Convection–diffusion equation Puns

I currently am working with the reaction-convection-diffusion equation where I am modeling the concentration of a generic system that I will begin to make more and more complex. I am able to make quite a few simplifying assumptions to begin that reduce: RCD equation: https://imgur.com/a/DUey7

to

RC equation (no axial diffusion): https://imgur.com/a/5yBSk

My first problem in understanding this is why the source term is multiplied by a density (if that's what it is, the document doesn't specify what exactly rho (p) is, but I assume it is the density of products. If I'm not wrong, every group of terms in either equation must result in units of mol/(L*s), or more generally, concentration per time. If this is true, then rho must be dimensionless since R' is probably already in these units.

This problem carries on when I nondimensionalize the equation to: https://imgur.com/a/Uf4Iv

In this way, the first term on the right side cancels out (no axial diffusion still), but the source term is still multiplied by rho(p) * length/velocity, where length/velocity is equal to the residence time (Tau).

If it's not clear, my question is why is rho(p) in this equation, I've never seen it written like this. If it stands for density, then what are its units, as it seems from the above pictures that it doesn't have any units whatsoever.

Suppose you have a transient convection-diffusion equation with a source term. Do you model this system with the finite difference method or the finite volume method? What do you take into account when choosing between these two methods?

Edit: the problem is one-dimensional.

Using energy analysis on a control volume i have followed the development of the heat equation for convection in a metal rod. DT/dt =ad^2T/dx^2

I want to apply a similar analysis on a forced convection system where water of a known flow and temperature field flow over a heated plate or flow around a heated resistor

Can anyone suggest examples or resources that demonstrate this anysis

Thanks

Hi guys,

My question is simply what I have as the subject of this post. I'm in a CFD course, and my professor asked us to discuss the relevance of a set of simulations and how their grid peclet numbers relate to the simulation. I went on Wikipedia and did some research but I'm finding it hard to understand what this number truly means.

edit: I meant to say "upwinding"

Hi! i'm working on solving some equaion which looks like 1-D heat diffusion equation. I tryed to solve this equation in order to check if my solver works well, but I got spurious oscillations in my solution. My code uses a Crank-Nicholson scheme for time advancing and five point scheme for espatial derivation.

The equation i'm trying to solve

I set Neumann boundary condition dY/dt=0 and solved it on a 20 points grid deffined in, with 100 time step of 1/20 length. This was my result:

Solution time evolution in a central point.

Full solution mesh.

Actualization:

I modified the code to use a ' θ scheme' ( θ =0 explicit, θ =1 implicit, θ =.5 C-N) and continued proving. I noticed pair and odd time steps give me two very different solution sets (initial values at time step 1).

Pair time steps solution set is the weirdest one.

Pair time step solution set.

Second actualization:

I solved it, it appears to me related to operations like 'A\M' (where M is a matrix). I stoped using that operations, I instead calculated inv(M) (luckly it's needed to be calculated only once) and those oscilaions disappeared.

Thanks you all guys! ^^

Hi everyone! i'm asking for suggestions to do it. i'm trying to solve spherical diffusion equation with FDM. Lucky i'm only interested about radial direction.

https://preview.redd.it/5jh14lgd13o61.png?width=282&format=png&auto=webp&s=0b1b0a64813dc382d0bb68ff12952b1fddf74687

At first I tried by using Runge-Kutta IV, and a 5 points derivative scheme (I already wrote a code to get 1st and 2d derivates based on 5 points). It appears to work, but it runs too slow, so that I'm thinking about try to solve it using implicit methods, like Cranck-Nicholson, but I just realized I've neverd used implicit methods in spherical coordinates.

So, do you have any suggestion to me?

I've been on this problem for ages and it's due tomorrow but I can't see to get anywhere with it. This is the imgur link because I can't format very well

https://i.imgur.com/G5vJKzF.png?1

I'm really stressing over this so please don't be too harsh on me

I'm stuck on a homework problem. It says to find the temperature profile (as a function of y) for an infinitely large plate that at t=0 starts moving and changes its temperature from T_inf to T_s. I can do the temperature profile without the plate moving, but I don't know how to solve the temperature profile when the plate is moving. Can anyone point me in the right direction?

Hi everyone! I'm stuck with the math of this problem, can you please identify my error? Whenever I program this, the solution explodes in RK4. This is a problem of CFD or numerical methods.

Here are the equations I worked with. I have a vector φ with an initial solution (pointy gaussian). The dimension is 1x213.

And I want to compute the solution of the PDE of advection / convection equation 1st order using upwind with Forward Euler and RK4, however, RK4 seems to be bad programmed.
In the pictures you have my equations. Can you find the error?

