Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

First and second derivative matrix elements for the stretching, bending, and torsional energy

Abstract: Matrix elements for the first and second derivatives of the internal coordinates with respect to Cartesian coordinates are reported for stretching, linear, nonlinear, and out-of-plane bending and torsional motion. Derivatives of the energy with respect to the Cartesian coordinates are calculated with the chain rule. Derivatives of the energy with respect to the internal coordinates are straightforward, but the calculation of the derivatives of the internal coordinates with respect to the Cartesian coordinates can be simplified by the following two steps outlined in this article. First, the number of terms in the analytical functions can be reduced or will vanish when the derivatives of the bond length, bond angle, and torsion angle are reported in a local coordinate system in which one bond lies on an axis and an adjacent bond lies in the plane of two axes or is projected onto perpendicular planes for linear and out-of-plane bending motion. Second, a simple rotation transforms these derivatives to the appropriate orientation in the space-fixed molecular coordinate system. Functions of the internal coordinates are invariant with respect to translation and rotation. The translational invariance and the symmetry of the second derivatives for a system with L atoms are used to select L-1- and L(L-1)/2-independent first and second derivatives, respectively, of which approximately half of the latter vanish in the local coordinate system. The rotational invariance permits the transformation of the simplified derivatives in the local coordinate system to any orientation in space. The approach outlined in this article simplifies the formulas by expressing them in a local coordinate system, identifies the most convenient independent elements to compute, from which the dependent ones are calculated, and defines a transformation to the space-fixed molecular coordinate system.

Computation of three-dimensional hydrostatic menisci

Abstract: A numerical method is developed for computing the shape of an infinite three-dimensional hydrostatic meniscus originating from interior contact lines whose projection in the horizontal plane has a specified shape. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. The elevation of the contact lines is either prescribed or computed as part of the solution to ensure a specified contact angle. The method is applied to study the hydrostatic meniscus developing around an elliptical vertical cylinder using elliptic coordinates and between two parallel vertical circular cylinders using bipolar coordinates. The results illustrate variations in the contact angle or contact line distribution due to the boundary geometry and furnish numerical estimates for the capillary force and torque.