In the context of physics, I have the following equation, which is the Euler equation for a fluid:
$$\rho\bigg(\dfrac{\partial}{\partial t}+\vec{u}\cdot\nabla\bigg)\vec{u}=-\nabla P$$
where $\rho$ and $P$ is the density and pressure respectively (both scalars), and $\vec{u}$ is the velocity. I want to understand what this equation means, so I am trying to write the left hand side in terms of derivatives, instead of the nabla operator, since I believe this is three equations packed into one. However, I am having trouble understanding what $\vec{u}\cdot\nabla\vec{u}$ is. As far as I know, $\nabla\vec{v}$ is a scalar. So, is the following correct?
$$\bigg(\dfrac{\partial}{\partial t}+\vec{u}\cdot\nabla\bigg)\vec{u}=\dfrac{\partial\vec{u}}{\partial t}+\vec{u}\cdot\nabla\vec{u}=$$
$$=\underset{vector}{\underbrace{\bigg(\dfrac{\partial u_x}{\partial t},\dfrac{\partial u_y}{\partial t},\dfrac{\partial u_z}{\partial t}\bigg)}}+\underset{vector}{\underbrace{(u_x,u_y,u_z)}}\cdot\underset{scalar}{\underbrace{\bigg(\dfrac{\partial u_x}{\partial x}+\dfrac{\partial u_y}{\partial y}+\dfrac{\partial u_z}{\partial z}\bigg)}}$$
The dot used between the vector and the scalar leads me to believe that both should be vectors and that the dot represents a scalar product between vectors, but I don't understand how.
Best Answer
$$(\vec{u}\cdot\nabla)\vec{u}=\left(\sum_i u_i \vec{e}_i \cdot \sum_j \vec{e}_j \frac{\partial}{\partial x_j}\right) \sum_k u_k\vec{e}_k$$ Using $\vec{e}_i\cdot\vec{e}_j=\delta_{ij}$: $$(\vec{u}\cdot\nabla)\vec{u}=\left(\sum_i u_i \frac{\partial}{\partial x_i}\right) \sum_k u_k\vec{e}_k$$ Using $\frac{\partial \vec{e}_k}{\partial x_i}=0$: $$(\vec{u}\cdot\nabla)\vec{u}= \sum_k \vec{e}_k\left(\sum_i u_i \frac{\partial u_k}{\partial x_i}\right)$$ For example, $x$ component is $$\left((\vec{u}\cdot\nabla)\vec{u}\right)_x= \sum_i u_i \frac{\partial u_x}{\partial x_i}=u_x\frac{\partial u_x}{\partial x}+u_y\frac{\partial u_x}{\partial y}+u_z\frac{\partial u_x}{\partial z}$$