Flow Past a Spherical Obstacle

(because the fluid is incompressible.) The boundary conditions are

and

The latter constraint arises because the surface of the sphere is impenetrable, which implies that at .

Let us search for an axisymmetric solution of Equation (7.45) of the form

(7.48) |

It can be seen, by comparison with Equation (7.23), that the previous expression definitely solves Equation (7.45). Moreover, the expression also automatically satisfies the boundary condition (7.46) [because ]. The remaining boundary condition, (7.47), yields . Hence, we obtain

or

Because the solutions of Laplace's equation, subject to well-posed boundary conditions, are unique (Riley 1974), we can be sure that the previous axisymmetric solution is the most general solution to the problem.

It is clear, by comparison with Equations (7.28) and (7.42), that the velocity potential (7.49) is the superposition of that associated with uniform flow with velocity , parallel to the -axis, and a dipole point source of strength , located at the origin. Thus, making use of Equations (7.26) and (7.44), the associated stream function takes the form

(7.52) |

Figure 7.4 show the contours of this stream function.

Bernoulli's theorem yields (see Section 4.3)

(7.53) |

where is the uniform fluid mass density, and the fluid pressure at infinity. Thus, making use of Equations (7.50) and (7.51), the pressure distribution on the surface of the sphere can be written

(7.54) |

The net force exerted on the sphere by the fluid has the Cartesian components

where the integrals are over all solid angle. Thus, it follows that

(7.58) |

In other words, the fluid exerts zero net force on the sphere, in accordance with d'Alembert's paradox. (See Section 4.5.)