Physics - Classical Mechanics - Viscosity and Turbulence

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Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to get into Viscosity and Turbulence.

So, without further ado, let's get straight into it!

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Real Fluid

In the idealized models we ignored two very important properties of fluid flow, which are viscosity and turbulence. Real fluids differentiate from ideal fluids in exactly these phenomena. Viscosity can be thought of as the internal friction of fluid and turbulence as the mixing and swirling of the various layers caused mainly by obstruction and high speed. Turbulence, more or less, occurs due to the fluid's viscosity, and so these topics are very closely related.

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Viscosity

Viscosity is basically the fluid's resistance to flow. The parallel layers of the fluid resist the motion of the fluid layers next to them, as well as the motion of solids that come in contact with them. As such, viscous but non-turbulent flow occurs in layers of different speed. If there was no viscosity the speed would be the same across the whole fluid.

It's due to this resistance of the fluid, also called drag, that the speed at the bottom is much less than the speed at the top. In the context of tubes, due to viscosity, the speed at the walls of the tube is basically zero, and maximum at the center.

If an obstruction comes up or the speed reaches a critical point, the layers start mixing and the flow becomes turbulent.

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Measuring Viscosity

In order to end up with a measure of viscosity, we start by placing fluid between two parallel plates. The bottom plate is fixed (v = 0), whilst the top plate moves to the right with a velocity v, dragging the fluid along with it. The layers of fluid that are in contact with the plates don't move relative to these plates. So, the top layer moves at the same speed v as the top plate, and the bottom layer remains at rest.

A visible skew will occur as each layer tries to drag along the next layer and the speed variates from v to 0. The flow is carefully kept laminar, so that the layers don't mix. A continuous shearing motion thus takes place. Because fluids have zero shear strength the rate at which they are sheared is related to the geometrical factors A and L.

The shear stress between the layers is equal to the force F by the area A. The rate of shearing, also called strain rate or velocity gradient, is the velocity v by the length L. As such, we can now define viscosity as the ratio of these two quantities. Viscosity is represented by the Greek later η (eta) and given by:

Fluids that satisfy this equation are called Newtonian Fluids, and are a subset of real fluids.

Solving the equation for force gives us:

So, the greater the viscosity, the greater the force required.

The S.I. unit of viscosity is the N ⋅ s / m² or Pa ⋅ s, but its more common to use a quantity related to cgs, the poise, which equals 1 dyn ⋅ s / cm².

There are actually two types of viscosity. The one measured in poise is also called dynamic viscosity. Dividing it by the density gives us kinematic viscosity which is represented by the Greek letter ν (nu). Kinematic viscosity is measured in stokes (St) which equals 1 cm² / s. The S.I. unit of 1 m² / s is rarely used.

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Poiseuille’s Law

As we already know, what causes flow is pressure difference. By also taking into account the resistance of fluids to flow, we can derive the following relationship:

As flow rate we consider the volume flow rate Q = dV / dt. Flow resistance is anything expect pressure that affects flow rate, such as viscosity. It's represented by capital R. As such, if p₁ and p₂ are the pressures at two points in a tube, the following will be true:

French scientist Poiseuille proved that the resistance to laminar flow in an incompressible fluid with viscosity η through a horizontal tube of radius r and length L is:

Combining these equations yields Poiseuille’s Law:

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Stokes' Law

Again skipping the proof, let's mention yet another useful equation, which is Stokes' law. It gives us the drag force on a falling sphere:

When the sphere is fully submerged and falling with constant velocity, we can derive its viscosity. The buoyant force and drag force are equal to the weight, which gives:

Depending on what's known it's possible to calculate the viscosity (maybe even kinematic) or density of a sphere that is submerged into a liquid.

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Turbulence and Reynolds number

Lastly, let's also get into an indicator of turbulence. For a uniform tube, Reynolds number N_R is given by:

This quantity is dimension-less but experiments revealed that the flow is laminar for values below 2000, and turbulent for values above 3000. In-between values show unstable and chaotic behavior.

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RESOURCES:

References

1. https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/14-7-viscosity-and-turbulence/
2. https://physics.info/viscosity/

Images

1. https://pxhere.com/en/photo/1045542

Mathematical equations used in this article, where made using quicklatex.

