Control volume Eulerian

Reynolds Transport Theorem The purpose of the Reynolds transport theorem is to provide the relationship between a system (for which the conservation law is written) and an Eulerian control volume that is coincident with a system at an instant in time. The control volume remains fixed in space, with the fluid flowing through it. The Reynolds transport theorem states that... 

Consider the system and control volume as illustrated in Fig. 2.2. The Eulerian control volume is fixed in an inertial reference frame, described by three independent, orthogonal, coordinates, say z,r, and 9. At some initial time to, the system is defined to contain all the mass in the control volume. A flow field, described by the velocity vector (t, z,r, 9), carries the system mass out of the control volume. As it flows, the shape of the system is distorted from the original shape of the control volume. In the limit of a vanishingly small At, the relationship between the system and the control volume is known as the Reynolds transport theorem. 

The fundamental relationship between a flowing system and an Eulerian control volume, which are coincident at an instant in time, is 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. 

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... 

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... 

Fig. 1.1. (A) Finite Eulerian control volume fixed in space with the fluid moving through it. (B) Finite Lagrangian control volume moving with the fluid such that the same fluid particles are always in the same control volume (i.e., a material control volume). (C) Finite general Lagrangian control volume moving with an arbitrarily velocity not necessarily equal to the fluid velocity. The sohd line indicate the control volume surface (C5) at time t, while the dashed line indicate the same CS at time t + dt.

The mixture mass M in a macroscopic Eulerian control volume V, at time t... 

We end this subsection deriving the convective mass flux terms over an infinitesimal Eulerian control volume element, intending to provide improved understanding of the physics and mathematical concepts involved. Note that even though we consider convective mass fluxes only at this point, the mathematical flux concepts are general. That is, all convective and diffusive fluxes can be derived in a similar manner. 

Fig. 1.2. A physical model of an infinitesimal Eulerian control volume fixed in space indicating the convective mass fluxes through the various faces of the volume element, used for derivation of the continuity equation.

To transform the system balance into an Eulerian control volume balance a particular form of the transport theorem applicable to single-phase systems of species c is required [149]. Let / be any scalar, vector or tensor field. The transport theorem for species c can be expressed as ... 

Again, the system balance can be transformed into an Eulerian control volume balance by use of the transport theorem (1.10). Hence,... 

This is the Eulerian control volume form of the momentum conservation equation. 

The mixture momentum associated with an Eulerian control volume E, at... 

Infinitesimal Eulerian control volume stress analysis... 

For somebody the physical meaning of the surface stresses might be intuitive and best explained formulating a stress balance over an infinitesimal Eulerian control volume. 

Until this point we have limited our thermodynamic description to simple (closed) systems. We now extend our analysis considering an open system. In this case the material control volume framework might not be a convenient choice for the fluid dynamic model formulation because of the computational effort required to localize the control volume surface. The Eulerian control volume description is often a better choice for this purpose. 

To convert the system balance description into an Eulerian control volume formulation the transport theorem is employed (i.e., expressed in terms of the... 

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 ... 

It is seen that the governing system equation (3.51) is transformed into a generic Eulerian control volume formulation (3.81) consisting of a volume integral determining the microscopic transport equations for the bulk phases and a surface integral determining the jump balance at the interface. 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

To transform the system balance into an Eulerian control volume balance a particular form of the transport theorem applicable to single-phase systems of species c is... 

Infinitesimal Eulerian Control Volume Stress Analysis... 

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