You just add the diffusion along the other dimensions. This superposition from orthogonal directions makes some sense, as they are independent.
So, going by wikipedia for Fick's second law of diffusion in 1D:
$$
\frac{\partial \psi}{\partial t}
=
D \frac{\partial^2 \psi}{\partial x^2}
$$
We extend it to 2d as:
$$
\frac{\partial \psi}{\partial t}
=
D \frac{\partial^2 \psi}{\partial x^2} + D \frac{\partial^2 \psi}{\partial y^2}
$$
The second derivative is called the "Laplacian operator", and for vector calculus (more than 1D) you may see it notated as $\nabla^2$. When applied to a scalar value, as here, it represents the sum of the partial differentials with respect to each dimension.
You need to "discretize" this. (I might add details here later.)
Math, discretization and Python code for 1D diffusion (step 3)
and for 2D diffusion (step 7)
I think once you've seen the 2D case, extending it to 3D will be easy.
HOWEVER This diffusion won't be very interesting, just a circle (or sphere in 3d) with higher concentration ("density") in the center spreading out over time - like heat diffusing through uniform metal. Although I've addressed your specific question about multi-dimensional diffusion here, from "ink released from one side of a vessel" and the #advection-diffusion tag it may be that you're wanting to simulate diffusion through a fluid, which requires a fluid simulation, which is of course more complex.
In that case, I'd suggest looking at the full 12 steps of the above, as a start.
