Volume-averaging theorems

Equations (5.116), (5.117), and (5.119) characterize the volume-averaging theorems of derivatives [Slattery, 1967b Whitaker, 1969],... 

The volume-averaged continuity equation for dispersed laminar multiphase flows is given by applying the volume-averaging theorems to Eq. (5.13) as... 

Applying the volume-averaging theorems to Eqs. (5.14) and (5.18), verify that the volume-averaged momentum and energy equations for phase k in a multiphase flow can be given by Eqs. (5.123) and (5.126), respectively. 

The first, second and third terms in (3.122) have to be reformulated using the conventional volume averaging theorems . The first theorem one makes use of relates the spatial average of a time derivative to the time derivative of a spatial average, and is called the Leibnitz rule for volume averaging ... 

The volume-averaging theorem can be used with the average of the gradient in equation 1.104 in order to obtain... 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

A special case of (3.128) is the theorem for the volume average of a divergence ... 

Area averaging can be considered to be a limiting case of local volume averaging [43, 47, 189]. Thus the phrase limiting form refers to the modified forms of the averaging theorems which are applicable to the governing 3D equations to derive a set of equations valid for ID problems. 

Gray WG, Lee PCY (1977) On the Theorems for Local Volume Averaging of Multiphase Systems. Int J Multiphase Flow 3 333-340... 

A number of results from the general transport theorem and the volume-averaging techniques identified in Gray (1975), Soo (1989) and others identified earlier are useful here ... 

Of course, both statements can be proved from the theorem of uniqueness for the attraction field. In addition, it is appropriate to comment a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist. 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

Consequently, the mean element tends toward a configuration having a maximum surface per unit volume, and since the crushed element tends to be equidimensional, the average element is an equidimensional irregular polyhedron whose form approaches that of a cube. This is Lienau s theorem. 

Space averages of e also may be defined by integrating over a fixed volume V of the chamber. Use of the divergence theorem in equation (12) shows that... 

Integration of eqn (8.201) over an atomic volume for which the integral of V p(r) vanishes yields, term for term, the atomic virial theorem for a time-dep>endent system (eqn (8.193)) or for a stationary state (eqn (6.23)). Thus, eqn (8.201) is, in terms of its derivation and its integrated form, a local expression of the virial theorem. The atomic virial theorem provides the basis for the definition of the average energy of an atom, as discussed in Chapter 6. 

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