Figures of revolution

Most of the situations encountered in capillarity involve figures of revolution, and for these it is possible to write down explicit expressions for and R2 by choosing plane 1 so that it passes through the axis of revolution. As shown in Fig. II-7n, R then swings in the plane of the paper, i.e., it is the curvature of the profile at the point in question. R is therefore given simply by the expression from analytical geometry for the curvature of a line... 

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... 

The method is independent of the angle of contact between the liquid and the plate, provided that this is not so variable as to distort the bubbles or drop seriously from the form of a figure of revolution about a vertical axis, to which alone the calculations apply. With bubbles under a plate, it is almost necessary to use very slightly concave plates, or it becomes impossible to retain the bubble in position.6 A measuring microscope with a very good vertical travel is desirable for the measurement of h, and it should also be capable of horizontal... 

The exact calculation of the weight of liquid lifted, in terms of the surface tension and density, is difficult and requires usually special solutions of the fundamental equation of Capillarity, for figures which often are not figures of revolution. The pull may reach a maximum some distance before the object is completely detached and the measurement of this maximum is considered more satisfactory than that of the pull at the moment of detachment.7 In most cases, however, the pull is applied by means of a torsion balance, and the upward motion of the object cannot be checked after the maximum pull is past, so that the detachment takes place almost immediately the maximum pull is reached. 

The parameter P is positive for oblate figures of revolution (for instance, a sessile drop or a meniscus in a capillary) but negative for prolate ones, like a pendent drop. In the absence of gravity P = 0 and the profile is spherical. 

A different approach was developed by Bashforth and Adams (1883) and extended by Sugden (1921). When the meniscus at the bottom of the capillary rise is the figure of revolution, both the radii of curvature must be equal at the apex. Denoting this radius of... 

The parameter fi is positive for oblate figures of revolution, i.e. for the meniscus in a capillary, a sessile drop, and a bubble under a plate, and is negative for prolate figures, i.e. for a pendant drop or an adjacent bubble. Bashforth and Adams (1883) reported their results as tables. For more detailed information, see for example in Adamson (1967). 

Since it is a very difficult task to measure d experimentally in a capillary tube, we need a relation between d and the experimentally accessible radius of the capillary tube, r. This relation can be derived by considering gravity and surface tension effects by applying fundamental Newton mechanics the complete proof is given in Section 6.1. In the case of a figure of revolution, where Rx = R2 = d, when the elevation of a general point on the surface is denoted by z, the fundamental equation is given as... 

An Important possibility to consider Is that, near P°, Kelvin condensation occurs In ripple and dimple features, so that the surface actually consists of multitudinous patches or lakes" of liquid adsorbate. The model, for a single dimple. Is shown In Figure 9. The dimple Is taken to be a figure of revolution, of... 

In 1951 Wolter analyzed mirrors having concentric figures of revolution, i.e., paraboloids, hyperboloids, and ellipsoids [4.38]. He showed that in order to achieve a true image over an extended field of view the X-rays have to undergo two successive reflections from either a paraboloid/hyperboloid or paraboloid/ellipsoid combination which are mounted in a coaxial and confocal arrangement (see Fig. 4.48). 

The relaxation time of the rotation of a molecular direction is independent of azimuth only if the molecule is a figure of revolution about the said direction. In all other cases the rotational relaxation time of a direction may be defined by considering that all positions of the molecule obtained by rotation about the given direction enter with equal weight. Such will often be the case with molecules in solution where complete randomness of orientation prevails. The change in orientation of a molecular direction depends on rotations about two axes perpendicular to the said direction. If the frictional coefficients of the rotation about these directions are /, and the relaxation time of the k direction (normal to and j) is given by... 

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