase k in Fig. 5.7 to be fixed in space while Ak varies as a function of time as a result of phase change. Applying Eq. (5.8) to the system in Fig. 5.7, with d/dt = 3/31, we have... 

The next task in our model derivation is to transform the system description (3.55) into an Eulerian control volume formulation by use of an extended form of the generalized transport theorem (see App A). For phase k the generalized Leibnitz theorem is written ... 

We can now combine the generalized transport theorem (3.70) and the Gauss theorem (3.73) to obtain ... 

The three dimensional Leibnitz theorem is also referred to as the general transport theorem. 

Let r,t,v denote the position vector, time and the velocity vector respectively, and take the density p(r,t) of the body S(t) to be a non-negative field. Conservation of mass requires that the mass be constant over time. Using the General Transport Theorem ([28], p. 19) to differentiate the integral, we thus must have... 

The normal component of the velocity of Va(t) is denoted by w n, and for this moving deforming control volume there exists a kinematlcal theorem known as the general transport theorem. This can be expressed as... 

A number of results from the general transport theorem and the volume-averaging techniques identified in Gray (1975), Soo (1989) and others identified earlier are useful here ... 

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