Viscosity

Viscosity is a property of fluids that measures their resistance to deformation during flow. When neighboring fluid layers move at different velocities, viscous stresses oppose their relative motion. In simple shear flow, dynamic viscosity relates shear stress to the velocity gradient perpendicular to the layers. Both liquids and gases possess viscosity; its value depends on the material and its physical conditions. The study of flow and deformation, including viscosity, belongs to rheology. (goldbook.iupac.org)

Definition and physical meaning

Consider a fluid between two parallel plates, one stationary and the other moving steadily. Moving the upper plate requires a tangential force because the fluid resists shearing. The shear stress, $\tau$, is the force divided by the plate area. For a Newtonian fluid, it is proportional to the shear rate, $\dot{\gamma}$:

Here, $u$ is velocity parallel to the plates, $y$ is distance perpendicular to them, and $\mu$, also written $\eta$, is dynamic viscosity. The derivative $du/dy$ describes how rapidly velocity changes across the fluid. A larger viscosity therefore requires greater stress to maintain the same shear rate. (grc.nasa.gov)

Newtonian behavior means that viscosity is independent of shear rate at fixed temperature and pressure. It does not mean that viscosity remains constant when these conditions change. More generally, Newtonian constitutive relations connect the components of the stress tensor linearly to spatial derivatives of velocity. (old.goldbook.iupac.org)

Dynamic and kinematic viscosity

In the International System of Units, dynamic viscosity is measured in pascal-seconds, Pa·s, equivalent to kg·m$^{-1}$·s$^{-1}$. The older unit poise, P, equals 0.1 Pa·s; one centipoise, cP, equals one millipascal-second, mPa·s. These conversions are especially useful for liquid-property tables. (old.iupac.org)

Kinematic viscosity, $\nu$, is dynamic viscosity divided by density, $\rho$:

Its SI unit is m²/s. The traditional unit stokes, St, equals $10^{-4}$ m²/s, and one centistokes, cSt, equals one mm²/s. Dynamic viscosity characterizes the stress associated with deformation; kinematic viscosity expresses viscous transport relative to the fluid’s density. Materials with equal dynamic viscosities can consequently have different kinematic viscosities. (grc.nasa.gov)

As a familiar scale, the dynamic viscosity of water at 20 °C is approximately 1 mPa·s. A quoted viscosity should therefore be accompanied by its temperature and, where relevant, pressure and measurement conditions. (nvlpubs.nist.gov)

Molecular origins and temperature dependence

In a dilute gas, viscosity arises from the transport of momentum between regions moving at different average velocities. Random molecular motion carries momentum across these regions, while collisions redistribute it. This process can be understood as a form of momentum diffusion. Elementary ideal-gas kinetic theory predicts that dynamic viscosity increases with temperature and is approximately independent of pressure within the dilute, continuum regime. These approximations fail when molecular travel distances become comparable to the container dimensions. (farside.ph.utexas.edu)

Liquid viscosity is strongly influenced by interactions between closely spaced molecules and by molecular shape. Stronger intermolecular attractions generally increase resistance to rearrangement. Heating usually lowers liquid viscosity because increased molecular kinetic energy facilitates movement past neighboring molecules. This contrasts with the usual temperature trend in dilute gases. These are general tendencies rather than universal formulas applicable to every material and state. (ch301.cm.utexas.edu)

Non-Newtonian behavior

For a non-Newtonian fluid, a single constant viscosity may not describe its response. An apparent shear viscosity can be defined as $\tau/\dot{\gamma}$, but its value may vary with shear rate or deformation history. In shear thinning, steady-flow viscosity decreases as shear rate increases; in shear thickening, it increases. Some materials also exhibit a yield stress, represented in constitutive models as a threshold below which sustained flow does not occur. (nvlpubs.nist.gov)

Time-dependent behavior must be distinguished from shear-rate dependence. Thixotropic materials exhibit a reversible, time-dependent reduction in viscosity under shearing, with recovery when shearing is reduced or stopped. Consequently, rotational tests may characterize apparent viscosity, shear thinning, and thixotropy separately rather than reporting one material constant. (store.astm.org)

Measurement

A viscometer determines viscosity from a measured flow or mechanical response. Capillary instruments relate flow through a narrow tube to the driving pressure or elapsed flow time. Rotational instruments shear a sample between surfaces and measure torque and rotational speed; controlled geometries permit conversion to shear stress and shear rate. They are particularly useful for concentrated suspensions, gels, and pastes. (nvlpubs.nist.gov)

Falling-ball methods infer viscosity from a sphere’s motion through a liquid: under comparable conditions, greater viscosity produces slower descent. Temperature control is essential because viscosity changes with temperature. For non-Newtonian samples, results also depend on the deformation imposed by the measurement method. (openstax.org)

Role in fluid motion

Viscosity enters the Navier–Stokes equations through viscous stresses governing momentum transport. Its importance relative to inertial effects is expressed by the Reynolds number:

where $U$ and $L$ are characteristic velocity and length scales. This dimensionless ratio allows flows of different sizes and fluids to be compared. (www1.grc.nasa.gov)

Near a solid surface, viscosity helps establish a boundary layer in which velocity changes rapidly with distance from the surface. Its thickness, development, and possible separation affect aerodynamic forces. Thus, even when inertial effects dominate much of a flow, viscosity can remain essential near boundaries and in determining drag. (grc.nasa.gov)

References

1. Viscosity — NASA Glenn Research Centergrc.nasa.gov
2. Absolute Viscosity of Water at 20° Cnvlpubs.nist.gov
3. Viscosity — University of Texas at Austinfarside.ph.utexas.edu
4. Liquid Properties — University of Texas at Austinch301.cm.utexas.edu
5. NIST Recommended Practice Guide: The Use of Nomenclature in Dispersion Science and Technologynvlpubs.nist.gov
6. D2196: Standard Test Methods for Rheological Properties of Non-Newtonian Materials by Rotational Viscometerstore.astm.org
7. 2 Properties of Liquids — OpenStax Chemistryopenstax.org
8. Reynolds Number — NASA Glenn Research Centergrc.nasa.gov