# Conservative form of the vector diffusion equation

For some reason I am unable to find a source on the internet about this. I think I have an answer, but I want to be doubly sure about this. All I could find (here), is that for an incompressible fluid, the viscous stress term is written in conservative form as: $$\nabla\cdot\left[\mu\left(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^{\top}\right)\right].$$ Knowing that this viscous stress acts as diffusion, I assumed that I could simply replace $$\boldsymbol{u}$$ by any vectorial quantity and $$\mu$$ by any diffusion coefficient. However, when working this out in spherical coordinates for the MHD induction equation, I find that I should get instead: $$\nabla\cdot\left[\eta\left(\nabla\boldsymbol{B}-(\nabla\boldsymbol{B})^{\top}\right)\right].$$ I'm confused. I can't seem to find any mistake I made which would explain the minus sign. Is one of them wrong or are they somehow both right?

At the same time, the advection term in the Navier-Stokes equation reads: $$-\nabla\cdot (\rho\boldsymbol{u}\otimes\boldsymbol{u}),$$ while in the MHD induction equation, we have instead: $$-\nabla\cdot (\boldsymbol{u}\otimes\boldsymbol{B}-\boldsymbol{B}\otimes\boldsymbol{u}).$$ I'm not sure I understand why they are both different, if they represent the same physics (advection), even though the flow field obviously advects itself.

• A few questions: How do you get anything conservative that includes diffusion? Navier-Stokes does not include magnetic fields, so should you not get different results (this all also depends upon the assumptions you used to construct these equations)? Jan 18, 2022 at 20:49
• 1. well, if you can write diffusion under a divergence, then you have it in conservative form. 2. I‘m talking about the MHD induction equation, but I compare the terms to the Navier-Stokes terms. Normally, if a physical phenomena is universal, it should be written the same way regardless of the quantity you are interested with (momentum, magnetic field). Although I suppose that since these quantities are vectorial, it complicates things. Jan 19, 2022 at 12:11
• I assumed that I could simply replace $u$ by any vectorial quantity... Why? ...if they represent the same physics... Do they? Jan 29, 2022 at 16:38