Bashforth-Adams equation

In Equation (83) R is given by Equation (36). Expression (83) is known as the Bashforth-Adams equation. It is conventional to express this equation in dimensionless form by expressing all distances relative to the radius at the apex b ... 

The Bashforth-Adams equation —the composite of Equations (36) and (84) —is a differential equation that may be solved numerically with /5 and 4> as parameters. Bashforth and Adams solved this equation for a large number of values between 0.125 and 100 by compiling values of x/b and z/b for 0° < 4> < 180°. Their tabular results, calculated by hand before the days of computers, were published in 1883. Other workers subsequently extended these... 

What is the Bashforth-Adams equation How would you use it to determine the surface tension of a liquid and the contact angle of a liquid with a surface ... 

At z=0 AP = 2y/b.) Defining /3=Apgb2/y° and substituting into the Young-Laplace equation yields one form of the Bashforth-Adams equation ... 

Surface tensions of poljuneric fluids or melts may be measured by a variety of direct techniques, with those based upon analysis of the shape of axisymmetric fluid drop profiles (3,72) the most versatile and popular. Discussions of the relative advantages and disadvantages of each technique are presented in Reference 3. Several drop geometries are possible as shown in Figure 18. The shape of these drops is governed by the Bashforth-Adams equation... 

Interfacial tension was measured using an axisymmetric drop shape analysis (ADSA) method [24]. In this method, interfacial tension is obtained by analyzing the change in the shape of a pendant drop of one liquid suspended in a second liqnid. The method is based on the Bashforth-Adams equation, which relates drop shape geometry to interfacial tension [25,26]. A schematic of a pendant drop with the appropriate dimensions for nse in the Bashforth-Adams equation is illustrated in fig. 13.6. 

In 2011, Srinivasan et al. [20] systematically assessed the contact angle measurement of superhydrophobic surfaces using a perturbation solution of the Bashforth-Adams equation. The uncertainty in the calculated contact angle is determined by the uncertainty in the drop height measurement. For example, assuming the resolution limit of the imaging camera is 10 jm, for a 0.7 pL water drop of 153° contact angle calculated from Bashforth-Adams equation, will result in an uncertainty of 3° in the... 

The Bashforth-Adams tables provide an alternate way of evaluating 7 by observing the profile of a sessile drop of the liquid under investigation. If, after all, the drop profiles of Figure 6.15 can be drawn using 0 as a parameter, then it should also be possible to match an experimental drop profile with the (3 value that characterizes it. Equation (85) then relates 7 to 0 and other measurable quantities. This method is claimed to have an error of only 0.1%, but it is slow and tedious and hence not often the method of choice in practice. 

Several additional points might be noted about the use of the Bashforth-Adams tables to evaluate 7. If interpolation is necessary to arrive at the proper (3 value, then interpolation will also be necessary to determine (x/bl. . This results in some loss of accuracy. With pendant drops or sessile bubbles (i.e., negative /3 values), it is difficult to measure the maximum radius since the curvature is least along the equator of such drops (see Figure 6.15b). The Bashforth-Adams tables have been rearranged to facilitate their use for pendant drops. The interested reader will find tables adapted for pendant drops in the material by Padday (1969). The pendant drop method utilizes an equilibrium drop attached to a support and should not be confused with the drop weight method, which involves drop detachment. 

One can measure the maximum pressure that can be applied to a gas bubble at the end of a vertical capillary, of radius r and depth t in a liquid, before it breaks away (Figure 3.13) [143], Before break-away the bubble has the shape of a sessile drop and is described by the equation of Bashforth and Adams. The pressure in the tube is the sum of the hydrostatic pressure (Apgt) and the pressure due to surface tension. Equations have been published which allow calculation of surface tension using Bashforth-Adams and density and depth data. 

Fig. 2.6 Images of sessile water drops (a) 3.4 pL, (b) 8.7 pL, (c) 20.7 pL, and (d) 33.2 pL on a completely nonwetting hydrophobic surface made from TFE powder. The curves are theoretical profiles calculated from the Bashforth and Adams equation (Reproduced with permission from [18], Copyright 2010 The American Chemical Society)...

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... 

