I have read this question and the accompanying answers - Linear Vs Non Linear Differential Equation as well as the useful comments and my question is related. I am in fluid dynamics and I deal with the material derivative on a daily basis and I recently read that the advective term

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"Note that the advection of momentum $(V · \nabla)V$ is non-linear and this is the term that leads to much of the interesting behaviour in fluid mechanics."

i.e $$ \mathbf u.\nabla A $$

where u is the flow velocity and A is any vector field is non linear and as a result the whole material derivative is non linear due to the presence of a single non linear term.

What exactly is the reason for the non linearity in the advective term of the material derivative ?

Is it because of the product of a tensor ( $ \nabla A$) and a vector ?


No, the nonlinearity comes specifically from having the same vector field in both slots in the original advection term $V\cdot(\nabla V)$. Note, in particular, that $(cV)\cdot\nabla(cV) = c^2\,V\cdot(\nabla V)$ (and when you replace $V$ with $V+W$, you likewise get cross-terms in the resulting formula).

  • $\begingroup$ thanks so much for your answer. So from the definition of nonlinearity can I conclude that the homogeneity property is satisfied but not the additivity or superposition property ? Hence non linear. $\endgroup$ – gansub Dec 23 '16 at 0:33
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    $\begingroup$ No, homogeneity is violated too. Linearity should pull out $c^1$, not $c^2$. $\endgroup$ – Ted Shifrin Dec 23 '16 at 0:56
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    $\begingroup$ quick clarification on this answer If the advector and advectee are not the same then is the product u nabla A linear ? Your answer is clear enough that the nonlinearity is only when the same vector field is in both slots in the original advection term. $\endgroup$ – gansub Jun 1 '17 at 5:30

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