Initial Steps

Intermediate Steps

Final steps

Comparison between Forward Euler Upwind with RK4 Upwind ; CFL=1

Note: Forward Euler upwind with CFL=1 gives the exact solution.
The code that I put in matlab is:

n= time step; j= space step; dt=delta time; CFL= adt/dx;
K_1(n,j) = -(CFL/dt)*(phi(n,j)-phi(n,j-1));

K_2(n,j) = phi(n,j) - 0.5*CFL* (phi(n,j)-phi(n,j-1));

K_3(n,j) = phi(n,j) - 0.5*CFL* (K_2(n,j)-K_2(n,j-1));

K_4(n,j) = phi(n,j) - CFL* ( K_3(n,j) - K_3(n,j-1) );

phi(n+1,j) = phi(n,j)+ (dt/6)*(K_1(n,j) +2*K_2(n,j) +2*K_3(n,j) + K_4(n,j));

A graphic is available here. Many thanks

I've been watching the lectures over and over and I asked my professor for help, but I'm still so lost. I can solve the IVPs with the Fourier Series, but I can't understand how to derive the formula for the eigenvalues and eigenfunctions with Dirichlet and Neumann, let alone for Robin since you have to do it graphically. Can anybody help?

I need help with the following problem:

Determine the temperature distribution u(x,t) for an infinite 1D rod using the diffusion equation if it's initial temperature is given by f(x);

where

f(x)={T_0 if x<0 and 0 if x⪖0

I know I start by saying that u(x,t)=X(x)T(t) and obtaining the expressions for X(x) and T(t), or at least that's what I think, but don't know where to continue from there.

I'd really appreciate any amount of help I could get. Thanks! ​ ​

Hello all, I am having trouble with a newton solver that I have been working on over the last week. It is in the context of modelling 2D random with proliferation walks via column averaging but that's beside the point. The PDE that describes this interaction is where D is the diffusion (migration) terma and lambda is the nonlinear (proliferation).

N_t = D * N_xx + lambda * N * (1 - N)

I have checked the equations used for the JAcobian and the f vector a dozen times to the notes in class so I'm 99% sure that's not the issue.

An implicit Euler method is used for those interested.

Newton's method fails to converge when proliferation is 'turned on' (pp > 0). The code works fine for pp = 0 (compared to the random walk data). Whenever I try pp = 0.01 i get the error('Newton failed to converge\n') message.

Any help would be greatly appreciated :D, code below

%% 1D Non-Linear Diffusion Equaltion Solver using Newton's Method
clear all
clc

% Set up variables
deltax = 0.1;
tau = 0.1; % Time step
xlims = 20; % X boundaries
xIC = 10; % Lattice Sites with the IC

pp = 0.01; % Probablity to proliferate
pm = 0.5;  % Probability for a migration


Newtol = 0.01; % newton tolerance
xarr = linspace(-xlims,xlims,2*xlims/deltax+1) + xlims; % x array

I = length(xarr); % number of lattice point
f = zeros(I,1); % the f vector used in newton iterations
J = zeros(I); % Jacobian for newton iterations
N = zeros(I,1); % Vector with averaged particle position
deltaN = N; % deltax vector for newton iterations
N(round(xlims/deltax+1)-xIC:round(xlims/deltax+1)+xIC) = 40; % Set up casona IC [-inf,0] = 1; (0,Inf] = 0


Nold = N; % used for newton iterations

curtol = Newtol + 1; % current tol, inital value must be larger than newtol
numsteps = 0; % newton steps
maxnewsteps = 10; % max newton steps
cond = 1; % convergence condition

D = pm * deltax^2/(4*tau); % diffusion constant
lam = pp / tau; % Nonlinear constant


% Set Tmax and the timestep array
Tmax = 100;

% for all the timesteps
for i = 1:(Tmax / tau)
    cond = 1; % reset convergence condition
    numsteps = 0; % reset newton step count

    % while newton has not converged
    while cond

        % Calculate the Jacobian and the f vector

        % Left Boundary
        J(1,1) = -1;
        J(1,2) = 1;
        f(1) = N(2) - N(1);

        % Right Boundary

I just watched a video on granular convection (the "hazelnut effect"), and it brought up the question of what the implications of it there are with respect to diffusion.

Why would granular material being agitated tend to clump objects based on size (I'm looking at a case where you have a tray of objects, as opposed to a column, so gravity isn't the cause), VS diffusion spreading everything evenly?

First time posting here but I’m completely stuck on figuring out how to either input this equation into a scientific calculator, or to solve it accurately.

I’m doing a chemistry course involving food and I’ve reached the topic of random walks and the diffusion equation.

The example in the course is measuring how long a mixture of a droplet of juice and alginate will take a develop a 1mm thick shell.

The formula given is:

D’s value is - 8*10^-10 m^2/sec

L^2 = 4Dt t = L^2 / 4D t = (10^-3 m)^2 / 4(D)

With the answer averaged at; t = 300 sec = 5 minutes

I’ve tried putting it into a scientific calculator and solving it step by step but I’m not getting the correct answer which should be averaged at 300 seconds.

Solutions to the diffusion equation with Dirichlet (fixed) boundary conditions

Is it part of the source function? I've had a hard time breaking down the components in the RTE to specifically look at the direct and diffuse components. If anyone knows a good source (that I can access) that would be extremely helpful.

One can corelate it with Fick’s first law of diffusion on corelating ‘chemical potential gradient’ with ‘concentration gradient’. Can anyone please explain with/without proper equations?

Suppose you have a transient convection-diffusion equation with a source term. Do you model this system with the finite difference method or the finite volume method? What do you take into account when choosing between these two methods?

Edit: the problem is one-dimensional.