CFD Online Discussion Forums - Help discretising diffusive terms

Hi,

I want to check that the method I'm attempting to implement is correct.

When evaluating the diffusive terms in the incompressible Navier-Stokes equations, (using the finite volume method on structured grids), for the U-momentum equation we have:

$\text{diffusive terms} = \frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial U}{\partial y}\right) + \underbrace{\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right)}$

(hopefully no mistakes so far!!)

Now, do I treat the underbraced terms explicitly as a source, or implicitly?

Assuming the former, do I compute the value of the underbraced terms at the cell center and then multiply by the cell volume?

i.e. is there anything wrong with the following:

$\frac{\partial}{\partial x} \left( \mu \frac{\partial U}{\partial x}\right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial V}{\partial x}\right) = \mu \frac{\partial}{\partial x} \left( \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} \right) + \frac{\partial U}{\partial x}\frac{\partial \mu}{\partial x} + \frac{\partial V}{\partial x}\frac{\partial \mu}{\partial y}$

...Can I simply evaluate the above at the cell center (taking the first expression on the RHS as zero due to the continuity condition) and multiply by the volume?

Thanks very much in advance,