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53

Summary: I find a formula for the diameter of a bubble large enough to support one human and plug in known values to get $d=400\,{\rm m}$. I'll have a quantitative stab at the answer to the question of how large an air bubble has to be for the carbon dioxide concentration to be in a breathable steady state, whilst a human is continuously producing carbon ...


27

Maybe I should turn the comment to an answer. The physics of the situation is the same as when one can upturn a water glass with the water not falling out. The atmospheric pressure keeps it in. There exist the diving bells with open bottoms . As they are lowered the pressure in the air goes up to balance the water pressure, because the lower in the water ...


18

There are two mechanisms for mixing at a liquid-liquid interface, firstly diffusion and secondly physical agitation. Diffusion is negligably slow in liquids, it takes days for solutes to travel a few centimetres, so the mixing is dominated by physical agitation e.g. wave action, convention currents, wind mixing etc. In this particular case it's hard to ...


18

Mammalian sense of smell is in general exquisitely keen: even though we think of ourselves as an animal having a dull smell sense comapared to that of, say, a dog, a pig or a rat, receptors for certain scents are still triggered by molecules counted in the tens. So the outgassing of volatile wood oils from, say, a table, can still be miniscule and well ...


16

This is to summarize some of the excellent comments made previously by participants of this discussion, and to emphasize a couple of important points. 1) The original question implied that gas exchange between the bubble and surrounding water may be enough to sustain indefinitely breathing organisms inside. However this does not seem to be possible ...


15

Let us try to rewrite the equation in approximate form of finite differences: $$\frac{A(x,t+\Delta t)-A(x,t)}{\Delta t} = C_3\frac{A(x+h,t)+A(x-h,t)-2A(x,t)}{h^2} +$$ $$+ C_2 \frac{v(x+h,t)A(x+h,t)-v(x-h,t)A(x-h,t)}{2h} + C_1 A(x,t)+C_0$$ Where $\Delta t$ -- is a time step, and $h$ -- space step. The expression becomes your PDE, in the limit $\Delta t\to0, h\...


12

The fog you are seeing is condensation of atmospheric water, not sublimed $CO_2$. The water fog is made very near the boiling surface, and then sinks slowly, exactly as it does in rainclouds. Therefore, just because you can see fog gathering on the floor does not mean that the $CO_2$ is confined there. The $CO_2$ molecules have a speed, in random directions,...


11

As Ted Bunn said, the linear concentration profile is only a steady state if there is a steady inflow at one end and a steady outflow at the other. This net flow is what preserves the concentration gradient. With the "closed box" boundary condition instead, there is indeed an error in your reasoning because the linear profile is no longer a steady state. So,...


11

Lenses and glass bottles are transparent. As you quoted above, the different has to do with diffusion. Here is an example of an image through a transparent object: Here is an example of a translucent object: This is an example of how diffusion causes translucency: As light passes through a translucent object, it either enters or exists a rough ...


11

Convection is the collective motion of particles in a fluid and actually encompasses both diffusion and advection. Advection is the motion of particles along the bulk flow Diffusion is the net movement of particles from high concentration to low concentration We typically describe the above two using the partial differential equations: \begin{align} \...


7

Living human bodies typically have a thin envelope of air warmed by conduction, and this warm air forms a rising plume. Classic photo: the plume of warm air above a human body, visualized by Schlieren Photography: ALWAYS SILENT, SOMETIMES DEADLY! Unless your subject is nude, any ejected gas will inflate their clothing and then leak promptly from ...


7

1) The answer depends on what you mean by rigorous; obviously the 1-d derivation on wikipedia is not exactly rigorous. It also depends on what you would like to use as a micro-physical starting point, and how much detail you desire. Hundreds of text books have been written on deriving diffusive laws from (quantum) kinetic theory, linear response theory, etc. ...


6

There are several ways I can interpret the question, so my main focus is going to be on the autocorrelation of an Ornstein-Uhlenbeck (O-U) process. So what is an O-U process and how is it different from regular Brownian diffusion? Brownian diffusion The stochastic differential equation (SDE) for Brownian diffusion of a particle can be written as $$\mathrm{...


5

Nobody has thus far touched on the probability that freshwater at a river/ocean interface is quite likely to be muddy. What does this mean? It means that the water is likely to contain a stable suspension of silicate micro- or nanoparticles, which are unable to aggregate due to short range electrostatic repulsion. This is what is called a colloid. The ...


