Water Systems — when can I use buckets of water to simulate an ODE?

Question

It is quite common to use physical systems to perform calculations (see here and here). This is for a number of reasons: sometimes the physical system is efficient, sometimes it helps us understand the general principles of the physical system, and sometimes, because it can be a good way of demonstrating how a formal system works.

In this case, I am interested in the latter case. I want to find a physical system that demonstrates how a system of ODEs works. Specifically, I'm interested in ODEs of a particular form:

$$\frac{\partial x_i}{\partial t} = \sum_j A_{ij} x_j + B_i$$

I have a specific case in mind, but the general case is interesting. In particular I have been thinking about water models, which can model a subset of these equations, and about which I have some questions (see below).

The following is an instantiation of the system:

$$ \frac{\partial x}{\partial t} = A - Bx$$ $$ \frac{\partial y}{\partial t} = Bx - Cy$$

with appropriately chosen $A\propto a$, $B \propto b$ etc.

Here is another system, choosing the appropriate $k \propto K$ (one can easily add a constant into this by changing the relative heights):

$$\frac{\partial x}{\partial t} = k(y-x)$$ $$\frac{\partial y}{\partial t} = k(x-y)$$

My questions:

1. Show whether the system $$ \dot{x} = a - bx + cy$$ $$\dot{y} = bx - dy$$ can be instantiated using pumps, taps, and holes (I'm fairly sure it cannot).

2. More generally, using pumps, taps and holes, what are the constraints on $[A_{ij}]$ and $[B_i]$.

3. Assuming that the equation in (1) cannot be instantiated, what physical modification could be used to make it possible. (there are quite a lot of possibilities, for example, this)

So far:

For 2: The way I'm thinking of going about it is to define a "water system" inductively. Let $\mathcal{W}$ be the set of "water systems", made of a set of differential equations and a logical condition $\mathcal{C}$ under which they apply. This may or may not be correct...

1. The pair containing $n$ diffential equations $\{\dot{x_i} = 0 \;|\; i=1\ldots n\}$ and $\mathcal{C}=T$ is a water system.

2. Joining basins: If $\{\dot{x}_i = f_i(x_1 \ldots x_n)\}\in\mathcal{W}$, then systems transformed by $$f_i\rightarrow f_i + k(x_i - x_j + \Delta h)$$ $$f_j \rightarrow f_j + k(x_j - x_i - \Delta h)$$ $$C \rightarrow C \wedge (something??) $$ also belongs to $\mathcal{W}$.

3. Leaks: If $\{\dot{x}_i = f_i(x_1 \ldots x_n)\}\in\mathcal{W}$, then systems transformed by $f_i\rightarrow f_i - k(x_i - h)$ is also in $\mathcal{W}$.

4. Other stuff

Anyway, not sure this is the right way of going about it.

Answer 1

I think I have something for question 1. It has to be divided into two cases: $c<d$ and $c>d$.

If $c<d$, then going from your first picture, you open up another hole in the bottom tank, with width $d-c$. You rerout the $c$-stream directly (or if you like your gravity working predictably, via another reservoir and a high-prerformance pump capable of keeping said reservoir dry) to the upper tank, and you let the $d-c$ go into the reservoir.

If $c>d$, you're in trouble, because then the $cy$ water that enters the upper tank would be more than the $dy$ water leaving the lower. I believe that this scenario is impossible without sensors and automaticly adjusting pumps and holes (trying to output the difference through pump $a$, for instance), and that defeats the whole idea if using tanks and pumps to make it easier to visualize.