What is Diffusion equation: Definition and 120 Discussions
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of convection–diffusion equation, when bulk velocity is zero.
Suppose there is a non-stationary diffusion process in 2D rectangular plane. Component diffuses from the outside through all four faces of the plane.
When I think about the simulations of the non-stationary diffusion in Matlab for example (finite difference numerical solution), I remember how...
I have a discussion with a colleague of mine.
We have a thin cuboid sample whose two dimensions are similar to each other and are both much bigger than the sample thickness. I'm doing an experiment in which the diffusion of some species is induced and its diffusion profile is measured in one of...
For a novice research problem, I am approximating a system as a spherical reactor of homogenized natural uranium and heavy water, reflected by infinite graphite. I was attempting to find the critical mass and dimensions for it (very similarly to Lamarsh 3e Ex.6.7-8). To do so, I need to...
Hello,
I want to model the thermal behaviour of a moving heat transfer fluid in 1D, with convective exchanges with the walls. I have obtained the following equation (1 on the figure). I have performed a second order spatial discretization with decentred schemes at the extremities (y = 0 and H)...
U=U(x,t)
Ut=DUxx; 0<=x<=L, t>0
U(x,0)=0 0<x<=L
U(0,t)=a(t); t>0 *a(t) is known function*
(dU/dx)=0 for x=L
I have looked into many ways but not one is usable for diffusion equation with this boundary conditions.
I've tried to show b) by using the sine Fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)##
Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx##
These are equal therefore it is an orthonomal basis.
I'm not sure if this is correct so...
Hi everyone,
I am trying to solve the 1 dimensional diffusion equation over an interval of 0 < x < L subject to the boundary conditions that C = kt at x = 0 and C = 0 at x = L. k is a constant. The diffusion equation is
\frac{dC}{dt}=D\frac{d^2C}{dx^2}
I am using the Laplace transform method...
Hi,
I understand the underlying concept of changing variables in PDEs (so that we can reduce it to a simpler form), however, I am just not completely clear on the mathematics of it so I have a quick question about it.
For example, if we have the transmission line equation \frac{\partial...
Good morning,
I'm trying to workout the time for an element to diffuse at set distance in microns.
I have the distance, the diffusion coefficient, just unsure which equation I actually use.
X= sqrt DT or the other one x= sqrt 2DT.
I can't seem to figure out when you use one and not the...
So the normal diffusion equation looks like
\frac{\partial c}{\partial t} = k\frac{\partial}{\partial x}\left(\frac{\partial c}{\partial x}\right)
I know how to get explicit and implicit solutions to this equation using finite differences. However, I am trying to do the same for an equation of...
I'm trying to solve the diffusion equation in spherical co-ordinates with spherical symmetry. I have included the PDE in question and the scheme I'm using and although it works, it diverges which I don't understand as Crank-Nicholson should be unconditionally stable for the diffusion. The code...
Hello everyone,
I wish to know if someone could help me with the adjoint multigroup diffusion equation. In particular with the terms that make up the macroscopic removal cross section. Below, both the multigroup diffusion equation and its adjoint are shown, but I'm not sure about the latter. I...
Both the heat equation and the diffusion equation describe processes which are irreversible, because the equations have an odd time derivative. But how can these equations describe the real world when we know that all processes in nature are reversible, information is always conserved? But these...
Homework Statement
Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique.
Homework Equations
Starting equation integrated over t:
int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1)
Result...
Dear all,
I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modeled in two regimes depending on the heat flux. For low heat fluxes ##\dot{q}##, the temperature gradient...
I have a working Matlab code solving the 1D convection-diffusion equation to model sensible stratified storage tank by use of Crank-Nicolson scheme (without εeff in the below equation).
As indicated by Zurigat et al; there is an additional mixing effect having a hyperbolic decaying form...
Well, I was solving the 3D diffusion equation:
##\displaystyle \frac{1}{c}\frac{\partial \phi(\mathbf{r},t)}{\partial t}-D\nabla^2 \phi(\mathbf{r},t)=q(\mathbf{r},t)##.
