Euler's Fluid Equations: Gradient of a Vector

[stormyweathers]

Hey guys,
I'm not sure how to interpret euler's fluid equations

$$\rho (\partial / \partial t + {\bf U} \cdot ∇) {\bf U} + ∇p = 0$$

I'm not sure what the meaning of $${\bf U} \cdot ∇ {\bf U}$$ is.
am I able to simply evaulate the dot product as $$U_{x}\partial_{x} + U_{y}\partial_{y}+ U_{z}\partial_{z}$$, and then use this to scale the vector U?

 

[tiny-tim]

hey stormyweathers!

'm not sure what the meaning of $${\bf U} \cdot ∇ {\bf U}$$ is.
am I able to simply evaulate the dot product as $$U_{x}\partial_{x} + U_{y}\partial_{y}+ U_{z}\partial_{z}$$, and then use this to scale the vector U?

(i'm not sure what you mean by "scale", but …)

yes, (U.∇)A = (U_x∂_x + U_y∂_y + U_z∂_z)A, for any vector A