Capillary Bashforth-Adams tables

Essentially, these steps are similar to those for capillary rise. For (i) the Bashforth-Adams tables and their modern variants can be used. Regarding (ii) laser-optical reflection techniques can nowadays yield profiles with great precision, so that accurate y values can be obtained. We shall now discuss some of the main features, leaving the numerous technical details to the specialized literature, except for noting that modern image analyses and automatization render steps (i) and (ii) less tedious, if not obviating parts of them. 

The method is very old, dating back to Simon in 1851. After further developments by Cantor ), Sugden ) investigated it thoroughly, thereby modifying the Bashforth-Adams tables to the case at hand, which is essentially a vemiant of [1.4.6]. At the bottom of the bubble R = R =b, as before, so there the capillary pressure is 2y/b. To this the hydrostatic pressure pgz must be added to obtain p. Here z is the vertical distance between the apex and the external liquid level. If the pressure is measured as a hydrostatic head h, it follows that... 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

Equation (478) is the exact analytical geometry expression of capillary rise in a cylindrical tube having a circular cross section, which considers the deviation of the meniscus from sphericity, so that the curvature corresponds to (AP = A pgy) at each point on the meniscus, where y is the elevation of that point above the flat liquid level (y = z+h). Unfortunately, this relation cannot be solved analytically. Numerous approximate solutions have been offered, such as application of the Bashforth and Adams tables in 1883 (see Equation (476)) derivation of Equation (332) by Lord Rayleigh in 1915 a polynomial fit by Lane in 1973 (see Equation (482)) and other numerical methods using computers in modern times. 

For intermediate values of rja, or for tubes of intermediate size, no general formula has been given. Bashforth and Adams have however published tables from which the form of any capillary surface may be calculated, and with the aid of these Sugden has further calculated a table of values of rjh for all values of rja, between 0 and 6. h is here the radius of curvature at the crown... 

Application to the capillary height method. Sugden1 has pointed out that, when the contact angle is zero, the ratio r/6 of the radius of the tube to the radius of the lowest point of the meniscus is the x/b of Bashforth and Adams s tables. By (6)... 

As with the rise in the capillary tube, all such formulae can only be accurate for moderately narrow tubes. A more satisfactory plan has been adopted by Sugden,2 who applied Bashforth and Adams s tables as follows. 

The cu-chetype numerical tabulations of x(z) profiles date back to Bashforth and Adams (almost to 1853 )). These tables were used for more than a century, either in their original form or after modification, until their significance waned with the advent of modem computers. The tables do not only apply to liquid menisci in capillaries, but also to cross-sections of sessile drops, pendent drops, etc. They cdso contain such information as the diameters and heights of sessile drops and contact angles. They give x/b and z/b as a function of 0 for various closely-spaced P values. Their application requires successive approximation because P can only be established if y is known. A starting Vcilue of y could, for instance, be obtained from one of the simpler equations, say from [1.3.2 or 6). 

F. Bashforth, J.C. Adams, An Attempt to Test the Theories of Capillary Action. Cambridge Unlv. Press, Cambridge (UK) (1883). (Extended tables giving the profile x(z) in eq. [ 1.3.14] as a function of (j) for various closely spaced values of p. Boucher et al. noted that these tables Involved painstaking efforts of C. Powalky. In addition, several students, working with hand-driven calculators, contributed. The computations were completed as early as 1853.)... 

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). 

The capillary rise for the interface between these metals with a tube 1.0 cm. in diameter and a contact angle of 0° was calculated as about 0.0064 mm., and so is negligible for our purposes. When the contact angle is other than zero, the rise is even smaller. In a tube with radius 0.5 mm., calculations using the tables of Bashforth and Adams—i.e.. Equation 3—showed that for this system Equation 2 yielded a result that was at most in error by 7% for a 0° contact angle. If the tube is smaller or the contact angle is not 0°,the error is less than 7%. Since the measurement error was of the order of a few per cent. Equation 2 could be taken as a satisfactory approximation to Equation 4, for the difference in capillary rise. 

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