[Fluid Mechanics] Negative sign on the viscous work term

[WeiShan Ng]

Homework Statement

I am revising on the derivation of the differential equation of energy (White's Fluid Mechanics 7th ed) and I'm having trouble understanding the sign convention used in the viscous work term.

The textbook first define an elemental control volume and list out the inlet viscous work flux and outlet viscous work flux passing through the x faces, y faces and z faces. The rate of work done by viscous stresses equals the product of the stress component, its corresponding velocity component, and the area of the element face.

My confusions:
1. Why is it the viscous work rate has a negative sign on it:
$$w_x = -u\tau_{xx} - v\tau_{xy} - w\tau_{xz}$$
instead of :
$$w_x = u\tau_{xx} + v\tau_{xy} + w\tau_{xz}$$

2. And why do we subtract the outlet terms from the inlet terms instead of the opposite? Since we are finding the net work done by the system, I would think of doing the opposite...

Really appreciated if anyone could clarify my doubt. Thanks!

 

[Chestermiller]

The equation for the rate of doing work as you have written it (with the + sign) is the rate of work being done by the surroundings on the system within the control volume. The equation for the rate of doing work as they have written it (with the - sign) is the rate of work being done by the system in the control volume on its surroundings. The form of the open system version of the first law that they are using calls for the latter.