Substantial derivatives

Chemistry as it was realized substantially derives from the interae-tion of electrons. The electronic theory of chemistry, particularly of organic chemistry, emerged, explaining the great richness of chemical observations and transformation, as expressed by Ingold, Robinson, Hammett, and many others following in their footsteps. 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

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... 

The derivative Df/Dt is called the substantial derivative or the derivative following the motion. It describes the time rate of change of a property J x,y,z,t) as would be noted by an observer floating along with the fluid with the local velocity v. Analytically it is expressed as... 

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 objective of this section is to establish a relationship between the time rate of change of an extensive property of a system and the behavior of the associated intensive property within a control volume that surrounds the system at an instant in time. This kinematic relationship, described in terms of the substantial derivative, is central to the derivation of conservation equations that describe fluid mechanics. 

The differential operator in the integrand represents the substantial derivative for a flowing system, although in this form it is somewhat disguised. 

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... 

This equation provides the relationship between the rate of change of an extensive property N for a system (a specific, but possibly flowing, mass) and the substantial derivative of the associated intensive variable r) in an Eulerian control volume 8V that is fixed in space. 

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

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 Eulerian acceleration derivative has a special name, called the substantial derivative. It is defined using a capital D 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,... 

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. 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector... 

Begin with the general vector form of the substantial derivative, as stated below ... 

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. 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

For a system that occupies a fixed-in-space control volume at an instant in time, the substantial derivative provides the required linkage to the control volume... 

Ironically, since the mass-continuity equation was already used in the derivation of the substantial-derivative form of Eq. 3.2, it is not directly useful for deriving the continuity equation itself. Its application simply returns a trivial identity. Instead, we begin with the integral form as stated in Eq. 2.30 to yield... 

Written out in cylindrical coordinates, using the definition of the substantial derivative, the continuity equation is given as... 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

In cylindrical coordinates, after we expand out the substantial derivative, the species mass conservation equation becomes... 

It is interesting to note that if the starting point had been Eq. 3.124, a trivial result would have been obtained because the overall mass-continuity equation has already been invoked through the introduction of the substantial derivative. The summation of Eq. 3.124 would simply reveal that zero equals zero. 

One important purpose of the energy equation is to describe and predict the fluid temperature fields. The energy equation will be closely coupled to the Navier-Stokes equations, which describe the velocity fields. The coupling comes through the convective terms in the substantial derivative, which, of course, involve the velocities. The Navier-Stokes equations are also coupled to the energy equation, since the density and other properties usually depend on temperature. Chemical reaction and molecular transport of chemical species can also have a major influence on the thermal energy of a flow. 

This equation represents the rate of change of the system s total stored energy in terms of the substantial derivative for a flowing system applied to an Eulerian control volume fixed in space. Differentiating the definition of total energy yields an expression for the substantial derivative of the total energy... 

Using this definition of h, we can expand the substantial derivative as... 

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