Change in liquid pressure in container with inclined walls upon heating

Question

This is not a homework question. Instead, I want to touch on the concepts of comparing height changes for a fluid in a restricted medium, such as this container, with other simple containers like that of a cylinder. Here's a problem that asks a question touching those concepts. If you don't feel like giving the exact answer, you don't have to. But my aim is to get a starting point, or something conceptual that can be told would be extremely helpful to me.

Both the containers contain liquid up to the same height, and they're connected by a tube. Assume that the shapes and sizes of the containers are not affected by the heating of liquid.

If the liquid in the container $A$ is heated, which direction should the water flow in?

And similarly, if the container $B$ is heated, which direction will the liquid flow then?

Assume that heat doesn't affect the containers.

What I've been thinking is this: Consider a cylindrical container containing a liquid. It is easy to see that weight of liquid divided by the area of the base of the container is pressure at the bottom. Since neither the area nor the weight of the liquid changes on heating, the pressure remains constant. And since $P = \rho g h$, it means that here $\rho h$ remained constant, density decreases, but height increases.

But for containers that aren't shaped like cylinders, I'm facing problems to see how the change in height would react with the change in density. Because this is not free expansion, and is restricted by the container too, I thought, it wouldn't be correct to assume the change in height as a linear relation for both the containers. But if I can compare the required change in height, to keep the pressure same, against the actual change in height for each container, then I'll get the result to whether the pressure in the container will reduce or increase. But how can I approach doing that? I'm also unsure about the answer to this problem, although the answer given is that in both cases, liquid flows from $B$ to $A$.

Answer 1

In the container with the larger neck, a thermally expanding liquid rises less than it would in a cylindrical container because the newly wetted cross-sectional area is larger and can accommodate more liquid volume per unit height.

Since the pressure at the container bottom scales with both the density (lower) and the depth (higher, but less than the inverse relationship seen with a cylinder), that bottom pressure decreases, resulting in inward flow (to A).

In the container with the smaller neck, the liquid rises more than it would in a cylinder because the newly wetted cross-sectional area is smaller.

Since the density decrease drives a height increase that exceeds the inverse relationship, the bottom pressure increases, resulting in outward flow (again to A).

Answer 2

Think about what’s going on here when you heat the liquid in either of these unusual containers. When you add heat, the liquid becomes less dense—the molecules spread out a bit, which means it takes up more space per unit of mass. Normally, if this were just an open container, this would push the liquid level up. But here, we’ve got two containers connected by a tube, and they’re fixed in shape, so the expansion isn’t straightforward.

Now, for a plain cylindrical container, when you heat it, the height of the liquid can simply rise to make up for the decrease in density. This keeps the pressure at the bottom pretty stable. But with these containers, the cross-sectional area changes with height, which complicates things. Since the shape varies, the height change needed to balance pressure doesn’t follow a simple rule as it would in a cylinder.

When we heat container $A$, the liquid in it gets lighter per unit volume. For A to maintain the same pressure at the point where it connects to the tube, the height of the liquid should ideally increase. However, because of $A$’s shape, it can’t rise enough to fully make up for the loss in density. This means the pressure at the bottom of $A$ ends up a bit lower than in $B$, so liquid will flow from $B$ to $A$ to even things out.

The same kind of thing happens if you heat container $B$. The density of the liquid in B drops, but its shape doesn’t allow the height to increase enough to compensate, so again, the pressure at the bottom of $B$ ends up lower than that of $A$. And once again, liquid moves from $B$ to $A$.

So in both scenarios, because the containers have shapes that limit how much the height can change to match the pressure shifts, liquid flows from $B$ to $A$ every time.