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1answer
33 views

Concentration of pollutant in the pool

I'm a beginner in the Mathematical Modeling. I'm trying to solve one problem from our book, where the point is to compute the concentration of pollutant in the pool. I tried to solve it by Fourier ...
0
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0answers
30 views

Calculation of charged sphere distribution near a wall in Cartesian coordinates

I am following a similar derivation as found in the beginning of this paper "Quantitative aspects of the growth of (charged) silica spheres" by A.P. Philipse. This paper calculates the growth of a ...
3
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0answers
60 views

Understanding various types of motion

In classical statistical mechanics, given a system of particles, one often goes about classifying various dynamics (or types of motion) the system may exhibit on different time scales, but studying ...
1
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0answers
90 views

How to derive equation for time it takes photons to diffuse through the Sun

I am wanting to use the Rosseland radiative heat flux equation to find the time it takes for photons to diffuse through the sun. The answer I am wanting to derive is: $$\tau_D~\frac{\rho \bar C_p ...
1
vote
1answer
32 views

Physical reason why Prandtl number is order unity for gases?

Is there a physical reason behind the fact that for gases the thermal diffusivity is on the same order of magnitude as kinematic viscosity (and as such a Prandtl number of order unity) and if so what ...
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0answers
16 views

Copper vs Tungsten (Diffusion)

I've been studying on how to compare the activation energy for (void-) diffusivity of Copper versus Tungsten. I ended up finding $E_a[cu] = 1.1[eV]$ and $E_a[W] = 5.2[eV]$ in literature where the ...
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0answers
16 views

Overdamped Fokker-Planck equation with general force field (gradient + ratation)

I am looking for general formulation of the overdamped Fokker-Planck equation (='Smoluchowski eq.') for a diffusing particle in general force field $F = -\nabla U + \nabla\times A$ . So far I have ...
5
votes
1answer
68 views

Diffusion coefficient for asymmetric (biased) random walk

I want to obtain a Fokker-Planck like equation by taking the continuous limit of a discrete asymmetric random walk. Let the probability of taking a step to the right be $p$, and the probability of ...
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0answers
19 views

Fick's Law for material extraction in a spherical material

I'm looking for a text which talks about how to extract a material from some object (spherical to be easier), (for example, water from the food) and learn about these formulation using the Fick's Law ...
3
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0answers
26 views

How does air going through silica gel change its humidity

Assume that the properties(temperature,humidity,velocity and so on) of the incoming air flow stay constant. The air flow goes through a silica gel plate. Humidity of the air will decrease, but how to ...
0
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0answers
28 views

What happens when a semipermeable membrane allows only one solute to pass through?

I'm essentially a medical student where we deal a lot with osmosis. But when we are taught, it is done generally with only a single solute in consideration. What if two different solutes are used on ...
2
votes
1answer
42 views

Deriving a diffusion equation from a given mass flux density [closed]

I am given that the mass flux density $q$ of a substance in a medium obeys the following physical law. $$q=-D\frac{n^3}{n-n_0}\frac{\partial n}{\partial x}+nv,$$ where $n(x,t)$ is the ...
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0answers
17 views

Connecting the diffusion coefficient in 2-dimensions and 3-dimensions?

Say the diffusion coefficient of the concentration of a particle in a fluid in 3-dimensions is $D_{3\textrm{d}}$. Can we estimate the diffusion coefficient of the same particle in the same fluid, in a ...
0
votes
1answer
29 views

Diffusion (2 species) versus advection (1 species)

Why do we see diffusion with 2 species but advection with a single species if both cases involve molecules spreading to increase entropy? To take a physical example: Let's say I have a box with two ...
0
votes
2answers
57 views

When I open a window to air out the room, how does the smell disperse?

Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature. The quickest way to right the world that comes to mind is to open a window. When I open a window, how do the ...
0
votes
1answer
46 views

Dependance of diffusion coefficient on size?

What's the dependance of the diffusion coefficient on size? More explicitly, suppose I have a particles of characteristic length $l$, dissolved in a liquid. How does $D$, the diffusion coefficient of ...
0
votes
1answer
87 views

1-D Fick's first law - partial derivative?

