The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.^{[1]}
Viscosity can be conceptualized as quantifying the frictional force that arises between adjacent layers of fluid that are in relative motion. For instance, when a fluid is forced through a tube, it flows more quickly near the tube's axis than near its walls. In such a case, experiments show that some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow through the tube. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion: the strength of this force is proportional to the viscosity.
A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity;^{[2]}^{[3]} such fluids are technically said to be viscous or viscid. A fluid with a relatively high viscosity, such as pitch, may appear to be a solid.
Viscosity  

A simulation of liquids with different viscosities. The liquid on the right has higher viscosity than the liquid on the left.  
Common symbols  η, μ 
Derivations from other quantities  μ = G·t 
The word "viscosity" is derived from the Latin "viscum", meaning mistletoe and also a viscous glue made from mistletoe berries.^{[4]}
In materials science and engineering, one is often interested 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 rate of change of the deformation 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 (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 at the bottom to at the top.^{[5]} 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 at the top. Moreover, the magnitude of the force acting on the top plate is found to be proportional to the speed and the area of each plate, and inversely proportional to their separation :
The proportionality factor is the viscosity of the fluid, with units of (pascalsecond). The ratio is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates (see illustrations to the right). If the velocity does not vary linearly with , then the appropriate generalization is
where , and 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 . It is a special case of the general definition of viscosity (see below), which can be expressed in coordinatefree form.
Use of the Greek letter mu () for the viscosity is common among mechanical and chemical engineers, as well as physicists.^{[6]}^{[7]}^{[8]} However, the Greek letter eta () is also used by chemists, physicists, and the IUPAC.^{[9]} The viscosity is sometimes also referred to as the shear viscosity. However, at least one author discourages the use of this terminology, noting that can appear in nonshearing flows in addition to shearing flows.^{[10]}
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.^{[11]} (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as
where is a viscosity tensor that maps the strain rate tensor onto the viscous stress tensor .^{[12]} Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" in total. However, due to spatial symmetries these coefficients are not all independent. For instance, for isotropic Newtonian fluids, the 81 coefficients can be reduced to 2 independent parameters. The most usual decomposition yields the standard (scalar) viscosity and the bulk viscosity :
where is the unit tensor, and the dagger denotes the transpose.^{[13]}^{[14]} 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 is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies and so the term containing drops out. Moreover, is often assumed to be negligible for gases since it is in a monoatomic ideal gas.^{[13]} One situation in which 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.
It is worth emphasizing that the above expressions are not fundamental laws of nature, but rather definitions of viscosity. As such, their utility for any given material, as well as means for measuring or calculating the viscosity, must be established using separate means.
In fluid dynamics, it is common to work in terms of the kinematic viscosity (also called "momentum diffusivity"), defined as the ratio of the viscosity μ to the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν) and has units :
Consistent with this nomenclature, the viscosity is frequently called the dynamic viscosity or absolute viscosity, and has units force × time/area.
Transport theory provides an alternate 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.^{[15]} To see this, note that in Newton's law of viscosity, , the shear stress has units equivalent to a momentum flux, i.e. momentum per unit time per unit area. Thus, can be interpreted as specifying the flow of momentum in the 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 onedimensional forms are given here:
where is the density, and are the mass and heat fluxes, and and are the mass diffusivity and thermal conductivity.^{[16]}
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.
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 . Its form is motivated by experiments which show that for a wide range of fluids, 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 nonNewtonian fluids that significantly deviate from this behavior. For example:
The Trouton ratio or Trouton's ratio is the ratio of extensional viscosity to shear viscosity.^{[17]} For a Newtonian fluid, the Trouton ratio is 3.^{[18]}
Shearthinning liquids are very commonly, but misleadingly, described as thixotropic.^{[19]}
Even for a Newtonian fluid, the viscosity usually depends on its composition and temperature. 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.
The viscous forces that arise during fluid flow must not be confused with the elastic forces that arise 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, 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.^{[20]} Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelastic.
