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Viscosity I: Liquid Viscosity

Michael Fowler

Introduction: Friction at the Molecular Level

Viscosity is, essentially, fluid friction.  Like friction between moving solids, viscosity transforms kinetic energy of (macroscopic) motion into heat energy.  Estrus is energy of random motion at the molecular level, so to accept whatsoever understanding of how this energy transfer takes place, it is essential to have some picture show, withal crude, of solids and/or liquids sliding past each other equally seen on the molecular scale.

To begin with, we'll review the molecular picture of friction between solid surfaces, and the significance of the coefficient of friction μ  in the familiar equation F = μ Northward .    Going on to fluids, nosotros'll give the definition of the coefficient of viscosity for liquids and gases, give some values for different fluids and temperatures, and demonstrate how the microscopic picture tin can requite at least a qualitative understanding of how these values vary: for case, on raising the temperature, the viscosity of liquids decreases, that of gases increases.  Also, the viscosity of a gas doesn't depend in its density!  These mysteries can only be unraveled at the molecular level, but there the explanations turn out to exist quite simple.

Every bit volition become clear later, the coefficient of viscosity η  can exist viewed in two rather different (just of grade consistent) ways: it is a measure of how much heat is generated when faster fluid is flowing past slower fluid, but information technology is also a measure of the charge per unit of transfer of momentum from the faster stream to the slower stream.  Looked at in this second way, it is analogous to thermal electrical conductivity, which is a measure of the rate of transfer of heat from a warm identify to a libation place.

Quick Review of Friction Between Solids

First, static friction: suppose a book is lying on your desk-bound, and yous tilt the desk.  At a certain angle of tilt, the volume begins to slide.  Before that, it's held in identify by "static friction".  What does that mean on a molecular level?  In that location must be some sort of attractive force between the volume and the desk to concord the volume from sliding.

Let'south expect at all the forces on the book: gravity is pulling it vertically down, and there is a "normal force" of the desk surface pushing the book in the management normal to the desk surface.  (This normal forcefulness is the springiness of the desktop, slightly compressed past the weight of the book.)   When the desk-bound is tilted, information technology's all-time to visualize the vertical gravitational strength every bit fabricated up of a component normal to the surface and one parallel to the surface (downhill).  The gravitational component perpendicular to the surface is exactly balanced by the normal force, and if the book is at rest, the "downhill" component of gravity is balanced by a frictional force parallel to the surface in the uphill management.  On a microscopic calibration, this static frictional force is from fairly short range attractions betwixt molecules on the desk and those of the book.

Question :  but if that's true, why does doubling the normal forcefulness double this frictional forcefulness? (Think F = μ Northward ,  where Due north  is the normal force, F  is the limiting frictional force just before the book begins to slide, and μ  is the coefficient of friction.  By the way, the first advent of F  being proportional to North  is in the notebooks of Leonardo da Vinci.)

Answer :  Solids are well-nigh always rough on an atomic calibration: when ii flat looking solid surfaces are pushed together, in fact only a tiny fraction of the common surface is really in contact at the atomic level. The stresses within that tiny surface area are large, the materials distort plastically and there is adhesion.  The picture show tin can be very circuitous, depending on the materials involved, but the bottom line is that in that location is but atom-cantlet interaction between the solids over a small area, and what happens in this minor area determines the frictional strength.  If the normal strength is doubled (by adding another volume, say) the tiny area of contact between the bottom book and the desk volition also double — the true area of atomic contact increases linearly with the normal forcefulness — that's why friction is proportional to N .   Inside the expanse of "true contact" actress force per unit area makes little difference.  (Incidentally, if two surfaces which actually are flat at the atomic level are put together, there is bonding.  This tin be a real claiming in the optical telecommunications industry, where wavelength filters (called etalons) are manufactured by having extremely apartment, highly parallel surfaces of transparent material separated by distances comparable to the wavelength of light.  If they touch, the etalon is ruined.)

