fluid mechanics

Submarine Explosion
-----------------------------

A large mass of incompressible, inviscid fluid contains a spherical bubble obeying Boyle's Law:

p V = constant

At great distances from the bubble, the pressure is zero.
Neglecting body forces, show that the radius R(t) of the bubble at time t satisfies the equation:

(d/dt) (R^2 (d/dt)(R) ) - (1/2) R ((d/dt)(R))^2 = k/R^2

for a constant k

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

    Here spherical symmetry applies and so:

    phi(r, t) = (F(t)/r) + G(t)

    Then we consider our boundary condition.
    A unit normal to the boundary between the bubble and the fluid is e_r, and so:

    grad(phi) dot n = (d/d r)(phi)

    The expanding bubble represents a moving boundary, given by R = R(t), with velocity:

    U = (d/dt)( R(t)) e_r

    Applying the boundary condition at r=R, we have:

    u(R,t) = (d/dt)(phi(R,t)) = - F(t)/R(t)^2 = (d/dt)(R(t))

    so that F(t) = - R^2 (d/dt)(R) and

    phi(r, t) = - (R(t)^2)/r (d/dt)(R(t)) + G(t), v(r, t) = R(t)^2/(r^2) (d/dt)(R(t)) e_r


    Bernoulli's equation for unsteady irrotational flow under zero body forces takes the form:

    p(r, t)/rho + (1/2)( R(t)^4 / r^4) (d/dt)(R(t)) - (1/r)(d/dt)( R(t)^2) (d/dt) (R(t)) + G'(t) = H(t)

    Writing this as r goes to infinity, where the fluid is at zero pressure, leads to:
    G'(t) = H(t).

    Thus Bernoulli's equation reduces to:

    p(r, t) / rho + (1/2) (R(t)^4 / r^4) ((d/dt)(R(t)))^2 - (1/r) (d/dt)( R(t)^2 (d/dt)(R(t)) ) = 0,

    to hold everwhere.
    In particular when r == R, i.e. at the boundary, we have:

    p(R, t) / rho + (1/2)*((d/dt)(R(t)))^2 - (1/R(t)) (d/dt)(R(t)^2 (d/dt)(R(t)) ) = 0

    Now at r == R, Boyle's law holds: pV = C, a constant:

    p(R, t) = C/( (4/3) pi R^3 ),

    so that

    p(R, t)/ rho = k/ R^3,

    for some constant k, and:

    k / R(t)^3 = (1/R(t)) (d/dt) (R(t)^2 (d/dt)(R(t))) - (1/2) ( (d/dt)(R(t)) )^2.

    Multiplying across by R, we obtain the required nonlinear equation, which we may solve numerically.

    equation1 := (3/2) R(t) ( (d/dt)(R(t)) )^2 + R(t)^2 ((d/dt)^2(R(t)) ) - 10/R(t)^2 = 0

    -----------------------
    Maple code:
    -----------------------

    > restart: with(plots) : with(Detools):
    k := 10
    eq1 := diff( R(t)^2* diff(R(t), t) ,t) - 1/2*R(t)* (diff(R(t),t))^2 - k/R(t)^2 = 0;
    p := dsolve( {eq1, R(0)=1, D(R)(0) = 0, R(t), type=numeric, range = 0..10} ):
    odeplot(p);

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