Flow through a Surface
The Reynold’s Transport Theorem requires that the fluid velocity normal to the control surface be known. This requires that the fluid velocity be broken into tangential and normal components.
The figure shows a control volume that encloses a region having a volume, R. The control surface is identified by the symbol, S. An arbitrary differential area, dS, has an outward unit normal, n. The fluid velocity on dS is designated by the vector, V.
The velocity vector can be broken into components that are normal and tangent to the surface. The component of the velocity in the direction normal to the surface, Vn, is obtained by taking the dot product of the velocity vector with the unit outward normal. The component of the velocity vector that is tangent to the surface, Vt, is obtained by subtracting the normal component, Vn, from the velocity vector, V. The tangential component of the velocity carries fluid past the surface element, but it doesn’t enter or exit. The velocity normal to the surface can be entering or leaving the surface.
The sign of the normal vector is very important. Flow is exiting through the surface if Vn is positive – going in the direction of the outward unit normal. Flow is entering the system if Vn is negative – going in the direction opposite the outward unit normal.
The net flow into or out of the control volume is obtained by integrating the normal component of the flow over the entire surface.
Demonstrate how to compute the velocity of fluid exiting or entering a control volume.