Visualizations were made using draw.io.

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Previous articles of the series

Rectlinear motion

• Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
• Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
• Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
• Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

• Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
• Projectile motion as a plane motion -> missile/bullet motion as a plane motion
• Smooth Circular motion -> smooth circular motion theory
• Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

• Force and Newton's first law -> force, 1st law
• Mass and Newton's second law -> mass, 2nd law
• Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
• Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
• Contact forces and friction -> contact force, friction
• Dynamics of Circular motion -> circular motion dynamics, applications
• Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
• Contact force and friction examples -> exercises in force and friction
• Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
• Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

• Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
• Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
• Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
• Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
• Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
• Internal Energy and Work -> Internal Energy, Internal Work

Momentum and Impulse

• Conservation of Momentum -> Momentum, Conservation of Momentum
• Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
• Collision Examples -> Various Elastic and Inelastic Collision Examples
• Impulse -> Impulse with Example
• Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
• Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

• Angular motion basics -> Angular position, velocity and acceleration
• Rotation with constant angular acceleration -> Constant angular acceleration, Example
• Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
• Parallel Axis Theorem -> Parallel axis theorem with example
• Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
• Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
• Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
• Angular Momentum -> Angular Momentum and its conservation
• Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
• Exercises around Angular motion -> Angular motion examples

Equilibrium and Elasticity

• Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
• Force Couple System -> Force Couple System, Example
• Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
• Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
• Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
• Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
• Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
• Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples

Gravity

• Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
• Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
• Gravitational Fields -> Gravitational Field Mathematics and Visualization
• Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
• Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
• Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
• Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
• Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
• Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
• Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
• Black Holes and Schwarzschild Radius -> Black Holes (Creation, Types, How To "See" Them), Schwarzschild Radius

Periodic Motion

• Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example
• Energy in Simple Harmonic Motion -> Forms of Energy in SHM (Potential, Kinetic, Total and Maximum Energy, Maximum Velocity), Simple Example
• Simple Harmonic Motion Equations -> SHM Equations (Displacement, Velocity, Acceleration, Phase Angle, Amplitude)
• Simple Harmonic Motion and Reference Circle -> SHM and Smooth Circular Motion, Reference Circle
• Simple Harmonic Motion Exercises -> 2 Complete Examples on Simple Harmonic Motion
• Simple Pendulum -> Simple Pendulum (Return Force, Small Angle Approximations, More Accurate Period, Gravity Approximation)
• Physical Pendulum -> Physical Pendulum (Return Torque, Small Angle Approximations, Estimating Moment of Inertia)
• Exercises around Pendulums -> Complete Examples on the 2 types of Pendulums (Simple, Physical)
• Damped Oscillation -> Damping Force, Total Force and Differential Equation, Motion Equations, Special Cases
• Forced Oscillation and Resonance -> Forced Oscillation (Differential Equation, Amplitude, Resonance)
• Exercises around Damped and Forced Oscillation -> Complete Examples on Damped Oscillation and Forced Oscillation
• Chaos and Chaotic Oscillation -> Chaos, Unpredictability and Randomness, Chaotic Oscillation

Fluid Mechanics

• Density and Pressure -> Fluids and Fluid Mechanics, Density, Specific Gravity, Pressure
• Measuring Pressure in Fluids -> Pressure in Fluids (Variation with Depth), Absolute and Gauge Pressure, Measuring Pressure
• Pascal's Principle and Hydraulics -> Static Equilibrium, Pascal's Principle, Hydraulic Systems
• Archimedes' Principle and Buoyancy -> Buoyant Force, Archimedes' Principle, Relation with Density
• Surface Tension -> Surface Tension (Cohesive and Adhesive Forces, Unit, Definition), Capillarity
• Exercises on Fluid Statics (part 1) -> Various Density and Pressure Examples
• Exercises on Fluid Statics (part 2) -> Hydraulic Car Jack, Buoyancy Examples, Surface Tension Example
• Introduction to Fluid Dynamics -> Ideal Fluid, Flow Characteristics, Volume Flow Rate, Flow Continuity, Mass Flow Rate
• Bernoulli’s Equation -> Energy Conservation and Flow, Bernoulli's Equation and Principle

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Final words | Next up

And this is actually it for today's post!

Next time we will get into Exercises around Fluid Dynamics...

See ya!

Keep on drifting!

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