I wrote a program to do this. The problem concerns the diffusion of light. However, all this time I've been working with...
So I want to write a short code to solve the diffusion equation and I want to be lazy and use the gradient function for the spatial differences, so for the second order derivative:
\frac{f(i+1)-2*f(i)+f(i-1)}{h^{2}}=\textrm{gradient}(\textrm{gradient}(f,h),h)
So the code I wold use is...
Hello everybody!
For my water in nanoscaled-pores simulations with SPH I need a value for the characteristic velocity.
My planned approach is to estimate this value by attaining the propagation speed of a diffusion wave.
But I have problems with understanding this process since I find some...
Homework Statement
Gas with thermal conductivity κ fills the space between two coaxial cylinders
(inner cylinder radius a, outer cylinder inner radius b). A current I is passed through
the inner cylinder, which has resistivity ρ. Derive an expression for the equilibrium temperature of the inner...
Homework Statement
I have to calculate the stationary field inside a room.
Homework EquationsThe Attempt at a Solution
I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Eeknx+Fe-knx)cos(kny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary...
I'm tinkering with the Convection Diffusion Equation (a second order differential equation) to model a temperature behavior in proximity to a heat source in a water bath. Just to get going I solved the system for some arbitrarily chosen boundary conditions. The result is that the temperature at...
The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time.
xsize = 10; % Model size, m
xnum = 10; % Number of nodes
xstp =...
Homework Statement
I am trying to understand the derivation of the diffusion equation from the Master equation for a 1D chain. We have an endless 1D discrete chain. State from ##n## can jump to ##n+1## and ##n-1## with equal probabilities. The distance between chain links is ##a##.
Homework...
Hi there. I was trying to solve the time dependent diffusion equation in only one dimension. I derived a explicit scheme using a finite difference in the time variable. The equation I am trying to solve is:
##\displaystyle \frac{1}{c} \frac{\partial \phi(x,t)}{\partial t} -...
Homework Statement
We let a dye diffuses into an environment of dimension L. We inject that dye into a box by one face, at t = 0 on x = 0. The linear density c follows that equation :
with the conditions :
Homework Equations / questions[/B]
a. nondimensionalize the equations and the...
I have been preforming experiments to study the diffusion of Hydrogen through Molybdenum. According to Sievert's law diatomic molecules would diffuse as atoms. But according to my experiments I notice that the flux of hydrogen is directly proportional to the pressure of hydrogen and not to the...
Hi,
I'm a second year undergrad and we've covered the heat equation,
\begin{equation}
∇^{2}\Psi = \frac{1}{c^{2}}\frac{\partial^2 \Psi}{\partial t^2}
\end{equation}
and the wave equation,
\begin{equation}
D∇^{2}u= \frac{\partial u}{\partial t}
\end{equation}
in our differential equations...
I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem.
A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...
Homework Statement
$$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$
$$U(0,t)=U_0 \quad for \quad t>0$$
$$U(y,0)=0 \quad for \quad y>0$$
$$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$
Homework Equations
This is a diffusion problem on...
Homework Statement
[/B]
We are heating a semi-infinite slab with a laser (radius of a stream is ##a##), which presents us with a steady surface heating (at ##z=0##), everywhere else on the surface the slab is isolated.
How does the temperature change with time?
Look at the limit cases: at ##t...
Hi
In diffusion equation ,if we have a infinite slab of moderator with thickness ±a and the sources is
s(x)= cos(x)
i think the first boundary
flux( ±a ) = 0
what will be the second boundary condition ??
Consider I have a packed column of length L filled with known characteristic adsorbent. I am putting a mixture of N components in it and I am solving for concentration of each component in mobile phase at the outlet of the column. The equations which are to be generalised are as follows: An...
Disclaimer: This is a homework problem
I need to analytically solve the diffusion equation for a 1d 1 group slab with width a, and source distribution Se^(-k(x+a/2))
I've gone through the math, and come up with my homogeneous and particular solution and attempted to apply the boundary...
Homework Statement
Hello, I don't understand the solution of an exercise
Let P(x) be a continuous function such that |P(x)|≤Ceax² .
