Elliptical profiles

The first stage, called dynamic stage, is the period during which a spherical droplet is flattened and deformed into a planetary ellipsoid with its major axis perpendicular to the flow direction as a result of the external pressure distribution. The eccentricity of the elliptical profile changes with time. 

Fig. 3.1. Cells flowing past illuminating beams of different profiles. A beam with an elliptical profile (lower beam) allows cells to pass into and out of the beam quickly (avoiding the coincidence of two cells in the beam at the same time). In addition, it provides more equal illumination if cells stray from the center of the beam. The small circular beam at the top does not illuminate cells equally if they are at the edge of the stream core. The larger circular beam (in the middle) illuminates cells equally, but often includes multiple cells in the beam at the same time.

Figure 6.9. Volume fraction profiles of an end-grafted polystyrene brush, of relative molecular mass 105 000, imder various solvent conditions (O, toluene at 21 C and cyclohexane at A, 53.4 °C , 31.5 °C o, 21.4 °C and A, 14.6 "C), deduced from neutron reflectivity measurements (all the solvents are deuterated). Toluene at 21 °C is a good solvent and the solid line is the classical parabolic profile. The theta temperature for d-cyclohexane is 34 °C and the dashed line is the elliptical profile predicted by analytical self-consistent field theory for theta conditions. After Karim et al. (1994).

Fig. 8.7 The Grilfith crack model for fracture. The flat, elliptical profile in the center of the plate represents a crack...

Figure 8.21 Elliptical profile of melt-grown crystal (130°C) of linear polyethylene with steps along the two long axes.

Lopez et al. [104] observed that the mold temperature produces signiflcant differences in the macroscopic morphology and properties of the injected SPS samples. In particular, samples molded at high temperatures had higher resistance to the organic solvent than samples molded at low temperatures. The core of the molded samples analyzed by TEM appeared spherulitic [104]. However, the spherulites were not fully developed but appeared sheaf structures type with an elliptical profile. The intermediate region presented lamellar crystals oriented perpendicular to the flow direction. 

For the fundamental modes on a circular fiber, is independent of 0. Consequently —tiy and p = p in Eq. (18-21). In other words, the propagation constants for the circular and elliptical fibers are identical for slight eccentricity, provided the core areas are equal [6]. The latter condition is equivalent to requiring equal profile volumes, as is clear from Eq. (17-13). Hence the present result is consistent with the more general result of Section 17-3, which showed that, within the Gaussian approximation, P = on an arbitrary, elliptical-profile fiber of slight eccentricity, provided the profile volumes are equal. 

Design an airplane wing for an elliptical lift/unit span distribution (this implies an elliptical profile to the wing as a function of distance from the plane). The area under the lift/unit span curve should be... 

Figure 4 shows simulation results for an elliptical profile. = 2 was chosen for the elliptical channel in the die. Simulation was performed for different Reynolds number (Re) ranging from 0.00001 to 1. The change of Re did not affect the shape. It was found that the major axis was increased by 8.35%, but the minor axis was increased by 22.7%, resulting in net shape change of the ellipse. The amount of shape change due to Newtonian die swell is comparable to the surface tension effect at f = 0.1. ... 

Fracture mechanics analysis requires the determination of the mode I stress intensity factor for a surface crack having a circular section profile. Here the circular section flaw will be approximated by a semi-elliptical flaw. 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Recently, an interesting correlation between the laser pulse polarization and the ellipticity of the electron beam profile has been observed [71]. However, no major influence of laser polarization on the efficiency of the electron acceleration processes has been observed so far, nor this influence has been predicted by theory and simulations, differently from the proton acceleration. For proton acceleration, a great improvement on bunch charge and quality are expected by using circularly polarized laser pulses focused on thin foils at ultra-high intensities [72-74]. 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 86). Multiple path, molecular diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral spreading. Consequently, the initially circular spots become progressively elliptical in the direction of flow. Efficiency and resolution are thus impaired. Elution must be halted before the solvent front reaches the opposite edge of the plate as the distance it has moved must be measured in order to calculate the retardation factors (Rf values) of separated components (p. 86). 

The mass flux in the spray scales with liquid metal flow rate. Gas pressure tends to narrow the spray whereas melt superheat tends to flatten the spray)3] By changing the process parameters and/or manipulating the configuration and/or motion of the spray, the mass distribution profile can be tailored to the desired shape. For example, a linear atomizer produces a relatively uniform mass distribution in the spray. The mass flux distribution in the spray generated with a linear atomizer has been proposed to follow the elliptical form of the Gaussian distribution)178]... 

Fig.22a,b. Bead density profiles in a brush of grafted chains in a theta solvent with N=200, and two different values of the grafting point densities. The smooth curves correspond to the elliptical theoretical prediction. Reprinted with permission from [199]. Copyright (1993) American Chemical Society... 

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

Fig. 14. One-dimensional cross section of an elliptic weighting filter. The characteristic length is defined as the section length when the relative weight has dropped to 2/a. The filter shape corresponds to the deformation profile of an elastic material under distributed load in a circle of radius Z./2.

Lens A lens is a means of changing the shape of a beam of light. In flow cytometry, lenses are used to narrow the laser beam to a small profile at the stream. Some lenses produce a beam with a circular cross-sectional shape others produce beams with an elliptical configuration. Lenses are also used in a flow cytometer to collect scattered light and fluorescence and then to transmit them to an appropriate photodetector. 

Figure 14 Illustration of the general procedure used to locate the initial relaxation direction (IRD) toward the possible decay products, (a) General photochemical relaxation path leading (via conical intersection decay) to three different final structures, (b) Potential energy surface for a model elliptic conical intersection plotted in the branching plane, (c) Corresponding energy profile (as a function of the angle a) along a circular cross section centered on the conical intersection point and with radius d.

The classical models of spiral galaxies were constructed using rotation velocities. In contrast, the models of elliptical galaxies were found from luminosity profiles and calibrated using central velocity dispersions or motions of companion galaxies. An overview of classical methods to construct models of galaxies is given by Perek (1962). 

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