I've recently been reviewing some concepts, including diffusion. Fick's 1st law: $$J = -D\frac{\partial C(x,t)}{\partial x}$$ as I understand it, applies to the steady state. For 1-D diffusion and ...
4
votes
1answer
48 views

Derivation of the diffusion coefficient?

The diffusion coefficient relates the particle flux $J$ to the gradient in the number density (of the 'labelled' particles) $\frac{\partial \bar n}{\partial z}$ such that; $$J=-D \frac{\partial \bar ...
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0answers
49 views

Does nature really follow the heat equation?

I think the heat equation says that the first derivative of temperature with respect to time in a stationary solid varies as the negative of the second derivative of temperature with respect to ...
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0answers
25 views

can 3D diffusion be interpreted as superposition of 1D diffusion

Is diffusion of a substance in water independent in the X, Y and Z directions? In other words, if I want to solve how a substance diffuses through a volume, can I separately solve equations for its ...
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0answers
23 views

Extension of the advection diffusion equation with diffusion perpendicular to the flow

Assume you model particle flow in fluid in a cylinder by an advection-diffusion equation, thus taking into account both diffusion and bulk flow. What would happen to the model if you would, at some ...
1
vote
1answer
124 views

Rigorous derivation of Fick's first law

I am looking for a rigorous derivation of Fick's law, i.e. that the current density $\mathbf{j}$ satisifies $\mathbf{j} = - D \nabla u$ where $u$ is e.g. some concentration and $D$ the diffusion ...
0
votes
1answer
19 views

Thermal diffusion equation with heat per unit length?

Let us say we have a metal rod. Along that rod there is a rate of heat generation of H per unit length. If we assume we are in the steady state then I would expect us (from the thermal diffusion ...
1
vote
2answers
109 views

Heat Losses from water pan due to vaporisation before boiling

I'm studying heat transfer from an electric heater to a cooking pan (Stainless steel) filled with water; specifically, my problem is related to water vaporisation, as I've already found good models to ...
1
vote
0answers
24 views

Waterproof Case Time Limits [closed]

A lot of cellphone cases are rated safe for one hour at two meter depth. I assume this is due to water molecule diffusion along case seams (like the zipped zipper in a ziplock bag), right? Is ...
0
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0answers
26 views

trying to figure out an expansion in Brownian motion derivation

In the derivation for the diffusion equation on the wikipedia article for Brownian motion, they have these equations: I can't figure out how $\rho(x+\Delta,t)$ gets expanded, though. For a Taylor ...
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0answers
33 views

The change in time of a concentration in a fluid can be described by Reynolds' theorem. Is that the whole story?

Let $d\in\left\{2,3\right\}$ and $\Omega_t\subseteq\mathbb R^d$ be the bounded set occupied by a fluid at time $t\ge 0$. Moreover, let $\eta_t:\Omega_t\to[0,\infty)$ be the concentration of imaginary ...
0
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0answers
18 views

Fluid flow in porous medium with moving boundary

Everyone, I would like to numerically model axisymmetric fluid flow in a porous medium with a moving boundary, where the fluid pressure is specified on the boundary, whose radius increases with time. ...
2
votes
1answer
31 views

The validity of constitutive diffusive fluxes

In transport phenomena the diffusive fluxes for mass, energy and momentum are the constitutive laws: $$\boldsymbol{j}_c=-D\boldsymbol{\nabla}c \quad \boldsymbol{j}_T=-k\boldsymbol{\nabla}T \quad ...
3
votes
2answers
105 views

More intuition in understanding diffusion

This is a short conceptual question: (I think I'm confusing some very key concepts, here's an attempt to try and resolve it) Systems whose dynamics are governed by diffusion, for example a particle ...
0
votes
2answers
51 views

Why do we need Diffusion currents to explain semiconductor current flow?

Why do we need the idea of carrier concentrations to explain current flow? Can we simply not associate the disparity in carrier concentrations between two samples to a disparity in relative charge ...
0
votes
1answer
63 views

Speed of spontaneous mixing of different gases

Suppose we have a rectangular box divided into two equal cubic parts by a vertical impenetrable wall. Part 1 of the box contains a standard state mixture of $(1-x)$ mole of gas $A$ (e.g. Oxygen) and ...
3
votes
1answer
70 views

Heuristics behind Dirac delta function in Master equation for probability?