Indeed, some authors have claimed that amorphous solids, such as glass and many polymers, are actually liquids with a very high viscosity (greater than 10^{12} Pa·s). ^{[21]} However, other authors dispute this hypothesis, claiming instead that there is some threshold for the stress, below which most solids will not flow at all,^{[22]} and that alleged instances of glass flow in window panes of old buildings are due to the crude manufacturing process of older eras rather than to the viscosity of glass.^{[23]}
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.^{[24]}
Viscosity is measured with various types of viscometers and rheometers. A rheometer is used for those 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. Close temperature control of the fluid is essential to acquire accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C.
For some fluids, the viscosity is constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (nonNewtonian fluids) cannot be described by a single number. NonNewtonian 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. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.^{[25]}
Also used in coatings, a Stormer viscometer uses loadbased rotation in order 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.
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.
The SI unit of dynamic viscosity is the pascalsecond (Pa·s), or equivalently kilogram per meter per second (kg·m^{−1}·s^{−1}). The CGS unit is called the poise^{[26]} (P), named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in ASTM standards, as centipoise (cP) since the latter is equal to the SI multiple millipascal seconds (mPa·s).
The SI unit of kinematic viscosity is square meter per second (m^{2}/s), whereas the CGS unit for kinematic viscosity is the stokes (St), named after Sir George Gabriel Stokes.^{[27]} In U.S. usage, stoke is sometimes used as the singular form. The submultiple centistokes (cSt) is often used instead.
The reciprocal of viscosity is fluidity, usually symbolized by or , 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.
Nonstandard units include the reyn, a British unit of dynamic viscosity. In the automotive industry the viscosity index is used to describe the change of viscosity with temperature.
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).^{[28]} 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.^{[29]}
In general, the viscosity of a system depends in detail on how the molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. 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 1985.^{[30]} 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. On the other hand, much more progress can be made for a dilute gas. Even 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 coarsegraining 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.^{[31]}
Momentum transport in gases is generally mediated by discrete molecular collisions, and in liquids by attractive forces which bind molecules close together.^{[15]} Because of this, the dynamic viscosities of liquids are typically much larger than those of gases.
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 and density gives
where is the Boltzmann constant, the molecular mass, and a numerical constant on the order of . The quantity , the mean free path, measures the average distance a molecule travels between collisions. Even without a priori knowledge of , this expression has interesting implications. In particular, since is typically inversely proportional to density and increases with temperature, 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. Note that this behavior runs counter to common intuition regarding liquids, for which viscosity typically decreases with temperature.^{[15]}^{[32]}
For rigid elastic spheres of diameter , can be computed, giving
In this case is independent of temperature, so . For more complicated molecular models, however, depends on temperature in a nontrivial 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 realworld gases.^{[33]}
A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of .^{[31]} It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions.^{[34]} As such, their technique allows accurate calculation of for more realistic molecular models, such as those incorporating intermolecular attraction rather than just hardcore repulsion.
It turns out that a more realistic modeling of interactions is essential for accurate prediction of the temperature dependence of , which experiments show increases more rapidly than the trend predicted for rigid elastic spheres.^{[15]} Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model,^{[35]} which describes rigid elastic spheres with weak mutual attraction. In such a case, the attractive force can be treated perturbatively, which leads to a particularly simple expression for :
where is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as
where is the viscosity at temperature . If is known from experiments at and at least one other temperature, then can be calculated. It turns out that expressions for obtained in this way are accurate for a number of gases over a sizable range of temperatures. On the other hand, Chapman and Cowling^{[31]} argue that this success does not imply that molecules actually interact according to the Sutherland model. Rather, they interpret the prediction for as a simple interpolation which is valid for some gases over fixed ranges of temperature, but otherwise does not provide a picture of intermolecular interactions which is fundamentally correct and general. Slightly more sophisticated models, such as the Lennard–Jones potential, may provide a better picture, but only at the cost of a more opaque dependence on temperature. In some systems the assumption of spherical symmetry must be abandoned as well, as is the case for vapors with highly polar molecules like H_{2}O.^{[36]}^{[37]}
In the kineticmolecular picture, a nonzero bulk viscosity arises in gases whenever there are nonnegligible relaxational timescales governing the exchange of energy between the translational energy of molecules and their internal energy, e.g. rotational and vibrational. As such, the bulk viscosity is for a monatomic ideal gas, in which the internal energy of molecules in negligible, but is nonzero for a gas like carbon dioxide, whose molecules possess both rotational and vibrational energy.^{[38]}^{[39]}
In contrast with gases, there is no simple yet accurate picture for the molecular origins of viscosity in liquids.
