I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through this gas film. The mathematical model is created by discretizing the volume from $V_0$ to $V_1$. Where $V_0$ represents the outer volume of the bubble, and $V_1$ represents the outer volume of the gas film (surrounding the bubble). The discretized scheme consists of $N$ equally spaced volume elements from $V_0$ to $V_1$ (finite difference method): \begin{equation} \Delta V = \frac{V_1 - V_0}{N} \end{equation} The volume of the bubble changes as a function of the amount of ozone leaving the bubble, which in turn changes as a function of time. The volume of the bubble and the amount of ozone inside the bubble is linked by the ideal gas law: \begin{equation} V_0 = \frac{n_\text{total}(t) \cdot R \cdot T}{P} \end{equation} $n_\text{total}$: the total amount of gas inside the bubble, $T$: the temperature, $R$: the gas constant, and $P$: the pressure.

The bubble does not only contain ozone. It also contains inert gases, so that: \begin{equation} n_\text{total} = n_\text{ozone}(t) + n_\text{inert} \end{equation} $n_\text{inert}$ will remain constant and only $n_\text{ozone}(t)$ will change over time.

There should be a limit for how much the volume can change before numerical errors will start to occur. Beyond this limit, the discretization scheme should break down and cause errors. How do I express this limit?

Is the limit given by: \begin{equation} Ratio = \frac{V_{0,initial}}{V_{1,initial}} \end{equation}

So that the volume change must not exceed the ratio of the two initial volumes of the bubble and the gas film?


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