Free transport definition

If the realizability condition in Eq. (8.52) is satisfied, found from Eq. (8.53) is guaranteed to be realizable. Using the method described above for the PBE (Eq. (8.35)), it is straightforward to extend Eq. (8.53) to second-order time-stepping. The extension to multiple velocity components v in multiple spatial dimensions is a bit more complicated. As described in Yuan Fox (2011) and in Section B.3 of Appendix B, when the CQMOM is used to constmct the multivariate quadratures all permutations must be used in a consistent manner in order to get the correct kinetic energy fiuxes. Nevertheless, for each permutation of the CQMOM, the basic time-stepping formula in Eq. (8.53) is used to update the transported moment set by modifying the definition of K , to include the multivariate moments in the optimal-moment set. Readers interested in more details on multidimensional free transport should consult Yuan Eox (2011) and Section B.3 of Appendix B. 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

In summary, advances in our understanding of the nutritional and hormonal requirements of cells in culture and of the role played by attachment factors and transport proteins have led to the possibility that any cell can be cultured in the absence of serum. The type of serum substitute used and the means of achieving a serum-free culture system should be determined by the ultimate purpose that the culture system is to serve. The advantages of serum-free culture are manifest both in the practical realm of providing an inexpensive and simple starting material for the purification of cell-secreted proteins, such as antibodies, and for providing a more precise definition of the in vitro environment in order to model more... 

The temperature of the fluid t F far away from the wall, appears in (1.23), the definition of the local heat transfer coefficient. If a fluid flows around a body, so called external flow, the temperature t F is taken to be that of the fluid so far away from the surface of the body that it is hardly influenced by heat transfer, i) F is called the free flow temperature, and is often written as diDC. However, when a fluid flows in a channel, (internal flow), e.g. in a heated tube, the fluid temperature at each point in a cross-section of the channel will be influenced by the heat transfer from the wall. The temperature profile for this case is shown in Figure 1.8. i) F is defined here as a cross sectional average temperature in such a way that t F is also a characteristic temperature for energy transport in the fluid along the channel axis. This definition of F links the heat flow from the wall characterised by a and the energy transported by the flowing fluid. 

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