Finding velocity field of spherical bubble in incompressible fluid I have the following question:
A spherical bubble in an incompressible fluid of density $\rho$ has radius $a(t)$. Write
down an expression for the velocity field at a radius $R > a$.
The pressure far from the bubble is $p_{\infty}$. What is the pressure at radius $R$?
What I know:


*

*I know the velocity is spherically symmetric, so let $u=u(r)$ in the radial direction.

*Then $u=\frac{da}{dt}$ on $r=a$ (1), and $u \to 0$ as $r\to\infty$ (2)
The problems I'm having:


*

*What are we solving here? Is this a potential flow (and why)? If so, then we need to find the velocity potential $\phi$ such that $\nabla^2 \phi = 0$. How do I then translate (2) into a boundary condition on $\phi$?

*Furthermore, I cannot visualise what is happening physically. We have a bubble increasing or decreasing in size – what could cause this?
 A: What I believe to be true:
The bubble changes shape with spherical symmetry, hence the flow is irrotational, and there is a velocity potential.
We solve this problem using conservation of mass at the boundary $r=R>a$, and at $r=a$.
By conservation of mass:
$4\pi r^2*u(r)=4\pi a^2 *\dot{a}$
$\implies u(r)=\frac{a^2 \dot{a}}{r^2}$
Then the velocity potential is:
$\phi=-\frac{a^2 \dot{a}}{r}$
and we use Bernoulli's theorem for steady flow at $r=R>a$ and at infinity to find an expression for pressure.
With regard to the bubble shrinking or expanding (neglecting the bubble rising), it is obvious why this may happen if we have a bubble in a container and compress or expand the size of the container. This changes the pressure inside the container, and so the bubble shrinks or expands accordingly to balance the pressure forces from the pressure outside the container. Here, we are saying a similar thing happens with the bubble responding to $p_\infty$, the pressure at infinity, and in fact for certain conditions, the pressure at the bubble surface can equal $p_\infty$ for all time.
