Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Properties of matter → Fluid mechanics ↓


Continuity equation and the Venturi effect


Introduction

In fluid mechanics, understanding fluid flow is crucial for everything from the design of pipelines to the functioning of aircraft. Two key concepts describing fluid behavior are the equation of continuity and the Venturi effect. These principles describe how fluid speed, pressure, and cross-sectional area are related under the law of conservation of mass, allowing engineers and scientists to predict how fluids will behave under different conditions.

Equation of continuity

The equation of continuity is a mathematical statement of the conservation of mass in a fluid flow system. It tells us that for an incompressible fluid flowing in a pipe, the mass flow rate must remain constant from one cross-section to the next.

Understanding the concept

Imagine water flowing through a hose. If the hose is compressed at some point, creating a narrower passage, the speed of the water at this narrower portion will increase. The equation of continuity explains this behavior.

The principle of conservation of mass states that mass cannot be created or destroyed. In the context of fluid flow, this principle indicates that the amount of fluid entering a system must be equal to the amount of fluid leaving it, as long as the fluid is incompressible.

Mathematical formulation

For steady flow, the rate of flow of the mass of the fluid passing through any section remains constant. This can be expressed mathematically as:

A₁v₁ = A₂v₂

Here, A₁ and A₂ are the cross-sectional areas at points 1 and 2, respectively, and v₁ and v₂ are the flow velocities at these points. The equation tells us that the product of the cross-sectional area and velocity at any point along the stream must remain constant.

This implies that if you reduce the cross-sectional area (A), the flow velocity (v) must be increased to maintain the same flow rate, and vice versa.

A₁ A₂ V₂ V₁

The figure above shows a simplified pipe with different cross-sectional areas. At the widest part, the area is A₁, and the fluid flows with velocity v₁. At the narrowest part, the area is A₂, and the velocity is v₂. According to the equation of continuity, A₁v₁ = A₂v₂.

Venturi effect

The Venturi effect is a phenomenon in which the velocity of a fluid increases as it passes through a constriction in a pipe, causing a decrease in static pressure. This effect is an application of Bernoulli's principle, which states that an increase in fluid velocity is accompanied by a decrease in pressure or a decrease in the potential energy of the fluid.

How it works

When a fluid enters a narrow section of a pipe, its speed increases and the pressure decreases. This happens because the fluid has to conserve energy; as the kinetic energy increases (due to increased speed), the pressure energy decreases.

Venturi effect in real life

The Venturi effect is used in many industries. Some of the most common uses include carburetors in vehicles, where this effect helps mix air with fuel efficiently, and in medical applications such as the Venturi mask, which delivers controlled oxygen rates to patients.

In the visual example above, we can see a pipe with a narrow section in the middle. The fluid accelerates as it passes through the narrow section, which reduces its pressure. This change in fluid dynamics due to a change in cross-sectional area is a demonstration of the Venturi effect.

Bernoulli's principle

The Venturi effect can also be understood from the Bernoulli equation, which is as follows:

P₁ + 0.5 * ρ * v₁² + ρgh₁ = P₂ + 0.5 * ρ * v₂² + ρgh₂

Here:

  • P is the pressure.
  • ρ is the density of the fluid.
  • v is the velocity of the fluid.
  • g is the acceleration due to gravity.
  • h is the height.

This equation shows us how pressure energy, kinetic energy, and potential energy change during fluid flow. As the fluid moves through a constricted section in the Venturi effect, the velocity term v² becomes larger and so the pressure P decreases.

Practical example

Consider a simple garden hose with a nozzle that reduces the diameter of the outlet. Applying the continuity equation, the speed of water increases in the narrow nozzle due to the reduced area.

Suppose the internal diameter of the garden hose is 2 cm and the nozzle reduces it to 1 cm. If the speed of water in the hose is 2 m/s, what will be the speed in the nozzle?

Given: D₁ = 2 cm → A₁ = π(1 cm)² D₂ = 1 cm → A₂ = π(0.5 cm)² v₁ = 2 m/s Find v₂. Using A₁v₁ = A₂v₂: π(1)² * 2 = π(0.5)² * v₂ => v₂ = (1)² * 2 / (0.5)² => v₂ = 8 m/s.

Here, as the area is reduced by a factor of four, the velocity increases by the same amount, demonstrating both aspects of the continuity equation and the Venturi effect, since the increase in velocity is correlated with the change in pressure.

Conclusion

The equation of continuity and the Venturi effect are fundamental concepts in fluid mechanics. They describe how fluids move in different cross-sectional areas and how these changes affect properties such as velocity and pressure. These concepts are useful in many practical applications in a variety of fields, making them valuable knowledge not only for those studying physics but also for engineering, meteorology, and other fields.


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Continuity equation and the Venturi effect

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