Deriving Shear Stress on a Conical Bore Fluid Element

[makethings]

I would like to derive a relation for shear stress on a conical bore shaped fluid element. Essentially, I have flow going through a converging nozzle. I know variables in this include length of the nozzle, inlet and outlet diameters and pressures and nozzle angle.

I've done a force balance on a cylindrical shaped fluid element. All I want to know is if I repeat the same procedure I used on that, how do I treat the shear forces that are acting on the angled part of the element? Is it simply taking the component acting along the same direction as the pressures?

 

[member 553083]

The shear force acting on a conical bore shaped fluid element is given by the equation: F = (pressure1 – pressure2)*A*tan(angle)where A is the area of the conical bore, pressure1 and pressure2 are the pressures on either side of the conical bore and angle is the angle of the conical bore. The shear stress acting on the conical bore is equal to the shear force divided by the area of the conical bore:τ = (pressure1 – pressure2)*tan(angle)