1.2 Viscosity

Viscosity measures a fluid's resistance to flow under applied stress. It governs how fluids behave in everything from blood vessels to oil pipelines, and it directly affects flow patterns, pressure drops, and energy losses. This guide covers the definition and physical origin of viscosity, the factors that influence it, how it's measured, and the key models used to predict it.

Definition of viscosity

Viscosity quantifies how strongly a fluid resists deformation when you apply a shear stress to it. Physically, it arises from internal friction between adjacent fluid layers sliding past each other at different velocities. Think of dragging your hand through honey versus water: honey has much higher viscosity, so it resists your motion far more.

This property shows up everywhere in fluid dynamics. It determines the pressure drop in pipelines, the drag on aircraft surfaces, and the energy lost to heat in bearings and other mechanical systems.

Factors affecting viscosity

Temperature effects on viscosity

Temperature is the single biggest factor controlling viscosity, but it works in opposite directions for liquids and gases.

• Liquids: Rising temperature weakens the cohesive (attractive) forces between molecules, letting them slide past each other more easily. Viscosity decreases with temperature. Motor oil, for example, flows much more freely when hot than when cold.
• Gases: Higher temperature increases molecular kinetic energy and collision frequency, which actually increases viscosity. Gas molecules transfer more momentum between layers, creating greater internal friction.

This opposite behavior is one of the most common points of confusion, so keep it straight: liquids get thinner when heated, gases get thicker.

Pressure effects on viscosity

Pressure has a much smaller effect on viscosity than temperature does, especially under normal conditions.

• Liquids: Viscosity increases slightly with pressure because compression reduces intermolecular spacing, increasing friction between molecules. At extremely high pressures (like those in hydraulic systems or gear contacts), this effect becomes significant and can raise viscosity by several orders of magnitude.
• Gases: Under normal conditions, gas viscosity is essentially independent of pressure. The mean free path between molecular collisions changes with pressure, but this is offset by a proportional change in molecular density, so the net effect on viscosity is negligible.

Newtonian vs non-Newtonian fluids

Characteristics of Newtonian fluids

A Newtonian fluid has a constant viscosity regardless of how fast you shear it. The relationship between shear stress and shear rate is perfectly linear:

$\tau = \mu \dot{\gamma}$

where $\tau$ is shear stress, $\mu$ is dynamic viscosity, and $\dot{\gamma}$ is shear rate. On a plot of $\tau$ vs. $\dot{\gamma}$, a Newtonian fluid gives a straight line through the origin, and the slope equals $\mu$.

Water, air, and most simple fluids under normal conditions behave as Newtonian fluids.

Types of non-Newtonian fluids

Non-Newtonian fluids have a viscosity that changes depending on the applied shear. There are several categories:

• 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. Blood and polymer solutions also behave this way.
• Shear-thickening (dilatant): Viscosity increases with shear rate. A cornstarch-water mixture (oobleck) is the go-to example: it flows when you handle it gently but stiffens when you hit it.
• Yield stress fluids (Bingham plastics): These require a minimum shear stress (the yield stress) before they begin to flow at all. Below that threshold, they behave like a solid. Toothpaste and mayonnaise are everyday examples. Once the yield stress is exceeded, they may flow as either Newtonian or non-Newtonian fluids.

Shear rate vs shear stress

These two quantities are central to understanding viscosity:

• Shear rate $\dot{\gamma}$ is the velocity gradient perpendicular to the flow direction. It tells you how quickly the fluid is being deformed: $\dot{\gamma} = \frac{dv}{dy}$
• Shear stress $\tau$ is the tangential force per unit area applied to the fluid: $\tau = \frac{F}{A}$

For Newtonian fluids, plotting $\tau$ against $\dot{\gamma}$ gives a straight line with slope $\mu$. For non-Newtonian fluids, the curve is nonlinear, and the slope at any point (the apparent viscosity) varies with shear rate.

Measurement of viscosity

Viscometer types and principles

Several instrument types measure viscosity, each suited to different fluids and conditions:

• Capillary viscometers (Ostwald, Ubbelohde): The fluid flows through a narrow calibrated tube under gravity. You measure the time it takes, then relate that to viscosity using the Hagen-Poiseuille equation. Best for low-viscosity Newtonian fluids.
• Rotational viscometers (cone-and-plate, parallel plate, concentric cylinder): A known torque is applied and the resulting angular velocity is measured (or vice versa). These can characterize both Newtonian and non-Newtonian fluids because you can vary the shear rate.
• Falling sphere viscometers (Hoeppler): A sphere of known density and size falls through the fluid. You measure its terminal velocity and calculate viscosity using Stokes' law.

