Space gradient

Fig. 7. (A) The WEFT sequence in this sequence the tt pulse is applied to rotate all of the magnetization (i.e. both solute and solvent) to the -z-axis. A delay (I>np) of sufficient length is used to allow the water magnetization to relax to the origin ( >np = InfZ) ) whilst during the same period, by virtue of faster longitudinal relaxation, the solute resonances have reached thermal equilibrium. An excitation pulse (represented here as a tj/2 pulse) is then applied and an almost water-free spectrum is acquired. However, in the presence of radiation damping the water quicldy returns nonexponentially to the equilibrium position at a similar rate to the solute nuclei (see Fig. 2). However, if during D p a series of n very weak and evenly spaced gradient pulses are applied so as to inhibit the effects of radiation damping, the water relaxes according to its natural spin-lattice relaxation rate. This is the basis of the Water-PRESS sequence (B). An example of a spectrum obtained with Water-PRESS is shown in Fig. IB and Fig. 6.

When performing the Reynolds averaging, the internal-coordinate vector is not affected. Thus, the Reynolds average commutes with time, space, and phase-space gradients. See Fox (2003) for a discussion of this topic. 

The mixed advection term in Eq. (B.4) involves both a spatial and a phase-space gradient of the NDF, and thus it is more complicated than the pure advection term described in Section B.3 of this appendix. Nevertheless, by working with the transport equation for the moments, we can show that it can be treated like an advection term with special properties. 

Process with w, 2 iscalled/tomogeneoMiifaprocess withQ w,Q (2 foreacha > 0 exists in the universe (e.g., in uniform systems, i.e those without space gradient of properties, changing the mass a-times we change the work and heat distribution in... 

Such a form of entropy inequality (1.42) and likewise the energy balance (1.5) will be used (in fact by further simplifications) in Chap. 2 where uniform systems without space gradients are treated The process is a time sequence of the states and we may expect the validity of (1.5), (1.42) for arbitrarily close time instants. Therefore we formulate these basic laws for the rate (time derivative) of the state functions (entropy, energy) with heatings (rate of heat exchange) and power, cf. (2.1),... 

We discuss the level of description on uniform fluid models from Chap. 2. Here the space gradients have no influence (observer s space scales are much less than the natural space scales) and therefore we discuss only the time scales. Different constitutive models may be applied on the same physical system, say models A, B, C, D in Sect. 2.2 on a uniform, closed fluid body. This corresponds to the use of various observer s scales relative to the same natural time scales which are in this... 

As a result, we obtain If scalar a is objective (3.55) then its material derivative a and space gradient grada are objective while Gradfl and da/dt are not. If a is an objective vector, diva, a.a = a are objective, while material derivative a is not. Ultimately, with objective (second order) tensor A, the vector divA and the scalars trA, detA are objective. 

Using (4.6), (4.4) and the following definitions of space gradients of densities and temperature... 

Chapters 2 through 4 develop modem continuum (rational) thermodynamics in its standard and most elaborated form. The most simple example or model— uniform systems (without space gradients of properties)—is discussed in Chap. 2 which also serves as a basic and relatively simple introduction to the methodology (Sects. 2.1 and 2.2 in this chapter) which is not complicated by the description of spatial distribution. Four examples—models of uniform materials with increasing... 

The two representatives are the duals of one another. The other dual quantities, we are going to use frequently here, are the outward unit normal vectors n and N of o- and X, respectively also, grad X = F and grad X = F denote material and space gradients of motion. 

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