Further useful ideas

The chapter discusses several further aspects of the physics and mathematics that prove very useful in practice. First we define 4-velocity, 4-momentum and 4-acceleration. Then we introduce the tetrad and show how it can be used to relate a given 4-momentum to the energy and momentum observed in a LIF (local inertial frame). Then we define covariant version of the vector operators div, grad, curl, and obtain simplified expressions for the divergence of a vector and an antisymmetric tensor. The generalized Gauss divergence theorem is then presented.

Derivation of the differential continuity equation in an introductory engineering fluid mechanics course

The differential continuity equation is elegantly derived in advanced fluid mechanics textbooks using the divergence theorem of Gauss, where the surface integral of the mass flux flowing out of a finite control volume is replaced by the volume integral of the divergence of the mass flux within the control volume. To avoid the need for introducing the Gauss divergence theorem in an introductory fluid mechanics course, introductory textbooks in fluid mechanics have opted to use a more simple approach, which depends on the consideration of an infinitesimal control volume and the use of Taylor series expansion. This approach, however, involves a first order truncation of the Taylor series expansion and the use of approximate equality signs which may imply to undergraduate students that the derived continuity equation is an approximate equation. The present study proposes an alternative derivation of the differential continuity equation using a finite control volume and is based on the simple concept of the antiderivative function and the fundamental theorem of calculus. The proposed derivation eliminates the need to formally introduce the Gauss divergence theorem in an introductory engineering fluid mechanics course while avoiding the use of truncated Taylor series expansion and approximate equality signs, hence providing a more simple and sound understanding of the derivation of the differential continuity equation to undergraduate engineering students.

Numerically stable form factor of any polygon and polyhedron

Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.

Application of Cubic B-Spline Curves for Hull Meshing

This paper describes the development of a regular hull meshing code using cubic B-Spline curves. The discretization procedure begins by the definition of B-Spline curves over stations, bow and stern contours of the hull plan lines. Thus, new knots are created applying an equal spaced subdivision procedure on defined B-spline curves. Then, over these equal transversal space knots, longitudinal B-spline curves are defined and subdivided into equally spaced knots, too. Subsequently, new transversal knots are created using the longitudinal equally spaced knots. Finally, the hull mesh is composed by quadrilateral panels formed by these new transversal and longitudinal knots. This procedure is applied in the submerged Wigley hulls Series 60 Cb=0.60. Their mesh volumes are calculated using the divergence theorem, for mesh quality evaluation.