This relation can be obtained from the fundamental equation (Chap. I (4)) for the form of a liquid surface under gravity and surface tension. Unfortunately this equation cannot be solved in finite terms. Approximate solutions have been obtained in several ways which are outside the scope of this book. Sufficient account must, however, be given of the methods of Bashforth and Adams,1 to enable the reader to use their tables of numerical results, which are the most complete and accurate ever compiled. Some other important approximate formulae will also be given, for applications of the fundamental equation to special cases. 

Poisson gave the first three terms of Rayleigh s equation (9), in 1831, and Laplace, the first two terms on the right-hand side of equation (10) for wide tubes, in 1805. Bosanquet1 has calculated the corrections for moderately wide tubes, using a different method of approximation from that of Bashforth and Adams. Porter8 has given further calculations of the corrections. 

When the pressure is a maximum, h is a maximum, and, by (17), IjX and rjX will be maxima hence a maximum value of rjX corresponds to the maximum pressure in the bubble. Only two of the fractions in equations (18) and (19) are independent. Sugden has taken various values of ft for each of a series of values r/a, and calculated the corresponding values of rjb from (19). Using Bashforth and Adams s tables, he found the values of and zjb for these values of rjb and / thence by (18) the values of rjX were determined. Examination of these showed what are the maximum values of r/X for each value of r/a, and a table was constructed it is reproduced by permission from the Journal of the Chemical Society. 

Naidich and Grigorenko 1992, Passerone and Ricci 1998) is making the sessile drop method more and more reliable and accurate. However, it should be noted that substantial effort has been required historically to derive a surface energy by curve fitting, and many authors have suggested simplifications which require far fewer measurements. Thus, Bashforth and Adams, (1883), rewrote equation (3.6) in the form ... 

Bashforth and Adams generated tables of ft and x / z for cp = 90° as well as tables of the ratios x/b and z/b for differing values of b and (p. By measurement of x90 and Z90 at cp = 90°, b and / can be determined from these tables and the liquid surface energy can then be derived from equation (3.8) with an accuracy of about 2% for / > 2 if the droplet coordinates are measured with an accuracy better than 0.1% (Sangiorgi et al. 1982). A method is presented in Appendix D allowing calculation of the mass of a droplet for an optimised [Pg.121]

Several graphical curve-fitting techniques have been developed (see Padday [53] for details) that can be used in conjunction with the numerical integration of the Laplace equation by Bashforth and Adams (and by subsequent workers) to determine d and to obtain y v. Smolders [54,55] used a number of coordinate points of the profile of the drop for curve fitting. If the surface tension of the liquid is known and if 0 > 90, a perturbation solution of the Laplace equation derived by Ehrlich [56] can be used to determine the contact angle, provided the drop is not far from spherical. Input data are the maximum radius of the drop and the radius at the plane of contact of the drop with the solid surface. The accuracy of this calculation does not depend critically on the accuracy of the interfacial tension. 

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). 

In 30 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. It is the first time in the literature that these methods are applied for the efficient solution of the resonance problem of the Schrodinger equation. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton methods. In particular, they are based on the fourth order Adams-Bashforth scheme (as predictor) and on the fifth order Adams-Moulton scheme (as corrector). More... 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

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. 

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. 

Contact angles were calculated from the observations of the edges of the menisci in the two tubes, as follows With reference to Figure 1, the measured height difference is Ah. Since the radius of the small tube is much less than that of the large tube. Ah may be used as an approximation for h in Equation 2, and this allows a preliminary value of 6 to be calculated. This 6 then is used to obtain approximate values of z for each tube, using the Bashforth and Adams tables for the 1-mm. tubes and the approximation formulas given by Blaisdell [4] for the 10-mm. tubes. 

The two equations, [53] and [55], form a system of coupled ODEs with the variable z playing the role of the independent variable. Given initial conditions at a point Zq these equations can be solved by standard numerical routines such as those discussed in the previous section. Because much computational effort is required to evaluate each p, at each increment of the independent variable z, a method that does not require too many evaluations of the right hand side of the iterative equation is desirable. Usually, a simple forward Euler routine is quite adequate for these purposes. If a multistep algorithm is used, the Adams-Bashforth method has been recommended by Kubicek and Marek the first-order Adams-Bashforth algorithm is, in fact, equivalent to the simple forward Euler algorithm. 

For example, the fourth-order Adams-Bashforth method applied to (2.38) generates the following fourth-order finite-difference equation ... 

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