At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions.^{[40]}
Building on this visualization, a simple theory can be constructed in analogy with the discrete structure of a solid: groups of molecules in a liquid are visualized as forming "cages" which surround and enclose single molecules.^{[41]} These cages can be occupied or unoccupied, and stronger molecular attraction corresponds to stronger cages. Due to random thermal motion, a molecule "hops" between cages at a rate which varies inversely with the strength of molecular attractions. In equilibrium these "hops" are not biased in any direction. On the other hand, in order for two adjacent layers to move relative to each other, the "hops" must be biased in the direction of the relative motion. The force required to sustain this directed motion can be estimated for a given shear rate, leading to


(1) 
where is the Avogadro constant, is the Planck constant, is the volume of a mole of liquid, and is the normal boiling point. This result has the same form as the widespread and accurate empirical relation


(2) 
where and are constants fit from data.^{[41]}^{[42]} One the other hand, several authors express caution with respect to this model. Errors as large as 30% can be encountered using equation (1), compared with fitting equation (2) to experimental data.^{[41]} More fundamentally, the physical assumptions underlying equation (1) have been extensively criticized.^{[43]} It has also been argued that the exponential dependence in equation (1) does not necessarily describe experimental observations more accurately than simpler, nonexponential expressions.^{[44]}^{[45]}
In light of these shortcomings, the development of a less adhoc model is a matter of practical interest. Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is IrvingKirkwood theory.^{[46]} On the other hand, such expressions are given as averages over multiparticle correlation functions and are therefore difficult to apply in practice.
In general, empirically derived expressions (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids.^{[47]}
The same molecularkinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the ChapmanEnskog approach the viscosity of a binary mixture of gases can be written in terms of the individual component viscosities , their respective volume fractions, and the intermolecular interactions.^{[48]} As for the singlecomponent gas, the dependence of on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in terms of elementary functions. To obtain usable expressions for which reasonably match experimental data, the collisional integrals typically must be evaluated using some combination of analytic calculation and empirical fitting. An example of such a procedure is the Sutherland approach for the singlecomponent gas, discussed above.
As for pure liquids, the viscosity of a blend of liquids is difficult to predict from molecular principles. One method is to extend the molecular "cage" theory presented above for a pure liquid. This can be done with varying levels of sophistication. One useful expression resulting from such an analysis is the LedererRoegiers equation for a binary mixture:
where is an empirical parameter, and and are the respective mole fractions and viscosities of the component liquids.^{[49]}
Since blending is an important process in the lubricating and oil industries, a variety of empirical and propriety equations exist for predicting the viscosity of a blend, besides those stemming directly from molecular theory.^{[49]}
In a suspension of solid particles (e.g. micronsize spheres suspended in oil), an effective viscosity can be defined in terms of stress and strain components which are averaged over a volume large compared with the distance between the suspended particles, but small with respect to macroscopic dimensions.^{[50]} Such suspensions generally exhibit nonNewtonian behavior. However, for dilute systems in steady flows, the behavior is Newtonian and expressions for can be derived directly from the particle dynamics. In a very dilute system, with volume fraction , interactions between the suspended particles can be ignored. In such a case one can explicitly calculate the flow field around each particle independently, and combine the results to obtain . For spheres, this results in the Einstein equation:
where is the viscosity of the suspending liquid. The linear dependence on is a direct consequence of neglecting interparticle interactions; in general, one will have:
where the coefficient may depend on the particle shape (e.g. spheres, rods, disks).^{[51]} Experimental determination of the precise value of is difficult, however: even the prediction for spheres has not been conclusively validated, with various experiments finding values in the range . This deficiency has been attributed to difficulty in controlling experimental conditions.^{[52]}
In denser suspensions, acquires a nonlinear dependence on , which indicates the importance of interparticle interactions. Various analytical and semiempirical schemes exist for capturing this regime. At the most basic level, a term quadratic in is added to :
and the coefficient is fit from experimental data or approximated from the microscopic theory. In general, however, one should be cautious in applying such simple formulas since nonNewtonian behavior appears in dense suspensions ( for spheres),^{[52]} or in suspensions of elongated or flexible particles.^{[50]}
There is a distinction between a suspension of solid particles, described above, and an emulsion. The latter is a suspension of tiny droplets, which themselves may exhibit internal circulation. The presence of internal circulation can noticeably decrease the observed effective viscosity, and different theoretical or semiempirical models must be used.^{[53]}
In the high and low temperature limits, viscous flow in amorphous materials (e.g. in glasses and melts)^{[55]}^{[56]}^{[57]} has the Arrhenius form:
where Q is a relevant activation energy, given in terms of molecular parameters; T is temperature; R is the molar gas constant; and A is approximately a constant. The activation energy Q takes a different value depending on whether the high or low temperature limit is being considered: it changes from a high value Q_{H} at low temperatures (in the glassy state) to a low value Q_{L} at high temperatures (in the liquid state).