Viscosity units and conversions

There are two types of viscosity, and mixing them up is a common mistake:

• Dynamic viscosity $\mu$: SI unit is Pascal-seconds (Pa·s). In practice, centipoise (cP) is widely used, where $1 \text{ cP} = 0.001 \text{ Pa·s}$. Water at 20°C has a dynamic viscosity of about 1 cP.
• Kinematic viscosity $\nu$: This is dynamic viscosity divided by density: $\nu = \frac{\mu}{\rho}$. SI unit is $\text{m}^2/\text{s}$, but centistokes (cSt) is common, where $1 \text{ cSt} = 10^{-6} \text{ m}^2/\text{s}$.

To convert between them, you need the fluid's density at the relevant temperature and pressure: $\mu = \nu \rho$.

Viscosity in fluid dynamics equations

Navier-Stokes equations and viscosity

The Navier-Stokes equations govern the motion of viscous fluids. They account for viscosity, pressure gradients, and body forces (like gravity). Within these equations, the viscous stress tensor is proportional to velocity gradients in the fluid, with $\mu$ as the proportionality constant.

One major consequence of viscosity in these equations is the formation of boundary layers: thin regions near solid surfaces where the fluid velocity transitions from zero at the wall (the no-slip condition) to the free-stream velocity farther away.

Reynolds number and viscous effects

The Reynolds number quantifies the ratio of inertial forces to viscous forces in a flow:

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

where $\rho$ is fluid density, $v$ is a characteristic velocity, $L$ is a characteristic length, and $\mu$ is dynamic viscosity.

• Low Re (below ~2300 in pipes): Viscous forces dominate. The flow is laminar, with smooth, parallel streamlines and predictable behavior.
• High Re (above ~4000 in pipes): Inertial forces dominate. The flow is turbulent, with chaotic velocity fluctuations and enhanced mixing.
• Transition region (Re ~2300 to 4000): The flow can be either laminar or turbulent depending on disturbances, surface roughness, and geometry.

The Reynolds number is one of the most important dimensionless groups in fluid mechanics because it tells you which flow regime to expect.

Viscous dissipation and energy loss

Viscous dissipation is the conversion of mechanical (kinetic) energy into heat due to internal friction during fluid deformation. The rate of dissipation depends on viscosity and the square of velocity gradients, described by the dissipation function:

$\Phi = \mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)^2$

• In laminar flow, viscous dissipation causes a steady pressure drop along the flow direction. The Hagen-Poiseuille equation for pipe flow is a direct consequence of this.
• In turbulent flow, dissipation occurs primarily at the smallest scales of motion (Kolmogorov microscales), where velocity gradients are steepest. This leads to significantly higher energy losses and pressure drops compared to laminar flow at the same flow rate.

Applications of viscosity

Viscosity in lubrication and bearings

Lubrication depends on the viscosity of oils or greases to keep moving surfaces separated, reducing friction and wear. In hydrodynamic lubrication (used in journal bearings and thrust bearings), the viscous forces within the lubricant generate a pressure field that supports the applied load and maintains a thin fluid film between the surfaces.

Choosing the right lubricant viscosity involves a trade-off: it must be high enough to maintain the separating film under load, but low enough to minimize viscous dissipation and the associated energy losses and heat generation.

Viscosity in pipe flow and pressure drop

Viscosity directly determines how much pumping power you need to move fluid through a pipe.

For laminar flow, the Hagen-Poiseuille equation gives the pressure drop:

$\Delta P = \frac{8 \mu L Q}{\pi R^4}$

where $L$ is pipe length, $Q$ is volumetric flow rate, and $R$ is pipe radius. Notice that pressure drop is directly proportional to $\mu$ and inversely proportional to $R^4$, so even a small reduction in pipe radius dramatically increases the required pressure.

For turbulent flow, the Darcy-Weisbach equation applies:

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

where $f$ is the Darcy friction factor (which itself depends on Re and surface roughness), $D$ is pipe diameter, and $v$ is average velocity. Here, both density and viscosity matter (viscosity enters through the friction factor's dependence on Re).

Viscosity in aerodynamics and drag reduction

Viscosity contributes to the skin friction drag on objects moving through fluids. The wall shear stress is:

$\tau_w = \mu \left.\frac{\partial u}{\partial y}\right|_{y=0}$

This is the viscous shear stress right at the surface, and integrating it over the entire surface gives the total skin friction drag.

Engineers use several boundary layer control techniques to reduce this drag:

• Laminar flow airfoils are shaped to delay the transition to turbulence, keeping the boundary layer laminar (and lower drag) over a larger portion of the surface.
• Riblets are tiny grooved surfaces that reduce turbulent skin friction.
• Polymer additives injected into the flow can suppress turbulence and reduce drag in pipelines.

At very high speeds, viscous dissipation also causes aerodynamic heating, where kinetic energy converts to heat and raises the surface temperature of the object (a major concern for re-entry vehicles and hypersonic aircraft).

Viscosity of common fluids

Gases and vapors

Gases have viscosities roughly 1,000 times lower than typical liquids, usually in the range of $10^{-5}$ to $10^{-6}$ Pa·s at standard conditions. Unlike liquids, gas viscosity increases with temperature.

