Fluid dynamics – problems and solutions

Fluid dynamics – problems and solutions

Torricelli’s theorem

1. A container filled with water and there is a hole, as shown in the figure below. If acceleration due to gravity is 10 ms^-2, what is the speed of water through that hole?

Known :

Height (h) = 85 cm – 40 cm = 45 cm = 0.45 meters

Acceleration due to gravity (g) = 10 m/s²

Wanted : The speed of water (v)

Solution :

Torricelli’s theorem states that the water leaves the hole with the same speed as an object free fall from the same height. Height (h) = 85 cm – 40 cm = 45 cm = 0.45 meters

Velocity of water is calculated using the equation of the free fall motion :

v_t² = 2 g h

v_t² = 2 g h = 2(10)(0.45) = 9

v_t = √9 = 3 m/s

2. A container filled with water and there is a hole, as shown in figure below. If acceleration due to gravity is 10 ms^-2, what is the speed of water through that hole.

Known :

Height (h) = 1.5 m – 0.25 m = 1.25 meters

Acceleration due to gravity (g) = 10 m/s²

Wanted : The speed of water (v)

Solution :

v_t² = 2 g h = 2(10)(1.25) = 25

v_t = √25 = 5 m/s

3. A container filled with water and there is a hole, as shown in figure below. If acceleration due to gravity is 10 ms^-2, what is the speed of water through that hole.

Known :

Height (h) = 1 m – 0.20 m = 0.8 meter

Acceleration due to gravity (g) = 10 m/s²

Wanted : The speed of water (v)

Solution :

v_t² = 2 g h = 2(10)(0.8) = 16

v_t = √16 = 4 m/s

4. A container filled with water and there is a hole, as shown in figure below. If acceleration due to gravity is 10 ms^-2, what is the speed of water through that hole.

Known :

Height (h) = 20 cm = 0.2 meters

Acceleration due to gravity (g) = 10 m/s²

Wanted: The speed of water (v)

Solution :

5. A container filled with water and there are two holes, as shown in the figure below. What is the ratio of x₁ to x₂?

Solution

Time interval of the water free fall from hole 1 :

h = 1/2 a t²

0.8 = 1/2 (10) t²

0.8 = 5 t²

t² = 0.8 / 5 = 0.16

t = 0.4 seconds

Time interval of the water free fall from hole 2 :

h = 1/2 a t²

0.5 = 1/2 (10) t²

0.5 = 5 t²

t² = 0.5 / 5 = 0.1

t = √0.1 second

The horizontal distance (x) :

x₁ = v₁ t₁ = (2)(0.4) = 0.8 meters

x₂ = v₂ t₂ = (√10)(√0.1) = (10)(0.1) = 1 meter

The ration of x₁ to x₂ :

x_{1 :} x₂ = 0.8 : 1 = 8 : 10 = 4 : 5

The equation of continuity

6. Water flows through a pipe of varying diameter, A to B and then to C. The ratio of A to C is 8 : 3. If the speed of water in pipe A is v, what is the speed of water in pipe C.

Known :

Area of A (A_A) = 8

Area of C (A_C) = 3

The speed of water in pipe A (v_A) = v

Wanted: The speed of water in pipe C (v_C)

Solution :

The equation of continuity :

A_A v_A = A_C v_C

8 v = 3 v_C

v_C = 8/3 v

7. If the speed of water in pipe with a diameter of 12 cm is 10 cm/s, what is the speed of water in a pipe with a diameter of 8 cm?

