Defining Equation for the Velocity Potential
This development parallels that for two-dimensional, planar flows in Chapter 4. If the flow is assumed to be irrotational, then the curl of the velocity vector is zero, Vx V = 0 . From this, it follows that a velocity potential, ф, exists such that V = Уф. Expanding the gradient operator in cylindrical coordinates, the velocity vector is given by:
from which it follows that in axisymmetric flow:
For incompressible flow, the continuity equation is given by V • V = 0 so that for irro – tational flow V-(Vp) = V^ = 0. Writing the Laplacian operator in cylindrical coordinates, it follows that:
which for an axisymmetric flow reduces to:
Э2ф Э2ф і Эф —2 = 0:
dx2 dr2 r dr
where the velocity potential ф = ф (x, r ). Thus, as in the two-dimensional planar flow case, the velocity potential for axisymmetric flow satisfies the Laplace’s Equation. Because the Laplace’s Equation is linear, superposition techniques may be used to construct solutions.