Principal radii

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). 

Figure B3.6.6. Illustration of the two principal radii of curvature for a membrane.

FIG. 1 The figure illustrates a piece of surface with non-positive Gaussian curvature. / and / 2 are the principal radii. The Gaussian K) and the mean (H) curvatures are expressed in terms of the principal radii as follows H — j (2/ i) -h 1/(2/ 2), K = l/(/ i/ 2). If R = —Ri at every point, the surface is called minimal. This implies that K is non-positive at every point. 

It can be shown generally that, if the surface has two principal radii of curvature i, / 2, then ... 

Figure 1. Piece of a hyperbolic surface. Ri and R2 are the principal radii.

At equilibrium (dt/) v n. = 0, which leads to the equilibrium condition for pressure expressed in terms of the two principal curvatures or alternatively in terms of the two principal radii of curvature ... 

The local surface curvature is determined by construction of a vector normal to the surface and drawing of two orthogonal planes through the normal vector (Figure 9.4). The location of the planes is chosen according to a requirement that the principal radii, r, and r2, of curvature of lines formed by intersection of the planes with the surface have the minimum and the maximum values. In inverse proportion to them are the principal surface curvatures, g1 = Hrl and g2= l/r2. 

When the volume dV2 of the liquid evaporates, the volume of the vapor increases by dVt the two partial differentials refer to the same mass of substance. Thus (3 V2/d Vl)P2 = —Pi ip2, Pi and p2 being the densities of the two phases. Integration of the equation (3p1/3p2)K1 = P1/P2 affords- p0 = (p,/p2) (p2 - Pa)-The pressure p0 is that on both sides of a plane liquid surface. Pressure p2 is different from p0 whenever the liquid surface is curved. If its two principal radii of curvature are/ and/ 2, then... 

This equation is exact, and therefore the determination of the principal radii of curvature at two points on the interface enables one to estimate Yij and Rq. However, it is also clear that the photographic image if analyzed, as such, is not accurate enough to give a satisfactory accuracy. 

This result is the Laplace equation for a single, spherical interface. In general, that is for any curved interface, this relationship expands to include the two principal radii of curvature, Pi and Rr-... 

Two curved regions on the surface of a water droplet have principal radii of 0.2, 0.67 and 0.1, 0.5 cm. What is their difference in vertical height ... 

Figure 2.28 An element of a curved surface with principal radii R and R2. Reprinted, by permission, from J. F. Padday, in Surface and Colloid Science, E. Matijevic, ed., Vol. 1, p. 79. Copyright 1969 by John Wiley Sons, Inc.

The pressure difference may also be numerically zero in the instance where the two principal radii of curvature lie on opposite sides of the surface, such as in the case of a saddle. 

This technique is based on the determination of the shape of a pendant drop that is formed at the tip of a capillary. The classical form of the Young and Laplace equation relates the pressure drop (Ap) across an interface at a given point to the two principal radii of curvature, r( and r2, and the interfacial tension (Freud and Harkins, 1929) ... 

R and Ro are the two principal radii of curvature. AP is also called Laplace pressure. Equation (2.5) is also referred to as the Laplace equation. 

Q q R i l, i 2 Rb Rd Rg RP Ro r rc S Electric charge (As), heat (J), quality factor of a resonator Heat per unit area (J m-2), integer coefficient Radius of a (usually) spherical object (m), gas constant Two principal radii of curvature (m) Radius of a spherical bubble (m) Radius of a spherical drop (m) Radius of gyration of a polymer (m) Radius of a spherical particle (m) Size of a polymer chain (m) Radius (m), radial coordinate in cylindrical or spherical coordinates Radius of a capillary (m) Entropy (J K-1), number of adsorption binding sites per unit area (mol m-2), spreading coefficient (Nm-1)... 

As a consequence of surface tension, there is a balancing pressure difference across any curved surface, the pressure being greater on the concave side. For a curved surface with principal radii of curvature rj and r2 this pressure difference is given by the Young-Laplace equation, Ap = y(llrx + l/r2), which reduces to Ap = 2y/r for a spherical surface. 

In a further theory, the pores are considered to be open-ended cylinders. Condensation will commence on the pore walls, for which the principal radii of curvature are the pore radius and infinity, and continue until the pore is filled with condensed liquid. Evaporation must take place from the concave liquid surfaces at the ends of the pore, for which (assuming zero contact angle) the principal radii of curvature are both equal to the pore radius. 

Interfacial tension causes a pressure difference to exist across a curved surface, the pressure being greater on the concave side (i.e., on the inside of a droplet or bubble). Consider an interface between phase A, in a droplet or bubble, and phase B, surrounding the droplet or bubble. These will have pressures pA and pB. If the principal radii of curvature are Rx and R2 then,... 

A Models to describe microparticles with a core/shell structure. Diametrical compression has been used to measure the mechanical response of many biological materials. A particular application has been cells, which may be considered to have a core/shell structure. However, until recently testing did not fully integrate experimental results and appropriate numerical models. Initial attempts to extract elastic modulus data from compression testing were based on measuring the contact area between the surface and the cell, the applied force and the principal radii of curvature at the point of contact (Cole, 1932 Hiramoto, 1963). From this it was possible to obtain elastic modulus and surface tension data. The major difficulty with this method was obtaining accurate measurements of the contact area. 

If instead of a spherical surface, any interface having two principal radii of curvature r, and r2 is considered, then... 

Here, v represents the surface tension of a smooth film, whereas r, and r2 are the principal radii of the formed hole. Suh (1983) has shown that for the case of a tellurium film, energy barrier to restore a planar surface is small and pit formation ensues for a melted textured surface. 

Spontaneous contraction of a liquid surface. The fundamental property of liquid surfaces is that they tend to contract to the smallest possible area. This tendency is shown in the spherical form of small drops of liquid, in the tension exerted by soap films as they tend to become less extended, and in many other properties of liquid surfaces. Plateau1 has undertaken a prolonged study of the forms assumed by liquid surfaces, under conditions when the disturbing effect of gravity is absent he showed that the surfaces always assume a curvature such that, if and are the principal radii of curvature at any point,... 

Let the radius of curvature of the surface at P, in the plane of the paper, be p the radius perpendicular to this is PC, since PC is the normal, and C being on the axis of revolution, P remains on the curve when PC rotates about OC and P moves perpendicular to the paper. The principal radii... 

An accurately circular cross-section is not necessary some workers have laid unnecessary stress on the attainment of this. The height of rise is proportional to the sum of the reciprocals of the principal radii of curvature of the meniscus, which will be nearly equal to the major and minor semi-axes of the tube, rx and r2, if the section is elliptical the error in the... 

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