Question

Annihilation of a sphere
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A sphere of radius a is surrounded by an infinite mass of liquid modeled as an ideal fluid of mass density rho.
The pressure at infinity is Pi.
The sphere is suddenly annihilated at t==0.
Show that the pressure at distance r greater than a from the center falls to:

Pi (1- (r/a) )

Answer 1

Solution 1.
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Similarly to the previous example, Bernoulli's equation for unsteady incompressible potential flow under zero body forces takes the form:

p(r, t)/ rho + (1/2) (F(t)/r^2)^2 + F'(t)/r + G'(t) == H(t)

Letting r go to infinity, we find that:

Pi / rho + G'(t) == H(t),

so that it reduces to:

p(r, t)/rho + (1/2) (F(t)/r^2)^2 + F'(t)/r = Pi /rho .

At t==0, we have:

u(r, 0) = F(0)/r = 0, so that

p(r, 0)/rho + F'(0)/r = Pi /rho.

At t == 0, r ==a, the surface of the sphere is suddenly put at zero pressure:
p(a, 0)= 0, and the equation above then gives:

F'(0) = (Pi a )/ rho

We conclude that:

p(r, 0) = Pi (1 - r/a)