Hydrodynamics

1. Let’s review some basic properties of fluid flow.

Solution:

Steady flow is such a motion of a fluid (gas) in which each particle moves in the same direction and with the same velocity.

It holds:

a.) Volumetric flow rate: Q=S⋅vQ = S \cdot v

b.) Mass flow rate: Qm=S⋅v⋅ρQ_m = S \cdot v \cdot \rho

c.) Continuity equation: S1⋅v1=S2⋅v2S_1 \cdot v_1 = S_2 \cdot v_2

d.) Bernoulli’s equation:

p1+12ρv12=p2+12ρv22p_1 + \tfrac{1}{2} \rho v_1^2 = p_2 + \tfrac{1}{2} \rho v_2^2 p1=ρgh1– hydrostatic pressurep_1 = \rho g h_1 \quad \text{– hydrostatic pressure} p2=ρgh2– hydrostatic pressurep_2 = \rho g h_2 \quad \text{– hydrostatic pressure} p1′=12ρv12– hydrodynamic pressurep_1' = \tfrac{1}{2} \rho v_1^2 \quad \text{– hydrodynamic pressure} p2′=12ρv22– hydrodynamic pressurep_2' = \tfrac{1}{2} \rho v_2^2 \quad \text{– hydrodynamic pressure}

e.) Velocity of liquid outflow (Torricelli’s formula)

v=2ghv = \sqrt{2 g h} d=v2Hgd = v \sqrt{\tfrac{2H}{g}}

  • h – distance of the opening in the vessel from the liquid surface

  • H – distance of the opening in the vessel from the bottom

f.) Flow around a real body (Newton’s law)

Valid also for gases (air):

F=12C⋅S⋅ρv2F = \tfrac{1}{2} C \cdot S \cdot \rho v^2

  • CC – drag coefficient (Tables)

 

2.A pipeline with variable cross-section carries 5 liters of water per second. What is the velocity of flowing water at sections with areas

  • a.) S1 = 20 cm2
  • b.) S2 = 100 cm2

Solution:

Analysis:

Q = 5 l.s-1 = 5 dm3.s-1 = 0.005 m3.s-1, S1 = 20 cm2 = 0.002 m2, S2 = 0.01 m2

physics-hydrodynamics-2.gif 

The velocities of flowing water are v1 = 2.5 m.s-1 and v2 = 0.5 m.s-1.

3.A small hydroelectric plant uses the energy of water flowing into a turbine from a height of 4 m. At what volumetric flow will the turbine have a power output of 600 kW if its efficiency is 75%?

Solution:

Analysis:

h = 4 m, P = 600·103 W, η = 0.75, ρ = 1000 kg.m-3, g = 10 m.s-2

physics-hydrodynamics-3.gif

The turbine will have the required power at a volumetric flow Q = 20 m3.s-1

4.A pump delivers 300 l of water in 1 minute. The inlet pipe has a diameter of 80 mm, through the outlet pipe the water flows with a velocity of 8 m.s-1. Determine the velocity of water in the inlet pipe and the diameter of the outlet pipe.

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5.In the wider part of a tube, water flows with a velocity v1 = 10 cm.s-1. What is the velocity in its narrower part, which has half the radius?

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6.Diesel fuel (ρ = 830 kg.m-3) is transported through a pipe of diameter 40 cm at velocity 1.5 m.s-1. Determine

  • a.) the hydrodynamic pressure in the pipe
  • b.) the mass of diesel transported in 1 hour
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7.In a horizontal pipe with diameter d1 = 5 cm, water flows at velocity v1 = 2 m.s-1 and pressure p1 = 2·105 Pa. What is the pressure in the narrower part of the pipe with diameter d2 = 2 cm?

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8.In a water pipe, water flows with velocity 2.24 m.s-1 and has a pressure of 0.1 MPa. What is the velocity in the constricted part where the pressure is 0.09 MPa?

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9.In the wall of a container filled with water, there is a hole 45 cm below the free surface. The container stands at the edge of a table so that the hole is 80 cm above the floor. At what distance from the table’s edge will the water jet hit the floor if the hole is opened?

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10.A cylindrical container is filled with 18.84 liters of water per minute. At the same time, water flows out through a hole at the bottom with diameter 1 cm. At what height will the water level stabilize assuming ideal outflow of the liquid?

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11. What drag hydrodynamic force acts on a sphere with radius r = 2.5 cm, if water flows around it with velocity 1.8 m.s-1. C = 0.48

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12. With what velocity does a raindrop fall if its mass is 0.005 g, radius 2.26 mm. ρ (air) = 1.3 kg.m-3, C = 0.4

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13.A submarine is subjected to a drag hydrodynamic force of 3600 N. The submarine has a frontal cross-section of 15 m2 and moves with velocity 14.4 km.h-1. Determine the drag coefficient C!

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14.Calculate the terminal velocity of a parachutist with total mass 80 kg, if a hemispherical parachute has radius 5 m, drag coefficient 1.33, and air density 1.3 kg.m-3

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15.

What is the radius of a glass bead (ρS = 2500 kg.m-3), if it falls in water (ρ = 1000 kg.m-3) with constant velocity v = 2 m.s-1. C = 0.48

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