The continuity equation and the divergence

[wuwei]

according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the center, but if I think that at the origin the density of water is unchanged and so the first term of continuity equation is 0 so divergence J is 0, too. but apparently it's wrong. what's the problem?

 

[boneh3ad]

It's really tough to say, exactly, because most of your post was an indecipherable run-on sentence. Would you mind starting over and restating the problem a bit more clearly?

 

[sweet springs]

Your case says
$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf{j}=C$$ at the center, otherwise
$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf{j}=0$$

If density is constant
$$\nabla\cdot\mathbf{j}=C$$ at the center, otherwise
$$\nabla\cdot\mathbf{j}=0$$

Anything wrong with it ?

 

[Chestermiller]

Your case says
$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf{j}=C$$ at the center, otherwise
$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf{j}=0$$

If density is constant
$$\nabla\cdot\mathbf{j}=C$$ at the center, otherwise
$$\nabla\cdot\mathbf{j}=0$$

Anything wrong with it ?

Yes. The divergence is zero at the center also, unless you have a point source at the center, in which case, the divergence is a spherical delta function.