The Law of Continuity

• In this incarnation our body has become an imaginary surface. Fluid flows through this surface either entering or leaving the enclosed volume. The flow is described mathematically by velocity vectors drawn at every point in 3-D space. Since we wish to relate this vector field of velocity to the scalar pressure field we will treat it as three scalar fields, one for each of the velocity components Vx, Vy and Vz.

Control Surface

Region

Control Volume

The Law of Continuity

Integral Archimedes' Principle Continuity Principle

Region P Vx : Vy : Vz

Surface -(P)i : -(P)j : -(P)k V

Volume -grad(P) grad(Vx) : grad(Vy) : grad(Vz)

Since the two fields have quite different requirements care is needed in choosing the corresponding terms.

This table shows the substitutions we use to move from Archimedes' principle to the Continuity Principle.