Figure P4.67 illustrates a system and fixed control volume attime t and the system at a

Obtain a photograph/image of a situation in which a fluid is flowing. Print this photo and draw a control volume through which the fluid flows. Write a brief paragraph that describes how the fluid flows into and out of this control volume.

In the region just downstream of a sluice gate, the water may develop a reverse flow region as is indicated in Fig. P4.59 and Video V10.9. The velocity profile is assumed to consist of two uniform regions, one with velocity Va = 10 fps and the other with Vb = 3 fps. Determine the net flowrate of water across the portion of the control surface at section (2) if the channel is 20 ft wide.

In the region just downstream of a sluice gate, the water may develop a reverse flow region as is indicated in Fig. P4.59 and Video V10.9. The velocity profile is assumed to consist of two uniform regions, one with velocity Va = 10 fps and the other with Vb = 3 fps. Determine the net flowrate of water across the portion of the control surface at section (2) if the channel is 20 ft wide.

From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration: d dt x2(t) x1(t) f (x, t)dx = x2 x1 f t dx + f (x2, t) dx2 dt f (x1, t) dx1 dt Discuss how this formula is related to the time derivative of the total amount of a property in a system and to the Reynolds transport theorem.

Air enters an elbow with a uniform speed of 10 m/s as shown in Fig. P4.62. At the exit of the elbow, the velocity profile is not uniform. In fact, there is a region of separation or reverse flow. The fixed control volume ABCD coincides with the system at time t = 0. Make a sketch to indicate (a) the system at time t = 0.01 s and (b) the fluid that has entered and exited the control volume in that time period.

A layer of oil flows down a vertical plate as shown in Fig. P4.63 with a velocity of V = (V0h2 ) (2hx x2 )j where V0 and h are constants. (a) Show that the fluid sticks to the plate and that the shear stress at the edge of the layer (x = h) is zero. (b) Determine the flowrate across surface AB. Assume the width of the plate is b. (Note: The velocity profile for laminar flow in a pipe has a similar shape. See Video V6.13.)

Figure P4.64 shows a fixed control volume. It has a volume V0 = 1.0 ft3 , a flow area A = 1.0 ft2 , and a length 0 = 1.0 ft. Position x represents the center of the control volume where the fluid velocity V0 = 1.0 ft/s and the density 0 = 1.800 slug/ft3 . Also, at position x the fluid density does not change locally with time but decreases in the axial direction at the linear rate of 0.25 slug/ft4 . Use the system or Lagrangian approach to evaluate dp/dt. Compare this result with that of the material derivative and flux terms.

Find DV/Dt for the system in Problem 4.64.

Water enters a 5-ft-wide, 1-ft-deep channel as shown in Fig. P4.66. Across the inlet the water velocity is 6 ft/s in the center portion of the channel and 1 ft/s in the remainder of it. Farther downstream the water flows at a uniform 2 ft/s velocity across the entire channel. The fixed control volume ABCD coincides with the system at time t = 0. Make a sketch to indicate (a) the system at time t = 0.5 s and (b) the fluid that has entered and exited the control volume in that time period.

Figure P4.67 illustrates a system and fixed control volume at time t and the system at a short time t later. The system temperature is T = 100 F at time t and T = 103 F at time t + t, where t = 0.1 s. The system mass, m, is 2.0 slugs, and 10 percent of it moves out of the control volume in t = 0.1 s. The energy per unit mass u is cvT, where cv = 32.0 Btu/slug F. The energy U of the system at any time t is mu. Use the system or Lagrangian approach to evaluate DU/Dt of the system. Compare this result with the DU/Dt evaluated with that of the material time derivative and flux terms.

The wind blows across a field with an approximate velocity profile as shown in Fig. P4.68. Use Eq. 4.16 with the parameter b equal to the velocity to determine the momentum flowrate across the vertical surface AB, which is of unit depth into the paper.

Water flows from a nozzle with a speed of V = 10 m/s and is collected in a container that moves toward the nozzle with a speed of Vcv = 2 m/s as shown in Fig. P4.69. The moving control surface consists of the inner surface of the container. The system consists of the water in the container at time t = 0 and the water between the nozzle and the tank in the constant diameter stream at t = 0. At time t = 0.1 s what volume of the system remains outside of the control volume? How much water has entered the control volume during this time period? Repeat the problem for t = 0.3 s.