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Summary: I find a formula for the diameter of a bubble large enough to support one human and plug in known values to get $d=400m$. 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 dioxide ...

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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 ...

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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 ...

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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, ... 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. ... 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 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 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 ... 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$, where ... 4 As an alternative to Christian Blatter's heat interpretation,$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. ... 3 I initially suspected that the picture here is one of a sand bar next to deeper water, not of two "seas" not mixing, where the light-colored water is light because it is shallow, and we are seeing the sand below, and the dense region is dark because it is too deep to see the bottom, and the light is absorbed rather than reflecting back.The foam we see at the ... 3 There are two aspects to be considered here: Does the wall reflect or absorb the radiation? If the wall reflects, is this specular or diffuse reflection? For both of these sub-questions, the answer depends on the material that the wall is made of and on the wavelength of the infrared radiation. At wavelengths close to the visible (for example, at 0.8 ... 3 Diffusion is an stochastical process where a single particle can move in each direction with the same probability. Another description of the diffusion coefficient is the 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 seen as the statistical ... 3 Can a small amount of smoke be dense enough to stay in the air keeping its shape for a minute or so? Or does it always dissipate quickly? I read this and think to myself "optimization problem". Firstly, you should know the following, which is the law of diffusion: $$\frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}$$ For ... 3 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 ... 3 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 ... 3 There are two phenomena present diffusion, which happens due to inhomogeneity in concentration. Particles "want to" go from areas of higher concentration to the lower ones. One can write this in the form of diffusion current $$J_{diff}(x) = - D \nabla \rho(x)$$ where$\rho(x)$is the concentration. This expression is known as Fick's law but it's actually ... 3 Here is an answer in the first case (unbounded domain). As already noted, the difference between the positions of the two particles performs a Brownian motion starting from point$(d,0,0)$with diffusion coefficient$D=D_1+D_2$and one is interested in the time$\tau$of the first hitting of the ball$B(r)$around zero with radius$r=r_1+r_2$when ... 3 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 ... 3 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 ... 3 The diffusion equation is a partial differential equation. The unknown quantity is a function$C(x,t)$. To complete the problem statement you need to specify an initial condition (at$t=0$) and boundary conditions. I'm guessing that your boundary conditions are at infinity, so we take $$C(x,t) \rightarrow 0,\ x\rightarrow \pm \infty.$$ We take a delta ... 3 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$\vec{\Gamma}(x,t)= -D\nabla N(x,t)$, ... 3 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 ... 2 Think about it physically. How much gas is in the tube? There are three sources: The gas coming in at$x=0$which was already in solution. The gas being carried away by the fluid at the other end,$x=L$Diffusion across the walls of the tube. These three terms are represented in the equation$\$ \frac{d}{dt} \left( A \int_{0}^{L} U(x,t) dt \right) = ...

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It's just the usual conservation law. When something is conserved then its amount in some observed region can't disappear but must be flowing in or out. So this means that time derivative of the integral of the quantity in a given volume (the LHS) must be equal to total flux associated with the quantity through a surface of that volume (the RHS). In your ...

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This was intended to be a comment, but is too long so I will post it as an answer. First of all, a disclaimer, I am a physict and all that I know about quantitative finance comes from self-learning, so please feel free of correcting me if I am mistaken (also in the physics stuff, of course!). I have been doing a little of research, and perhaps you are right ...

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