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How to make a variable of 3rd order derivatives?
Posted 14 avr. 2022, 09:52 UTC−4 Fluid & Heat, General, Parameters, Variables, & Functions 1 Reply
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Summary
I am trying to solve Navier-Stokes with a forcing term which is a function of the third derivatives of another problem. I am trying to find the best way to do that.
Full problem
I am considering the Navier-Stokes equations coupled with a non-linear PDE.
The forcing term 
is defined in terms of the variables 
and 
, and is given by
where
Issue
As the forcing term contains third order derivatives, I have to compute it seperately in its own PDE. I am trying to figure out the best way to do that. I have set up a general PDE which is of the form
where the matrix 
is in terms of the second order derivatives of and . But the issue that I am having is the boundary conditions. I have tried zero flux (but that doesn't work since does not contain ). Dirichlet boundary conditions also do not work because of the third order derivatives. I would like some advice on how to compute .
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I found the solution, it was to define the functions 
which were subject to Dirichlet conditions dependent on 
. These were a part of the 
PDE (which went from 2 dependent variables to 6).