Gas Interaction in a Box: Force and Pressure Dynamics

[etotheipi]

There is a box $\mathcal{B} \subseteq \mathbb{R}^3$ with a partition coinciding with the plane $z=0$. The gas in the region $z<0$ is initially at $p_1$ and the gas in the region $z > 0$ is initially at $p_2$, where $p_1 \neq p_2$. At time $t=0$, the partition is removed (details not important). Questions:

- at exactly $t=0$, what is the magnitude of the force that gas $1$ exerts on gas $2$ (and vice versa, by NIII)?
- how to determine the function $p(\mathbf{x},t)$, for all $\mathbf{x} \in \mathcal{B}$ and all $t \geq 0$?

 

[kuruman]

There is a box $\mathcal{B} \subseteq \mathbb{R}^3$ with a partition coinciding with the plane $z=0$. The gas in the region $z<0$ is initially at $p_1$ and the gas in the region $z > 0$ is initially at $p_2$, where $p_1 \neq p_2$. At time $t=0$, the partition is removed (details not important). Questions:

- at exactly $t=0$, what is the magnitude of the force that gas $1$ exerts on gas $2$ (and vice versa, by NIII)?
- how to determine the function $p(\mathbf{x},t)$, for all $\mathbf{x} \in \mathcal{B}$ and all $t \geq 0$?

I am not sure it makes sense to ask about the force that the $z<0$ gas exerts on the $z>0$ gas. Before you go to NIII, what about NI? The simplest model is the ideal gas in which the gas molecules occupy only a tiny fraction of the volume of $\mathcal{B}$ which is another way of saying that they rarely undergo collisions with each one another. The force $pA$ that the gas exerts on the walls, as you know, comes as a result of elastic collisions with the immovable walls.

Thus, while the partition is in place there is an average $pA$ force acting in each of its sides. If it's free to move, it will do so according to NI and NII as more impulse per unit time is delivered on one side than the other. You might even be able to define the force that the partition exerts on the gas as the NIII counterpart of $pA$. However when the partition is removed at $t=0$, a $z<0$ gas molecule that was about to collide with it will keep on going according to NI until it collides with the far wall in $z>0$. That is why the first question does not make sense to me.

I see the second question also as iffy but to a lesser extent. Pressure is a macroscopic property of the gas so asking for $p(\mathbf{x},t)$ can lead into trouble. Suppose you subdivide your box into elements $\Delta V$. For pressure to make sense, every element must contain at least one molecule, preferably more and that sets a lower limit for $\Delta V$. In other words, I don't think that the transition from $\Delta V$ to $dV$ to geometric point $\mathbf{x}$ can be made because of the "granularity" of the gas molecules.

 

[etotheipi]

I think making the transition to a continuous description of the fluid is fine. In Landau & Lifshitz they explained that when you speak of infinitely small elements of volume, it just means small compared to the body as a whole but large compared to the inter-molecular distance. Thus you can abstract away any details about the molecules and just treat it as a continuous medium.

And further, the macroscopic state functions of the fluid ($\rho, p, T,$ etc) are in general functions of the position coordinate $\mathbf{x}$ and the time $t$, as well as things like e.g. the velocity field $\mathbf{v}(\mathbf{x}, t)$.

I think you are right in that asking for the force between the two gases at $t=0$ is not going to yield much meaningful results. But maybe interesting to determine how the distribution of pressure $p(\mathbf{x},t)$ evolves after removing the boundary; except I can't figure out how to setup the equation(s)!

 

[Fred Wright]

When you remove the barrier the gas of higher concentration (higher pressure) diffuses into the gas of lower concentration. I suggest solving Fick's second law,
$$
\frac{\partial c(z,t)}{\partial t}=D\frac{\partial ^2 c}{\partial z^2}
$$
where c is the concentration. The concentration can be related to the ideal gas law.

 

[Chestermiller]

When you remove the barrier, the force exerted by gas 1 on gas 2 (also equal to the force gas 2 exerts on gas 1) will lie somewhere between the force that gas 1 had exerted on the barrier previously and the force that gas 2 had exerted on the barrier. The gases will each be experiencing rapid irreversible deformations that involve viscous stresses and discontinuous pressure waves. The forces and deformations involved can be quantified using the equations of gas dynamics.

