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 sin�nt� 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 .

(a) A reasonable assumption for the form of the velocity field, q, of the fluid is that it has a radial component only. This means that the fluid motion is purely in the radial direction towards or away from the center of the sphere.

(b) To find the velocity, v, of the boundary, we can use the relation between the velocity potential, F, and the velocity vector, v. The velocity vector is given by the gradient of the velocity potential, v = ∇F. In this case, since the fluid motion is purely radial, the velocity vector can be written as v = v(r) r̂, where v(r) is the magnitude of the velocity.

To find v(r), we can differentiate the velocity potential, F, with respect to r. Since F = -a n sin(nt) r, where n is a constant, the derivative of F with respect to r is given by dv/dr = -an sin(nt). Therefore, the velocity of the boundary is given by v = -an sin(nt) r̂.

(c) To find f(t), we need to consider the boundary condition. The boundary condition for fluid at infinity is that the pressure is constant and equal to P∞. The pressure at any point is related to the velocity potential by the Bernoulli's equation for unsteady flow, which states that P + 0.5ρv^2 = constant, where P is pressure, ρ is fluid density, and v is the velocity magnitude.

At the boundary of the sphere, the velocity magnitude is given by v = -an sin(nt), and using the Bernoulli's equation, we have P + 0.5ρ(-an sin(nt))^2 = P∞ + 0.5ρ(0)^2. Simplifying this equation gives P = P∞ - 0.5ρa^2n^2sin^2(nt).

Since the fluid is incompressible, the density, ρ, is constant. Therefore, we can replace ρa^2n^2 with a new constant, C = ρa^2n^2. Thus, the equation for pressure at the boundary becomes P = P∞ - 0.5Csin^2(nt).

Comparing this equation with the velocity potential F = -an sin(nt) r, we find that f(t) = -0.5Csin^2(nt).

(d) To determine the pressure at any point on the sphere, we can use Bernoulli's equation for unsteady flow again. Bernoulli's equation states that P + 0.5ρv^2 = constant.

Since the fluid motion is purely radial, the velocity vector can be written as v = v(r) r̂. The velocity magnitude, v, is given by v = -an sin(nt). Therefore, the pressure at any point on the sphere is given by P + 0.5ρ(-an sin(nt))^2 = constant.

Simplifying this equation gives P = constant - 0.5ρa^2n^2sin^2(nt). Using the same approach as in part (c), we replace ρa^2n^2 with the constant C = ρa^2n^2. Thus, the pressure at any point on the sphere is P = constant - 0.5Csin^2(nt).

(e) To find the maximum pressure attained on the sphere, we need to find the maximum value of sin^2(nt). The maximum value of sin^2(nt) is 1, which occurs when nt = π/2 + kπ/2, where k is an integer.

Substituting this into the equation for pressure, we have P = constant - 0.5C(1) = constant - 0.5C.

Considering the boundary condition that the pressure at infinity is P∞, we have P = P∞ - 0.5C.

Since C = ρa^2n^2, we can substitute C back into the equation and write the maximum pressure as:
P = P∞ - 0.5ρa^2n^2

Finally, we can substitute n = 2π/T, where T is the period of the motion, to get that the maximum pressure attained on the sphere is:
P = P∞ - 0.5ρa^2(2π/T)^2 r^2

Simplifying further, we have:
P = P∞ + (ρa^2π^2)/T^2 r^2.

This equation matches the given expression: P = P∞ + (A/3a)π^2 r^2, where A = ρa^2π^2/T^2.