8.5 Viscosity

Definition of viscosity

Viscosity characterizes a fluid's resistance to deformation under applied stress. It quantifies the internal friction within a fluid, describing how easily layers of fluid slide past one another. In statistical mechanics, viscosity bridges the gap between microscopic molecular behavior and the macroscopic flow properties you can observe and measure.

Fluid resistance to flow

Think of viscosity as a measure of how much a fluid "fights back" when you try to make it move. Honey has a high viscosity and requires significant force to pour, while water has a low viscosity and flows readily. Several factors influence viscosity:

• Temperature and pressure both affect how strongly molecules interact
• Molecular structure determines the strength and type of intermolecular forces
• Fluids with higher viscosity dissipate more energy as heat during flow

Shear stress vs. strain rate

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

$\eta = \frac{\tau}{\dot{\gamma}}$

where $\eta$ is the dynamic viscosity, $\tau$ is the shear stress (force per unit area applied parallel to the fluid surface), and $\dot{\gamma}$ is the strain rate (how fast adjacent fluid layers slide past each other).

For Newtonian fluids, this ratio is constant: double the shear rate, and the shear stress doubles proportionally. Many common fluids like water and air behave this way. Non-Newtonian fluids break this linear relationship, as covered later.

Microscopic origins

Viscosity isn't just a macroscopic label. It emerges directly from what molecules are doing at the microscopic level. The kinetic theory framework lets you trace viscosity back to molecular collisions and forces.

Molecular interactions

Intermolecular forces are a major contributor to viscosity, especially in liquids:

• Stronger interactions (hydrogen bonding, dipole-dipole forces) produce higher viscosity. Water, with its extensive hydrogen bonding network, is more viscous than many nonpolar liquids of similar molecular weight.
• Van der Waals forces dominate in nonpolar fluids like hydrocarbons, which generally have lower viscosities.
• Molecular size and shape matter too. Long-chain molecules get tangled and resist flow more than compact, spherical molecules.

Mean free path

The mean free path ($\lambda$) is the average distance a molecule travels between successive collisions. It connects directly to transport properties like viscosity.

• In gases, $\lambda$ is relatively large (on the order of tens of nanometers at standard conditions). Increasing pressure reduces $\lambda$ because molecules are packed more tightly.
• In liquids, $\lambda$ is extremely short since molecules are nearly in constant contact with their neighbors. This contributes to liquids having much higher viscosities than gases.
• The relationship between mean free path and viscosity is more nuanced than "shorter $\lambda$ means higher viscosity." In gases, viscosity actually increases as molecular collisions become more frequent, because momentum transfer between layers becomes more efficient.

Kinetic theory of viscosity

Kinetic theory provides the quantitative link between molecular motion and viscosity. The central idea: viscosity arises because molecules carry momentum from one fluid layer to another through their random thermal motion and collisions.

Maxwell-Boltzmann distribution

The Maxwell-Boltzmann speed distribution describes the probability of finding a molecule with a given speed in a gas at thermal equilibrium:

$f(v) = 4\pi\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \, e^{-mv^2 / 2k_B T}$

where $m$ is the molecular mass, $k_B$ is Boltzmann's constant, and $T$ is temperature.

This distribution tells you the range of speeds molecules have at a given temperature. At higher temperatures, the distribution broadens and shifts toward higher speeds. Since viscosity depends on how fast molecules move and how often they collide, this distribution is a key input for calculating transport coefficients.

Momentum transfer

Here's the physical picture of how viscosity works in a gas:

1. Imagine two adjacent layers of gas moving at slightly different velocities.
2. Molecules randomly cross from one layer to the other due to thermal motion.
3. A molecule moving from the faster layer into the slower layer carries extra forward momentum with it, speeding up the slower layer.
4. A molecule crossing the other way carries less momentum, slowing down the faster layer.
5. This net transfer of momentum between layers acts as an effective friction force, which is viscosity.

For an ideal gas, kinetic theory gives the viscosity as:

$\eta \approx \frac{1}{3} n m \bar{v} \lambda$

where $n$ is the number density, $m$ is molecular mass, $\bar{v}$ is the mean molecular speed, and $\lambda$ is the mean free path. This result has a surprising consequence for gases, discussed next.

Temperature dependence

Viscosity responds to temperature very differently in liquids and gases. This contrast reveals fundamentally different microscopic mechanisms at work.

