fluid mechanics

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) )

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  1. Solution 1.
    -------------------

    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)

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