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Radii of curvature

This expression, known as the Euler-Bernoulli equation, is standard in texts on strength of materials. [Pg.774]


The encircling probe was characterised with its mirror in water. As we did not own very tiny hydrophone, we used a reflector with hemispherical tip with a radius of curvature of 2 mm (see figure 3c). As a result, it was possible to monitor the beam at the tube entrance and to measure the position of the beam at the desired angle relatively to the angular 0° position. A few acoustic apertures were verified. They were selected on an homogeneous criteria a good one with less than 2 dB of relative sensitivity variations, medium one would be 4 dB and a bad one with more than 6 dB. [Pg.823]

The preceding conclusion is easily verified experimentally by arranging two bubbles with a common air connection, as illustrated in Fig. II-2. The arrangement is unstable, and the smaller of the two bubbles will shrink while the other enlarges. Note, however, that the smaller bubble does not shrink indefinitely once its radius equals that of the tube, its radius of curvature will increase as it continues to shrink until the final stage, where mechanical equilibrium is satisfied, and the two radii of curvature are equal as shown by the dotted lines. [Pg.5]

If the first plane is rotated through a full circle, the first radius of curvature will go through a minimum, and its value at this minimum is called the principal radius of curvature. The second principal radius of curvature is then that in the second plane, kept at right angles to the first. Because Fig. II-3 and Eq. II-7 are obtained by quite arbitrary orientation of the first plane, the radii R and R2 are not necessarily the principal radii of curvature. The pressure difference AP, cannot depend upon the manner in which and R2 are chosen, however, and it follows that the sum ( /R + l/f 2) is independent of how the first plane is oriented (although, of course, the second plane is always at right angles to it). [Pg.6]

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. [Pg.8]

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... [Pg.14]

The case of very large drops or bubbles is easy because only one radius of curvature (that in the plane of the drawings) is considered. Equation 11-12 then becomes... [Pg.29]

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. [Pg.285]

Consider the case of two soap bubbles having a common septum. The bubbles have radii of curvature Ri and R2, and the radius of curvature of the common septum is R. Show under what conditions R would be zero and under what conditions it would be equal to R2. [Pg.527]

Most instruments are configured with a fixed value for the radius of curvature, r, so changing the value of B selectively passes ions of particular values of momentum, mv, tlirough tlie magnetic sector. Thus, it is really the momentum that is selected by a magnetic sector, not mass. We can convert this expression to one involving the accelerating potential. [Pg.1333]

Here p/p° is the relative pressure of vapour in equilibrium with a meniscus having a radius of curvature r , and y and Vi are the surface tension and molar volume respectively, of the liquid adsorptive. R and T have their usual meanings. [Pg.113]

At the middle of the capillary where the eflect of the walls on chemical potential is negligible, the radius of curvature will be equal to r as calculated by the Kelvin equation (3.20) but it will become progressively larger as the wall is approached. [Pg.125]

Fig. 3.21 The effect of meniscus curvature on surface tension. Plot of /) against r . y is the surface tension of the meniscus having the mean radius of curvature and y that of a plane surface of liquid, according to Melrose. The value of y/y was calculated by the equation V = /x(l - with a . = 3 a. Fig. 3.21 The effect of meniscus curvature on surface tension. Plot of /) against r . y is the surface tension of the meniscus having the mean radius of curvature and y that of a plane surface of liquid, according to Melrose. The value of y/y was calculated by the equation V = /x(l - with a . = 3 a.
Unless extremely high potentials are to be used, the intense electric fields must be formed by making the radius of curvature of the needle tip as small as possible. Field strength (F) is given by Equation 5.1 in which r is the radius of curvature and k is a geometrical factor for a sphere, k = 1, but for other shapes, k < 1. Thus, if V = 5000 V and r = 10 m, then, for a sphere, F = 5 x 10 V/m with a larger curvature of, say, Iff m (0.1 mm), a potential of 500,000 V would have to be applied to generate the same field. In practice, it is easier to produce and apply 5000 V rather than 500,000 V. [Pg.23]

E = electric potential (voltage) between the inner and outer ESA plates R= radius of curvature of the ion trajectory... [Pg.177]

A large electric potential applied to a needle provides a very intense field at its tip, where the radius of curvature is small. [Pg.386]

Similarly, a sharp edge (razor blade) or a very sharp curve can also provide an intense electric field. For any given electric potential, as the radius of curvature of a tip or edge becomes smaller, the electric field becomes increasingly stronger. [Pg.386]

By growing thin whiskers along a sharp edge or thin wire, the ends of the whiskers become regions of very small radius of curvature and, consequently, provide very intense electric fields. [Pg.386]

The high potential and small radius of curvature at the end of the capillary tube create a strong electric field that causes the emerging liquid to leave the end of the capillary as a mist of fine droplets mixed with vapor. This process is nebulization and occurs at atmospheric pressure. Nebulization can be assisted by use of a gas flow concentric with and past the end of the capillary tube. [Pg.390]

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where... Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...
The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... [Pg.236]


See other pages where Radii of curvature is mentioned: [Pg.302]    [Pg.303]    [Pg.6]    [Pg.53]    [Pg.261]    [Pg.297]    [Pg.665]    [Pg.851]    [Pg.1333]    [Pg.1693]    [Pg.113]    [Pg.119]    [Pg.125]    [Pg.153]    [Pg.25]    [Pg.150]    [Pg.238]    [Pg.240]    [Pg.292]    [Pg.292]    [Pg.147]    [Pg.525]    [Pg.535]    [Pg.2]    [Pg.445]    [Pg.259]    [Pg.478]    [Pg.488]    [Pg.255]    [Pg.322]   
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Border radius of curvature

Curvature radius

Curvatures

Global radius of curvature

Minimum radius of curvature

P Radius of curvature

Rate constants and radius of curvature

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