Properties Classification of Fluids: Newtonian, Non-Newtonian, Viscosity, Compressible, Incompressible Fluids

Introduction to Fluids

In engineering, a fluid is defined as a substance that deforms continuously under the action of a shear force, no matter how small. This means fluids cannot resist shear stress on a static basis; they will flow. This fundamental characteristic distinguishes fluids from solids. Fluids encompass both liquids and gases. Understanding the properties of fluids is crucial for analyzing their behavior in various engineering applications, such as pipe flow, pumps, turbines, and aerodynamic systems.

Classification of Fluids

Fluids can be broadly classified based on their behavior under stress and their compressibility. The primary classifications we will focus on are:

• Newtonian vs. Non-Newtonian Fluids
• Compressible vs. Incompressible Fluids

Newtonian vs. Non-Newtonian Fluids

The distinction between Newtonian and Non-Newtonian fluids lies in their response to shear stress, specifically how their viscosity behaves. This relationship is described by Newton's law of viscosity.

Newtonian Fluids

A Newtonian fluid is one where the shear stress is directly proportional to the rate of shear strain (velocity gradient). This means the viscosity of a Newtonian fluid remains constant regardless of the applied shear stress or the rate at which it is deformed. For these fluids, the relationship can be expressed mathematically.

The shear stress ($\tau$) is given by:

$\tau = \mu \frac{du}{dy}$

Where:

• $\tau$ is the shear stress (force per unit area)
• $\mu$ is the dynamic viscosity of the fluid
• $\frac{du}{dy}$ is the velocity gradient or rate of shear strain

In simpler terms, if you double the force applied to make a Newtonian fluid flow, it will flow twice as fast. The "thickness" or resistance to flow (viscosity) doesn't change.

Examples of Newtonian Fluids:

• Water
• Air
• Most common gases
• Certain oils (like mineral oil at moderate temperatures)
• Alcohol

Most simple liquids and gases behave as Newtonian fluids under normal conditions.

Non-Newtonian Fluids

A Non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity. For these fluids, the shear stress is not linearly dependent on the rate of shear strain. Their viscosity changes with the applied shear stress or the shear rate. This behavior can be quite complex and leads to several sub-classifications.

The apparent viscosity ($\mu_{app}$) of a Non-Newtonian fluid is a function of the shear rate ($\dot{\gamma} = \frac{du}{dy}$):

$\tau = \mu_{app}(\dot{\gamma}) \dot{\gamma}$

The viscosity can either increase or decrease with shear rate, or it can depend on time as well.

Types of Non-Newtonian Fluids:

1. Shear-Thinning Fluids (Pseudoplastic): The apparent viscosity decreases as the shear rate increases. These are the most common types of Non-Newtonian fluids. They become "thinner" and easier to flow when stirred or agitated.
   • Examples: Ketchup, paint, blood, shampoo, nail polish, polymer solutions.
2. Shear-Thickening Fluids (Dilatant): The apparent viscosity increases as the shear rate increases. They become "thicker" and more resistant to flow when subjected to higher shear rates.
   • Examples: A mixture of cornstarch and water (oobleck), wet sand.
3. Bingham Plastics: These fluids behave like a solid at low stresses but flow like a fluid once a certain minimum stress, called the yield stress ($\tau_y$), is exceeded.
   • Examples: Toothpaste, mayonnaise, drilling mud, some paints.
   The relationship is: $\tau = \tau_y + \mu_p \frac{du}{dy}$ for $\tau > \tau_y$ where $\mu_p$ is the plastic viscosity.
4. Time-Dependent Fluids: The viscosity of these fluids changes with the duration of shear.
   • Thixotropic Fluids: Viscosity decreases with time under constant shear rate. They become thinner the longer they are stirred. Examples: Yogurt, some paints, certain inks.
   • Rheopectic Fluids: Viscosity increases with time under constant shear rate. These are rare. Example: Gypsum paste.

Viscosity

Viscosity is a fundamental property of fluids that measures their resistance to deformation. It is essentially the "thickness" or "internal friction" of a fluid. A fluid with high viscosity flows slowly, while a fluid with low viscosity flows easily.

Types of Viscosity

There are two main types of viscosity: dynamic viscosity and kinematic viscosity.

Dynamic Viscosity ($\mu$)

Also known as absolute viscosity, dynamic viscosity is the measure of the fluid's internal resistance to shear stress. It is the proportionality constant in Newton's law of viscosity.

Units:

• SI units: Pascal-second (Pa·s) or N·s/m²
• CGS units: Poise (P), where 1 P = 0.1 Pa·s. The centipoise (cP) is more commonly used, where 1 cP = 0.01 P = 0.001 Pa·s. Water at 20°C has a dynamic viscosity of approximately 1 cP.

