# 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. Heat is energy of random motion at the molecular level, so to have any understanding of how this energy transfer takes place, it is essential to have some picture, however crude, of solids and/or liquids sliding past each other as 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 $\mu $ in the familiar equation $F=\mu N.$ Going on to fluids, we’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 can give at least a qualitative understanding of how these
values vary: for example, 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 be
quite simple.

As will become clear later, the coefficient of viscosity $\eta $ can be viewed in two rather different (but of
course consistent) ways: it is a measure of how much heat is generated when faster
fluid is flowing by slower fluid, but it is also a measure of the *rate of transfer of momentum* from the
faster stream to the slower stream.
Looked at in this second way, it is analogous to thermal conductivity,
which is a measure of the rate of transfer of heat from a warm place to a
cooler place.

### Quick Review of Friction Between Solids

First, *static*
friction: suppose a book is lying on your desk, and you tilt the desk. At a certain angle of tilt, the book begins to
slide. Before that, it’s held in place by
“static friction”. What does that mean
on a molecular level? There must be some
sort of attractive force between the book and the desk to hold the book from
sliding.

Let’s look 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 direction normal to the desk surface. (This normal force is the springiness of the desktop, slightly compressed by the weight of the book.) When the desk is tilted, it’s best to visualize the vertical gravitational force as made 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 direction. On a microscopic scale, this static frictional force is from fairly short range attractions between molecules on the desk and those of the book.

*Question*:
but if that’s true, why does doubling the normal force double this
frictional force? (Recall $F=\mu N,$ where $N$ is the normal force, $F$ is the limiting frictional force just before
the book begins to slide, and $\mu $ is the coefficient of friction. By the way, the first appearance of $F$ being proportional to $N$ is in the notebooks of Leonardo da Vinci.)

*Answer *: Solids
are almost always *rough* on an atomic
scale: when two 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 area are large, the materials distort plastically
and there is adhesion. The picture can
be very complex, depending on the materials involved, but the bottom line is
that there is only atom-atom interaction between the solids *over a small area*, and what happens in
this small area determines the frictional force. If the normal force is doubled (by adding
another book, say) the tiny area of contact between the bottom book and the
desk will also double$\u2014$the true
area of atomic contact *increases linearly
with the normal force*$\u2014$that’s why
friction is proportional to $N.$ Within
the area of “true contact” extra pressure makes little difference. (Incidentally, if two surfaces which really *are* flat at the atomic level are put
together, there is bonding. This can be
a real challenge in the optical telecommunications industry, where wavelength
filters (called etalons) are manufactured by having extremely flat, 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 force turns
out to have a limit$\u2014$the book
begins to slide. But there’s still some friction:
experimentally, the book does not have the full acceleration the component of gravity
parallel to the desktop should deliver.
This must be because in the area of contact with the desk the two sets
of atoms are constantly colliding, loose bonds are forming and breaking, some atoms
or molecules fall away. This all causes
a lot of atomic 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’s kinetic energy. This is the familiar dynamic friction you use
to warm your hands by rubbing them together in wintertime. It’s often called* kinetic *friction. Like
static friction, it’s proportional to the normal force: $F={\mu}_{K}N$. The proportionality to the normal force is
for the same reason as in the static case: the kinetic frictional drag force also
comes from the tiny area of true atomic contact, and this *area* is proportional to the normal force.

A full account of the physics of friction (known as
tribology) can be found, for example, in *Friction
and Wear of Materials*, by Ernest Rabinowicz, Second Edition, Wiley, 1995.

### Liquid Friction

What happens if instead of two solid surfaces in contact, we
have a solid in contact with a liquid?
First, there’s no such thing as static friction between a solid and a
liquid. If a boat is at rest in still
water, it will move in response to the slightest force. Obviously, a tiny force will give a tiny
acceleration, but that’s quite different from the book on the desk, where a
considerable force gave no acceleration at all.
But there *is* dynamic liquid
friction$\u2014$even
though an axle turns a lot more easily if oil is supplied, there is still *some* resistance, the oil gets warmer as
the axle turns, so work is being expended to produce heat, just as for a solid
sliding across another solid.

One might think that for solid/liquid friction there would be some equation analogous to $F={\mu}_{K}N:$ perhaps the liquid frictional force is, like the solid, proportional to pressure? But experimentally this turns out to be false$\u2014$there is little dependence on pressure over a very wide range. The reason is evidently that since the liquid can flow, there is good contact over the whole common area, even for low pressures, in contrast to the solid/solid case.

