= Point-source spherical thermal similarity
{title2=$z\propto B_0^{1/4}t^{1/2}$}
In the <Boussinesq approximation>, a spherical <buoyant thermal> with conserved total buoyancy $B_0$ obeys $\dot b=\alpha u$ and $d(Vu)/dt=B_0$, with $V=4\pi b^3/3$. A zero-volume, zero-impulse point source gives $b=\alpha z$, $Kz^3\dot z=B_0t$, where $K=4\pi\alpha^3/3$. Hence $z^4=2B_0t^2/K$ and $u=\sqrt{B_0/(2K)}/z$. The zero-time singularity represents an ideal source limit, not a finite initially regular sphere.
Back to article page
