# How to compute this weird derivative?

#### bearcat

How to compute this simple but weird derivative?

Hello,

I am working on numerical solution of incompressible fluid dynamics problem. I need to compute the Jacobian of a viscous flux. I have to deduct some derivatives like this

$$\displaystyle \frac{\partial (\frac{\partial U}{\partial X})}{\partial U}=?$$

U, X are independent to each other.

Some people say the result is absolutely not zero, but I still could not find an analytical answer. I wonder this will turn out into a derivative operator rather than an expression of fixed value. If an analytical expression is not possible, how to handle this in a finite difference method?

Thank you and have a good day!

Bearcat

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#### Anonymous1

Hello,

I am working on numerical solution of incompressible fluid dynamics problem. I need to compute the Jacobian of a viscous flux. I have to deduct some derivatives like this

$$\displaystyle \frac{\partial (\frac{\partial U}{\partial x})}{\partial U}=?$$

Some people say the result is absolutely not zero, but I still could not find an analytical answer. I wonder this will turn out into a derivative operator rather than an expression of fixed value. If an analytical expression is not possible, how to handle this in a finite difference method?

Thank you and have a good day!

Bearcat
I've seen something like this before...

What you want to do is:

(1)Take the partial of U w.r.t. x. Denote the result by L.

(2)Transform L so that it is in terms of U. (This is the hard part)

(3)Take the partial of Transform(L) w.r.t. U.

Can you post what U is?

#### bearcat

In my case, U is the unknown velocity in N-S equation, x is the coordinate. they are independent to each other. What I am looking for is an analytical expression for my coding. I am sure it's solvable in some way as I am not the first to encounter this problem.

#### bearcat

I suspect it's possibly $$\displaystyle \frac{1}{\delta _{x}}$$.

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