Show that formula (8) for the solution of the diffusion equation makes sense for 0 < t < 1/(4ak), but not necessarily for larger t.
Homework Equations
Equation...
Hi guys,
I have functioning MATLAB code for my solution of the 3D Diffusion equation (using a 3D Fourier transform and Crank-Nicolsen) that runs just from the command window and automatically plots the results. However, it seems like my solution just decays to zero regardless of what initial...
Hello. I have some questions regarding the equation:
k\frac{\partial}{\partial x}\left( u\frac{\partial (u-r(x,t))}{\partial x} \right) = \kappa \frac{\partial u}{\partial t}.
u is positive. r(x,t) is given as an input.
I have implemented this non-linear diffusion equation using backward Euler...
Homework Statement
Exercise: Hexachlorobenzene (C6Cl6) is a highly toxic waste product of pesticide manufacturing. It is resistant to biodegradation. Sediments at the bottom of a reservoir in the Upper Mississippi River catchment have been found to contain high C6Cl6 concentrations. The...
This is my attempt at the solution. I have been told that the given function is a solution. I just need to prove it. As you can see that I am stuck. What am I doing wrong?
Hello,
I am trying to solve the following simple diffusion equation using the method of lines:
du/dt=D(du2/dx2)function k=func1()
k = [0, 0, 0, 0, 0, 0, 0.4380, 0, 1e5]; %k(9) initial value of the molecules
end %k(7) D/dx2
function...
Homework Statement
Find the solution for:
({\partial{}_t}^2 -D \Delta^2)G(\vec{r},t;\vec{r}_o,t_o)=\delta(\vec{r}-\vec{r}_o)\delta(t-t_o)
In two dimensions.
Homework EquationsThe Attempt at a Solution
Am I supposed to use bessel eqs? I'm kind of stuck in starting the problem :L
Hi,
I want to solve the following diffusion equation:
(d/dt) C(r,z,t)=D*∇^2 C(r,z,t)
where C is the concentartion and D is the coefficient of diffusivity (constant)
with initial condition C(r,z,0)=C0 (constant)
and boundary condition dc/dr=0 at r=0; (dc/dz) at z=-L equal to (dc/dz)...
Homework Statement
Plot the transient conduction of a material with k = 210 w/m K, Cp = 350 J/kg K, ρ = 6530 kg/m3
Where the material is a cylinder, with constant cross sectional area and is well insulated. The boundary conditions for the cylinder:
T(0,t) = 330K
T(l,t) = 299K...
Homework Statement
A uniform rod of length l has an initial (at time t = 0) temperature distribution given by u(x, 0) = sin(\frac{πx}{l}), 0 \leq x \leq l.
The temperature u(x, t) satisfies the classical one-dimensional diffusion equation, ut = kuxx
The ends of the rod are...
I have a general question about the solution to the Diffusion equation using the explicit finite difference method. Now, it is known the solution is stable when D*dt/dx^2 is less than 0.5, based on the choice of time and space steps. However, how does the choice of the time and space steps...
I'm trying to implement a numerical code for the diffusion equation with moving boundaries. I have no problems with the numerical implementation, but with the transformation of coordinates. In spherical coordinates, the diffusion equation is
\frac{\partial c}{\partial t} = D...
Hello... My main problem is to find the Green's function of this pde:
\frac{∂T(x,t)}{∂t}=\frac{∂}{∂x}(D(x)\frac{∂T(x,t)}{∂x}) .
This is the well-known diffusion equation, but here, the diffusion coefficient is coordinate-dependent.
My main focus is to find the Green's function, but i know...
"Consider a solution of the diffusion equation ##u_{t} = u_{xx}## in {0 ≤ x ≤ L, 0 ≤ t ≤ ∞}.
a) Let M(T) = the maximum of u(t,x) in the closed rectangle {0 ≤ x ≤ L, 0 ≤ t ≤ T}. Does M(T) increase or decrease as a function of T?
b) Let m(T) = the minimum of u(t,x) in the closed rectangle {0 ≤ x ≤...