8.6 Viscosity

Definition of viscosity

Viscosity describes a fluid's internal resistance to flow. When you pour honey versus water, the difference you notice is viscosity at work. More precisely, viscosity measures how strongly a fluid resists deformation when layers of that fluid slide past each other.

This resistance comes from intermolecular forces and collisions between molecules within the fluid. Higher viscosity means you need more force to get the fluid moving and keep it flowing. It also shows up as drag force on objects moving through the fluid.

Shear stress vs. strain rate

Viscosity is formally defined as the ratio of shear stress to strain rate:

• Shear stress is the force per unit area acting parallel to fluid layers (think of it as layers of fluid being pushed sideways past each other).
• Strain rate is the velocity gradient, meaning how quickly the fluid velocity changes as you move perpendicular to the flow direction.

If the relationship between shear stress and strain rate is linear, the fluid is called Newtonian. If it's nonlinear, the fluid is non-Newtonian. This distinction matters a lot for predicting how a fluid will behave.

Types of fluids

Newtonian fluids

A Newtonian fluid has a constant viscosity no matter how much shear stress you apply. Double the shear rate, and the shear stress doubles right along with it. Water, air, and most simple oils are Newtonian fluids.

Their behavior follows Newton's law of viscosity (covered in the math section below), which makes calculations much simpler. Most of the fluid dynamics problems you'll see in an intro course assume Newtonian behavior.

Non-Newtonian fluids

Non-Newtonian fluids change their effective viscosity depending on how hard or fast you shear them. There are several subtypes:

• Shear-thinning (pseudoplastic): Viscosity decreases as shear rate increases. Ketchup is a classic example. It resists flowing in the bottle, but once you shake it and apply shear, it flows easily.
• Shear-thickening (dilatant): Viscosity increases with shear rate. Cornstarch mixed in water does this. Push on it slowly and it flows; hit it hard and it resists like a solid.
• Bingham plastics: These behave like a solid until a minimum shear stress (the yield stress) is exceeded, then they flow. Toothpaste and mayonnaise are common examples.

Modeling non-Newtonian fluids requires more complex equations than Newton's law of viscosity.

Viscosity coefficients

Dynamic viscosity

Dynamic viscosity (also called absolute viscosity) is the most direct measure of a fluid's resistance to shear flow. It's defined as:

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

where $\tau$ is shear stress and $\frac{du}{dy}$ is the strain rate. The symbol is $\mu$ (mu), and the SI unit is pascal-seconds (Pa·s), equivalent to $\text{kg/(m·s)}$.

Dynamic viscosity tells you the force needed to slide one layer of fluid past another at a given rate.

Kinematic viscosity

Kinematic viscosity factors out the fluid's density:

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

where $\rho$ is the fluid density. The symbol is $\nu$ (nu), and the SI unit is $\text{m}^2/\text{s}$.

Why bother with a second type? In many flow problems (especially those involving gravity-driven flow or free surfaces), density and viscosity always appear together as this ratio. Using kinematic viscosity simplifies those equations.

Factors affecting viscosity

Temperature effects

For liquids, viscosity decreases as temperature rises. Higher thermal energy means molecules move more freely and intermolecular forces weaken. Think about heating maple syrup: it pours much more easily when warm.

For gases, the opposite happens. Viscosity increases with temperature because gas molecules move faster and collide more frequently, transferring more momentum between layers.

The Arrhenius equation (see the math section) describes the temperature dependence of liquid viscosity quantitatively. Temperature is one of the biggest practical factors in lubrication and heat transfer systems.

Pressure effects

Pressure has a relatively small effect on liquid viscosity under normal conditions, though at very high pressures (deep-sea environments, hydraulic systems), liquids do become noticeably more viscous. Gases show a more significant viscosity increase under pressure. For most intro-level problems, you can treat liquid viscosity as pressure-independent.

Composition effects

A fluid's molecular structure and composition directly affect its viscosity. Heavier, longer molecules generally produce higher viscosity. Dissolved substances, suspended particles, and polymer chains all change how a fluid flows. Polymer solutions, for instance, often exhibit shear-thinning behavior and viscoelastic properties. This is why viscosity control matters so much in formulating paints, cosmetics, and food products.

Measurement of viscosity

Viscometers

Several types of viscometers exist, each suited to different situations:

• Capillary viscometers measure the time it takes for fluid to flow through a narrow tube between two marked points. They work well for low-to-medium viscosity Newtonian fluids.
• Rotational viscometers measure the torque needed to spin an object (like a cylinder or disk) inside the fluid. These are versatile and can measure viscosity across different shear rates, making them useful for non-Newtonian fluids too.
• Falling ball viscometers track the terminal velocity of a sphere sinking through the fluid. This method is based on Stokes' law and works best with transparent Newtonian fluids.

Rheometers

Rheometers are more advanced instruments that measure both viscous and elastic properties of fluids. They can run oscillatory tests, creep tests, and stress relaxation tests. If you need to fully characterize a non-Newtonian or viscoelastic material, a rheometer is the tool for the job.

Viscosity in fluid dynamics

Laminar vs. turbulent flow

Viscosity is central to whether flow is laminar (smooth, orderly layers) or turbulent (chaotic, mixing motion).