For intermediate temperatures, varies nontrivially with temperature and the simple Arrhenius form fails. On the other hand, the twoexponential equation
where , , , are all constants, provides a good fit to experimental data over the entire range of temperatures, while at the same time reducing to the correct Arrhenius form in the low and high temperature limits. Besides being a convenient fit to data, the expression can also be derived from various theoretical models of amorphous materials at the atomic level.^{[57]}
In the study of turbulence in fluids, a common practical strategy is to ignore the smallscale vortices (or eddies) in the motion and to calculate a largescale motion with an effective viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of energy in the smallerscale flow (see large eddy simulation).^{[58]}^{[59]} In contrast to the viscosity of the fluid itself, which must be positive by the second law of thermodynamics, the eddy viscosity can be negative.^{[60]}^{[61]}
Observed values of viscosity vary over several orders of magnitude, even for common substances. For instance, a 70% sucrose (sugar) solution has a viscosity over 400 times that of water, and 26000 times that of air.^{[63]} More dramatically, pitch has been estimated to have a viscosity 230 billion times that of water.^{[62]}
The viscosity of water is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature, the viscosity can be estimated using the semiempirical relation:
where A = 2.414×10^{−5} Pa·s, B = 247.8 K, and C = 140 K.
Experimentally determined values of the viscosity at various temperatures are given below.
Temperature (°C)  Viscosity (mPa·s) 

10  1.3059 
20  1.0016 
30  0.79722 
50  0.54652 
70  0.40355 
90  0.31417 
Under standard atmospheric conditions (25 °C and pressure of 1 bar), the viscosity of air is 18.5 μPa·s, roughly 50 times smaller than the viscosity of water at the same temperature. Except at very high pressure, the viscosity of air depends mostly on the temperature.
Substance  Viscosity (mPa·s)  Temperature (°C) 

Benzene  0.604  25 
Water^{[63]}  1.0016  20 
Mercury  1.526  25 
Whole milk^{[64]}  2.12  20 
Olive oil^{[64]}  56.2  26 
Honey^{[65]}  200010000  20 
Ketchup^{[a]}^{[66]}  500020000  25 
Peanut butter^{[a]}^{[67]}  10^{4}10^{6}  
Pitch^{[62]}  2.3×10^{11}  1030 (variable) 
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity.
Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity.Drag forces always decrease fluid velocity relative to the solid object in the fluid's path.
Fluid mechanicsFluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical,
civil, chemical and biomedical engineering, geophysics, astrophysics, and biology.
Fluid Mechanics can also be defined as the science which deals with the study of behaviour of fluids either at rest or in motion.
