# math

posted by
**paul gokool** on
.

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,

= ¥ + +

A

3a

p P n a

2

r 2 .