Gauss-Laplace equation

The methods used in this study are mainly the drop and bubble shape technique and the torsion pendulum rheometry. The drop and bubble shape technique, which is based on fitting the shape of an axisymmetric liquid meniscus to the respective Gauss Laplace equation, allows to continuously monitor the... 

The Gauss-Laplace equation describing liquid menisci in general was discussed in detail by Padday Russel (1960) and Padday et al. (1975). The profile of an axisymmetric drop can be calculated in dimensionless co-ordinates from the following equation (Rotenberg et al. 1983),... 

Beside the fitting of drop profile co-ordinates to the Gauss-Laplace equation, based on least square algorithms, a relation exists which allows the surface tension to be calculated from the characteristic diameters and Dj (Andreas et al. 1938, Girault et al. 1984, Hansen Rodsrud 1991). 

Four parameters have to be adjusted in order the fit the experimental coordinates of the drop shape the localisation of the drop apex, Y , the radius of curvature R , and the parameter B. A software package called ADSA does the detection of the drop edge coordinates and the fitting of the Gauss-Laplace equation to these data. A suitable algorithm to solve the Gauss Laplace equation is given in Appendix 5F. 

Fig. S.I6 Procedure of fitting the Gauss-Laplace equation to experimental drop shape coordinates (o) the family of curves represent different theoretical drop shapes calculated for different parameters B...

Eqs(5.19), (5.20). The accuracy of measurements with the pendent drop depends on the algorithm used for the drop shape analysis. While the accuracy is of the order of 1 mN/m when using characteristic drop diameters only, the analysis of the full drop profile by fitting the data to the Gauss-Laplace equation gives values with an accuracy of 0.1 mN/m. 

Appendix 5F Numerical Algorithm to Solve the Gauss-Laplace Equation... 

In brief, via the CCD camera (1) with objective (2) and the frame grabber (3), an image of the shape of a drop (9) is transferred to a computer, where by using the ADSA software the coordinates of this drop are determined and compared to profiles calculated from the Gauss-Laplace equation of capillarity. The only free parameter in this equation, die interfaeial tension Y, is obtained at optimum fitting of the drop-shape coordinates. The dosing system (7) allows one to change the drop volume and hence the drop surface area. This possibility is used in dilational relaxation experiments as outlined in Sec. VI. 

To describe a liquid meniscus and hence to obtain the interfacial tension from the profile coordinates the Gauss - Laplace equation is used. This equation represents the mechanical equilibrium for two homogeneous fluids separated by an interface (Neumann and Spelt 1996). It relates the pressure difference across a curved interface to the surface tension and the curvature of the interface... 

Drop and bubble shape tensiometry is a modem and very effective tool for measuring dynamic and static interfacial tensions. An automatic instrument with an accurate computer controlled dosing system is discussed in detail. Due to an active control loop experiments under various conditions can be performed constant drop/bubble volume, surface area, or height, trapezoidal, ramp type, step type and sinusoidal area changes. The theoretical basis of the method, the fitting procedure to the Gauss-Laplace equation and the key procedures for calibration of the instrument are analysed and described. 

The Laplace equation is written for each point and the resulting matrix equations for all points are solved using Gauss elimination. If we set... 

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

The classical theories of Young,1 Laplace,2 Gauss,8 and Poisson4 all led to the fundamental equation (3) and to some others. It is not proposed... 

This equation was first developed by Laplace and Gauss on the basis of mechanics. 

Finally, note that equation (2.30) was derived under the condition R H, which implies that tube diameters are small compared to the capillary length (R k ). In practice, this means that diameters are in the submillimeter range. When this condition is not met, correction terms must be introduced in equation (2.31) [the first such correction term for H(R) is of order R]. These correction terms were worked out in detailed mathematical developments first by Laplace, and subsequently by Poisson, Gauss, and Rayleigh in the 19th century. 

Important theoretical contributions to the study of fluid surfaces were made by Carl Friedrich Gauss in 1830 and by Simeon Denis Poisson in 1831. Gauss re-derived the Laplace-Young equation by examining the energy of the fluid surface and obtained an expression for the angle of contact at the boundary. Poisson introduced the concept that the density of the fluid in the region of the surface was different from that of the bulk fluid. 

The computation of the incomplete Gamma functions is a vital part of evaluating of the ERIs for all integral methods except the Rys-Gauss quadrature. In the evaluation of Fm(T ) two formulae can be used depending on the value of the argument, T. Firstly, there is the asymptotic formula in which the Laplace formula (see equation 13) is utilized. 

---