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 ↓


Viscosity and Poiseuille's law


In the study of fluid mechanics, understanding the behavior of fluids in motion is paramount. Viscosity and Poiseuille's law are two fundamental concepts that help us understand how fluids move, especially in confined environments such as pipes. This text discusses these concepts in depth, providing an accessible explanation for undergraduate physics students.

Viscosity: the internal friction of fluids

Viscosity is the property of a fluid that resists the force that tends to flow the fluid. Imagine honey and water; honey is thick and flows slowly because of its high viscosity, while water flows more easily.

Viscosity is basically the internal friction in a fluid moving along different layers. When layers of fluid move relative to each other, the viscosity of the fluid determines how easy or difficult it is for these layers to move over each other.

In mathematical terms, viscosity (η) is defined as the ratio of shear stress (τ) to the velocity gradient (rate of shear) perpendicular to the direction of shear.

η = τ / (du/dy)

Where:

  • τ is the shear stress.
  • du/dy is the velocity gradient perpendicular to the layer direction.

Types of flow: laminar vs. turbulent

Before we explore further, it's important to distinguish between laminar and turbulent flow, since viscosity plays a different role in both:

  • Laminar flow: Smooth and orderly fluid motion in parallel layers with no disruption between the layers.
  • Turbulent flow: Chaotic and irregular fluid motion where eddies, vortices, and instabilities prevail.

Viscosity primarily governs laminar flow. An example of laminar flow can be seen when syrup is poured onto a pancake, which flows steadily and uniformly. In turbulent flow, viscosity still affects the resistance to flow, but the chaotic nature makes the predictability of motion less likely.

Understanding Poiseuille's law

Poiseuille's law, also sometimes known as the Hagen-Poiseuille law, describes the volume flow rate of a Newtonian fluid with laminar flow in a long cylindrical pipe. This law gives us a mathematical relationship that shows how a fluid flows through a pipe.

Q = (πR⁴ΔP) / (8ηL)

Where:

  • Q is the volume flow rate (the amount of fluid passing through the pipe per unit time).
  • R is the radius of the pipe.
  • ΔP is the pressure difference between the two ends of the pipe.
  • η is the dynamic viscosity of the fluid.
  • L is the length of the pipe.

Key points from Poiseuille's law

Many important features of fluid dynamics can be derived from Poiseuille's law. It is worth noting that:

  • The flow rate is proportional to the fourth power of the pipe's radius. This means that even a small increase in the pipe's radius causes a significant increase in the flow rate. For example, doubling the radius will increase the flow rate by 16 times.
  • Flow rate is directly proportional to the pressure difference. If you increase the pressure difference in the pipe, more fluid will flow through it.
  • Flow rate is inversely proportional to viscosity. Fluids with higher viscosity flow more slowly than fluids with lower viscosity under the same conditions.
  • Flow rate is inversely proportional to the length of the pipe. The longer the pipe, the slower the flow due to frictional losses within the fluid.

Practical applications

Both viscosity and Poiseuille's law find practical applications in various fields including engineering, biology, and medicine. For example, understanding blood flow in capillaries or designing pipelines for oil transportation requires acknowledging these principles.

Example problem: blood flow through a capillary

Let us consider blood flowing through a capillary of radius 0.0015 cm and length 0.5 cm with viscosity 0.0027 poise and pressure difference 2000 dyne/cm².

Use of Poiseuille's law:

Q = (π * (0.0015)⁴ * 2000) / (8 * 0.0027 * 0.5)

Calculating these values will give the flow rate, which will help us understand how efficiently blood moves through microcapillaries in biological systems.

SVG illustration

Let's represent a simple flow through a pipe with SVG:

Flow direction

The blue rectangle represents a pipe, while the red line indicates the direction of fluid flow. The relationship between flow rate and pipe diameter can be seen by changing this setup: using Poiseuille's law a wider blue pipe results in a longer red line (higher flow rate).

Factors affecting viscosity

Viscosity is not constant for all fluids and can vary with temperature and pressure:

  • Temperature: For liquids, viscosity decreases as temperature increases because the fluid molecules move faster, overcoming intermolecular forces more easily. For gases, however, viscosity increases with temperature because greater kinetic energy causes more frequent molecular collisions.
  • Pressure: In most practical situations under normal conditions, the effect of pressure on the viscosity of a liquid is minimal. However, for gases, increased pressure can increase the viscosity because the density of the gas increases.

Example: Engine oil viscosity

Consider engine oil, which must maintain optimum viscosity across temperature ranges. During cold starts in the winter, you need oil that is not overly thick and still provides lubrication. However, as the engine warms up, you want to avoid oil that becomes too thin and loses its protective properties.

Manufacturers design oils with additives to ensure that their viscoelastic properties remain effective under a variety of conditions, reflecting the dependence of viscosity on temperature.

Concluding remarks on viscosity and Poiseuille's law

Understanding viscosity and Poiseuille's law enables us to predict fluid behavior in complex systems, balancing flow rate, pressure, and viscosity. These concepts are fundamental to applications in science and engineering, providing an important foundation for further exploration in fluid mechanics.

An understanding of the role of viscosity in resisting flow and the accurate calculations provided by Poiseuille's law provide valuable tools for students and professionals dealing with real-world fluid systems.


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