PROBLEM 6: As shown in fig a horizontal jet of water exits a nozzle with a uniform speed of , strikes a vane and is turned through an angle ‘θ ‘. Determine the anchoring force needed to hold the vane stationary if gravity and viscous effects are negligible.

SOLUTION:

We select a control volume that includes the vane and a portion of water as shown in fig (c) and apply the linear momentum equation in this fixed control volume. The only portions of the control volume through which water flows are section (1) and section (2). Hence the x and z components of linear momentum equations are

(as flow is steady)

And

(as flow is steady)

Or

……………………….. (1)

…………….……….. (2)

 

Where, and and are the net components of force in the x and z direction acting on the contents of the control volume. Depending on the particular flow situation being considered and the co-ordinate system chosen, the x and z components of velocity u and w, can be positive, negative or zero. In this example the flow is in the positive directions at both the inlet and the outlets.

The water enters and leaves the control volume as a free jet at atmospheric pressure. Hence there is atmospheric pressure surrounding the entire control volume, and the net pressure force on the control volume surface is zero. If we neglect the weight of the water and the vane, the only forces applied to the control volume contents are the horizontal and vertical components of the anchoring forces, and , respectively.

With negligible gravity and viscous effects and since , the speed of the fluid remains constant so that .Hence at section (1) and at section (2) .

By using this information equation (1) and (2) can be written as

…………………… (3)

……………..………. (4)

Equations (3) and (4) can be simplified by using conservation of mass, which states that for this incompressible fluid flow , or since . Thus

………………… (5)

……………..............................................…….. (6)

With the given data we obtain

And,