# Advection term for a matrix equation

How can I calculate a quantity like

$$(\vec{v} \cdot \nabla) M$$

where $$\vec{v}$$ is the velocity vector, and $$M$$ is some 3x3 matrix? (if one wants, assume $$M$$ is a tensor) This would be the advective term in a matrix equation that involves the material derivative. Further, how does one calculate this in cylindrical coordinates? An advective term like

$$(\vec{v} \cdot \nabla) \vec{c}$$

where $$\vec{c}$$ is a vector makes sense to me, calculating one specifically for a matrix is throwing me off.

• Related : Nabla commutation in electromagnetism. Commented Jun 15, 2022 at 21:53
• May be : $$\left[\left(\boldsymbol{\upsilon\cdot\nabla}\right)\mathbf M\vphantom{\dfrac{a}{b}}\right]_{ij}=\boldsymbol{\upsilon\cdot\nabla}\mathrm M_{ij} \tag{A}\label{A}$$ Commented Jun 15, 2022 at 22:08

I have figured out the correct answer for all interested:

If we represent $$M$$ in cylindrical coordinates as

$$M = M_{rr}(\hat{r} \otimes \hat{r}) + M_{r\phi}(\hat{r} \otimes \hat{\phi}) + M_{rz}(\hat{r} \otimes \hat{z}) + ...$$

where $$M_{rr} = M_{rr}(r,\phi,z)$$, and similar for $$M_{r\phi}, M_{rz},$$ etc., then one can use the relation

$$(\vec{v} \cdot \nabla)f(r,\phi,z) = v_{r}\frac{\partial f}{\partial r} + \frac{v_{\phi}}{r}\frac{\partial f}{\partial \phi} + v_{z}\frac{\partial f}{\partial z}$$

and

$$(\vec{v} \cdot \nabla)\vec{u} = [(\vec{v} \cdot \nabla)u_{r}-\frac{v_{\phi}u_{\phi}}{r}]\hat{r} + [(\vec{v} \cdot \nabla)u_{\phi}+\frac{v_{\phi}u_{r}}{r}]\hat{\phi} + [(\vec{v} \cdot \nabla)u_{z}]\hat{z}$$

where $$u_{r} = \vec{u}(r,\phi,z) \cdot \hat{r}$$ and the same for $$u_{\phi},u_{z}$$ to calculate directly

$$(\vec{v} \cdot \nabla)M = (\vec{v} \cdot \nabla)[M_{rr}(\hat{r} \otimes \hat{r}) + M_{r\phi}(\hat{r} \otimes \hat{\phi}) + ...] = ((\vec{v} \cdot \nabla)M_{rr})(\hat{r} \otimes \hat{r}) + M_{rr}((\vec{v} \cdot \nabla)(\hat{r} \otimes \hat{r})) + ...$$