So I have been trying to get my head around the derivation given by Frank White in his Fluid Mechanics book on pressure work (page 164,. He states that for mass going into the control volume, $$d\dot{W}_p=-(pdA)V_{n,in}=-p(-\mathbf{V\cdot n})dA.$$
I can't figure out where the minus comes from. If I try to look at the velocity going into the CV, I always get $d\dot{W}_p=dF(V_{in}cos\theta)$, where $\theta$ is the angle between the pressure force pushing into the CV ($dF$), and $V_{in}$ is the velocity of the flow going into the CV. Note that for fluid going in, the angle is always less than 90 degrees. In vector form, this then becomes equal to $d\dot{W}_p=\mathbf{dF\cdot V}$. If we take into account that $\mathbf{dF}=p(-\mathbf{dA})$ (because the pressure force is acting exactly opposite to the local area vector), then the work formula becomes $d\dot{W}_p=-p(\mathbf{V\cdot n})dA$ which is actually positive as $\mathbf{V \cdot{n}}<0$ when fluid goes into the CV.
I am therefore trying to figure out where he got the extra minus sign. Or is he using the fact that work done ON the system is negative according to the sign convention?