### Newton’s Analysis of Viscous Drag

Isaac Newton was the first to attempt a quantitative definition of a coefficient of viscosity. To make things as simple as possible, he attempted an experiment in which the fluid in question was sandwiched between two large parallel horizontal plates. The bottom plate was held fixed, the top plate moved at a steady speed ${v}_{0},$ and the drag force from the fluid was measured for different values of ${v}_{0},$ and different plate spacing. (Actually Newton’s experiment didn’t work too well, but as usual his theoretical reasoning was fine, and fully confirmed experimentally by Poiseuille in 1849 using liquid flow in tubes.)

Newton assumed (and it has been amply confirmed by 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 space between the plates the speed of the fluid increases linearly with height, so that, for example, the fluid halfway between the plates is moving at ${\scriptscriptstyle \frac{1}{2}}{v}_{0}:$

Just as for kinetic friction between solids, to keep the top
plate moving requires a steady force. Obviously, the force is proportional to the
total amount of fluid being kept in motion, that is, to the total area of the
top plate in contact with the fluid. The
significant parameter is the *horizontal
force per unit area of plate*, $F/A,$ say. This clearly has the same dimensions as
pressure (and so can be measured in Pascals) although it is physically completely
different, since in the present case the force is *parallel* to the area (or rather to a line within it), not
perpendicular to it as pressure is.

(*Note for experts only*:
Actually, viscous drag and pressure are not completely unrelated$\u2014$as we
shall discuss later, the viscous force may be interpreted as a rate of transfer
of momentum into the fluid, momentum parallel to the surface that is, and
pressure can *also* be interpreted as a
rate of transfer of momentum, but now perpendicular to the surface, as the
molecules bounce off. Physically, the
big difference is of course that the pressure doesn’t have to do any work to
keep transferring momentum, the viscous force does.)

$F/A=\eta {v}_{0}/d$

defines the *coefficient
of viscosity* $\eta .$ The SI units
of $\eta $ are **Pascal.seconds**,
or **Pa.s**.

A convenient unit is the milliPascal.second, mPa.s. (It
happens to be close to the viscosity of water at room temperature.) Confusingly, there is another set of units out
there, the poise, named after Poiseuille$\u2014$usually
seen as the centipoise, which happens to equal the millipascal.second! And, there’s *another* viscosity coefficient in common use: the kinetic viscosity,
$\nu =\mu /\rho ,$ where $\rho $ is the fluid density. This is the relevant parameter for fluids
flowing downwards gravitationally. But we’ll
almost always stick with $\eta $.

Here are some values of $\eta $ for common liquids:

Liquid | Viscosity in mPa.s |
---|---|

Water at 0℃ | 1.79 |

Water at 20℃ | 1.002 |

Water at 100℃ | 0.28 |

Glycerin at 0℃ | 12070 |

Glycerin at 20℃ | 1410 |

Glycerin at 30℃ | 612 |

Glycerin at 100℃ | 14.8 |

Mercury at 20℃ | 1.55 |

Mercury at 100℃ | 1.27 |

Motor oil SAE 30 | 200 |

Motor oil SAE 60 | 1000 |

Ketchup | 50,000 |

Some of these are obviously ballpark $\u2013$ the others probably shouldn’t be trusted to be better that 1% or so, glycerin maybe even 5-10% (see CRC Tables); these are quite difficult measurements, very sensitive to purity (glycerin is hygroscopic) and to small temperature variations.

To gain 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 can be pictured as made up of many layers, like a pile of printer
paper, each sheet moving a little faster than the sheet below it in the pile,
the top sheet of fluid moving with the plate, the bottom sheet at rest. This is
called *laminar flow*: laminar just
means sheet (as in laminate, when a sheet of something is glued to a sheet of
something else). If the top plate is
gradually speeded up, at some point laminar flow becomes unstable and
turbulence begins. We’ll assume here
that we’re well below that speed.

So where’s the friction?
It’s not between the fluid and the plates (or at least very little of it
is$\u2014$the
molecules right next to the plates mostly stay in place) it’s *between the individual sheets*$\u2014$throughout
the fluid. Think of two neighboring
sheets, the molecules of one bumping against their neighbors as they pass. As they crowd past each other, on average the
molecules in the faster stream are slowed down, and those in the slower stream
speeded up. Of course, *momentum* is always conserved, but the
macroscopic kinetic energy of the sheets of fluid is partially lost$\u2014$transformed
into heat energy.

*Exercise*: Suppose a mass $m$ of fluid moving at ${v}_{1}$ in the $x$-direction mixes with a mass $m$ moving at ${v}_{2}$ in the $x$-direction.
Momentum conservation tells us that the mixed mass $2m$ moves at ${\scriptscriptstyle \frac{1}{2}}\left({v}_{1}+{v}_{2}\right).$ Prove
that the total kinetic energy has *decreased*
if ${v}_{1},{v}_{2}$ are unequal.

