U-shaped curves

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Aromatic polyesters that do not contain any flexible structural units are often nonmeltable or extremely high melting polymers that cannot be processed. Copolymerization is a way to obtain processable wholly aromatic polyesters The Tm versus copolyester composition curve is a U-shaped curve exhibiting a minimum that is generally well below the Tm of corresponding homopolymers. Liquid crystalline aromatic polyesters, for instance, are usually copolymers.72 An example is Ticona s Vectra, a random copolyester containing 4-oxybenzoyl and 6-oxy-2-naphthoyl units in ca. 70 30 mol ratio. This copolymer melts at ca. 

The relationship between dose and SLA was characterized by an inverted U-shaped curve for the phencyclinoids tested and dextroamphetamine. Geometric increases in cumulative IV doses produced nearly linear increases followed by decreases in SLA for each... 

U-shaped curve, we have mixtures that can be ignited for a sufficiently high spark energy. From Equation (4.25) and the dependence of the kinetics on both temperatures and reactant concentrations, it is possible to see why the experimental curve may have this shape. The lowest spark energy occurs near the stoichiometric mixture of XCUi =9.5%. In principle, it should be possible to use Equation (4.25) and data from Table 4.1 to compute these ignitability limits, but the complexities of temperature gradients and induced flows due to buoyancy tend to make such analysis only qualitative. From the theory described, it is possible to illustrate the process as a quasi-steady state (dT/dt = 0). From Equation (4.21) the energy release term represented as... 

Other dose-response relationships may also be seen for certain compounds such as, e.g., essential metals, where symptoms of dehciency may occur if the intake is too low, whereas toxic symptoms may occur if the intake is too high. For such compounds, the dose-response curve is generally U-shaped as illustrated in Figure 4.2. It should be noted that the right part of the U-shaped curve representing the toxic effects in reality is the typical S-shape observed for toxic effects in general. 

In relation to hormesis (Section 4.12), the dose-response curve can be an inverted U-shaped curve (see Figure 4.3) or a J-shaped curve (see Figure 4.4). 

Tubular type equipment Long and slender straight pipe or a coiled pipe or a U-shaped curved pipe. The flow direction is considered to be only along the axial direction. 

For the two catalyst systems NdV/DIBAH/EASC and NdV/TIBA/EASC the ratios of Ai/ Nd were varied over a broad range from 5 to 50. It was shown that kp depends on the nature of the cocatalyst (TIBA and DIBAH) and on the respective concentrations of these cocatalysts. As kp is not constant the authors suggest that it is better considered as an apparent rate constant ka. For the cocatalyst TIBA the dependence of ka on nA /nN< results in a S-shaped curve with a maximum in ka = 311 L mol-1 min 1 at TiBA/ Ndv = 50 while activation with DIBAH leads to an inverse u-shaped curve with a maximum ka 220 L -mol 1 min 1 at Dll Aii/ NdV = 35 (Fig. 11 in Sect. 2.2.8) [179]. 

Why Do Microphases Form The most simple quantum mechanical model of a metal is a box with only one finite dimension. The walls are infinite barriers, so an electron inside has no chance to escape. This model accepts only those electron energies which correspond to electronhalf-wavelengths which are simple fractions of the box size. Metal electrons may fill part of these energy levels with two electrons per level. The upper level of energy reached is called the Fermi level. A smooth U-shaped curve of energy vs. wave number includes all acceptable energies in this model. 

As is seen from Fig. 3.13.1, there exists a critical B value, Bc, for which x - x" - 1/2 for B < Bc no phase separation is encountered. The critical value, Be, which separates the U—shaped curves from those that have minima away from x — 1/2, is determined from the requirements... 

