Page 476 - Introduction to Continuum Mechanics

460 Problems

(a) Determine the linear momentum and rate of increase of linear momentum in this control
volume.
(b) Determine the outflux of linear momentum.
(c) Determine the net resultant force that is acting on the material contained in the control
volume.
7.13. Do Problem 7.12 for the same velocity field, with p = — and the cylindrical control
*i
volume bounded byjcj = 1 andjcj = 3.
7.14. Consider the flow field v = jcej - y*2 withp = constant. For a control volume defined
by jc = 0,jc = 2,)' = 0,)> = 2,z = 0,z = 2 determine the net resultant force and couple that
is acting on the material contained in this volume.

7.15. Do Problem 7.14 for the control volume defined by x = 2,y = 2,xy = 2 ,z — Q^z = 2.
7.16. For Hagen-Poiseuille flow in a pipe

calculate the momentum flux across a cross-section. For the same flow rate, if the velocity is
assumed to be uniform, what is the momentum flux across a cross section ? Compare the two
results.
7.17. A pile of chain on a table falls through a hole from the table under the action of gravity.
Derive the differential equation governing the hanging length x.
7.18. A water jet of 5 cm. diameter moves at 12 m/sec, impinges on a curved vane which deflects
it 60° from its direction. Neglect the weight. Obtain the force exerted by the liquid on the
vane.
7.19. A horizontal pipeline of 10 cm. diameter bends through 90°, and while bending, changes
its diameter to 5 cm. The pressure in the 10 cm. pipe is 140 kPa. Estimate the resultant force
on the bends when 0.05 m/sec of water is flowing in the pipeline.
7.20. Figure P7.1 shows a steady water jet of area A impinging onto the flat wall. Find the
force exerted on the wall. Neglect weight and viscosity of water.

721. Frequently in open channel flow, a high speed flow "jumps" to a low speed flow with an
abrupt rise in the water surface. This is known as a hydraulic jump. Referring to Fig.P7.2, if
the flow rate is Q per unit width, find the relation between yi and y% Assume the flow before
and after the jump is uniform and the pressure distribution is hydrostatic.

7.22. If the curved vane of Example. 7.6.2 moves with a velocity v<v 0 in the same direction as
the oncoming jet, find the resultant force exerted on the vane by the jet.
3
7.23. For the half-arm sprinkler shown in Fig.P7.3, find the angular speed if Q ~ 0.566 m /sec.
Neglect friction.