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

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 sheet jostle past those in the slower sheet, remember they are all jiggling about with thermal energy. The jiggling helps break them loose if they get jammed temporarily against each other, so as the temperature increases, the molecules jiggle more furiously, unjam more quickly, and the fluid moves more easily$\u2014$the viscosity goes down.

This drop in viscosity with temperature is dramatic for
glycerin. A glance at the molecule suggests that the zigzaggy shape might cause
jamming, but the main cause of the stickiness is that the outlying H’s in the
OH groups readily form hydrogen bonds (see *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 just bounce off each other (but even that bouncing randomizes their direction, converting macroscopic kinetic energy to heat). Note that mercury is a little more viscous than water: this can only be understood properly with some detailed chemical analysis, mercury is a metal so the bonds are metallic, meaning delocalized electron states. Water molecules attract with hydrogen bonds. Unfortunately, we can't investigate this further here.

Another mechanism generating viscosity is the diffusion of
faster molecules into the slower stream and vice versa. As discussed below, this is far the dominant
factor in viscosity of gases, but 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 too runny (that is, low viscosity) it will not keep the metal surfaces from grinding against each other; if it gets too thick, more energy will be needed to turn the axle. “Viscostatic” oils have been developed: the natural decrease of viscosity with temperature (“thinning”) is counterbalanced by adding polymers, long chain molecules at high temperatures that curl up into balls at low temperatures.

### Oiling a Wheel Axle

The simple linear velocity profile pictured above is actually a good 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 $\eta =200\text{mPa}\text{.s}\text{.}$ (Note: mPa, millipascals, not Pascals! 1Pa = 1000mPa.)

If the total cylindrical area is, say, 100 sq cm., and the
speed is 1 m.s^{-1}, the force per unit area (sq. m.)

$$F/A=\eta {v}_{0}/d=200\cdot {10}^{-3}\cdot 1/{10}^{-3}=200N/{\text{m}}^{\text{2}}.$$

So for our 100 sq.cm bearing the force needed to overcome
the viscous “friction” is $2N.$ At the
speed of 1 m sec^{-1}, this means work is being done at a rate of 2
joules per sec., or 2 watts, which is heating up the oil. (This heat must be conducted away, or the oil
continuously changed by pumping, otherwise it will get too hot.)

### *Viscosity: Kinetic Energy Loss and Momentum Transfer

So far, we’ve viewed the viscosity coefficient $\eta $ as a measure of friction, of the dissipation
into heat of the energy supplied to the fluid by the moving top plate. But $\eta $ 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 flow between two parallel plates, the bottom plate at rest and the top one moving, a steady force per unit area $F/A$ in the $x$-direction applied to the top plate is needed to maintain the flow.

From Newton’s
law $F=dp/dt,$ $F/A$ *is the
rate at which momentum in the **$x$**-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
up momentum to keep moving with the plate (these molecules also constantly *lose* momentum by bouncing 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
flow? Where does the momentum fed in by the moving top plate go?

*Answer*: the $x$-direction momentum supplied at the top passes
downwards from one layer to the next, ending up at the bottom plate (and
everything it’s attached to). Remember
that, unlike kinetic energy, momentum is always conserved$\u2014$it can’t
disappear.

So, there is a steady flow *in the **$z$**-direction*
of $x$-direction momentum. Furthermore, the left-hand side of the equation

$F/A=\eta {v}_{0}/d$

is just this momentum flow rate. The right hand side is the coefficient of viscosity multiplied by the gradient in the $z$-direction of the $x$-direction velocity.

Viewed in this way, $F/A=\eta {v}_{0}/d$ is a *transport
equation*. It tells us that the rate
of transport of $x$-direction momentum downwards 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 can express this slightly differently
by noting that the rate of change of $x$-direction *velocity*
is proportional to the rate of change of $x$-direction *momentum
density*.

Recall that we mentioned earlier the so-called *kinetic* viscosity coefficient, $\nu =\mu /\rho .$ Using that in the equation

$F/A=\eta {v}_{0}/d=\nu \rho {v}_{0}/d,$

replaces the velocity gradient with an $x$-direction *momentum*
gradient. To abbreviate a clumsy phrase,
let’s call the $x$-direction momentum density ${\pi}_{x},$ and the current of this in the $z$-direction ${J}_{z}\left({\pi}_{x}\right).$ Then our equation becomes

$${J}_{z}\left({\pi}_{x}\right)=\nu \frac{d{\pi}_{x}}{dz}.$$

The current of ${\pi}_{x}$ in the $z$-direction is proportional to how fast ${\pi}_{x}$ is changing in that direction.

This closely resembles heat flowing from a hot spot to a cold spot: heat energy flows towards the place where there 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 (see later). Here, the ${\pi}_{x}$ momentum flow is analogous: it too flows to where there is less of it, and the kinetic viscosity coefficient corresponds to the thermal conductivity.