Stokes Flow--Sphere -- from Eric Weisstein's World of Physics

Stokes Flow--Sphere

For fluid motion around a solid sphere, symmetry yields and in the Navier-Stokes equations with no inertia and force terms, which then simplify in spherical coordinates Eric Weisstein's World of Math to

(1)

(2)

(3)

(4)

The boundary conditions are

(5)

(6)

(7)

(8)

where is the free stream velocity, a is the radius of the sphere, and is the angle from the free stream flow vector. The solution is

(9)

(10)

(11)

where is the ambient pressure. The velocity field and pressure in a planar slice through the sphere are illustrated above.

Stokes Flow, Stokes Flow--Cylinder, Stokes Velocity

References

Lamb, S. H. §337-338 in Hydrodynamics, 6th ed. New York: Dover, pp. 597-604, 1945.

Stokes, G. G. "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums." Cambridge Philos. Trans. 9, 8-106, 1851. Reprinted in Mathematical and Physical Papers, 2nd ed., Vol. 3. New York: Johnson Reprint Corp., p. 1, 1966.