Radius of interaction when adding an object into a boiling pool? When you put something, like a spoon for example, into a volume of boiling water, it is well-known that the water will stop boiling, and will not start boiling again until the spoon reaches the 100 degree temperature.
My question is rather large : if you consider an infinitely large pool of boiling water (with maybe heat being added at infinite or at the bottom) and if you add a cold object inside the pool, is there a way to estimate the size of the zone where the boiling stops ? Maybe using a toy model with a lot of symmetries ?
 A: This requires solving the heat-equation
$$\partial_t u=\alpha \nabla^2 u,$$
with $u$ the local temperature and $\alpha$ the heat diffusivity. Finding time-dependent solutions in three dimensions is often quite difficult. You pose a variety of problems: I will consider the suspension of a cold sphere in an infinite boiling bath. In the limit of large and small heat-capacity of the sphere we can use two classical solutions.
When the heat capacity of the sphere is very large (such that the heat in the sphere is much larger than the total heat flux times experimental time scale), we can assume temperature field is in quasi-equilibrium and hence $\partial_tu\simeq 0$, in which case the diffusion equation becomes the Laplace equation $\nabla^2u=0$, which with boundary condition at $u(R)=u_{sphere}$  (where $R$ denotes the spheres surface) and a bath temperature $u(\infty)=u_{water}$ yields $$u(r)=u_{water}- (u_{water}-u_{sphere})(R/r),$$  with $r$ the radial position. Hence there is no inherent length-scale for the decay of the temperature: if $u_{water}$ is exactly 100 C the water would stop boiling everywhere.
In the opposite limit of an infinitesimal heat capacity, the temperature near the sphere at $r=0$ would have an amount of heat $\Delta q$ extracted while the temperature in the rest of the pool would not have changed. In this case we need to solve the time-dependent diffusion condition $\partial_tu=\alpha\nabla^2u$ with boundary condition $u(t=0,\mathbf{r})-u_{water}=(\Delta q/C) \delta(\mathbf{r})$ with $\delta(\mathbf{r})$ the Dirac delta function and $C$ the water heat capacity per unit volume. The solution to this problem is well-known and given by the Gaussian $$u(t,r)-u_{water}=\frac{\Delta q \exp[-r^2/(12 \alpha t)]}{C (12\pi \alpha t)^{3/2} }.$$ This equation decays super-exponentially with a length scale $l=\sqrt{\alpha t}$, and is strictly only valid when $l\gg R$. Again for an initial water temperature of exactly 100 C the water everywhere would eventually stop boiling.
In summary: interestingly enough there is no intrinsic length scale and the water everywhere would stop boiling. For more realistic cases where the water temperature is slightly above boiling you could use the supplied equations to plot the boiling/non-boiling regions and how they change in time. Again this would yield scaleless results, which is common for diffusion problems.
A: A simple argument is that the influence of the cold spoon cannot spread much faster than the speed of sound in the water (c=1543 m/s at 100 C). The heat is after all in the form of molecules bouncing around.
So we would get a region of non-boiling water expanding fast. But now the kinetic view also tells us that this region will be constantly invaded by fast-moving molecules from outside. It will vanish when the temperature gets close enough to boiling: while ideal water has an exact boiling temperature, non-ideal water has a bit of a spread $\sigma$ (say a tenth of a degree). So the total heat in a hemisphere of radius $ct$ will be $2\pi c^3t^3 C T/3$ so the temperature difference will scale as $T(t)\propto 1/t^3$ and reach $\sigma$ very quickly, at which point boiling starts again in the region.
