Viscosity and the flow of liquids

Layers that rub

Stir a cup of honey, and then a cup of water. The honey resists. When a liquid flows, we can think of it as thin layers that slide over each other. The layer next to a wall does not move. The layers farther from the wall move faster. A layer is dragged forward by the faster one above it and held back by the slower one below it. This internal friction is viscosity.

Take two plates with a liquid between them. The bottom plate is at rest, and the top plate, of area $A$, is moved at a speed $v$. The distance between the plates is $z$. To keep the top plate moving, we must pull with a force $F$. Isaac Newton found that

The ratio $\dfrac{v}{z}$ is the velocity gradient, the change of speed from one layer to the next. The constant $\eta$ is the coefficient of viscosity, or the viscosity. A thick liquid has a large $\eta$.

The unit of viscosity

From $\eta = \dfrac{Fz}{Av}$, the SI unit is $\dfrac{\text{N m}}{\text{m}^2\ \text{m/s}} = \text{N s m}^{-2} = \text{Pa s} = \text{kg m}^{-1}\ \text{s}^{-1}$. The older unit is the poise: $1\ \text{poise} = 0.1\ \text{Pa s}$. A smaller unit is $1\ \text{cP} = 1\ \text{mPa s}$, and water at $20\ ^\circ\text{C}$ is about $1\ \text{cP}$.

Glycerol has three O-H groups in each molecule, so it forms many hydrogen bonds, and its molecules cannot slide past each other easily. Its viscosity is $1400$ times that of water. Long chain molecules, such as the molecules in oil, also tangle and give a high viscosity.

Temperature

The viscosity of a liquid falls as it gets hotter. At a higher temperature, the molecules have more energy to jump out of their neighbours' grip and slide by. Honey flows well when it is warmed. The change follows a pattern like the Arrhenius equation:

For water, $E \approx 15.6\ \text{kJ/mol}$. A graph of $\ln\eta$ against $1/T$ is nearly a straight line with slope $E/R$.

The viscosity of a gas rises as it gets hotter, which is the opposite. In a gas, the layers rub because molecules jump between them and carry momentum. At a higher temperature the molecules move faster and carry more.

Poiseuille's law: flow in a tube

When a liquid flows in a narrow tube, it flows in layers, which is laminar flow. The speed is highest at the middle and zero at the wall. The profile is a parabola. Jean Poiseuille found that the volume that passes each second is

where $\Delta P$ is the pressure difference between the ends, $r$ the radius and $L$ the length of the tube. The flow rate depends on the fourth power of the radius. If the radius is halved, the flow falls to $\tfrac{1}{16}$. The reason is that a narrower tube both has less area and slows the middle layers more. This explains why a blocked artery has such a big effect on blood flow. The law holds for laminar flow, which for a tube means a Reynolds number $Re = \dfrac{\rho v d}{\eta}$ below about $2000$.

For water with $\eta = 1.002\ \text{mPa s}$ in a tube of radius $0.40\ \text{mm}$ and length $15\ \text{cm}$, under a pressure difference of $1.5\ \text{kPa}$:

Stokes' law: a ball in a liquid

A small sphere of radius $r$ moving slowly at speed $v$ in a liquid feels a drag $F = 6\pi\eta r v$, which is Stokes' law. A ball released in a liquid speeds up until the forces balance. The weight, which acts down, equals the buoyancy plus the drag, which act up. The ball then falls at a steady speed, the terminal speed:

Notice $r^2$, the difference of the densities and the viscosity in the denominator. Stokes' law holds when the Reynolds number is below about $1$.

A glass bead of radius $0.50\ \text{mm}$ and density $2500\ \text{kg/m}^3$ falls in glycerol, with $\eta = 1.41\ \text{Pa s}$ and $\rho = 1260\ \text{kg/m}^3$. Its steady speed is $\dfrac{2 \times (5.0 \times 10^{-4})^2 \times 1240 \times 9.81}{9 \times 1.41} = 4.79 \times 10^{-4}\ \text{m/s}$, about half a millimetre each second. If we measure this speed, we find $\eta$. That is a falling ball viscometer.

Set $r = 2\ \text{mm}$, $\eta = 1.4\ \text{Pa s}$ and $\rho_s = 7800\ \text{kg/m}^3$. Double the radius, and the speed grows by four times. Double the viscosity, and the speed halves.

The Ostwald viscometer

An Ostwald viscometer is a U-shaped glass tube with a narrow part. The time that a fixed volume of liquid takes to flow through between two marks is measured. The liquid flows under its own weight, so the driving pressure is proportional to its density. By Poiseuille's law the time is proportional to $\dfrac{\eta}{\rho}$. For two liquids,

If water, with $\eta = 1.002\ \text{mPa s}$ and $\rho = 998\ \text{kg/m}^3$, takes $72\ \text{s}$, and a liquid of density $1100\ \text{kg/m}^3$ takes $132\ \text{s}$, then $\eta = 1.002 \times \dfrac{1100 \times 132}{998 \times 72} = 2.02\ \text{mPa s}$.

Practise with new numbers

Try it in Python

The program fits a straight line to $\ln\eta$ against $1/T$ for water, finds the activation energy of flow, and predicts the viscosity at $50\ ^\circ\text{C}$.

Common mistakes

• Writing the viscosity of a liquid as rising with temperature. For a liquid it falls. Only for a gas does it rise.
• Forgetting that the flow rate goes as $r^4$. Halving the radius cuts the flow to a sixteenth, not a half or a quarter.
• Using the diameter in Stokes' law. The formula has the radius.
• Leaving out the buoyancy. The steady speed depends on $\rho_s - \rho$, not on $\rho_s$ alone.
• Mixing units. $1\ \text{poise} = 0.1\ \text{Pa s}$, and $1\ \text{mPa s} = 1\ \text{cP}$.
• Forgetting the density in an Ostwald viscometer. The time alone gives $\eta/\rho$.

Remember these

• $F = \eta A\dfrac{v}{z}$. The unit of $\eta$ is Pa s. $1\ \text{poise} = 0.1\ \text{Pa s}$ and $1\ \text{cP} = 1\ \text{mPa s}$.
• Strong forces between molecules and long molecules mean a high viscosity. For a liquid, $\eta$ falls as $T$ rises.
• Poiseuille: $Q = \dfrac{\pi\Delta Pr^4}{8\eta L}$.
• Stokes: drag $6\pi\eta rv$ and a steady speed $\dfrac{2r^2(\rho_s - \rho)g}{9\eta}$.
• Ostwald viscometer: $\dfrac{\eta_1}{\eta_2} = \dfrac{\rho_1t_1}{\rho_2t_2}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Change a viscosity of $8.5\ \text{poise}$ to Pa s.

2. Why does honey flow more easily when it is warmed?

3. The radius of a tube is halved, and the pressure difference and length are the same. By what factor does the flow rate change?

4. A ball falls at a steady speed in a liquid. Name the three forces on it, and say how they balance.

5. In an Ostwald viscometer, a liquid with the same density as water takes $150\ \text{s}$, and water takes $100\ \text{s}$. What is the viscosity of the liquid? The viscosity of water is $1.002\ \text{mPa s}$.

References

• OpenStax, Chemistry 2e, Chapter 10, section 10.2 (viscosity).
• The viscosities are standard values from the CRC Handbook of Chemistry and Physics. Poiseuille's and Stokes' laws are in OpenStax, University Physics Volume 1, Chapter 14.