On tilting the desk more, the static frictional strength turns out to have a limit — the book begins to slide.  But there's still some friction: experimentally, the book does not have the full dispatch the component of gravity parallel to the desktop should deliver.  This must exist because in the surface area of contact with the desk the ii sets of atoms are constantly colliding, loose bonds are forming and breaking, some atoms or molecules fall away.  This all causes a lot of diminutive and molecular vibration at the surface. In other words, some of the gravitational potential energy the sliding book is losing is ending up as heat instead of adding to the book'due south kinetic energy.  This is the familiar dynamic friction you use to warm your hands by rubbing them together in winter.  It'south often chosen kinetic friction.  Like static friction, it's proportional to the normal force: F = μ Thou N .  The proportionality to the normal force is for the same reason every bit in the static instance: the kinetic frictional drag force also comes from the tiny area of true diminutive contact, and this area is proportional to the normal forcefulness.

A total account of the physics of friction (known every bit tribology) can be plant, for example, in Friction and Wearable of Materials, by Ernest Rabinowicz, 2nd Edition, Wiley, 1995.

Liquid Friction

What happens if instead of ii solid surfaces in contact, nosotros accept a solid in contact with a liquid?  First, there'due south no such matter as static friction betwixt a solid and a liquid.  If a gunkhole is at balance in still water, information technology volition move in response to the slightest force.  Obviously, a tiny strength will give a tiny dispatch, but that's quite dissimilar from the book on the desk-bound, where a considerable force gave no dispatch at all.  Only there is dynamic liquid friction — even though an beam turns a lot more easily if oil is supplied, there is still some resistance, the oil gets warmer equally the axle turns, so work is being expended to produce heat, just as for a solid sliding across another solid.

Ane might think that for solid/liquid friction at that place would exist some equation analogous to F = μ K N :   maybe the liquid frictional force is, like the solid, proportional to pressure?  Just experimentally this turns out to exist false — in that location is little dependence on force per unit area over a very broad range.  The reason is evidently that since the liquid can flow, in that location is good contact over the whole common area, even for low pressures, in contrast to the solid/solid case.

Newton's Analysis of Gummy Drag

Isaac Newton was the starting time to attempt a quantitative definition of a coefficient of viscosity.  To make things equally uncomplicated equally possible, he attempted an experiment in which the fluid in question was sandwiched between two big parallel horizontal plates.  The bottom plate was held fixed, the top plate moved at a steady speed v 0 ,  and the elevate force from the fluid was measured for dissimilar values of v 0 ,  and unlike plate spacing.  (Actually Newton's experiment didn't work besides well, but as usual his theoretical reasoning was fine, and fully confirmed experimentally by Poiseuille in 1849 using liquid catamenia in tubes.)

Newton assumed (and it has been handsomely confirmed past experiment) that at least for low speeds the fluid settles into the flow pattern shown below.  The fluid in close contact with the bottom plate stays at rest, the fluid touching the top plate gains the same speed v 0  as that plate, and in the infinite betwixt the plates the speed of the fluid increases linearly with top, and then that, for case, the fluid halfway between the plates is moving at 1 2 v 0 :

Simply as for kinetic friction between solids, to keep the top plate moving requires a steady force.  Manifestly, the strength is proportional to the total corporeality of fluid beingness kept in motion, that is, to the full area of the top plate in contact with the fluid.  The pregnant parameter is the horizontal force per unit of measurement area of plate, F / A ,  say.  This clearly has the aforementioned dimensions as pressure (and then can be measured in Pascals) although it is physically completely different, since in the present case the strength is parallel to the expanse (or rather to a line inside it), not perpendicular to it as pressure is.

(Note for experts merely: Actually, viscous drag and pressure are not completely unrelated — as we shall discuss later, the viscous force may be interpreted every bit a charge per unit of transfer of momentum into the fluid, momentum parallel to the surface that is, and force per unit area can as well be interpreted as a charge per unit of transfer of momentum, only now perpendicular to the surface, as the molecules bounciness off.  Physically, the big difference is of form that the pressure doesn't have to do whatsoever piece of work to keep transferring momentum, the gluey force does.)

Newton conjectured that the necessary force F / A  would exist proportional to the velocity slope in the vicinity of the top plate.  In the simple geometry above, the velocity gradient is the same everywhere between the plates, v 0 / d ,  so

F / A = η v 0 / d

defines the coefficient of viscosity η .   The SI units of η  are Pascal.seconds, or Pa.south.

