Can we solve convection diffusion equation using pdetoolbox given a is constant vector?

Hi!

We would like to solve the following 2D convection diffusion equation. Would it be possible using vanilla MATLAB pdetoolbox?

where

or the flow is laminar in simpler words.

[Torsten]

Why v_x ? Shouldn't it be v_y ?

For r=R, velocity should be 0. This is not the case for your formula.

Are you sure diffusion in x-direction is relevant compared to convection in y-direction ?

In other words: Don't you think it suffices to solve

dc/dt = -v_y * dc/dy - f(c)

?

[Hashim]

Hopefully I can explain my system using the domain/geometry image above. So, for my system I have a flow incoming E4 and through E2. This flow is laminar and I want it to be explained via the equation given in my original post. Now at E3 I have a heterogenous reaction at the surface of an electrode. This involves the species in the flow getting oxidized at E3 via some kinetics which I denoted by .

In this case the velocity component in the x or axial direction in the direction of y/radial is important which is why the inclusion. This is perhaps because only the flux perpendicular to the E3 is crucial to our response. Now my understanding so far is that the only the gradient across the y or radial direction is important or so suggests the literature.

Okay I have now made a little correction to the original PDE as the diffusion into y is what is important as well the convective effect on the concentrartion profile of the species of interest in the x direction. Would this possible to do in the pde toolbox? or do I need external toolboxes which I do know that I will need if my flow is turbulent.

Answer 1

[SAI SRUJAN]

Hi Hashim,

I understand that you are trying to solve a non linear partial differential equation.

Typically, the Partial Differential Equations (PDEs) in MATLAB adhere to the following format,

c(x,t,u,du/dx).du/dt=(x^-m).d/dx[(x^-m)f(x,t,u,du/dx)]+s(x,t,u,du/dx)

Given a PDE, model the partial differential equation to make an analogy to the format specified above.To be more specific in resolving the issue, you can use the "pdepe" MATLAB function to further solve the issue by setting initial and boundary conditions.

For a comprehensive understanding of the "pdepe" function in MATLAB, please refer to the following documentation.

Solve System of PDEs - MATLAB & Simulink - MathWorks India

Solve 1-D parabolic and elliptic PDEs - MATLAB pdepe - MathWorks India

[Torsten]

The PDE above has 3, not 2 independent variables: t, x and y.

If the problem is not simplified (thus one dependency is neglected), it's not possible to solve it using "pdepe".