It can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved, and are best addressed by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Fuel oilFuel oil (also known as heavy oil, marine fuel or furnace oil) is a fraction obtained from petroleum distillation, either as a distillate or a residue. In general terms, fuel oil is any liquid fuel that is burned in a furnace or boiler for the generation of heat or used in an engine for the generation of power, except oils having a flash point of approximately 42 °C (108 °F) and oils burned in cotton or woolwick burners. Fuel oil is made of long hydrocarbon chains, particularly alkanes, cycloalkanes and aromatics. The term fuel oil is also used in a stricter sense to refer only to the heaviest commercial fuel that can be obtained from crude oil, i.e., heavier than gasoline and naphtha.
Small molecules like those in propane, naphtha, gasoline for cars, and jet fuel have relatively low boiling points, and they are removed at the start of the fractional distillation process. Heavier petroleum products like diesel fuel and lubricating oil are much less volatile and distill out more slowly, while bunker oil is literally the bottom of the barrel; in oil distilling, the only things denser than bunker fuel are carbon black feedstock and bituminous residue (asphalt), which is used for paving roads and sealing roofs.
Grease (lubricant)Grease is a semisolid lubricant. Grease generally consists of a soap emulsified with mineral or vegetable oil. The characteristic feature of greases is that they possess a high initial viscosity, which upon the application of shear, drops to give the effect of an oillubricated bearing of approximately the same viscosity as the base oil used in the grease. This change in viscosity is called shear thinning. Grease is sometimes used to describe lubricating materials that are simply soft solids or high viscosity liquids, but these materials do not exhibit the shearthinning properties characteristic of the classical grease. For example, petroleum jellies such as Vaseline are not generally classified as greases.
Greases are applied to mechanisms that can only be lubricated infrequently and where a lubricating oil would not stay in position. They also act as sealants to prevent ingress of water and incompressible materials. Greaselubricated bearings have greater frictional characteristics due to their high viscosity.
Heavy crude oilHeavy crude oil (or extra heavy crude oil) is highlyviscous oil that cannot easily flow to production wells under normal reservoir conditions.It is referred to as "heavy" because its density or specific gravity is higher than that of light crude oil. Heavy crude oil has been defined as any liquid petroleum with an API gravity less than 20°. Physical properties that differ between heavy crude oils and lighter grades include higher viscosity and specific gravity, as well as heavier molecular composition. In 2010, the World Energy Council defined extra heavy oil as crude oil having a gravity of less than 10° and a reservoir viscosity of no more than 10,000 centipoises. When reservoir viscosity measurements are not available, extraheavy oil is considered by the WEC to have a lower limit of 4° °API. In other words, oil with a density greater than 1000 kg/m3 or, equivalently, and a specific gravity greater than 1 and a reservoir viscosity of no more than 10,000 centipoises. Heavy oils and asphalt are dense nonaqueous phase liquids (DNAPLs). They have a low solubility and a viscosity lower than, and density higher than, water. Large spills of DNAPL will quickly penetrate the full depth of the aquifer and accumulate on its bottom.
HemodynamicsHemodynamics or hæmodynamics is the dynamics of blood flow. The circulatory system is controlled by homeostatic mechanisms, such as hydraulic circuits are controlled by control systems. Hemodynamic response continuously monitors and adjusts to conditions in the body and its environment. Thus hemodynamics explains the physical laws that govern the flow of blood in the blood vessels.
Blood flow ensures the transportation of nutrients, hormones, metabolic wastes, O2 and CO2 throughout the body to maintain celllevel metabolism, the regulation of the pH, osmotic pressure and temperature of the whole body, and the protection from microbial and mechanical harms.Blood is a nonNewtonian fluid, best studied using rheology rather than hydrodynamics. Blood vessels are not rigid tubes, so classic hydrodynamics and fluids mechanics based on the use of classical viscometers are not capable of explaining hemodynamics.The study of the blood flow is called hemodynamics. The study of the properties of the blood flow is called hemorheology.
KetchupKetchup is a sauce used as a condiment. Originally, recipes used egg whites, mushrooms, oysters, mussels, or walnuts, among other ingredients, but now the unmodified term usually refers to tomato ketchup. Various other terms for the sauce include catsup, catchup (archaic), ketsup, red sauce, tomato sauce, or, specifically, mushroom ketchup or tomato ketchup.