Figure 5 Theoretical solubility profile (thick line) and its component curves (thin lines) of a compound containing five acidic and three basic g roups. The pK values are 2, 2.7, 4.1, 5.3, 6.6 (acids) and 12.7, 9.9, 8.3 (bases). Thick dot atpH=4 represents reference solubility of 0.006 units. Broad U-shaped curve in the 2.5

In general, if A 7 is small, as is likely to be the case for homogeneous solutions, the free energy, plotted as a function of composition, forms a broad, shallow U-shaped curve, and the ranges of composition over which the single phases are... 

The relationship between the measured effect and steady-state plasma concentration sometimes yields bell-shaped (e.g., nortriptyline) or U-shaped (e.g., clo-nidine) curves. Several unusual concentration-effect curves, including the U-shaped curve describing the blood pressure lowering effect of clonidine, have been explained by Paalzow and associates as being a result of multiple receptor responses. The drug then acts on several different receptors that can have opposite effects and that are triggered at different concentrations. 

HOMO of a wide variety of olefins gave a U-shaped curve (Fig. 4-26). Clearly a single reagent, phenyl azide, has spanned the whole range of Fig. 4-23. 

Figure 3 Stylized dose-response curves reflecting the relative distribution of stimulatory dose ranges. Note the maximum stimulatory response is usually 130%—160% of the control value regardless of the width of the stimulatory dose range the inverted U-shaped curve was used for illustrative purposes only, whereas examples in the hormesis database include both inverted U- and U-(J-) shaped curves depending upon the endpoint measured. Modified from Calabrese (30).

The influence of various substituents on 5 Sn for series of tetra-coordinate tin(I V) compounds SnX4 n Yn (n = 0 ) can be graphically depicted as U-shaped curves. These substituent effects are not linear, but approximately pair-wise additive. Some representative Sn NMR data are given in Table 2.1.3 (tin hydrides). Table 2.1.4 (tetraorgano. tin). Table 2.1.5 (tin halides). Table 2.1.6 (tin chalcogenides), and Table 2.1.7 (compounds with Sn—N, Sn—P, Sn—As, Sn—Sb, and Sn—Bi bonds). 

Increasing the pH of sewage also has an effect on the rate of sulfide generation in the slime layer by SRB. The optimum environmental pH for SRB is around pH 7.0 and the rate of sulfate reduction follows an inverted U-shaped curve with the maximum rate centered on pH 7.0 (Pomeroy and Bowlus, 1946). Raising the pH beyond 7.0 has the effect of significantly... 

This condition can be fulfilled even if DM < ED, but in that case a U-shaped curve similar to that in Fig. 36 should not be obtained and the ratio IM //D should increase steadily with temperature. 

When the equilibrium constants were plotted again st the Hammett function for the substituent X, a U-shaped curve resulted, showing all substituents to enhance the stability relative to styrene. This indicated a double bond to present between the metal and olefin, with both the 7r and a components contributing equally. Equilibrium constants were of the order 0.03 -0.05. The relative stabilities of cis- and trans- olefin complexes with platinum(II) were found from a similar study 73>, following the reaction... 

CORIOLIS METERS. An object moving in a rotating system experiences a Coriolis force proportional to its mass and forward velocity and to the angular velocity of the system. This force is perpendicular to the object s direction of travel and to the direction of the angular velocity of the system. In a Coriolis meter (Fig. 8.26) the fluid is passed through two U-shaped curved tubes that are vibrated at their... 

The filtration of particles in a gas stream can be quite different from the filtration of the same particles in a liquid stream. The three mechanisms of aerosol particle retention may be illustrated from the data of Spurny et al20 in Figures 2.22 and 2.23. The U-shaped curves are characteristic of the efficiency of aerosol particle collection as a function of particle size. However, "capillary-pore" membranes have a deeper minimum in the curves than do "tortuous-pore membranes."... 

The variation of CLTE for PP/EPR alloys over the entire EPR concentration range is shown in Fig. 8.7. A U-shaped curve was observed. The value of CLTE... 

The U-Shaped Curve of Iodine Intake and Thyroid Disorders... 

---