A user-friendly unit of measurement is the milliPascal.second, mPa.s. (It happens to be close to the viscosity of water at room temperature.)  Confusingly, there is another ready of units out at that place, the poise, named after Poiseuille — usually seen as the centipoise, which happens to equal the millipascal.2d!  And, in that location'due south some other viscosity coefficient in mutual employ: the kinetic viscosity, ν = μ / ρ ,  where ρ  is the fluid density.  This is the relevant parameter for fluids flowing downwards gravitationally.  Simply nosotros'll nearly e'er stick with η .

Hither are some values of η  for common liquids:

Liquid Viscosity in mPa.s
Water at 0℃ ane.79
H2o at xx℃ 1.002
Water at 100℃ 0.28
Glycerin at 0℃ 12070
Glycerin at xx℃ 1410
Glycerin at 30℃ 612
Glycerin at 100℃ fourteen.viii
Mercury at 20℃ 1.55
Mercury at 100℃ one.27
Motor oil SAE 30 200
Motor oil SAE sixty grand
Ketchup 50,000

Some of these are obviously ballpark –  the others probably shouldn't be trusted to exist meliorate that 1% or and so, glycerin maybe fifty-fifty 5-10% (run into CRC Tables); these are quite difficult measurements, very sensitive to purity (glycerin is hygroscopic) and to minor temperature variations.

To proceeds some insight into these very different viscosity coefficients, we'll try to analyze what's going on at the molecular level.

A Microscopic Picture of Viscosity in Laminar Flow

For Newton's picture of a fluid sandwiched between two parallel plates, the bottom one at rest and the top one moving at steady speed, the fluid tin be pictured equally made up of many layers, similar a pile of printer paper, each canvass moving a little faster than the canvass below information technology in the pile, the top sheet of fluid moving with the plate, the bottom canvass at residual. This is called laminar menstruum: laminar but means canvass (every bit in laminate, when a canvass of something is glued to a sheet of something else).  If the top plate is gradually speeded upward, at some indicate laminar flow becomes unstable and turbulence begins.  We'll assume here that we're well below that speed.

So where'southward the friction?  It's non betwixt the fluid and the plates (or at least very niggling of it is — the molecules right next to the plates mostly stay in identify) it's between the private sheets — throughout the fluid.  Recall of ii neighboring sheets, the molecules of ane bumping against their neighbors as they pass.  As they oversupply by each other, on average the molecules in the faster stream are slowed downward, and those in the slower stream speeded upward.  Of grade, momentum is always conserved, but the macroscopic kinetic energy of the sheets of fluid is partially lost — transformed into heat energy.

Do :  Suppose a mass k  of fluid moving at v 1  in the 10 -direction mixes with a mass yard  moving at v 2  in the x -management.  Momentum conservation tells us that the mixed mass 2 m  moves at one two ( v 1 + v 2 ) .   Testify that the total kinetic energy has decreased if v i , v ii  are unequal.

This is the fraction of the kinetic energy that has disappeared into rut.

This molecular picture of sheets of fluids moving past each other gives some insight into why viscosity decreases with temperature, and at such different rates for different fluids.  As the molecules of the faster sail jostle by those in the slower sheet, remember they are all jiggling nigh with thermal energy.  The jiggling helps suspension them loose if they get jammed temporarily against each other, so equally the temperature increases, the molecules jiggle more than furiously, unjam more than quickly, and the fluid moves more easily — the viscosity goes downwardly.

This drop in viscosity with temperature is dramatic for glycerin. A glance at the molecule suggests that the zigzaggy shape might cause jamming, only the main cause of the stickiness is that the outlying H's in the OH groups readily form hydrogen bonds (encounter Atkins' Molecules, Cambridge).

For mercury, a fluid of round atoms, the drop in viscosity with temperature is small.  Mercury atoms don't jam much, they mainly merely bounce off each other (but fifty-fifty that bouncing randomizes their direction, converting macroscopic kinetic energy to heat).  Note that mercury is a niggling more than viscous than h2o: this can simply exist understood properly with some detailed chemical analysis, mercury is a metal so the bonds are metal, meaning delocalized electron states. H2o molecules attract with hydrogen bonds. Unfortunately, nosotros can't investigate this further hither.

Some other mechanism generating viscosity is the diffusion of faster molecules into the slower stream and vice versa.  Every bit discussed below, this is far the ascendant factor in viscosity of gases, simply is much less important in liquids, where the molecules are crowded together and constantly bumping against each other.

This temperature dependence of viscosity is a real problem in lubricating engines that must run well over a wide temperature range.  If the oil gets besides runny (that is, depression viscosity) it will not continue the metal surfaces from grinding against each other; if it gets too thick, more energy volition be needed to turn the axle.  "Viscostatic" oils accept been adult: the natural decrease of viscosity with temperature ("thinning") is counterbalanced by adding polymers, long concatenation molecules at high temperatures that curl up into balls at low temperatures.

Oiling a Wheel Axle

The uncomplicated linear velocity profile pictured above is actually a expert model for ordinary lubrication.  Imagine an axle of a few centimeters diameter, say about the size of a fist, rotating in a bearing, with a 1 mm gap filled with SAE 30 oil, having η = 200  mPa .southward .   (Notation: mPa, millipascals, non Pascals!  1Pa = 1000mPa.)

If the full cylindrical area is, say, 100 sq cm., and the speed is 1 m.southward-1, the strength per unit expanse (sq. m.)

F / A = η v 0 / d = 200 ⋅ 10 − 3 ⋅ 1 / 10 − 3 = 200 N / one thousand 2 .

So for our 100 sq.cm begetting the forcefulness needed to overcome the viscous "friction" is 2 Due north .   At the speed of 1 g sec-1, this means work is being done at a rate of two joules per sec., or 2 watts, which is heating upwards the oil.  (This heat must be conducted away, or the oil continuously changed by pumping, otherwise it will get likewise hot.)

*Viscosity: Kinetic Energy Loss and Momentum Transfer

Then far, we've viewed the viscosity coefficient η  as a measure of friction, of the dissipation into rut of the energy supplied to the fluid by the moving top plate.  But η  is also the key to understanding what happens to the momentum the plate supplies to the fluid.

For the picture above of the steady fluid catamenia between two parallel plates, the bottom plate at remainder and the top one moving, a steady force per unit of measurement area F / A  in the x -direction practical to the tiptop plate is needed to maintain the menses.

From Newton's constabulary F = d p / d t , F / A is the rate at which momentum in the 10 -direction is being supplied (per unit area) to the fluid.  Microscopically, molecules in the immediate vicinity of the plate either adhere to it or keep bouncing against it, picking upwardly momentum to keep moving with the plate (these molecules too constantly lose momentum by billowy off other molecules a little further away from the plate).

Question :  But doesn't the total momentum of the fluid stay the same in steady menses? Where does the momentum fed in by the moving top plate become?

Respond : the ten -direction momentum supplied at the elevation passes downwardly from i layer to the next, ending up at the bottom plate (and everything it'southward attached to).  Remember that, dissimilar kinetic energy, momentum is always conserved — it can't disappear.

So, in that location is a steady flow in the z -management of x -direction momentum.  Furthermore, the left-hand side of the equation

F / A = η five 0 / d

is merely this momentum menses charge per unit.  The right hand side is the coefficient of viscosity multiplied past the gradient in the z -direction of the x -direction velocity.

Viewed in this way, F / A = η v 0 / d  is a transport equation.  It tells us that the rate of transport of x -direction momentum down is proportional to the rate of change of x -direction velocity in that direction, and the constant of proportionality is the coefficient of viscosity.  And, we tin can limited this slightly differently past noting that the charge per unit of change of 10 -direction velocity is proportional to the charge per unit of change of x -direction momentum density.

Recall that we mentioned earlier the so-called kinetic viscosity coefficient, ν = μ / ρ .   Using that in the equation

F / A = η five 0 / d = ν ρ v 0 / d ,

replaces the velocity gradient with an x -management momentum gradient.  To abbreviate a clumsy phrase, allow's phone call the x -direction momentum density π ten ,  and the current of this in the z -direction J z ( π x ) .   So our equation becomes

J z ( π x ) = ν d π 10 d z .

The electric current of π x  in the z -management is proportional to how fast π ten  is changing in that management.

This closely resembles heat flowing from a hot spot to a common cold spot: oestrus free energy flows towards the place where at that place is less of it, "downhill" in temperature.  The rate at which it flows is proportional to the temperature gradient, and the constant of proportionality is the thermal conductivity (come across later).   Here, the π x  momentum catamenia is analogous: it besides flows to where at that place is less of it, and the kinetic viscosity coefficient corresponds to the thermal electrical conductivity.

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