- #1

- 1,847

- 89

## Main Question or Discussion Point

Hi PF!

I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Lets say I have some vector ##\vec{V} = v_x i + v_y j##. Then ##\vec{V} \otimes \vec{V} = (v_x i + v_y j)^2 = v_x v_x ii + v_x v_y ij+v_yv_xji+v_yv_yjj## (which is a 2 by 2 matrix). Now if I take $$\nabla \cdot (\vec{V} \otimes \vec{V}) = \partial_x (v_x v_x) (i\cdot i)i +\partial_x( v_x v_y)(i\cdot i)j+\partial_x (v_yv_x)(i\cdot j)i+\partial_x(v_yv_y)(i\cdot j)j+\\ \partial_y (v_x v_x) (j\cdot i)i +\partial_y( v_x v_y)(j\cdot i)j+\partial_y (v_yv_x)(j\cdot j)i+\partial_y(v_yv_y)(j\cdot j)j=\\

\partial_x (v_x v_x)i +\partial_x( v_x v_y)j+\partial_y (v_yv_x)i+\partial_y(v_yv_y)j$$ where I could then use the product rule, simplify, factor out ##\nabla \cdot \vec{V}## and set equal to zero by continuity (if ##\vec{V}## was velocity of an incompressible fluid). Do these operations look correct?

Again, I've never been showed this but it looks intuitive and feels correct.

I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Lets say I have some vector ##\vec{V} = v_x i + v_y j##. Then ##\vec{V} \otimes \vec{V} = (v_x i + v_y j)^2 = v_x v_x ii + v_x v_y ij+v_yv_xji+v_yv_yjj## (which is a 2 by 2 matrix). Now if I take $$\nabla \cdot (\vec{V} \otimes \vec{V}) = \partial_x (v_x v_x) (i\cdot i)i +\partial_x( v_x v_y)(i\cdot i)j+\partial_x (v_yv_x)(i\cdot j)i+\partial_x(v_yv_y)(i\cdot j)j+\\ \partial_y (v_x v_x) (j\cdot i)i +\partial_y( v_x v_y)(j\cdot i)j+\partial_y (v_yv_x)(j\cdot j)i+\partial_y(v_yv_y)(j\cdot j)j=\\

\partial_x (v_x v_x)i +\partial_x( v_x v_y)j+\partial_y (v_yv_x)i+\partial_y(v_yv_y)j$$ where I could then use the product rule, simplify, factor out ##\nabla \cdot \vec{V}## and set equal to zero by continuity (if ##\vec{V}## was velocity of an incompressible fluid). Do these operations look correct?

Again, I've never been showed this but it looks intuitive and feels correct.

Last edited: