Substantial derivative velocity vector

Here N is the extensive variable associated with the conservation law (e.g., the momentum vector P), p is the fluid s mass density, and t] is the intensive variable associated with N (e.g., the velocity vector V). The volume of the control volume is given as 6V. In a cartesian coordinate system (, y, z),SV = dxdydz. The operator D/Dt is called the substantial derivative. 

The reason to make this point is to contrast the situation for the substantial derivative of a vector field, as illustrated in the next section for the velocity vector. 

The extra terms appear because in noncartesian coordinate systems the unit-vector derivatives do not all vanish. Only in cartesian coordinates are the components of the substantial derivative of a vector equal to the substantial derivative of the scalar components of the vector. The acceleration in the r direction is seen to involve w2, the circumferential velocity. This term represents the centrifugal acceleration associated with a fluid packet as it moves in an arc defined by the 9 coordinate. There is also a G acceleration caused by a radial velocity. In qualitative terms, one can visualize this term as being related to the circumferential acceleration (spinning rate) that a dancer or skater experiences as she brings her arms closer to her body. 

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. 

The equations in this section retain some compact notation, including the substantial derivative operator D/Dt, the divergence of the velocity vector V-V, and the Laplacian operator V2. The expansion of these operations into the various coordinate systems may be found in Appendix A. 

The left side corresponds to the accumulation of internal energy in a control volume that moves at the local fluid velocity at each point on its surface. If the substantial derivative is expanded using vector notation, then there are actually two terms on the left side. 

---