Cherry flow

More generally, we will address the question of realizing any flow on the 2-torus = E /Z, which can and should be viewed as the space of tip positions in the plane E modulo the lattice periodicity W . The coexistence of drifting and pinning motions then becomes a corollary to the realization of Cherry flows on 2-tori, see [57], by spiral tip dynamics in lattice symmetries. 

Fig. 3.5. Cherry flows (a) irrational winding (b) rational winding with stable and unstable periodic drifts. Basins of A = pinning (grey), and drifting region (white).

Theorem 13.13. If the set N of Theorem 13.11 is quasiminimal with a rotation number P, then the flow on N is topologically conjugate to a special Cherry flow restricted to its quasiminimal set) with a rotation number / . 

The Cherry flow is a flow on a two-dimensional torus with two equilibrium states a saddle and an unstable node both unstable separatrices are stable one stable separatrix is a-limit to a node and the other lies in the closure of the unstable separatrices and it is P -stable [see Fig. 13.7.4(a)]. The closure of the unstable separatrices is a quasiminimal set which contains the saddle O and a continuum of unclosed P-stable trajectories. The rotation number for such flows is defined in the same way as for flows on a torus without equilibrium states. Since there is no periodic orbits in a Cherry flow. 

Fig. 13.7.4. (a) The Cherry flow on a torus, (b) A quasiperiodic flow obtained from the Cherry flow by identifying the sepciratrices of the saddle. 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

Carotenoids are lipid-soluble pigments responsible for many of the brilliant red, orange, and yellow colors in edible fruits (lemons, peaches, apricots, oranges, strawberries, cherries, etc.), vegetables (carrots, tomatoes, etc.), fungi (chanterelles), flow-... 

Processes controlling nuclide distributions. The general equations for onedimensional advective transport along a groundwater flow path of groundwater constituents, and the incorporation of water/rock interactions, are given in such texts as Freeze and Cherry (1979). The equations can be applied to the distribution in groundwater of each isotope I with a molar concentration Iw and parent with Pw to obtain... 

Chemical mass is redistributed within a groundwater flow regime as a result of three principal transport processes advection, hydrodynamic dispersion, and molecular diffusion (e.g., Bear, 1972 Freeze and Cherry, 1979). Collectively, they are referred to as mass transport. The nature of these processes and how each can be accommodated within a transport model for a multicomponent chemical system are described in the following sections. 

Nonionizable chemicals (e.g., hydrocarbons, ethers, alcohol) that sorb onto organic materials in an aquifer (i.e., organic carbon) are retarded in their movement in groundwater. The sorbing solute travels at linear velocity that is lower than the groundwater flow velocity by a factor of Ra, the retardation factor. If the Koc of a compound is known, the retardation factor may be calculated using the following equation from Freeze and Cherry (1974) for unconsolidated sediments ... 

Table 1.3 Range of hydrologic properties of Lower Yakima basalt flows and interbeds (Freeze and Cherry 1979)...

The Harvey Group consists of three units Harvey Mountain, Cherry Mountain, and York Mills Formations. The upper Harvey Mountain formation consists of felsic volcanic rocks, typically flow-banded massive rhyolitic lava accompanied by ash-fall tuff. Rocks of the middle Cherry Mountain formation are characterized by the predominance of ash-fall and ash-flow tuffs at the top and volcanogenic sedimentary rocks at the base. The lower York Mills formation is a sequence of interbedded clastic, volcanogenic and minor volcanic rocks, that latter consisting of felsic volcanic-clast conglomerate, volcanogenic sandstone, siltstone and shale, felsic crystal lithic lapilli tuff and flow-banded rhyolite. 

Flow and diffusion transport dissolved and mobile particulate arsenic in groundwater. The flow velocity (speed and direction) of groundwater is largely controlled by changes in the elevation of the water table with lateral distance, water pressure and density, and the permeability and other properties of the aquifer. In some circumstances, temperature gradients may also affect groundwater flow (Freeze and Cherry, 1979), 25. 

Fig. 2.18 A cross-section of a much-quoted model (following Freeze and Cherry, 1979, who cited Hubbert, 1940). The surface is described as undulating in a mode that can be expressed by a simple mathematical equation, and the water table is assumed to follow topography in a fixed mode. The stippled section describes a water system from a low-order divide to a nearby low-order valley the thick lines mark there impermeable planes that are an intrinsic part of the U-shape flow paths model, enlarged in Fig. 2.19. The cross-section emphasizes topographic undulations and disregards the location of the terminal base of drainage and the location of the main water divide.

A number of processes have been developed using continuous chums (97) and alternative systems similar to the Cherry-Burrell Gold n Flow process (98). The major disadvantage to churning, either batch or continuous, is that the resultant... 

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