Ketchup is a sweet and tangy sauce now typically made from tomatoes, sugar, and vinegar, with assorted seasonings and spices. The specific spices and flavors vary, but commonly include onions, allspice, coriander, cloves, cumin, garlic, and mustard; and sometimes include celery, cinnamon or ginger.The market leader in the United States (60% market share) and United Kingdom (82%) is Heinz. Hunt's has the second biggest share of the US market with less than 20%. In much of the UK, Australia and New Zealand ketchup is also known as "tomato sauce" (a term that means a fresher pasta sauce elsewhere in the world) or "red sauce" (especially in Wales).
Tomato ketchup is most often used as a condiment to dishes that are usually served hot and may be fried or greasy: french fries, hamburgers, hot dogs, chicken tenders, tater tots, hot sandwiches, meat pies, cooked eggs, and grilled or fried meat. Ketchup is sometimes used as the basis for, or as one ingredient in, other sauces and dressings, and the flavor may be replicated as an additive flavoring for snacks such as potato chips.
LiquidA liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, and plasma), and is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena. Water is, by far, the most common liquid on Earth.
The density of a liquid is usually close to that of a solid, and much higher than in a gas. Therefore, liquid and solid are both termed condensed matter. On the other hand, as liquids and gases share the ability to flow, they are both called fluids. Although liquid water is abundant on Earth, this state of matter is actually the least common in the known universe, because liquids require a relatively narrow temperature/pressure range to exist. Most known matter in the universe is in gaseous form (with traces of detectable solid matter) as interstellar clouds or in plasma from within stars.
LotionA lotion is a lowviscosity topical preparation intended for application to the skin. By contrast, creams and gels have higher viscosity, typically due to lower water content. Lotions are applied to external skin with bare hands, a brush, a clean cloth, or cotton wool.
While a lotion may be used as a medicine delivery system, many lotions, especially hand lotions and body lotions are meant instead to simply smooth, moisturize, soften and perhaps perfume the skin.Some skincare products, such as sunscreen and moisturizer, may be available in multiple formats, such as lotions, gels, creams, or sprays.
Motor oilMotor oil, engine oil, or engine lubricant is any of various substances comprising base oils enhanced with additives, particularly antiwear additive plus detergents, dispersants and, for multigrade oils viscosity index improvers. Motor oil is used for lubrication of internal combustion engines. The main function of motor oil is to reduce friction and wear on moving parts and to clean the engine from sludge (one of the functions of dispersants) and varnish (detergents). It also neutralizes acids that originate from fuel and from oxidation of the lubricant (detergents), improves sealing of piston rings, and cools the engine by carrying heat away from moving parts.In addition to the basic constituents noted in the preceding paragraph, almost all lubricating oils contain corrosion (GB: rust) and oxidation inhibitors. Motor oil may be composed of only a lubricant base stock in the case of nondetergent oil, or a lubricant base stock plus additives to improve the oil's detergency, extreme pressure performance, and ability to inhibit corrosion of engine parts.
Motor oils today are blended using base oils composed of petroleumbased hydrocarbons, that means organic compounds consisting of carbon and hydrogen, or polyalphaolefins (PAO) or their mixtures in various proportions, sometimes with up to 20% by weight of esters for better dissolution of additives.
MucokineticsMucokinetics are a class of drugs which aid in the clearance of mucus from the airways, lungs, bronchi, and trachea. Such drugs can be further categorized by their mechanism of action:
mucolytic agents
expectorants
surfactants
wetting agents (hypoviscosity agents)
abhesivesIn general, clearance ability is hampered by bonding to surfaces (stickiness) and by the viscosity of mucous secretions in the lungs. In turn, the viscosity is dependent upon the concentration of mucoprotein in the secretions.
Expectorants and mucolytic agents are different types of medication, yet both are intended to promote drainage of mucus from the lungs.
An expectorant (from the Latin expectorare, to expel or banish) works by signaling the body to increase the amount or hydration of secretions, resulting in more yet clearer secretions and as a byproduct lubricating the irritated respiratory tract. One expectorant, guaifenesin, is commonly available in many cough syrups. Often the term "expectorant" is incorrectly extended to any cough medicine, since it is a universal component.A mucolytic agent is an agent which dissolves thick mucus and is usually used to help relieve respiratory difficulties. It does so by dissolving various chemical bonds within secretions, which in turn can lower the viscosity by altering the mucincontaining components.