Some reference values at 20°C and atmospheric pressure:

• Air: $1.81 \times 10^{-5}$ Pa·s
• Helium: $1.96 \times 10^{-5}$ Pa·s
• Steam (100°C): $1.22 \times 10^{-5}$ Pa·s

Liquids and solutions

Liquids span a huge range of viscosities depending on molecular structure and intermolecular forces:

• Water at 25°C: $8.90 \times 10^{-4}$ Pa·s (~1 cP)
• Glycerin at 20°C: ~1.41 Pa·s
• Honey: 2 to 10 Pa·s (varies with type and temperature)

For solutions, dissolved substances can either raise or lower viscosity. Dissolving sugar in water increases viscosity (the sugar molecules interfere with flow), while adding alcohol to water decreases it.

Polymers and suspensions

Polymers and suspensions frequently exhibit non-Newtonian behavior because their microstructure responds to shear.

• Polymer melts and solutions (molten plastics, DNA solutions) are typically shear-thinning. As shear rate increases, long polymer chains align and disentangle, reducing resistance to flow.
• Particle suspensions (blood, cement paste) can be shear-thinning or shear-thickening depending on particle size, shape, concentration, and interactions.

These materials are often described using non-Newtonian models like the power law or Carreau model (covered below).

Temperature and pressure dependence

Arrhenius equation for temperature

For many liquids, the temperature dependence of viscosity follows the Arrhenius form:

$\mu = A \, e^{\frac{E_a}{RT}}$

where $A$ is a pre-exponential factor, $E_a$ is the activation energy for flow, $R$ is the universal gas constant, and $T$ is absolute temperature.

The physical picture: molecules need to overcome an energy barrier $E_a$ to slide past their neighbors. Higher temperature gives them more thermal energy to clear that barrier, so viscosity drops exponentially. The value of $E_a$ depends on the fluid's molecular structure; fluids with stronger intermolecular forces have higher activation energies and are more sensitive to temperature changes.

Pressure-viscosity coefficient

The Barus equation describes how viscosity increases with pressure:

$\mu = \mu_0 \, e^{\alpha P}$

where $\mu_0$ is viscosity at a reference pressure (usually atmospheric), $P$ is gauge pressure, and $\alpha$ is the pressure-viscosity coefficient.

For most liquids, $\alpha$ is positive, meaning viscosity rises with pressure. Under everyday conditions this effect is small, but it becomes critical in elastohydrodynamic lubrication (EHL), such as in rolling element bearings and gear teeth. Contact pressures in these applications can reach gigapascals, causing the lubricant viscosity to increase by several orders of magnitude and enabling it to support enormous loads.

Viscosity models and correlations

Sutherland's formula for gases

Sutherland's formula predicts how gas viscosity varies with temperature:

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

where $\mu_0$ is viscosity at reference temperature $T_0$, and $S$ is Sutherland's constant (specific to each gas). The formula comes from kinetic theory, modeling intermolecular interactions as a combination of hard-sphere collisions and a weak attractive potential.

It's accurate for most gases over a wide range, roughly 0°C to 1000°C, and is commonly used in aerodynamic and heat transfer calculations.

Andrade equation for liquids

The Andrade equation is a simpler empirical model for liquid viscosity:

$\mu = A \, e^{\frac{B}{T}}$

where $A$ and $B$ are fluid-specific constants and $T$ is absolute temperature. It has the same exponential form as the Arrhenius equation but treats the activation energy as a single lumped constant $B$. This works well as a first approximation for liquids with simple molecular structures.

Power law and Carreau models for non-Newtonian fluids

Power law model:

$\mu = K \dot{\gamma}^{n-1}$

where $K$ is the consistency index, $\dot{\gamma}$ is shear rate, and $n$ is the flow behavior index.

• $n < 1$: shear-thinning
• $n > 1$: shear-thickening
• $n = 1$: Newtonian (reduces to $\mu = K$)

The power law is simple and widely used, but it has a known limitation: it predicts infinite viscosity as $\dot{\gamma} \to 0$ and zero viscosity as $\dot{\gamma} \to \infty$, neither of which is physically realistic.

Carreau model:

$\mu = \mu_\infty + (\mu_0 - \mu_\infty)\left[1 + (\lambda \dot{\gamma})^2\right]^{\frac{n-1}{2}}$

where $\mu_0$ is the zero-shear viscosity, $\mu_\infty$ is the infinite-shear viscosity, $\lambda$ is a time constant, and $n$ is the flow behavior index.

The Carreau model fixes the power law's shortcomings. It predicts Newtonian plateaus at both very low and very high shear rates, with a shear-thinning transition region in between. This makes it much more realistic for polymers and complex fluids. The four parameters are determined by fitting to experimental rheological data from steady-shear or oscillatory tests.