Defining Equation for the Velocity Potential

Подпись: V = ux ex + urer + ueee: Defining Equation for the Velocity Potential Подпись: (7.13)

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:

Defining Equation for the Velocity Potential Подпись: Эф э! ’ Defining Equation for the Velocity Potential Подпись: Эф dr Подпись: (7.14)

from which it follows that in axisymmetric flow:

Подпись: д2ф + 1 _d_r дф dx2 r dr |_ dr Defining Equation for the Velocity Potential Подпись: (7.15)

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:

Подпись: (7.16)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.

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