Alternatively, attacking the affinity between secretions and the biological surfaces is another avenue, which is used by abhesives and surfactants.
Any of these effects could improve airway clearance during coughing.
An expectorant increases bronchial secretions and mucolytics help loosen thick bronchial secretions. Expectorants reduce the thickness or viscosity of bronchial secretions thus increasing mucus flow that can be removed more easily through coughing. Mucolytics break down the chemical structure of mucus molecules. The mucus becomes thinner and can be removed more easily through coughing.
Newtonian fluidA Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.
More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic (that is, its mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion, respectively.
Newtonian fluids are the simplest mathematical models of fluids that account for viscosity. While no real fluid fits the definition perfectly, many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical calculations under ordinary conditions. However, nonNewtonian fluids are relatively common, and include oobleck (which becomes stiffer when vigorously sheared), or nondrip paint (which becomes thinner when sheared). Other examples include many polymer solutions (which exhibit the Weissenberg effect), molten polymers, many solid suspensions, blood, and most highly viscous fluids.
Newtonian fluids are named after Isaac Newton, who first used the differential equation to postulate the relation between the shear strain rate and shear stress for such fluids.
NonNewtonian fluidA nonNewtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e. constant viscosity independent of stress. In nonNewtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a nonNewtonian fluid. Many salt solutions and molten polymers are nonNewtonian fluids, as are many commonly found substances such as custard, honey, toothpaste, starch suspensions, corn starch, paint, blood, and shampoo.
Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of nonNewtonian fluids is dependent on shear rate or shear rate history. Some nonNewtonian fluids with shearindependent viscosity, however, still exhibit normal stressdifferences or other nonNewtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a nonNewtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit timedependent viscosity. Therefore, a constant coefficient of viscosity cannot be defined.
Although the concept of viscosity is commonly used in fluid mechanics to characterize the shear properties of a fluid, it can be inadequate to describe nonNewtonian fluids. They are best studied through several other rheological properties that relate stress and strain rate tensors under many different flow conditions—such as oscillatory shear or extensional flow—which are measured using different devices or rheometers. The properties are better studied using tensorvalued constitutive equations, which are common in the field of continuum mechanics.
Poise (unit)The poise (symbol P; /pɔɪz, pwɑːz/) is the unit of dynamic viscosity (absolute viscosity) in the centimetre–gram–second system of units. It is named after Jean Léonard Marie Poiseuille (see Hagen–Poiseuille equation).
The analogous unit in the International System of Units is the pascalsecond (Pa⋅s):
The poise is often used with the metric prefix centi because the viscosity of water at 20 °C (NTP) is almost exactly 1 centipoise. A centipoise is one hundredth of a poise, or one millipascalsecond (mPa⋅s) in SI units (1 cP = 10^{−3} Pa⋅s = 1 mPa⋅s).
The CGS symbol for the centipoise is cP. The abbreviations cps, cp, and cPs are sometimes seen.
Liquid water has a viscosity of 0.00890 P at 25 °C and a pressure of 1 atmosphere (0.00890 P = 0.890 cP = 0.890 mPa⋅s).
RheologyRheology (; from Greek ῥέω rhéō, "flow" and λoγία, logia, "study of") is the study of the flow of matter, primarily in a liquid state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force. It is a branch of physics which deals with the deformation and flow of materials, both solids and liquids.The term rheology was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920, from a suggestion by a colleague, Markus Reiner. The term was inspired by the aphorism of Simplicius (often attributed to Heraclitus), panta rhei, "everything flows", and was first used to describe the flow of liquids and the deformation of solids.
It applies to substances that have a complex microstructure, such as muds, sludges, suspensions, polymers and other glass formers (e.g., silicates), as well as many foods and additives, bodily fluids (e.g., blood) and other biological materials or other materials that belong to the class of soft matter such as food.
