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?

 

[Nate810]

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)

 

[Moetasim]

To derive the relation for shear stress on a conical bore fluid element, we need to consider the forces acting on the element. Similar to the force balance on a cylindrical fluid element, we can use the Navier-Stokes equation to account for the pressure forces and viscous shear forces acting on the element. However, in this case, we also need to consider the component of the shear force acting along the direction of the nozzle angle.

To do this, we can break down the shear force into its components using trigonometry. The component acting along the direction of the nozzle angle can then be added to the pressure forces in the force balance equation.

Additionally, we need to consider the change in area along the conical bore. This will affect the velocity and therefore the shear stress on the element. By taking into account the change in area and using the continuity equation, we can relate the velocity to the area and pressure.

In summary, to derive the relation for shear stress on a conical bore fluid element, we need to consider the forces acting on the element, including the component of the shear force along the nozzle angle direction and the change in area along the conical bore. By incorporating these factors into the force balance equation and using the continuity equation, we can obtain the desired relation for shear stress.