Liquids vs. gases

Gases: Viscosity increases with temperature. Higher temperature means faster molecules ($\bar{v} \propto \sqrt{T}$), which transfer momentum more effectively between layers. Notably, for an ideal gas, viscosity is independent of pressure, because while increasing pressure raises the number density $n$, it decreases the mean free path $\lambda$ by the same factor, and the two effects cancel.

Liquids: Viscosity decreases with temperature. In liquids, flow requires molecules to overcome attractive forces from their neighbors. Higher temperature gives molecules more kinetic energy to escape these interactions, so the fluid flows more easily.

This opposite behavior is one of the clearest signatures that viscosity in gases is dominated by momentum transfer through collisions, while viscosity in liquids is dominated by intermolecular attractive forces.

Arrhenius equation

For many liquids, the temperature dependence of viscosity follows an Arrhenius-type relationship:

$\eta = A \, e^{E_a / RT}$

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

The activation energy represents the energy barrier a molecule must overcome to move past its neighbors. A large $E_a$ means viscosity drops steeply with increasing temperature. This equation works well for simple liquids over moderate temperature ranges, though it breaks down near the glass transition or for highly structured fluids.

Viscosity measurement techniques

Measuring viscosity accurately requires choosing the right method for the fluid type and viscosity range. Two broad categories cover most laboratory and industrial measurements.

Capillary viscometers

These instruments measure viscosity by timing how long a fluid takes to flow through a narrow tube under gravity or applied pressure.

• Poiseuille's law relates the volumetric flow rate through the capillary to the viscosity: $Q = \frac{\pi r^4 \Delta P}{8 \eta L}$, where $r$ is the tube radius, $\Delta P$ is the pressure drop, and $L$ is the tube length.
• Common types include Ostwald, Ubbelohde, and Cannon-Fenske viscometers.
• They directly measure kinematic viscosity ($\nu = \eta / \rho$). To get dynamic viscosity, multiply by the fluid's density.
• Best suited for Newtonian fluids with low to moderate viscosities.

Rotational viscometers

These measure the torque required to rotate an object (a cylinder, cone, or plate) immersed in the fluid.

• They can control and vary the shear rate, making them suitable for both Newtonian and non-Newtonian fluids.
• Common geometries: concentric cylinder (Couette), cone-and-plate, and parallel plate.
• They allow continuous measurement over a range of shear rates, which is essential for characterizing shear-dependent behavior.
• They cover a much wider viscosity range than capillary viscometers.

Types of viscosity

Different viscosity measures are used depending on the fluid and the application. Knowing which one to use prevents errors in calculations and comparisons.

Dynamic vs. kinematic viscosity

• Dynamic viscosity ($\eta$), also called absolute viscosity, is the ratio of shear stress to shear rate. Its SI unit is the Pascal-second (Pa·s). The older CGS unit is the poise (P), where $1 \text{ Pa·s} = 10 \text{ P}$.
• Kinematic viscosity ($\nu$) is dynamic viscosity divided by density: $\nu = \eta / \rho$. Its SI unit is $\text{m}^2/\text{s}$, and the CGS unit is the Stokes (St).
• Kinematic viscosity naturally appears in equations where gravitational or inertial forces matter, such as the Reynolds number: $Re = \frac{v L}{\nu}$.

Apparent viscosity

For non-Newtonian fluids, the ratio of shear stress to shear rate isn't constant. The apparent viscosity is simply that ratio evaluated at a specific shear rate:

$\eta_{\text{app}} = \frac{\tau}{\dot{\gamma}} \bigg|_{\text{at a given } \dot{\gamma}}$

This lets you compare non-Newtonian fluids (polymers, suspensions, emulsions) to Newtonian fluids under specific flow conditions. Always report the shear rate alongside the apparent viscosity, since the value changes depending on the conditions.

Non-Newtonian fluids

Non-Newtonian fluids have viscosities that depend on the applied shear rate or stress. They don't follow the simple linear relationship $\tau = \eta \dot{\gamma}$ with a constant $\eta$. Many real-world fluids fall into this category.

Shear-thinning fluids

These fluids become less viscous as shear rate increases. Common examples: ketchup, paint, and blood.

At the molecular level, shear-thinning often occurs because polymer chains or elongated particles align in the flow direction under shear, reducing resistance. The power-law model captures this behavior:

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

where $K$ is the consistency index and $n$ is the flow behavior index. For shear-thinning fluids, $n < 1$. The smaller $n$ is, the more dramatic the thinning effect.

Shear-thickening fluids

These fluids become more viscous as shear rate increases. The classic example is a concentrated cornstarch-water mixture: you can slowly stir it, but if you punch it, it resists like a solid.