• Laminar flow tends to occur at low velocities or in high-viscosity fluids. It's predictable and easier to model.
• Turbulent flow tends to occur at high velocities or in low-viscosity fluids. It's much harder to analyze.

The Reynolds number ($Re$) predicts which regime you're in. It's a dimensionless ratio of inertial forces to viscous forces:

$Re = \frac{\rho v L}{\mu}$

where $\rho$ is density, $v$ is flow velocity, $L$ is a characteristic length (like pipe diameter), and $\mu$ is dynamic viscosity. For pipe flow, $Re < 2000$ generally means laminar, and $Re > 4000$ generally means turbulent, with a transition zone in between.

Boundary layer effects

Near any solid surface, a thin boundary layer forms where the fluid velocity goes from zero at the surface (the no-slip condition) to the free-stream velocity farther away. Viscosity controls how thick this layer is and how velocity changes within it.

The boundary layer directly affects drag forces on objects moving through fluids. In aerodynamics, heat transfer, and pipe flow, understanding the boundary layer is essential for accurate predictions.

Applications of viscosity

Lubrication

Lubricant viscosity determines how well it protects mechanical parts. Too thin, and the lubricant film breaks down, allowing metal-to-metal contact. Too thick, and excess fluid friction wastes energy. The right viscosity depends on temperature, load, and speed. Viscosity index improvers are additives that help lubricants maintain a suitable viscosity across a wide temperature range.

Fluid transport

Moving fluid through pipes requires overcoming viscous resistance. Higher viscosity means larger pressure drops and more pumping energy. Engineers use viscosity data to size pumps, select pipe diameters, and design systems for oil pipelines, water distribution networks, and chemical processing plants.

Manufacturing processes

Many manufacturing steps depend on precise viscosity control: injection molding of plastics, coating and painting operations, food processing, and pharmaceutical production. If viscosity drifts out of spec, product quality suffers and processes become inconsistent.

Viscosity in everyday life

Liquids vs. gases

Liquids generally have much higher viscosities than gases. Among liquids, the range is enormous: water at 20°C has a viscosity of about 1 centipoise (cP), while honey can be thousands of times more viscous. Gas viscosities are comparatively tiny but still matter for things like air resistance in sports and vehicle aerodynamics.

Examples in nature

• Blood viscosity affects how hard the heart has to work to circulate blood. Abnormal blood viscosity is linked to cardiovascular problems.
• Magma viscosity controls whether a volcanic eruption is explosive (high viscosity, gas gets trapped) or effusive (low viscosity, lava flows freely).
• Plant sap viscosity influences how efficiently nutrients travel through trees.
• Viscous damping in joints and soft tissues provides natural shock absorption during movement.

Mathematical representations

Newton's law of viscosity

This is the foundational equation for Newtonian fluids:

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

• $\tau$ = shear stress (Pa)
• $\mu$ = dynamic viscosity (Pa·s)
• $\frac{du}{dy}$ = velocity gradient perpendicular to the flow direction (strain rate, in $\text{s}^{-1}$)

The equation says that shear stress is directly proportional to the strain rate, with viscosity as the constant of proportionality. This linear relationship is what defines a Newtonian fluid.

Viscosity equations

Arrhenius equation for temperature dependence of liquid viscosity:

$\mu = A \, e^{E_a / (RT)}$

• $A$ = pre-exponential factor
• $E_a$ = activation energy for viscous flow
• $R$ = universal gas constant
• $T$ = absolute temperature (K)

As $T$ increases, the exponential term decreases, so viscosity drops. This matches the everyday observation that hot liquids flow more easily.

Power law model for non-Newtonian fluids:

$\tau = K \dot{\gamma}^n$

• $K$ = consistency index
• $\dot{\gamma}$ = shear rate
• $n$ = flow behavior index ($n < 1$ for shear-thinning, $n > 1$ for shear-thickening, $n = 1$ recovers Newtonian behavior)

Sutherland's formula for gas viscosity as a function of temperature:

$\mu = \mu_0 \frac{T_0 + C}{T + C} \left(\frac{T}{T_0}\right)^{3/2}$

• $\mu_0$ = reference viscosity at temperature $T_0$
• $C$ = Sutherland's constant (specific to each gas)

Units and conversions

SI units

• Dynamic viscosity: pascal-seconds (Pa·s), where $1 \text{ Pa·s} = 1 \text{ kg/(m·s)}$
• Kinematic viscosity: $\text{m}^2/\text{s}$
• For smaller values, millipascal-seconds (mPa·s) are common. Note that $1 \text{ mPa·s} = 1 \text{ cP}$.

Other common units

• Poise (P) and centipoise (cP) for dynamic viscosity
  • $1 \text{ Pa·s} = 10 \text{ P} = 1000 \text{ cP}$
  • Water at 20°C ≈ 1 cP, which is a handy reference point
• Stokes (St) and centistokes (cSt) for kinematic viscosity
  • $1 \text{ m}^2/\text{s} = 10{,}000 \text{ St} = 1{,}000{,}000 \text{ cSt}$
  • $1 \text{ cSt} = 1 \text{ mm}^2/\text{s}$
• Saybolt Universal Seconds (SUS) are still used in the oil industry but are less common in academic settings.