Viscosity

Viscosity
Viscosity
	A simulation of liquids with different viscosities. The liquid on the left has lower viscosity than the liquid on the right.
Common symbols	η, μ
Derivations from; other quantities	μ = G·t
Dimension

In continuum mechanics, viscosity is a property of a fluid that quantifies the resistance force acting on fluids when there is relative motion between fluid parcels. This resistance force is caused by the stress in fluid parcels, which ideally is directly proportional to the strain rate (the time derivative of strain) that arises when fluid parcels are in relative motion, and the relative speed between the boundary between adjacent fluid parcels is zero. In liquids, viscosity arises from cohesive molecular forces, while in gases it results from molecular collisions.^[1]^[2] Except for the case of superfluidity, there is no fluid with zero viscosity, and thus all fluid flows involve viscous effects to some degree.^[1]

For liquids, it corresponds to the informal concept of thickness; for example, syrup has a higher viscosity than water.^[3] Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per metre squared, or pascal-seconds.^[2]

For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls.^[4] Some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity.

In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is negligible in certain cases. For example, the viscosity of a Newtonian fluid does not vary significantly with the rate of deformation.

Zero viscosity (no resistance to shear stress) is observed only at very low temperatures in superfluids; otherwise, the second law of thermodynamics requires all fluids to have positive viscosity.^[5]^[6] A fluid that has zero viscosity (non-viscous) is called ideal or inviscid.

For non-Newtonian fluids' viscosity, there are pseudoplastic, plastic, and dilatant flows that are time-independent, and there are thixotropic and rheopectic flows that are time-dependent.

Etymology

The word "viscosity" is derived from the Latin viscum ("mistletoe"). Viscum also referred to a viscous glue derived from mistletoe berries.^[7]

Definitions

Dynamic viscosity

In materials science and engineering, there is often interest in understanding the forces or stresses involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the deformation rate over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs.

Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.

In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed $u$ (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from $0$ at the bottom to $u$ at the top.^[8] Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.

In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to $u$ at the top. Moreover, the magnitude of the force, $F$ , acting on the top plate is found to be proportional to the speed $u$ and the area $A$ of each plate, and inversely proportional to their separation $y$ :

F=\mu A{\frac {u}{y}}.

The proportionality factor is the dynamic viscosity of the fluid, often simply referred to as the viscosity. It is denoted by the Greek letter mu ( $μ$ ). The dynamic viscosity has the dimensions $\mathrm {(mass/length)/time}$ , therefore resulting in the SI units and the derived units:

[\mu ]={\frac {\rm {kg}}{\rm {m{\cdot }s}}}={\frac {\rm {N}}{\rm {m^{2}}}}{\cdot }{\rm {s}}={\rm {Pa{\cdot }s}}=

pressure multiplied by time

=

energy per unit volume multiplied by time.

The aforementioned ratio $u/y$ is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction parallel to the normal vector of the plates (see illustrations to the right). If the velocity does not vary linearly with $y$ , then the appropriate generalization is:

\tau =\mu {\frac {\partial u}{\partial y}},

where $\tau =F/A$ , and $\partial u/\partial y$ is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what defines $\mu$ . It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form.

Use of the Greek letter mu ( $\mu$ ) for the dynamic viscosity (sometimes also called the absolute viscosity) is common among mechanical and chemical engineers, as well as mathematicians and physicists.^[9]^[10]^[11] However, the Greek letter eta ( $\eta$ ) is also used by chemists, physicists, and the IUPAC.^[12] The viscosity $\mu$ is sometimes also called the shear viscosity. However, at least one author discourages the use of this terminology, noting that $\mu$ can appear in non-shearing flows in addition to shearing flows.^[13]

Kinematic viscosity

In fluid dynamics, it is sometimes more appropriate to work in terms of kinematic viscosity (sometimes also called the momentum diffusivity), defined as the ratio of the dynamic viscosity ( $μ$ ) over the density of the fluid ( $ρ$ ). It is usually denoted by the Greek letter nu ( $ν$ ):

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

and has the dimensions $\mathrm {(length)^{2}/time}$ , therefore resulting in the SI units and the derived units:

[\nu ]=\mathrm {\frac {m^{2}}{s}} =\mathrm {{\frac {N{\cdot }m}{kg}}{\cdot }s} =\mathrm {{\frac {J}{kg}}{\cdot }s} =

specific energy multiplied by time

=

energy per unit mass multiplied by time.

