Gradients spatial

Restored parameters for the evaluation of PDSM, may be different PMF of material tensor of stresses or its invariants, spatial gradients of elastic features (in particular. Young s modulus E and shear modulus G), strong, technological ( hardness HRC, plasticity ), physical (density) and others. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Westerblad, H., Lee, J.A., Lamb, A.G., Bolsover, S.R., Allen, D.G. (1990). Spatial gradients of intracellular calcium in skeletal muscle during fatigue. Pfluegers Arch. 415, 734-740. 

The measurement of spatial gradients around roots at a resolution sufficient to provide an acceptable test of model predictions is very challenging and few studies have attempted it. Figure 7 shows the measured and predicted depletion of K for 4-day-old rape seedlings grown as a planar mat in contact with a column... 

Convection is mass transfer that is driven by a spatial gradient in pressure. This section presents two simple models for convective mass transfer the stirred tank model (Section II.A) and the plug flow model (Section n.B). In these models, the pressure gradient appears implicitly as a spatially invariant fluid velocity or volumetric flow rate. However, in more complex problems, it is sometimes necessary to develop an explicit relationship between fluid velocity and pressure gradients. Section II.C describes the methods that are used to develop these relationships. 

Diffusion is the movement of mass due to a spatial gradient in chemical potential and as a result of the random thermal motion of molecules. While the thermodynamic basis for diffusion is best apprehended in terms of chemical potential, the theories describing the rate of diffusion are based instead on a simpler and more experimentally accessible variable, concentration. The most fundamental of these theories of diffusion are Fick s laws. Fick s first law of diffusion states that in the presence of a concentration gradient, the observed rate of mass transfer is proportional to the spatial gradient in concentration. In one dimension (x), the mathematical form of Fick s first law is... 

SIGNAL RECEPTION AND ANALYSIS TEMPORAL ANALYSIS OF CHEMICAL SIGNALS TO DETERMINE SPATIAL GRADIENTS... 

The results demonstrate (i) that in turbulent odor plumes, purely chemical spatial gradients can be calculated when measuring with sensors scaled to lobster olfactory organs, (ii) that rapid odor access to the lobster s olfactory organs (under low ambient flow conditions) is accom-... 

The form of (1.15) is identical to the balance equation that is used in finite-volume CFD codes for passive scalar mixing.17 The principal difference between a zone model and a finite-volume CFD model is that in a zone model the grid can be chosen to optimize the capture of inhomogeneities in the scalar fields independent of the mean velocity and turbulence fields.18 Theoretically, this fact could be exploited to reduce the number of zones to the minimum required to resolve spatial gradients in the scalar fields, thereby greatly reducing the computational requirements. 

The CFD code must use a grid that also resolves spatial gradients in the mean velocity and turbulence fields. At some locations in the reactor, the scalar fields may be constant, and thus a coarser grid (e.g., a zone) can be employed. 

More precisely, the spatial gradient of the mean velocity is independent of position in a homogeneous turbulent shear flow. 

From the terms on the right-hand side of (5.425), it can be seen that residual mixture-fraction fluctuations are produced by spatial gradients in the filtered mixture fraction, and are destroyed by molecular dissipation at sub-grid scales. The multi-environment LES... 

All non-linear terms involving spatial gradients require transported PDF closures. Examples of such terms are viscous dissipation, pressure fluctuations, and scalar dissipation. 

More generally, by using the linear transformation given in (5.107) on p. 167, the mixing model can be decomposed into a non-premixed, inert contribution for and a premixed, 118 reacting contribution for y>rp. It may then be possible to make judicious assumptions concerning the joint scalar dissipation rate. For example, if the spatial gradients of and y>rp are assumed to be uncorrelated, then... 

Note that the turbulent diffusivity Tt(x, t) must be provided by a turbulence model, and for inhomogeneous flows its spatial gradient appears in the drift term in (6.177). If this term is neglected, the notional-particle location PDF, fx>, will not remain uniform when VTt / 0, in which case the Eulerian PDFs will not agree, i.e., i=- f0. 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

Each of these steps introduces numerical errors that must be carefully controlled (Jenny et al. 2001). In particular, taking spatial gradients in step (2) using noisy estimates from step (1) can lead to numerical instability. We will look at this question more closely when considering particle-field estimation below. 

Since the liquid is perfectly mixed, the density is the same everywhere in the tank it does not vary with radial or axial position i.e., there are no spatial gradients in density in the tank. This is why we can use a macroscopic system. It also means that there is only one independent variable, t. 

The types of water transport that can alter the major ion concentrations without significantly affecting their relative abimdances are listed in Table 3.8. Note that these are all physical phenomena. In other words, the spatial gradients in concentrations of conservative substances are largely controlled by physical, rather than chemical, processes. 

Gauss s theorem, which states that the rate of change in [C] with respect to time (0 and at some depth (z) is equal to the negative spatial gradient of the mass flux (F),... 

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