I'm reading this paper [Phys. Rev. Lett. 106, 160601 (2011)] and it studies simple diffusion where a particle stochastically resets to its initial position $x_0$ at a constant rate $r$. As you can ...
0
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0answers
32 views

Is it possible that my 750ml water bottle contains spent fuel from a space shuttle launch

Firstly I live in New Zealand which is very distant from USA which is where the space shuttle was originally launched from. So what is the possibility of molecules from the rocket engines spent ...
2
votes
1answer
61 views

alternative derivation of Einstein relation

Is there a derivation of the Einstein–Smoluchowski relation without the assumption of the Boltzmann distribution? Every time I see a derivation, it always assumes the Boltzmann distribution, such as ...
1
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0answers
16 views

Diffusion of magnetization (NMR)

In the context of MRI I'm trying to understand the mechanisms of magnetization diffusion. Consider a sample that is magnetized by an external magnetic field. We have displacement of magnetic moment ...
3
votes
1answer
119 views

Fick's first law inhomogeneous proof

I have seen Fick's first law of diffusion derived for a homogeneous material many times, however I am struggling to find a satisfactory proof for inhomogeneous, particularly for particle diffusion. ...
1
vote
2answers
97 views

diffusion velocity

i understand that diffusion is the movement of particles from high concentration areas to low concentration, but what is the cause of that movement atomically. And especially in the case of charge ...
0
votes
1answer
70 views

Brownian motion, net displacement, and diffusion - conceptual

I'm having trouble reconciling some conceptual issues of brownian motion. Let's say we have a box with two compartments separated by a membrane. Solute is at a high concentration on one side, and at ...
0
votes
1answer
124 views

Does diffusion current in semiconductor always exist?

According to what I understand, diffusion current is caused by the change in concentration of charge carriers in semiconductor (free electrons and holes) from higher concentration region to lower ...
0
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0answers
46 views

How long does it take for an Oxygen to diffuse around Earth?

We drill into the ground and find a huge pocket of oxygen. How long does it take for that oxygen to diffuse around Earth? I'd like to know for the first oxygen molecule how long it takes to effect the ...
0
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0answers
41 views

How can I get the boundary and initial conditions of the convection–diffusion equation consistant?

I want to solve the 1D convection–diffusion equation. The boundary conditions are a flux in from the bottom and a flux out on the top. Furthermore I want no concentration inside at the beginning. I ...
0
votes
2answers
160 views

Diffusion of gas into vaccum

I'm interested in solving the diffusion equation for gas in vacuum. I have a general question and a more specific questions. What I know: The Diffusion Equation: For density function ...
1
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0answers
74 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
0
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0answers
55 views

Confusion regarding interpretation of a diffusion related PDE?

So I was reading Keller & Segel paper related to mathematical model of cellular aggregation in slime molds. In the model (considering 2D assumption) we have an equation like below: ...
2
votes
2answers
137 views

Show that the boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$ [closed]

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by transformation techniques. Any help would be ...
4
votes
0answers
62 views

Stochastic process generating fractional diffusion

One way to generate Brownian motion is as follows: Define a waiting time probability distribution $\psi(t)$ and step length probability distribution $\lambda(x)$. Require also that $\langle \psi ...
1
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0answers
69 views

How do I apply a renormalization technique to estimate the fractal dimension of a diffusion limited aggregate?

Diffusion Limited Aggregation (DLA) is an interesting phenomena observed in nature and discussed here. From a theoretical view point, it'd be nice to know about the fractal dimension of a DLA formed ...
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3answers
506 views

How does the dissolution of salt affect the solution density?

Suppose you have a container of water as a solvent and you a certain amount of salt as a solute sitting at the bottom of the container that has yet to start dissolving. Supposing temperature and ...
0
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0answers
129 views

1D drift-diffusion equation with single absorbing boundary

If we have just the simple diffusion equation (in 1D): $$ \frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{\partial x^2} $$ with an absorbing boundary at x=0 and initial condition ...