5

Diffusion is a stochastic process where a single particle can move in each direction with the same probability. Another description of the diffusion coefficient is the following equation: $$D = x^2/(2t)$$ where $t$ is the time and $x^2$ is the mean squared displacement of the particles at this time. The mean squared displacement, $x^2$, can be interpreted ...


5

For Brownian motion, Langevin equation, Fokker-Planck equations, Stochastic process.. from the viewpoint of physicists, the following are standard references: Brownian Motion: Fluctuations, Dynamics, and Applications The Fokker-Planck Equation: Methods of Solutions and Applications Handbook of Stochastic Methods: for Physics, Chemistry and the Natural ...


5

I don't know a good answer to your first question (I'd be interested in a good text for that myself), but I can answer the second. It's easier to explain if we temporarily imagine $\phi$ represents the concentration of some dye made up of little particles suspended in the fluid. The convective term (aka advective term) is transport of $\phi$ due to the ...


5

This is covered in the standard convection-diffusion type of equation: $$ \frac{\partial C}{\partial t} + \vec{u} \cdot \nabla C= D \nabla^2 C$$ Where $C$ is the smoke concentration, and $D$ is the diffusion coefficient of smoke. While the air may be stagnant initially, it will come to move due to buoyancy effects, thus giving some non-zero air velocity $\...


5

I asked the oceanologist (Nikolai Koldunov) about this photo. Here is his answer: In the ocean even if the difference of density is small (e.g., of the order $0.1\,kg/m^3$) the process of mixing between two water masses is rather slow (without strong turbulence). The picture probably was taken close to the estuary of a big river. In this case ...


5

According to this website, the diffusion coefficient of methane (which is also produced in farts) is about 20 ${\rm m}^2$/s.


5

It's true. Special equipment and a long time is required to mix helium and nitrogen. According to one study, a mixture of 2.7% He, 93.3% N at 800 p.s.i.g. required a special cradle to repeatedly upend the cylinder, and 20.5 hours to reach equilibrated gas, which then remained mixed: http://pubs.acs.org/doi/abs/10.1021/je60005a002. The helium repeatedly ...


5

For a single-component fluid, the conservation of mass follows $$ \left(\begin{array}{c}\text{mass of fluid } \\ \text{in volume }\Delta V\end{array}\right)=\left(\begin{array}{c}\text{flux of fluid } \\ \text{in/out of volume }\Delta V\end{array}\right)+\left(\begin{array}{c}\text{sources or} \\ \text{sinks in }\Delta V\end{array}\right) $$ In terms of a ...


5

Your confusion comes from the fact that you are confusing two different ways of representing the random motion. These two ways go by the names stochastic differential equation and fokker-planck equation To establish a base case to reference later in the answer, let's first discuss what happens when there is no randomness. In this case, your differential ...


4

The differential is used to specify that the number is for a "differential range", which is a way to remind you that the notions involved are somewhat fuzzy. Let me give a purely mathematical example. Suppose I tell you that I am going to pick an arbitrary real number between 0 and 10, with the likely hood of a number being picked being proportional to the ...


4

The error is in your intuition. Your calculation is correct. One thing that might help your intuition is to think about what happens at the edges of the region under consideration. There must be a steady inflow from one end and a steady outflow from the other end. Fluid is continually flowing "downhill" (from high concentration to low), but the source at ...


4

Marek suggested I post my comment (which doesn't completely answer the question) as an answer. Here it is: Suppose you have two Brownian motions with diffusion coefficients $D_1$ and $D_2$, which start at the same point at $t=0$. Let $x_i$ be the average displacement vector for particle $i$ after time $t$. Then, $\langle x_i^2 \rangle = 2D_it$, where $x_i^...


4

As an alternative to Christian Blatter's heat interpretation, $A$ might describe the concentration of particles adsorbed onto a one-dimensional substrate surface (or a two-dimensional one, where we ignore one of the dimensions). New particles are adsorbed at rate $C_0$ per unit length. Adsorbed particles detach from the surface at rate $-C_1$ per particle. ...


4

If the drop is very much static (in still water) and of similar fluid properties to the water around it (so that the ink just labels some initial region), then this is the correct equation to use. If, however, you want to treat the ink as having distinct properties from the water, then you want the Navier-Stokes equations. Since you are interested in ...


4

Since the comment answered your question I'll just go ahead and set out a more generalised version. It's straightforward to simplify things back down to your case. Consider the following continuity equation: $$ \dot{N}(x,t) = -\nabla\cdot\vec{\Gamma}(x,t) + S(x,t), $$ where $\vec{\Gamma}(x,t)$ is the flux (in your case $\vec{\Gamma}(x,t)= -D\nabla N(x,t)$, ...



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