Divergence with Chain Rule

  • #1
16
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I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see

[tex] \frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2 [/tex]

K is a constant and T is a scalar field. It seems obvious that there is some way to use the chain rule on the middle term to get the left and right terms, but I frankly don't exactly understand the "rules" of how to use it with the divergence. I know you can't just factor out [tex] \frac{1}{T} [/tex] from the middle term, but I'm not sure how to actually simplify that middle expression. Any help is appreciated.
 

Answers and Replies

  • #2
Infrared
Science Advisor
Gold Member
773
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I think the relevant product rule is ##\nabla\cdot (fV)=(\nabla f)\cdot V+f (\nabla\cdot V)##. Try applying this with ##f=1/T## and ##V=\nabla T##.

You will also want to use the chain rule for ##\nabla (1/T)=-\frac{\nabla T}{T^2}##.
 

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