The mechanism typically involves hydrocluster formation or particle jamming. At high shear rates, particles are forced together faster than the liquid between them can flow out of the way, creating transient solid-like clusters. In the power-law model, shear-thickening fluids have $n > 1$.

Viscosity in statistical mechanics

Statistical mechanics provides the theoretical foundation for deriving viscosity from first principles, starting from molecular interactions and the Boltzmann equation.

Chapman-Enskog theory

This theory systematically derives transport coefficients (viscosity, thermal conductivity, diffusion) from the Boltzmann equation using a perturbation expansion.

1. Start with the Boltzmann equation, which describes how the molecular velocity distribution evolves due to streaming and collisions.
2. Assume the distribution function is close to the local Maxwell-Boltzmann equilibrium, with small deviations.
3. Expand the distribution function in powers of the Knudsen number (ratio of mean free path to macroscopic length scale).
4. Solve order by order. The first-order correction gives expressions for transport coefficients in terms of collision integrals, which depend on the intermolecular potential.

For a gas of hard spheres with diameter $d$, Chapman-Enskog theory gives:

$\eta = \frac{5}{16 d^2} \sqrt{\frac{m k_B T}{\pi}}$

This confirms that gas viscosity increases as $\sqrt{T}$ and is independent of pressure, consistent with the simpler kinetic theory result. The theory is highly accurate for dilute gases but needs modifications (e.g., Enskog corrections) for dense gases and liquids.

Correlation functions

The Green-Kubo relations provide an alternative, more general route to transport coefficients. For viscosity:

$\eta = \frac{V}{k_B T} \int_0^{\infty} \langle \sigma_{xy}(0) \, \sigma_{xy}(t) \rangle \, dt$

where $\sigma_{xy}$ is the off-diagonal component of the stress tensor, $V$ is the system volume, and the angle brackets denote an equilibrium ensemble average.

This expression says viscosity equals the time integral of the stress autocorrelation function. Physically, it measures how long stress fluctuations persist in the fluid. This approach is powerful because it works for any fluid (not just dilute gases) and can be evaluated directly from molecular dynamics simulations.

Applications in physics

Viscosity appears throughout physics whenever fluid flow and energy dissipation are involved.

Fluid dynamics

The Navier-Stokes equations are the governing equations for viscous fluid flow. Viscosity enters as the coefficient in the diffusive (dissipative) terms. The Reynolds number compares inertial forces to viscous forces:

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

• Low $Re$ (below ~2000 for pipe flow): viscous forces dominate, producing smooth laminar flow.
• High $Re$: inertial forces dominate, and the flow becomes turbulent.

Viscosity also governs energy dissipation in flows. In any real fluid, kinetic energy is continuously converted to heat through viscous friction.

Boundary layer theory

Near a solid surface, the fluid velocity transitions from zero (at the surface, due to the no-slip condition) to the free-stream value over a thin region called the boundary layer. Ludwig Prandtl introduced this concept in 1904.

• The boundary layer thickness grows with viscosity: more viscous fluids have thicker boundary layers.
• Within the boundary layer, viscous effects are significant even if the bulk flow is nearly inviscid.
• Boundary layer separation (where the layer detaches from the surface) causes drag increases and is central to aerodynamic design.

Viscosity in everyday life

Viscosity shows up constantly in engineering and daily experience, often in ways you might not immediately connect to molecular physics.

Lubrication

Lubricants work by maintaining a thin viscous film between moving surfaces, preventing direct metal-to-metal contact.

• The viscosity index measures how much a lubricant's viscosity changes with temperature. A high viscosity index means the oil stays relatively stable across a wide temperature range.
• Multigrade oils (like 5W-30) are engineered to behave as low-viscosity fluids at cold temperatures (easy starting) and higher-viscosity fluids at operating temperatures (adequate protection).
• Selecting the wrong viscosity leads to either excessive friction (too viscous) or inadequate film protection (too thin).

Food science applications

Viscosity directly affects the texture and mouthfeel of food products. A thick sauce "coats" your mouth differently than a thin broth, and that difference is viscosity at work.

• Thickeners like cornstarch, xanthan gum, and gelatin modify viscosity to achieve desired consistency.
• Rheological measurements during processing help predict how a product will behave during mixing, pumping, and packaging.
• Many food products are non-Newtonian: yogurt is shear-thinning (it flows more easily when stirred), which affects both processing and the consumer experience.