#$4.(15)$ Water flows in the annulus between a casing and a shaft. The cross$-$sectional area of the annulus is constant at A$.$ Rotating at Omega, the blade does work and increases the pressure of the flow $($P$\_(1)$ P$\_(2)).$ Calculate the required torque T to get Delta P$=$P$\_(2)-$P$\_(1).$ Note that only

thing rotating is the blade. For the simplicity, assume the following:

i$)1$D flow $($no radial and circumferential, only axial velocity exist$).$

ii$)$ The velocity does not change while passing through the blade.

iii$)$ Only the rotating blade does work on the flow.

iv$)$ The change in the temperature is negligible.

v$)$ Steady, incompressible, axisymmetric, and uniform flow.