Divergence Gauss theorem

Assume that there is a volume V that is enclosed by a surface A whose orientation is described by a normal outward-pointing unit vector n. The Gauss divergence theorem, which is used to relate surface integrals to volume integrals, is stated as 

In this equation a is a scalar, G is a vector, and n is the outward-pointing unit vector at the control surface. 

The Gauss theorem is also applicable to a second-order tensor field, such as that of the stress tensor  

As discussed in Section A.18, a vector r on a surface whose orientation is described by the unit vector n is determined from the tensor as 

The substantial derivative operator is stated as follows. However, be cautious when applying the substantial derivative operator to a vector, since, in general, the substantial derivative of a vector does not equal the substantial derivative of the scalar components of the vector. In the following, the velocity vector V is presumed to have components v, where i indicates the directions of the coordinates. 

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It is possible, and very useful, to write the surface integral in terms of a volume integral via the use of the Gauss divergence theorem, which states that... 

Using the Gauss divergence theorem, the Reynolds transport theorem (Eq. 2.27) can be rewritten as... 

Applying the Gauss divergence theorem (Section A. 15), the net force on the control volume can be represented in terms of a volume integral as... 

Take the results of the Gauss divergence theorem and evaluate the net force on the differential control volume using the divergence of the stress tensor,... 

The surface integral can be converted to a volume integral using the Gauss divergence theorem, Eq. 2.29,... 

Recognizing the terms in the parenthesis on the right-hand side as the divergence of the mass-flux vector and dV = rdrdOdz, it can be seen that the procedure has recovered the Gauss divergence theorem (Eq. 2.29). That is,... 

Explain why the control-volume volume dV — r2 sin 6d6drd

[Pg.138]

From the Gauss divergence theorem and the definition of enthalpy (h that = e+p/p), it follows... 

Since there is not a continuously differentiable relationship between the inlet and outlet flows, the Gauss divergence theorem (i.e., the V- operation) has no practical application. Recall that, by definition, the surface unit vector n is directed outward. The sign of the mass-fraction difference in Eq. 16.68 is set by recognizing that the inlet flow velocity is opposite the direction of n, and vice versa for the exit. The overall mass-continuity equation,... 

We have assumed that the geometric connection coefficients can be defined in some sensible way. To do this, we simply note that, in order to define conservation laws (i.e., to do physics) in a Riemannian space, it is necessary to be have a generalized form of Gauss divergence theorem in the space. This is certainly possible when the connections are defined to be the metrical connections, but it is by no means clear that it is ever possible otherwise. Consequently, the connections are assumed to be metrical and so gai, given in (3), can be written explicitly as... 

Substituting Eqs. 2.5-11 and 2.5-7 into Eq. 2.5-6, using the Gauss Divergence Theorem, we obtain ... 

Substituting Eq. (6.14) into Eq. (6.21) and by using integration by parts and the Gauss divergence theorem, we obtain... 

Equation (2.7) can be integrated by parts and in virtue of the Gauss divergence theorem we obtain... 

Additional groundwater manifestations of Gauss divergence theorem... 

Finally, for bulk flux. Gauss divergence theorem (5) becomes... 

We again use Gauss divergence theorem to convert eq. (1.12) from a surface to a volume integral and, since V is arbitrary, we find... 

Application of the Gauss divergence theorem, equation (3) can be rewritten in following... 

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