Known :

Diameter 1 (d₁) = 12 cm, radius 1 (r₁) = 6 cm

Diameter 2 (d₂) = 8 cm, radius 2 (r₂) = 4 cm

The speed of water 1 (v₁) = 10 cm/s

Wanted : The speed of water 2 (v₂)

Solution :

Area 1 (A₁) = π r² = π 6² = 36π cm²

Area 2 (A₂) = π r² = π 4² = 16π cm²

The equation of continuity :

A₁ v₁ = A₂ v₂

(36π)(10) = (16π) v₂

(36)(10) = (16) v₂

360 = (16) v₂

v₂ = 360/16

v₂ = 22.5 cm/s

8. Water flows through a pipe of varying diameter, as shown in figure below. If area 1 (A₁) = 8 cm², A₂ = 2 cm² and the speed of water in pipe 2 = v₂ = 2 m/s then what is the speed of water in pipe 1 = v₁.

Known :

Area 1 (A₁) = 8 cm²

Area 2 (A₂) = 2 cm²

Speed of water in pipe 2 (v₂) = 2 m/s

Wanted : the speed of water in pipe 1 (v₁)

Solution :

The equation of continuity :

A₁ v₁ = A₂ v₂

8 v₁ = (2)(2)

8 v₁ = 4

v₁ = 4 / 8 = 0.5 m/s

9. If the diameter of the larger pipe is 2 times the diameter of smaller pipe, what is the speed of fluid at the smaller pipe.

Known :

Diameter of the larger pipe (d₁) = 2

Radius of the larger pipe (r₁) = ½ d₁ = ½ (2) = 1

Area of the larger pipe (A₁) = π r₁² = π (1)² = π (1) = π

Diameter of the smaller pipe (d₂) = 1

Radius of the smaller pipe (r₂) = ½ d₂ = ½ (1) = ½

Area of the smaller pipe (A₂) = π r₂² = π (1/2)² = π (1/4) = ¼ π

The speed of fluid at the larger pipe (v₁) = 4 m/s

Wanted : The speed of fluid at the smaller pipe (v₂)

Solution :

The equation of continuity :

A₁ v₁ = A₂ v₂

π 4 = ¼ π (v₂)

4 = ¼ (v₂)

v₂ = 8 m/s

Bernoulli’s principle and equation

10. Water is pumped with a 120 kPa compressor entering the lower pipe (1) and flows upward at a speed of 1 m/s. Acceleration due to gravity is 10 m/s and water density is 1000 kg/m^-3. What is the water pressure on the upper pipe (II).

Known :

Radius of the lower pipe (r₁) = 12 cm

Radius of the lower pipe (r₂) = 6 cm

Water pressure in the lower pipe (p₁) = 120 kPa = 120,000 Pascal

The speed of water in the lower pipe (v₁) = 1 m.s^-1

The height of the lower pipe (h₁) = 0 m

The height of the upper pipe (h₂) = 2 m

Acceleration due to gravity (g) = 10 m.s^-2

Density of water = 1000 kg.m^-3

Wanted: Water pressure in pipe 2 (p₂)

Solution :

The speed of water in pipe 2 is calculated with the equation of continuity :

Water pressure in pipe 2 is calculated using the equation of Bernoulli :

11. A large pipe 5 meters above the ground and a small pipe 1 meter above the ground. The velocity of the water in a large pipe is 36 km/h with a pressure of 9.1 x 10⁵ Pa, while the pressure in the small pipe is 2.10⁵ Pa. What is the water velocity in the small pipe? Water density = 10³ kg/m³

Known :

Water pressure in the large pipe (p₁) = 9.1 x 10⁵ Pascal = 910,000 Pascal

Water pressure in the small pipe (p₂) = 2 x 10⁵ Pascal = 200,000 Pascal

Water speed in the large pipe (v₁) = 36 km/h = 36(1000)/(3600) = 36000/3600 =10 m/s

The height of the large pipe (h₁) = -4 meters

The height of the small pipe (h₂) = 0 meter

Acceleration due to gravity (g) = 10 m.s^-2

Density of water = 1000 kg/m³

Wanted: The speed of water in the small pipe (v₂)

Solution :

The speed of water in the small pipe (v₂) is calculated using the equation of Bernoulli :