General definition

In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity.^[14] (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as

\tau _{ij}=\sum _{k}\sum _{\ell }\mu _{ijk\ell }{\frac {\partial v_{k}}{\partial r_{\ell }}},

where $\mu _{ijk\ell }$ is a viscosity tensor that maps the velocity gradient tensor $\partial v_{k}/\partial r_{\ell }$ onto the viscous stress tensor $\tau _{ij}$ .^[15] Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" $\mu _{ijkl}$ in total. However, assuming that the viscosity rank-2 tensor is isotropic reduces these 81 coefficients to three independent parameters $\alpha$ , $\beta$ , $\gamma$ :

\mu _{ijk\ell }=\alpha \delta _{ij}\delta _{k\ell }+\beta \delta _{ik}\delta _{j\ell }+\gamma \delta _{i\ell }\delta _{jk},

and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus $\beta =\gamma$ , leaving only two independent parameters.^[14] The most usual decomposition is in terms of the standard (scalar) viscosity $\mu$ and the bulk viscosity $\kappa$ such that $\alpha =\kappa -{\tfrac {2}{3}}\mu$ and $\beta =\gamma =\mu$ . In vector notation this appears as:

{\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {v} +(\nabla \mathbf {v} )^{\mathrm {T} }\right]-\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {\delta } ,

where $\mathbf {\delta }$ is the unit tensor.^[13]^[16] This equation can be thought of as a generalized form of Newton's law of viscosity.

The bulk viscosity (also called volume viscosity) expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of $\kappa$ is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies $\nabla \cdot \mathbf {v} =0$ and so the term containing $\kappa$ drops out. Moreover, $\kappa$ is often assumed to be negligible for gases since it is $0$ in a monatomic ideal gas.^[13] One situation in which $\kappa$ can be important is the calculation of energy loss in sound and shock waves, described by Stokes' law of sound attenuation, since these phenomena involve rapid expansions and compressions.

The defining equations for viscosity are not fundamental laws of nature, so their usefulness, as well as methods for measuring or calculating the viscosity, must be established using separate means. A potential issue is that viscosity depends, in principle, on the full microscopic state of the fluid, which encompasses the positions and momenta of every particle in the system.^[17] Such highly detailed information is typically not available in realistic systems. However, under certain conditions most of this information can be shown to be negligible. In particular, for Newtonian fluids near equilibrium and far from boundaries (bulk state), the viscosity depends only space- and time-dependent macroscopic fields (such as temperature and density) defining local equilibrium.^[17]^[18]

Nevertheless, viscosity may still carry a non-negligible dependence on several system properties, such as temperature, pressure, and the amplitude and frequency of any external forcing. Therefore, precision measurements of viscosity are only defined with respect to a specific fluid state.^[19] To standardize comparisons among experiments and theoretical models, viscosity data is sometimes extrapolated to ideal limiting cases, such as the zero shear limit, or (for gases) the zero density limit.

Momentum transport

Transport theory provides an alternative interpretation of viscosity in terms of momentum transport: viscosity is the material property which characterizes momentum transport within a fluid, just as thermal conductivity characterizes heat transport, and (mass) diffusivity characterizes mass transport.^[20] This perspective is implicit in Newton's law of viscosity, $\tau =\mu (\partial u/\partial y)$ , because the shear stress $\tau$ has units equivalent to a momentum flux, i.e., momentum per unit time per unit area. Thus, $\tau$ can be interpreted as specifying the flow of momentum in the $y$ direction from one fluid layer to the next. Per Newton's law of viscosity, this momentum flow occurs across a velocity gradient, and the magnitude of the corresponding momentum flux is determined by the viscosity.

The analogy with heat and mass transfer can be made explicit. Just as heat flows from high temperature to low temperature and mass flows from high density to low density, momentum flows from high velocity to low velocity. These behaviors are all described by compact expressions, called constitutive relations, whose one-dimensional forms are given here:

{\begin{aligned}\mathbf {J} &=-D{\frac {\partial \rho }{\partial x}}&&{\text{(Fick's law of diffusion)}}\\[5pt]\mathbf {q} &=-k_{t}{\frac {\partial T}{\partial x}}&&{\text{(Fourier's law of heat conduction)}}\\[5pt]\tau &=\mu {\frac {\partial u}{\partial y}}&&{\text{(Newton's law of viscosity)}}\end{aligned}}

where $\rho$ is the density, $\mathbf {J}$ and $\mathbf {q}$ are the mass and heat fluxes, and $D$ and $k_{t}$ are the mass diffusivity and thermal conductivity.^[21] The fact that mass, momentum, and energy (heat) transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the microscopic level in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics.

