Motion in the Normal Direction

Abstract

In this chapter we discuss the motion of an interface under an internally generated velocity field for constant motion in the normal direction. This velocity field is defined by V↦ = a N↦ or V _n = a, where a is a constant. The corresponding level set equation (i.e., equation (4.4)) is

$$\phi t\, + \,a\left| {\nabla \phi } \right|\, = \,0,\,$$

((6.1))

where a can be of either sign. When a > 0 the interface moves in the normal direction, and when a < 0 the interface moves opposite the normal direction. When a = 0 this equation reduces to the trivial φ _t = 0, where φ is constant for all time. Figure 6.1 shows the evolution of a star-shaped interface as it moves normal to itself in the outward direction.