The first problem here will discuss a technique used by geologists to characterize stone. A sample of the stone to be studied is rotated at high speed in a bath of concentrated acid and the products of its dissolution are measured over time. From this information one can properties of the stone. To be specific, we will model a stone of calcium carbonate dissolving in a bath of concentrated hydrochloric acid.
What are the ingredients of the model? If we want to keep track of the position and amounts of the various ions then we need to model the fluid and how it responds to the spinning sample since it carries the ions along with it. Also, the ions are created at the surface of the stone as it interacts with the acid so we need some mathematics to deal with that as well.
As a bit of teaser let’s be a bit more precise about the pieces involved. Since the dissolution of the stone is taking place at its surface a good description of the velocity field will come from the von Karman similarity solution and then using this in a velocity in a convection-diffusion model for the ions.
For the fluid, we assume a steady state, \(\vec{u}_t=0\), and axial symmetry since no dependence on the angle \(\theta\) is expected. Taking cylindrical coordinates, the velocity \(\vec{u}(r,z) = (u_r(r,z),u_\theta(r,z),u_z(r,z))\), satisfies $$\begin{align}u_{r}\frac{\partial u_{r}}{\partial r} -\frac{u_\theta^2}{r} + u_{z}\frac{\partial u_{r}}{\partial z}&=-\frac{1}{\rho}\frac{\partial P}{\partial r}+\frac{\mu}{\rho}\left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u_{r}}{\partial r}\right) – \frac{u_{r}}{r^{2}} + \frac{\partial^2u_{r}}{\partial z^2}\right), \\ u_{r}\frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}u_{r}}{r} + u_{z}\frac{\partial u_{\theta}}{\partial z}&=\frac{\mu}{\rho}\left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u_{\theta}}{\partial r}\right) – \frac{u_{\theta}}{r^{2}} + \frac{\partial^2u_{\theta}}{\partial z^2}\right), \\ u_{r}\frac{\partial u_{z}}{\partial r} + u_{z}\frac{\partial u_{z}}{\partial z}&=-\frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\mu}{\rho}\left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u_{z}}{\partial r}\right)+ \frac{\partial^2u_{z}}{\partial z^2}\right),\end{align}$$ where the pressure \(P = P(r,z)\). Note that this is just the components of the Navier-Stokes equation \(\rho\left(\vec{u}_t+(\vec{u}\cdot\nabla)\vec{u}\right)=-\nabla P + \mu\Delta\vec{u}\). We also assume that the fluid is incompressible so $$\frac{1}{r}\frac{\partial}{\partial r}(ru_{r}) + \frac{\partial u_{z}}{\partial z} = 0.$$
What’s next? Nondimensionalize and search for a similarity solution… stay tuned!