Newtonian and non-Newtonian fluids

Newton's law of viscosity is not a fundamental law of nature, but rather a constitutive equation (like Hooke's law, Fick's law, and Ohm's law) which serves to define the viscosity $\mu$ . Its form is motivated by experiments which show that for a wide range of fluids, $\mu$ is independent of strain rate. Such fluids are called Newtonian. Gases, water, and many common liquids can be considered Newtonian in ordinary conditions and contexts. However, there are many non-Newtonian fluids that significantly deviate from this behavior. For example:

Shear-thickening (dilatant) liquids, whose viscosity increases with the rate of shear strain.
Shear-thinning liquids, whose viscosity decreases with the rate of shear strain.
Thixotropic liquids, that become less viscous over time when shaken, agitated, or otherwise stressed.
Rheopectic liquids, that become more viscous over time when shaken, agitated, or otherwise stressed.
Bingham plastics that behave as a solid at low stresses but flow as a viscous fluid at high stresses.

Trouton's ratio is the ratio of extensional viscosity to shear viscosity. For a Newtonian fluid, the Trouton ratio is 3.^[22]^[23] Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic.^[24]

Viscosity may also depend on the fluid's physical state (temperature and pressure) and other, external, factors. For gases and other compressible fluids, it depends on temperature and varies very slowly with pressure. The viscosity of some fluids may depend on other factors. A magnetorheological fluid, for example, becomes thicker when subjected to a magnetic field, possibly to the point of behaving like a solid.

In solids

The viscous forces that arise during fluid flow are distinct from the elastic forces that occur in a solid in response to shear, compression, or extension stresses. While in the latter the stress is proportional to the amount of shear deformation, in a fluid it is proportional to the rate of deformation over time. For this reason, James Clerk Maxwell used the term fugitive elasticity for fluid viscosity.

However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even granite) will flow like liquids, albeit very slowly, even under arbitrarily small stress.^[25] Such materials are best described as viscoelastic—that is, possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation).

Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The extensional viscosity is a linear combination of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers.

In geology, earth materials that exhibit viscous deformation at least three orders of magnitude greater than their elastic deformation are sometimes called rheids.^[26]

Measurement

Viscosity is measured with various types of viscometers and rheometers. Close temperature control of the fluid is essential to obtain accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. A rheometer is used for fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer.^[27]

For some fluids, the viscosity is constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (non-Newtonian fluids) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate.

One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.

In coating industries, viscosity may be measured with a cup in which the efflux time is measured. There are several sorts of cup—such as the Zahn cup and the Ford viscosity cup—with the usage of each type varying mainly according to the industry.

Also used in coatings, a Stormer viscometer employs load-based rotation to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.

Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method, the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy.^{[citation needed]}

Extensional viscosity can be measured with various rheometers that apply extensional stress.

Volume viscosity can be measured with an acoustic rheometer.

Apparent viscosity is a calculation derived from tests performed on drilling fluid used in oil or gas well development. These calculations and tests help engineers develop and maintain the properties of the drilling fluid to the specifications required.

Nanoviscosity (viscosity sensed by nanoprobes) can be measured by fluorescence correlation spectroscopy.^[28]

Units

The SI unit of dynamic viscosity is the newton-second per metre squared (N·s/m²), also frequently expressed in the equivalent forms pascal-second (Pa·s), kilogram per meter per second (kg·m⁻¹·s⁻¹) and poiseuille (Pl). The CGS unit is the poise (P, or g·cm⁻¹·s⁻¹ = 0.1 Pa·s),^[29] named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in ASTM standards, as centipoise (cP). The centipoise is convenient because the viscosity of water at 20 °C is about 1 cP, and one centipoise is equal to the SI millipascal second (mPa·s).

The SI unit of kinematic viscosity is metre squared per second (m²/s), whereas the CGS unit for kinematic viscosity is the stokes (St, or cm²·s⁻¹ = 0.0001 m²·s⁻¹), named after Sir George Gabriel Stokes.^[30] In U.S. usage, stoke is sometimes used as the singular form. The submultiple centistokes (cSt) is often used instead, 1 cSt = 1 mm²·s⁻¹ = 10⁻⁶ m²·s⁻¹. 1 cSt is 1 cP divided by 1000 kg/m^3, close to the density of water. The kinematic viscosity of water at 20 °C is about 1 cSt.

The most frequently used systems of US customary, or Imperial, units are the British Gravitational (BG) and English Engineering (EE). In the BG system, dynamic viscosity has units of pound-seconds per square foot (lb·s/ft²), and in the EE system it has units of pound-force-seconds per square foot (lbf·s/ft²). The pound and pound-force are equivalent; the two systems differ only in how force and mass are defined. In the BG system the pound is a basic unit from which the unit of mass (the slug) is defined by Newton's second law, whereas in the EE system the units of force and mass (the pound-force and pound-mass respectively) are defined independently through the second law using the proportionality constant g_c.

Kinematic viscosity has units of square feet per second (ft²/s) in both the BG and EE systems.

Nonstandard units include the reyn (lbf·s/in²), a British unit of dynamic viscosity.^[31] In the automotive industry the viscosity index is used to describe the change of viscosity with temperature.

