I was reading a book and he was measuring the rate of change of the volume of the fluid by the below integration formula :
$\frac { d }{ dt } volume(\Omega )=\iint _{ \partial \Omega } \overrightarrow { u } .\hat { n }$
It is the first time I see an integration like this ! Where is the differential operator for example (dt , dx ..etx) ?
What does it mean to integrate over the boundary $\partial \Omega$ ?
What does it mean to integrate the dot product of the velocity and the normal component $\hat {n}$ ?
I don't understand it at all so please help me !
I really believe $\vec{u}$ is understood to be the velocity vector. That was $\vec{u} \cdot \vec{n}$ is calculating the fluid moving in the direction normal to the surface. Think of the example where fluid flows at an angle $\theta$ relative to your surface $S$. If we allow $\vec{u}$ to be constant (i.e the velocity vector) then the black of water is a parallelepiped of volume $A \|\vec{u}_0\| \cos \theta$. If $\vec{n}$ is a vector normal to $S$ of length equal to the area of $A$, then we can write;
$$\textrm{Flow rate} = A \|\vec{u}_0\| \cos \theta = \vec{u_0} \cdot \vec{n}$$
In general the velocity field $\vec{u}$ is not constant and the surface $S$ may be curved. To compute the flow rate, we first parametrize $S$ using $G(u,v)$ and consider a small rectangle of size $\nabla u \times \nabla v$ that is mapped to a small patch $S_0$ of $S$. For any sample point $G(u_0,v_0)$ in $S_0$, the vector $\vec{n}(u_0,v_0) \nabla u \nabla v$ is a normal vector of length approximately equal to the area of $S_0$. The patch is nearly rectangular so we have;
$$\textrm{Flow rate through} \ S_0 \approx \vec{u}(u_0,v_0) \cdot \vec{n}(u_0,v_0) \nabla u \nabla v$$
Now if we sum up the flow over all the patches we have;
$$\\$$
$$\textrm{Flow rate through S} = \iint_S \vec{u} \cdot d\vec{S} = \iint_D \vec{u}(x,y) \cdot \vec{n}(x,y) \ dA$$