Newtonian fluids can be characterized by a single coefficient of viscosity for a specific temperature. Although this viscosity will change with temperature, it does not change with the strain rate. Only a small group of fluids exhibit such constant viscosity. The large class of fluids whose viscosity changes with the strain rate (the relative flow velocity) are called nonNewtonian fluids.
Rheology generally accounts for the behavior of nonNewtonian fluids, by characterizing the minimum number of functions that are needed to relate stresses with rate of change of strain or strain rates. For example, ketchup can have its viscosity reduced by shaking (or other forms of mechanical agitation, where the relative movement of different layers in the material actually causes the reduction in viscosity) but water cannot. Ketchup is a shear thinning material, like yogurt and emulsion paint (US terminology latex paint or acrylic paint), exhibiting thixotropy, where an increase in relative flow velocity will cause a reduction in viscosity, for example, by stirring. Some other nonNewtonian materials show the opposite behavior, rheopecty: viscosity going up with relative deformation, and are called shear thickening or dilatant materials. Since Sir Isaac Newton originated the concept of viscosity, the study of liquids with strain rate dependent viscosity is also often called NonNewtonian fluid mechanics.The experimental characterisation of a material's rheological behaviour is known as rheometry, although the term rheology is frequently used synonymously with rheometry, particularly by experimentalists. Theoretical aspects of rheology are the relation of the flow/deformation behaviour of material and its internal structure (e.g., the orientation and elongation of polymer molecules), and the flow/deformation behaviour of materials that cannot be described by classical fluid mechanics or elasticity.
Shear stressA shear stress, often denoted by τ (Greek: tau), is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.
Shear stress arises from shear forces, which are pairs of equal and opposing forces acting on opposite sides of an object.
ThixotropyThixotropy is a timedependent shear thinning property. Certain gels or fluids that are thick, or viscous, under static conditions will flow (become thin, less viscous) over time when shaken, agitated, sheared or otherwise stressed (time dependent viscosity). They then take a fixed time to return to a more viscous state.
Some nonNewtonian pseudoplastic fluids show a timedependent change in viscosity; the longer the fluid undergoes shear stress, the lower its viscosity. A thixotropic fluid is a fluid which takes a finite time to attain equilibrium viscosity when introduced to a steep change in shear rate. Some thixotropic fluids return to a gel state almost instantly, such as ketchup, and are called pseudoplastic fluids. Others such as yogurt take much longer and can become nearly solid. Many gels and colloids are thixotropic materials, exhibiting a stable form at rest but becoming fluid when agitated. Thixotropy arises because particles or structured solutes require time to organize. An excellent overview of thixotropy has been provided by Mewis and Wagner.Some fluids are antithixotropic: constant shear stress for a time causes an increase in viscosity or even solidification. Fluids which exhibit this property are sometimes called rheopectic. Antithixotropic fluids are less well documented than thixotropic fluids.
TurbulenceIn fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.
The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics.
ViscometerA viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a special type of viscometer. Viscometers only measure under one flow condition.
In general, either the fluid remains stationary and an object moves through it, or the object is stationary and the fluid moves past it. The drag caused by relative motion of the fluid and a surface is a measure of the viscosity. The flow conditions must have a sufficiently small value of Reynolds number for there to be laminar flow.
At 20 °C, the dynamic viscosity (kinematic viscosity × density) of water is 1.0038 mPa·s and its kinematic viscosity (product of flow time × factor) is 1.0022 mm2/s. These values are used for calibrating certain types of viscometers.
Elementary calculation of viscosity for a dilute gas 

Consider a dilute gas moving parallel to the axis with velocity that depends only on the coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density. Under these assumptions, the velocity of a molecule passing through is equal to whatever velocity that molecule had when its mean free path began. Because is typically small compared with macroscopic scales, the average velocity of such a molecule has the form
where is a numerical constant on the order of . (Some authors estimate ;^{[15]}^{[32]} on the other hand, a more careful calculation for rigid elastic spheres gives .) Now, because half the molecules on either side are moving towards , and doing so on average with half the average moleculer speed , the momentum flux from either side is The net momentum flux at is the difference of the two: According to the definition of viscosity, this momentum flux should be equal to , which leads to 
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