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The radius of a sphere immersed in an infinite ocean of incompressible fluid with density
according to the equation, R(t) = A + a cos(nt), where R(t) is a function of time, t, and A,
constants, with ‘a’ being positive. The fluid moves radially under no external forces and
pressure at infinity is P¥. If the velocity potential, F , for the motion of the fluid is given
where f (t) is a function of time, then address the following:
(a) What is a reasonable assumption for the form of the velocity field, q , of the fluid?
(b) Show that the velocity, v , of the boundary is, v = −an sinnt r.
(c) By considering the boundary condition, find f (t).
(d) Determine the pressure at any point on the sphere using Bernouilli’s equation for unsteady
(e) Show that the maximum pressure attained on the sphere is,

= ¥ + +
p P n a
r 2 .

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