 

[etotheipi]

Thanks! I tried to play around with it for an hour or so but I think in the end this problem comes down to solving the navier stokes equations, and I'm not sure if there's too much more insight to be gained except for maybe a bit of computational proficiency.

I wonder, @Chestermiller, maybe if there are some simplifying assumptions that might be made? If not then I don't think I'll spend too much more time worrying about actually solving it

 

[Chestermiller]

Thanks! I tried to play around with it for an hour or so but I think in the end this problem comes down to solving the navier stokes equations, and I'm not sure if there's too much more insight to be gained except for maybe a bit of computational proficiency.

I wonder, @Chestermiller, maybe if there are some simplifying assumptions that might be made? If not then I don't think I'll spend too much more time worrying about actually solving it

It depends on what you are trying to determine. Are you trying to get the final state of the system, or the initial force at the interface of the gases?

 

[etotheipi]

It depends on what you are trying to determine. Are you trying to get the final state of the system, or the initial force at the interface of the gases?

The initial plan was to figure out how the pressure in the container varies as a function of position and time, up until the system reaches the equilibrium state. But I figured that perhaps the only/most direct way to do that would be to actually program a simulation using the navier stokes equations.

Finding the final state of the system should not be difficult; we could just impose insulating wall boundary conditions so that $U$ is constant, and apply the ideal gas law to the initial and final states. The entropy change could be similarly found by adding together the $mR \ln(v_2/v_1)$ contributions from both gases.

 

[Fred Wright]

If you are interested in treating this as a diffusion problem see https://link.springer.com/content/pdf/bbm:978-3-319-24847-9/1.pdf on how to solve Ficks's second law. I get,
$$
P(z,t)=\left ( \frac{P_1V_1 + P_2V_2}{2(V_1+V_2)} \right )-\left ( \frac{P_1V_1 + P_2V_2}{2(V_1+V_2)} \right )erf(\frac{z}{2\sqrt{Dt}})
$$assuming the system is isothermal and where $P_1$ and $P_2$ are the initial pressures. This might be a naive approach but it would be interesting to see how it compares with the alternative you come up with.

 

[Chestermiller]

The initial plan was to figure out how the pressure in the container varies as a function of position and time, up until the system reaches the equilibrium state. But I figured that perhaps the only/most direct way to do that would be to actually program a simulation using the navier stokes equations.

Finding the final state of the system should not be difficult; we could just impose insulating wall boundary conditions so that $U$ is constant, and apply the ideal gas law to the initial and final states. The entropy change could be similarly found by adding together the $mR \ln(v_2/v_1)$ contributions from both gases.

A more interesting problem would be to allow the barrier to freely move rather than removing it entirely, and to treat the barrier as being insulated.

 

[etotheipi]

A more interesting problem would be to allow the barrier to freely move rather than removing it entirely, and to treat the barrier as being insulated.

That does seem more interesting!

I think the motion of the partition will be oscillatory [i.e. it will 'overshoot']. The problem I see in determining the equations of motion is that the force per unit area on (both faces of) the piston depends not only on the gas pressure near the piston but also viscous stresses in the gas. Meanwhile, both gases are undergoing irreversible adiabatic expansions and compressions.

Do you have any hints as to how to go about writing down the equations of motion?

 

[Chestermiller]

That does seem more interesting!

I think the motion of the partition will be oscillatory [i.e. it will 'overshoot']. The problem I see in determining the equations of motion is that the force per unit area on (both faces of) the piston depends not only on the gas pressure near the piston but also viscous stresses in the gas. Meanwhile, both gases are undergoing irreversible adiabatic expansions and compressions.

Do you have any hints as to how to go about writing down the equations of motion?

See my posts in this thread: https://www.physicsforums.com/threa...n-an-adiabatic-container.981529/#post-6272441

 

[etotheipi]

See my posts in this thread: https://www.physicsforums.com/threa...n-an-adiabatic-container.981529/#post-6272441

Awesome, thanks. I'll have a read through that thread

 

[Chestermiller]

Awesome, thanks. I'll have a read through that thread

I liked the name etotheipi better.

 

[vanhees71]

When considering adiabatic changes only, you can just use ideal hydrodynamics (i.e., assuming local thermal equilibrium at any time).