The conservation of mass in a region enclosing flow

[miniradman]

Homework Statement

In the equation representing the conservation of mass enclosing flow:

$\int\rho \underline{u} \cdot \underline{\widetilde{n}} dA = 0$

where:
$\underline{u}=$ velocity vector
$\underline{\widetilde{n}} =$ unit vector normal to surface A

a) what the meaning of the term $\underline{u} \cdot \underline{\widetilde{n}}$
b) what is the value of $\underline{u} \cdot \underline{\widetilde{n}}$ on a solid surface

The Attempt at a Solution

I find it hard to visualise a graphical representation of what a dot product actually is, however I know that it has something to do with how much one vector acts upon another. Using this fact, would the dot product of the normal vector and velocity vector give the velocity of fluid hitting the surface dA? I'm unsure about this.

part B, I have no idea what specific value could be derived from an expression like this one.

 

[Chestermiller]

Using this fact, would the dot product of the normal vector and velocity vector give the velocity of fluid hitting the surface dA? I'm unsure about this.

It's the component of the velocity perpendicular to the differential element of surface area dA. This times dA tells you the volumetric flow rate through dA (since the component of u parallel to dA does not result in any flow through dA).

part B, I have no idea what specific value could be derived from an expression like this one.

What is the volumetric flow rate per unit area through a solid surface?

Chet