Why is there a pressure loss with friction?
The greater the friction (and viscosity) the greater the resistance (opposition) to motion. As friction increases, the pressure drop must increase, and more work will have to be done on a fluid for a given flow rate.
The sign on the differentials depends on the coordinate system (reference) or orientation used, e.g. is the coordinate system (in a pipe flow) measured from the center of the flow outward, or from the pipe wall inward.
What does du/dx < 0 mean? Simply as the distance/position increases, then u is decreasing. If du/dx increases, then u is increasing, i.e. the fluid is accelerating with x.
dP/dx < 0 means a pressure drop as x increases, or dP/dx > 0 means pressure increases with x. Then one has to look at what causes the Pressure to increase or decrease.