Factors affecting dynamic viscosity:

• Temperature: For liquids, viscosity generally decreases as temperature increases because the intermolecular forces weaken. For gases, viscosity generally increases with temperature because the increased kinetic energy of molecules leads to more frequent collisions and momentum transfer.
• Pressure: Pressure has a minor effect on the viscosity of liquids but can have a more significant effect on gases, especially at high pressures.

Kinematic Viscosity ($\nu$)

Kinematic viscosity is the ratio of dynamic viscosity to density ($\rho$). It represents the ratio of viscous forces to inertial forces within the fluid. It describes how readily a fluid flows without the influence of external forces like gravity.

The formula is:

$\nu = \frac{\mu}{\rho}$

Units:

• SI units: Meter squared per second (m²/s)
• CGS units: Stokes (St), where 1 St = 10^-4 m²/s. The centistokes (cSt) is commonly used, where 1 cSt = 0.01 St = 10^-6 m²/s.

Kinematic viscosity is important in the study of fluid flow, particularly in determining the Reynolds number, which characterizes the flow regime (laminar or turbulent).

Measurement of Viscosity

Viscometers are instruments used to measure viscosity. Common types include:

• Capillary Viscometers: Measure the time taken for a fixed volume of fluid to flow through a narrow tube under gravity.
• Rotational Viscometers: Measure the torque required to rotate a spindle immersed in the fluid at a constant speed.
• Falling Sphere Viscometers: Measure the time taken for a sphere to fall through a fluid under gravity.

Compressible vs. Incompressible Fluids

This classification is based on how the fluid's density changes in response to changes in pressure.

Incompressible Fluids

An incompressible fluid is one whose density remains constant, regardless of changes in pressure. In reality, no fluid is perfectly incompressible, but many fluids can be treated as such under certain conditions.

For an incompressible fluid:

$\rho$ = constant

This assumption simplifies many fluid mechanics equations.

Conditions for Incompressibility:

• Liquids: Liquids are generally considered incompressible because their molecules are already closely packed, and their density changes very little with pressure. Even very high pressures cause only minuscule changes in liquid density.
• Gases: Gases can be treated as incompressible if the changes in pressure are very small relative to the absolute pressure. A common rule of thumb is that if the Mach number (the ratio of flow velocity to the speed of sound) is less than approximately 0.3, the gas can be considered incompressible. This is often true for low-speed airflow applications.

Examples:

• Water at typical engineering conditions.
• Hydraulic fluids.
• Air flowing at speeds below approximately 100 m/s.

Compressible Fluids

A compressible fluid is one whose density changes significantly with changes in pressure and temperature. All gases are compressible, and liquids can become significantly compressible at very high pressures or when cavitation occurs.

For a compressible fluid, density ($\rho$) is a variable and is often related to pressure ($P$) and temperature ($T$) through an equation of state, such as the ideal gas law:

$P = \rho R T$

Where $R$ is the specific gas constant.

When dealing with compressible fluids, especially gases at high speeds, engineers must account for changes in density, temperature, and pressure, which makes the analysis more complex. Concepts like shock waves, isentropic flow, and non-uniform velocity profiles become important.

Examples:

• Steam.
• Natural gas.
• Air at high speeds (e.g., in jet engines, supersonic flows).
• Liquids subjected to extreme pressures.

Summary of Fluid Properties and Classifications

Understanding these properties and classifications is fundamental to solving fluid mechanics problems.

Property/Classification	Description	Key Characteristics / Formulas	Examples
Newtonian vs. Non-Newtonian	Newtonian: Shear stress proportional to shear rate.	$\tau = \mu \frac{du}{dy}$ (Viscosity $\mu$ is constant)	Water, Air, Alcohol
	Non-Newtonian: Shear stress not proportional to shear rate.	$\tau = \mu_{app}(\dot{\gamma}) \dot{\gamma}$ (Viscosity $\mu_{app}$ varies with shear rate $\dot{\gamma}$)	Ketchup (Shear-thinning), Cornstarch/Water (Shear-thickening), Toothpaste (Bingham Plastic)
Viscosity	Dynamic ($\mu$): Resistance to shear flow.	Units: Pa·s or Poise. Decreases with T for liquids, increases for gases.	Water ≈ 1 cP (at 20°C)
	Kinematic ($\nu$): Ratio of dynamic viscosity to density.	$\nu = \frac{\mu}{\rho}$. Units: m2/s or Stokes.	Important for Reynolds number calculation.
Compressible vs. Incompressible	Incompressible: Density is constant.	$\rho$ = constant. Mach Number ($M$) < 0.3.	Liquids (generally), Gases at low speeds.
	Compressible: Density changes with pressure and temperature.	$\rho$ varies. Mach Number ($M$) > 0.3. Ideal Gas Law: $P = \rho R T$.	Gases at high speeds, liquids at very high pressures.