12. A pipe with a radius of 15 cm connected with another pipe with a radius of 5 cm. Both are in a horizontal position. The velocity of the water flow in the large pipe is 1 m/s at a pressure of 10⁵ N/m². What is the water pressure on the small pipe (1 g cm^-3)

Known :

Radius of the large pipe (r₁) = 15 cm = 0.15 m

Radius of the small pipe (r₂) = 5 cm = 0.05 m

The water pressure in the large pipe (p₁) = 10⁵ N m^-2 = 100.000 N m^-2

The speed of water in the large pipe (v₁) = 1 m s^-1

Acceleration due to gravity (g) = 10 m.s^-2

Water density = 1 gr cm^-3 = 1000 kg m^-3

Height difference (Δh) = 0.

Wanted: Pressure in the small pipe (p₂)

Solution :

The speed of water in pipe 2 is calculated using the equation of continuity :

The water pressure in the small pipe (p₂) is calculated using the equation of Bernoulli :

1. What is fluid dynamics?
   • Answer: Fluid dynamics is the branch of physics that studies the motion of fluids (liquids and gases) and the forces acting on them. It encompasses the principles and equations that describe how fluids flow, interact with solid boundaries, and affect one another.
2. What’s the difference between laminar and turbulent flow?
   • Answer: Laminar flow is characterized by smooth, parallel layers of fluid moving in orderly paths. Turbulent flow, on the other hand, is chaotic, with eddies, swirls, and rapid fluctuations. Turbulence generally occurs at high velocities or in irregularly shaped channels.
3. How is the concept of viscosity important in fluid dynamics?
   • Answer: Viscosity measures a fluid’s resistance to shear or flow. High-viscosity fluids (like honey) resist flow more than low-viscosity fluids (like water). In fluid dynamics, viscosity plays a crucial role in determining the nature of fluid flow, energy dissipation, and drag forces.
4. What is Bernoulli’s principle?
   • Answer: Bernoulli’s principle states that in a steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant. Specifically, where the fluid velocity is high, the pressure is low, and vice versa.
5. How does the principle of lift in aerodynamics relate to fluid dynamics?
   • Answer: The lift on an aircraft wing can be explained using Bernoulli’s principle and Newton’s third law. As air flows over the wing, it moves faster over the curved top surface than the bottom, creating a pressure difference. This difference in pressure, combined with the downward deflection of air by the wing, results in an upward force or lift.
6. What is the equation of continuity in fluid dynamics?
   • Answer: The equation of continuity states that the product of the cross-sectional area (A) of a flow and its velocity (v) remains constant along a streamline in a steady flow. Mathematically, A₁​v_1​=A₂​v₂​, where A_1​ and A₂​ are cross-sectional areas and v₁​ and v_2​ are the velocities at two points along the streamline.
7. What role does the Reynolds number play in fluid dynamics?
   • Answer: The Reynolds number is a dimensionless quantity that helps predict the flow regime (laminar, transitional, or turbulent) in fluid dynamics. It’s defined as the ratio of inertial forces to viscous forces and depends on factors like fluid velocity, characteristic length, and fluid properties.
8. How does the drag force act on objects moving in a fluid?
   • Answer: Drag force opposes the motion of an object through a fluid. It arises due to the viscous resistance of the fluid and the pressure differences around the object. The magnitude and nature of drag depend on factors like the object’s shape, roughness, speed, and the properties of the fluid.
9. What is the Venturi effect?
   • Answer: The Venturi effect refers to the decrease in fluid pressure that occurs when a fluid flows through a constricted section of a pipe. As the fluid’s velocity increases in the constricted section (due to the conservation of mass), its pressure decreases according to Bernoulli’s principle.
10. Why does fluid speed up when flowing through a narrow section of a pipe or channel?

• Answer: This behavior can be explained by the principle of conservation of mass. In a steady flow, the volume of fluid entering a section of a pipe must equal the volume leaving. If the pipe narrows, the fluid must speed up to allow the same volume to pass through in a given time.