1. Water enters a cylindrical tank through two pipes at rates of 250 and 100 gal/min (see Fig.). If the level of the water in the tank remains constant, calculate the average velocity of the flow leaving the tank through an 8-in. inside-diameter pipe.
2. Two rivers merge to form a larger river as shown in Fig. At a location downstream from the junction (before the two streams completely merge), the nonuniform velocity profile is as shown and the depth is 6 ft. Determine the value of V.
3. Air at standard conditions enters the compressor shown in Fig. at a rate of 10 ft3/s. It leaves the tank through a 1.2-in.-diameter pipe with a density of 0.0035 slugs/ft3 and a uniform speed of 700 ft/s. (a) Determine the rate (slugs/s) at which the mass of air in the tank is increasing or decreasing. (b) Determine the average time rate of change of air density within the tank.
4. A hypodermic syringe (see Fig.) is used to apply a vaccine. If the plunger is moved forward at the steady rate of 20 mm/s and if vaccine leaks past the plunger at 0.1 of the volume flowrate out the needle opening, calculate the average velocity of the needle exit flow. The inside diameters of the syringe and the needle are 20 mm and 0.7 mm
5. A 10-mm diameter jet of water is deflected by a homogeneous rectangular block (15 mm by 200 mm by 100 mm) that weighs 6 N as shown in Video V5.6 and Fig. Determine the minimum volume flowrate needed to tip the block.
6. Determine the magnitude and direction of the anchoring force needed to hold the horizontal elbow and nozzle combination shown in Fig. in place. Atmospheric pressure is 100 kPa(abs). The gage pressure at section (1) is 100 kPa. At section (2), the water exits to the atmosphere.
7. A converging elbow (see Fig.) turns water through an angle of 135° in a vertical plane. The flow cross section diameter is 400 mm at the elbow inlet, section (1), and 200 mm at the elbow outlet, section (2). The elbow flow passage volume is 0.2 m3 between sections (1) and (2). The water volume flowrate is 0.4 m3/s and the elbow inlet and outlet pressures are 150 kPa and 90 kPa. The elbow mass is 12 kg. Calculate the horizontal (x direction) and vertical (z direction) anchoring forces required to hold the elbow in place.
8. A Pelton wheel vane directs a horizontal, circular cross-sectional jet of water symmetrically as indicated in Fig. and Video V5.6. The jet leaves the nozzle with a velocity of 100 ft/s. Determine the x direction component of anchoring force required to (a) hold the vane stationary, (b) confine the speed of the vane to a value of 10 ft/s to the right. The fluid speed magnitude remains constant along the vane surface.
A vane slider assembly moves under the influence of a liquid jet as shown below.
The coefficient of kinetic friction for motion of the slider along the surface is μ = 0.49. Calculate:
a) The acceleration of the slider (in m/sec2) at the instant when U = 7.90 m/sec.
b) The terminal speed of the slider (in m/sec).
9. A 100-ft-wide river with a flowrate of 2400 ft3/s flows over a rock pile as shown in Fig. Determine the direction of flow and the head loss associated with the flow across the rock pile.
10. A water siphon having a constant inside diameter of 3 in. is arranged as shown in Fig. If the friction loss between A and B is 0.8V2/2g, where V is the velocity of flow in the siphon, determine the flowrate involved.
11. For the 180° elbow and nozzle flow shown in Fig. determine the loss in available energy from section (1) to section (2). How much additional available energy is lost from section (2) to where the water comes to rest?
12. Water is pumped from the large tank shown in Fig. The head loss is known to be equal to 4V2/2g and the pump head is hp= 20 - 4Q2, where hp is in ft when Q is in ft3/s. Determine the flowrate.
13. Two water jets collide and form one homogeneous jet as shown in Fig. (a) Determine the speed, V, and direction, θ, of the combined jet. (b) Determine the loss for a fluid particle flowing from (1) to (3), from (2) to (3). Gravity is negligible.
14. For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by the equation
v = 3xy + x2y
Determine the velocity component in the x direction so that the volumetric dilatation rate is zero
15. The velocity components of an incompressible, two-dimensional velocity field are given by the equations
u = y2 - x (1 + x)
v = y (2x + 1)
Show that the flow is irrotational and satisfies conservation of mass.
16. Determine the stream function corresponding to the velocity potential
Φ = x3 - 3 x y2
17. It is known that the velocity distribution for two-dimensional flow of a viscous fluid between wide parallel plates (Fig.) is parabolic; that is,
with v = 0. Determine, if possible, the corresponding stream function and velocity potential.
18. Water flows over a flat surface at 4 ft/s, as shown in Fig. A pump draws off water through a narrow slit at a volume rate of 0.1 ft3/s per foot length of the slit. Assume that the fluid is incompressible and inviscid and can be represented by the combination of a uniform flow and a sink. Locate the stagnation point on the wall (point A) and determine the equation for the stagnation streamline. How far above the surface, H, must the fluid be so that it does not get sucked into the slit?
19. One end of a pond has a shoreline that resembles a half-body as shown in Fig. A vertical porous pipe is located near the end of the pond so that water can be pumped out. When water is pumped at the rate of 0.08 m3/s through a 3-m-long pipe, what will be the velocity at point A? Hint: Consider the flow inside a half-body.
20. A Rankine oval is formed by combining a source-sink pair, each having a strength of 36 ft2/s and separated by a distance of 12 ft along the x axis, with a uniform velocity of 10 ft/s (in the positive x direction). Determine the length and thickness of the oval.
21. An ideal fluid flows past an infinitely long, semicircular "hump" located along a plane boundary, as shown in Fig. Far from the hump the velocity field is uniform, and the pressure is p0. (a) Determine expressions for the maximum and minimum values of the pressure along the hump, and indicate where these points are located. Express your answer in terms of ρ, U, and p0. (b) If the solid surface is the ψ = 0 streamline, determine the equation of the streamline passing through the point θ = π/2, r = 2a.
22. Determine the shearing stress for an incompressible Newtonian fluid with a velocity distribution of
V = (3xy2 - 4x3)î + (12x2y - y3)j.
23. A layer of viscous liquid of constant thickness (no velocity perpendicular to plate) flows steadily down an infinite, inclined plane. Determine, by means of the Navier-Stokes equations, the relationship between the thickness of the layer and the discharge per unit width. The flow is laminar, and assume air resistance is negligible so that the shearing stress at the free surface is zero.
24. An infinitely long, solid, vertical cylinder of radius R is located in an infinite mass of an incompressible fluid. Start with the Navier-Stokes equation in the θ direction and derive an expression for the velocity distribution for the steady flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity ω. You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity.