Substantial derivative operator

The term on the left-hand side, arising from the product of mass and acceleration, can be expanded using the expression for the substantial derivative operator... 

Establishing the connection between Eqs. 2.30 and 2.31 and the substantial-derivative operator is facilitated by using the mass-conservation equation, which is derived formally at the beginning of the next section. For the present the result is simply stated as... 

To give a concrete example of the general vector representation, the substantial-derivative operator can be expanded in cylindrical coordinates as... 

From Eq. 2.61, and the scalar substantial-derivative operator (Eq. 2.43), it is clear by inspection that the substantial derivative of a vector is not equivalent to the substantial derivatives of the vector s scalar components. Although there is a certain resemblance, there are extra terms that appear,... 

Discuss the pro s and con s of writing and using the spatial components of the substantial-derivative operator as either one of the two equivalent notations for either a scalar of vector field ... 

In cylindrical coordinates, apply the substantial-derivative operator... 

Using the definition of the substantial-derivative operator in its more general form (Eq. 2.31),... 

Using the fact that volume V = pm is the product of the mass density and the mass, expand the substantial-derivative operator in cartesian and in cylindrical coordinates. 

Use the overall mass-continuity equation to rewrite the species continuity equations, introducing the substantial-derivative operator. Discuss the differences between the two forms of the species-continuity equations. 

Based on a semidifferential control volume that spans the channel radius, develop the overall and species continuity equations for flow along the tube. Show how the species equations may be written is a form that uses the substantial-derivative operator. 

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. 

Batchelor [6] and Ghil and Childress [28] examine the transformation of the governing Eulerian equations for compressible flows from an inertial to a rotational frame. As it turns out, only the momentum equation is actually effected by this transformation because the material or substantial derivative operator of scalar functions is invariant to rotation. 

It should be obvious that this term is volumetric, which means that the accumulation rate process applies to the entire system contained within the control volume. The stipulation that the control volume be stationary simplifies the mathematics to some extent, but the final form of the force balance does not depend on details pertaining to the movement of the control volume. Possibilities for this motion and the appropriate time derivatives are summarized in Table 8-1. The substantial derivative operator... 

For a simple thermodynamic system the change in total eneigy, tffitotal. is given by the differential of total. whereas for a fluid dynamic CV 5 t denotes the substantial derivative operator applied on the total energy vtiriable. 

Based on the above analysis it follows that the relation (1.150) can be transformed into the Lagrangian framework considering the enthalpy property on the form hit, r) simply by replacing the differential operators by the substantial derivative operators[13] ... 

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