PARABOLIC ARC

Figure 1-3 Areas Under a Parabolic Arc Covering Two Subintervals of a Simpson s Rule Integration.

The area under a parabolic arc concave upward is bh, where b is the base of the figure and h is its height. The area of a parabolic arc concave downward is jh/t. The areas of parts of the figure diagrammed for Simpson s rule integration are shown in Fig. 1-3. The area A under the parabolic arc in Fig. 1-3 is given by the sum of four terms ... 

Colburn [Tran.s. Am. In.st. Chem. Eng., 35, 211 (1939)] has shown that when the equihbrium line is straight near the origin but curved slightly at its upper end, Nqc . u be computed approximately by assuming that the equihbrium curve is a parabolic arc of slope mo near the origin and passing through the point X, K X at the upper end. The Colburn equation for this case is... 

There are three forms of inducer camber lines in the axial direction. These are circular arc, parabolic arc, and elliptical arc. Circular arc camber lines are used in compressors with low pressure ratios, while the elliptical arc produces good performance at high pressure ratios where the flow has transonic mach numbers. 

Shinbrot and Muzzio (1998) suggested that intruder inertia (elfectively, its total mass rather than size or density alone) may play an important role in what they called a "reverse buoyancy" effect in shaken systems. They suggested that upon each shake, a heavy intruder thrown upward tends to continue on a parabolic arc, sustained by its larger inertia that provides more space below for the... 

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

Let 0 be angular deflection of a dipole from the symmetry axis of the potential 1/(0), let p be a small angular half-width of the well (p Ci/2), and let (/0 be the well depth its reduced value u Uo/(kgT) is assumed to be 1. Since in any microscopically small volume a dipole moment of a fluid is assumed to be zero, we consider that two such wells with oppositely directed symmetry axes arise in the interval [0 < 0 < 2ji]. For brevity we consider now a quarter-arc of the circle. The bottom of the potential well is flat at 0 < 0 parabolic dependence U on 0. The form factor/is defined as the ratio of this flat-part width to the whole width of the well. Thus, the assumed potential profile is given by... 

Continuum sources Continuum sources, unlike the line sources discussed above, offer the potential for multielemental analysis with a single excitation source. Eligh-pressure xenon-arc lamps have been most widely used as continuum sources for AFS because of their high intensity. Each lamp consists of a silica envelope that contains 10-30 atm of xenon and two electrodes to excite the xenon. Improved sensitivity is obtained by the use of a parabolic mirror to focus the emission into an intense beam. 

The oxidation of aluminium at room temperature is reported to conform to an inverse logarithmic equation for growth periods up to 5 years duration. At elevated temperatures, oxidation studies over shorter periods illustrate conformity to parabolic, linear and logarithmic relationships according to time and temperature. These kinetic variations arc attributed to different mechanisms of film formation . ... 

The German X-ray satellite telescope ROSAT was placed in orbit in 1990. It is equipped with a four-mirror system of parabolic and hyperbolic Zerodur cylinders nested into one another (see Fig. 4.66). The inner surfaces of these reflectors were ground and polished by Carl Zeiss, Germany. The residual surface roughness is less than 0.2 nm RMS. An angular resolution of 3.3 arc seconds was achieved for the telescope due to the precise fabrication and mounting of the mirrors. 

Figure 11.15 (a] Cross-sectional views of a circular and noncircular capillary occupied by a gas and a wetting liquid, [b) Liquid flow along the corners of a capillary tube, directed along the x-axis. The corner flow problem can be solved using the lubrication approximation so that the velocity profde adopts a parabolic shape, as depicted. The gas-liquid interface is a circular arc of radius o, which weakly depends on x. (Reprinted with permission from Ref. 83, copyright 1988, Elsevier.)... 

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