The reciprocal of viscosity is fluidity, usually symbolized by $\phi =1/\mu$ or $F=1/\mu$ , depending on the convention used, measured in reciprocal poise (P⁻¹, or cm·s·g⁻¹), sometimes called the rhe. Fluidity is seldom used in engineering practice.^{[citation needed]}

At one time the petroleum industry relied on measuring kinematic viscosity by means of the Saybolt viscometer, and expressing kinematic viscosity in units of Saybolt universal seconds (SUS).^[32] Other abbreviations such as SSU (Saybolt seconds universal) or SUV (Saybolt universal viscosity) are sometimes used. Kinematic viscosity in centistokes can be converted from SUS according to the arithmetic and the reference table provided in ASTM D 2161.

Molecular origins

Momentum transport in gases is mediated by discrete molecular collisions, and in liquids by attractive forces that bind molecules close together.^[20] Because of this, the dynamic viscosities of liquids are typically much larger than those of gases. In addition, viscosity tends to increase with temperature in gases and decrease with temperature in liquids.

Above the liquid-gas critical point, the liquid and gas phases are replaced by a single supercritical phase. In this regime, the mechanisms of momentum transport interpolate between liquid-like and gas-like behavior. For example, along a supercritical isobar (constant-pressure surface), the kinematic viscosity decreases at low temperature and increases at high temperature, with a minimum in between.^[33]^[34] Kostya Trachenko and Vadim Brazhkin [ru] provided a rough estimate for the value at the minimum, given by

\nu _{\text{min}}={\frac {1}{4\pi }}{\frac {\hbar }{\sqrt {m_{\text{e}}m}}}

where $\hbar$ is the Planck constant, $m_{\text{e}}$ is the electron mass, and $m$ is the molecular mass.^[34]

In general, however, the viscosity of a system depends in detail on how the molecules constituting the system interact, and there are no simple but correct formulas for it. The simplest exact expressions are the Green–Kubo relations for the linear shear viscosity or the transient time correlation function expressions derived by Evans and Morriss in 1988.^[35] Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of molecular dynamics computer simulations. Somewhat more progress can be made for a dilute gas, as elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the equations of motion of the gas molecules. An example of such a treatment is Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the Boltzmann equation.^[18]

Pure gases

Elementary calculation of viscosity for a dilute gas

Consider a dilute gas moving parallel to the $x$ -axis with velocity $u(y)$ that depends only on the $y$ coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density.

Under these assumptions, the $x$ velocity of a molecule passing through $y=0$ is equal to whatever velocity that molecule had when its mean free path $\lambda$ began. Because $\lambda$ is typically small compared with macroscopic scales, the average $x$ velocity of such a molecule has the form

$u(0)\pm \alpha \lambda {\frac {du}{dy}}(0),$

where $\alpha$ is a numerical constant on the order of $1$ . (Some authors estimate $\alpha =2/3$ ;^[20]^[36] on the other hand, a more careful calculation for rigid elastic spheres gives $\alpha \simeq 0.998$ .) Next, because half the molecules on either side are moving towards $y=0$ , and doing so on average with half the average molecular speed $(8k_{\text{B}}T/\pi m)^{1/2}$ , the momentum flux from either side is

${\frac {1}{4}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \left(u(0)\pm \alpha \lambda {\frac {du}{dy}}(0)\right).$ The net momentum flux at $y=0$ is the difference of the two: $-{\frac {1}{2}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \alpha \lambda {\frac {du}{dy}}(0).$

According to the definition of viscosity, this momentum flux should be equal to $-\mu {\frac {du}{dy}}(0)$ , which leads to

$\mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.$

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature $T$ and density $\rho$ gives

\mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},

where $k_{\text{B}}$ is the Boltzmann constant, $m$ the molecular mass, and $\alpha$ a numerical constant on the order of $1$ . The quantity $\lambda$ , the mean free path, measures the average distance a molecule travels between collisions. Even without a priori knowledge of $\alpha$ , this expression has nontrivial implications. In particular, since $\lambda$ is typically inversely proportional to density and increases with temperature, $\mu$ itself should increase with temperature and be independent of density at fixed temperature. In fact, both of these predictions persist in more sophisticated treatments, and accurately describe experimental observations. By contrast, liquid viscosity typically decreases with temperature.^[20]^[36]

For rigid elastic spheres of diameter $\sigma$ , $\lambda$ can be computed, giving

\mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{\text{B}}mT}}{\sigma ^{2}}}.

In this case $\lambda$ is independent of temperature, so $\mu \propto T^{1/2}$ . For more complicated molecular models, however, $\lambda$ depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.^[37]

Chapman–Enskog theory

A technique developed by

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