70

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 ...


69

This effect appears like a paradox, as dry soil makes a very bad water conductor. Two effects prevent water from infiltrating: Air in the soil pores cannot escape: dry soil includes lots of air bubbles in small to large pores. If you expect water to get in, how do you think the air could get out? Often it gets stuck, and no water can infiltrate anymore. ...


31

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 ...


19

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 ...


19

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 ...


19

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} \frac{...


19

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 ...


18

You are not missing anything. Rather I think you are placing too much emphasis on the scientific accuracy of something said for effect in a very chatty presentation. The spoken words almost instantly are not repeated in the headlines, whereas the rest of the quote is. And the presenter does not develop this idea of being able to smell thioacetone almost ...


16

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\...


15

Dry objects may tend to not break the surface tension, or "skin" of the water. With wet or damp objects the moisture in them will tend to merge with the water's skin. Just as if you were to spill a little water on a table, you can see it has an edge where it stops. If it meets another bit of water on the table they will merge easily, and not stay two ...


12

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 surface ...


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

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{...


10

The kinetic energy of a particle of mass $m$ and velocity $v$ is related to the temperature by $$ \frac{1}{2}mv^2~=~\mu kT, $$ for $\mu$ a dimensionless constant. The velocity $v~=~dx/dt$ means that the rate of diffusion is related to the mass of the particle $$ R_\textrm{diff}~\propto~\frac{1}{\sqrt m}. $$ For a molecule of a gram molecular weight, this ...


10

If $c$ and $\vec v$ is an arbitrary pair of functions, then the identity you wrote is false; instead it must read $$ \nabla \cdot (c\vec v) = (\vec v\cdot \nabla) c + c (\nabla \cdot \vec v), $$ which is easy to prove component-wise. If your text is disregarding the second term, then presumably they're working under conditions where $\nabla \cdot \vec v = ...


9

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 ...


8

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 ...


8

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 clothing ...


8

The Schrödinger equation is a wave equation, not a diffusion equation. While the equations look similar, the $i$ in Schrodinger equation differentiates them; that allows non-decaying oscillatory solutions, which diffusion equations do not allow. That said there are certainly relations between the two. The Schrödinger equation is analogous to the Fokker-...


8

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. ...


8

PREFACE After several edits, this answer provides a naive explanation of why your approach failed, how to fix it (naive-ish) and a completely different (but right) approach to solve the problem. Intro You are right: the diffusion coefficient should be $D=4pqD_0$, $D_0$ being the "normal one" (see below for derivations). I do not know precisely why your ...


8

The heat equation, as you've written it, models the flow of energy via thermal conduction (heat) through some region with well defined boundary conditions. You have yet to provide the specifics of the boundary region, so my answer will remain general and vague. The $\alpha$ is the "diffusion coefficient" which is the isotropic form (diagonal terms only) of ...


8

Diffusion and adhesion are different phenomena. Diffusion happens due to a concentration gradient, and this is how the water is slowly absorbed by the soil. This happens faster in dry soil per unit area per unit time than damp soil. On the other hand, adhesion is the ability of one material to stick to the other. Damp soil is more sticky to water than dry ...


7

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 ...


7

Yup, I wrote something like that in the wikipedia article... The correspondence [between the Navier-Stokes equation and the convection-diffusion equation] is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is: $$\frac{\partial \mathbf{M}}{\partial t} = \frac{\mu}{\rho} \nabla^2\mathbf{M} -\mathbf{v} \...


7

convection = diffusion + advection. That is, convection is the sum of fluid movement due to bulk transport of the media (like the water in a river flowing down a stream - advection) and the brownian/osmotic dispersion of a fluid constituent from high density to lower density regions (like a drop of ink slowly spreading out in a glass of water - diffusion).


7

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 ...


7

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 ...


7

Both Schrödinger and Wave Equation have plane wave solutions, that's right. The difference is the dispersion relation, which is quadratic for the Schrödinger equation and linear for the wave equation. This is important, because the Schrödinger equation was designed to correctly reproduce the quadratic dispersion relation that was observed for electrons. (...


7

It is a sticky question, and as van Kampen puts it, " no universal form of the diffusion equation exists, but each system has to be studied individually." https://link.springer.com/article/10.1007/BF01304217 (Unfortunately, I don't have full access to his paper, but you might be able to get it through your library.) Now, the main reason the question is ...


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