Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Fluid mechanics ↓


Fluid Dynamics and Bernoulli's Principle


Introduction to fluid dynamics

Fluid dynamics is the branch of physics that studies the behavior of fluids (liquids and gases) in motion. It is a part of fluid mechanics and provides important information about the behavior of fluids in structures, pipes, and natural systems. Understanding fluid dynamics is essential for fields as diverse as aerodynamics, hydrodynamics, engineering, meteorology, oceanography, and even medical science.

Basic properties of liquids

To understand fluid dynamics we need to understand some basic properties of fluids:

  • Density (ρ): This is the mass per unit volume of a fluid. It is usually measured in kilograms per cubic meter (kg/m³).
  • Pressure (P): Pressure is the force applied per unit area. It is usually measured in pascals (Pa).
  • Viscosity (μ): Viscosity measures a fluid's resistance to deformation or flow. It affects how a fluid moves through a pipe or around a solid object.
  • Flow rate: This is the volume of fluid passing a point in a given time. It is often expressed as cubic metres per second (m³/s).

Classification of fluid flow

Fluid flow may be classified based on various factors:

  • Laminar vs. Turbulent Flow:
    • Laminar flow: Smooth and orderly. Fluid particles move in parallel layers. 
      Reynolds number (Re) < 2000
    • Turbulent flow: chaotic and disordered, characterized by eddies and vortices. 
      Reynolds number (Re) > 4000
  • Incompressible vs. Compressible Flow:
    • Incompressible flow: Fluid density remains constant. Suitable for most fluids.
    • Compressible flow: The density of a fluid changes with pressure. Important in gas dynamics.

Continuity equation

The continuity equation is a fundamental principle derived from the conservation of mass. For steady flow of an incompressible fluid in a channel, the mass flow rate must remain constant. This leads to the continuity equation:

A₁V₁ = A₂V₂

Where:

  • A₁ and A₂ are the cross-sectional areas at two different points.
  • V₁ and V₂ are the flow velocities at these points.

This equation implies that if the area decreases, the velocity of the fluid must increase, and vice versa.

Bernoulli's principle

Bernoulli's principle is an important concept in fluid dynamics, named after Swiss physicist Daniel Bernoulli. It describes the conservation of energy in a fluid stream. The principle states that in a streamlined flow:

P + 0.5ρV² + ρgh = constant

Where:

  • P is the pressure energy per unit volume.
  • 0.5ρV² is the kinetic energy per unit volume.
  • ρgh is the potential energy per unit volume.
  • ρ is the density of the fluid.
  • V is the speed of the fluid.
  • g is the acceleration due to gravity.
  • h is the height above a reference point.

Understanding Bernoulli's principle

Bernoulli's principle essentially states that an increase in fluid speed is accompanied by a decrease in pressure or a decrease in the potential energy of the fluid.

Visually, for an incompressible, non-viscous fluid in a streamlined flow, Bernoulli's equation can be applied between two points in the fluid stream:

P₁ + 0.5ρV₁² + ρgh₁ = P₂ + 0.5ρV₂² + ρgh₂

Example 1: Flow through a narrow pipe

Let's consider a simple visual example: a pipe with varying diameters. Imagine a fluid flowing horizontally through a narrowed pipe, where the cross-section decreases, and then widens. According to Bernoulli's principle the following effects occur:

At the narrowest part of the pipe, the velocity of the fluid increases due to the reduction in the cross-sectional area. According to Bernoulli's principle, this increase in velocity leads to a decrease in pressure. Conversely, as the pipe widens again, the velocity decreases, and the pressure returns to its original state.

Example 2: Airplane wings

Bernoulli's principle also applies to the lift experienced by airplane wings. The shape of the wing causes air to travel faster over the top surface than the bottom surface. According to Bernoulli's principle, this results in lower pressure above the wing and higher pressure below it, creating lift.

In the figure, the curved path of the air over the wing is longer, so the velocity is higher, which leads to lower pressure than the flat area underneath.

Application of Bernoulli's theorem

The applications of Bernoulli's equation are very wide. Some classic examples include:

  • Venturi meter: An instrument that measures the rate of flow of a fluid. It uses the principle that pressure decreases as the velocity of the fluid increases in a constricted section of a pipe.
  • Pitot Tube: Used to measure the velocity of air flow. It combines intake static pressure and dynamic pressure to calculate fluid velocity based on differential pressure.
  • Atomizer: Common in spray bottles, atomizers use Bernoulli's principle to produce aerosols through differential pressure across their nozzle.

Here is a simple representation of a Venturi meter:

The narrowest part of the Venturi meter is called the throat. As the fluid passes through this part, its velocity increases, causing a decrease in pressure. By measuring this change in pressure, the flow rate can be determined.

Limitations of Bernoulli's principle

While Bernoulli's equation is incredibly useful, it has limitations and assumptions. It is derived under the assumptions that the flow is steady, along stream lines, incompressible, and inviscid (negligible viscosity). In real-world applications, these assumptions may not be entirely true.

For example, in highly viscous fluids or turbulent flows, energy losses due to viscosity and turbulence affect the accuracy of Bernoulli's principle. Thus, engineers often refine or replace Bernoulli's predictions with empirical corrections to account for such situations.

Conclusion

In the field of fluid mechanics, fluid dynamics and Bernoulli's principle are central concepts that describe how fluids behave when moving and what kind of forces they exert. Mastering these fundamental ideas is critical to understanding and designing systems in many fields, from aerospace engineering to everyday plumbing systems. By understanding how energy is conserved and controlled within fluid systems, one can predict, control, and use fluid behavior for technological and practical applications.


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Fluid Dynamics and Bernoulli's Principle

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