doPhysics.in
All chapters
doPhysics · Fluids

Viscosity

Start reading0%done
Why this chapter
Why this chapter exists

Honey pours slowly from a spoon, while water runs off at once. A ball bearing sinks through thick oil at a slow, steady pace. A drizzle drop seems to float down, and a cloud full of water hangs in the sky. A tiny turn of a hospital drip's clamp changes the flow of saline a great deal. All of these come from one property of real fluids: friction between their layers. This chapter measures that friction, follows it into pipes, and uses it to find the steady speed of falling drops and balls.

You will learn
By the end of this chapter you will
  1. Describe a flowing liquid as layers, and find the velocity gradient across a film.
  2. Use Newton's law of viscosity, \(F = \eta A\,dv/dy\), and give the unit and dimensions of \(\eta\).
  3. Explain why heating lowers the viscosity of a liquid but raises that of a gas.
  4. Describe the parabolic velocity profile in a pipe, and say where the viscous stress is largest.
  5. Find the Reynolds number of a flow, and decide whether it is streamline or turbulent.
  6. Derive the form of Stokes' law by dimensions, and use \(F = 6\pi\eta r v\).
  7. Name the three forces on a sphere falling in a liquid, find its starting acceleration, and derive and use its terminal velocity.
  8. Apply terminal velocity to drizzle and clouds, rising bubbles, merging drops and the falling-ball viscometer.
  9. Derive and use Poiseuille's law, \(Q = \pi P r^4/8\eta l\), and combine tubes by their fluid resistance.
  10. State when Stokes' law and Poiseuille's law hold.
Before you begin
Before you begin

From the chapter on Pressure in Fluids you should know pressure, density and \(P = P_0 + \rho g h\). From the chapter on Archimedes' Principle you should know the buoyant force, \(F_B = \rho_f V g\). From Dynamics you should know Newton's second law, the balance of forces on a body moving steadily, and kinetic friction. Stress and shear strain from the chapter on Stress, Strain and Hooke's Law will help. The chapter on Fluid Flow describes streamline and turbulent flow; this chapter puts numbers to them.

Already know this?

Lessons

1
Friction Inside a Fluid
A flowing liquid moves in layers, and each layer drags on its neighbours. Newton's law of viscosity puts a number to that drag, \(F = \eta A\,dv/dy\), and the coefficient of viscosity \(\eta\) has its own unit, the pascal second.
Layers and the velocity gradientNewtons law of viscosityUnits of viscosity
14 min0%
→
2
Viscosity in Liquids and Gases
Heating makes a liquid less viscous and a gas more viscous, for two different reasons. In a pipe the layers form a parabola, fastest on the axis and at rest at the wall. The Reynolds number tells whether a flow stays smooth or turns turbulent.
Viscosity and temperatureVelocity profile in a pipeReynolds number and turbulence
13 min0%
→
3
Stokes' Law and Terminal Velocity
The drag on a small sphere moving slowly through a fluid is \(6\pi\eta rv\), and its form follows from dimensions alone. A sphere let go in a liquid speeds up until weight, buoyancy and drag balance, then falls at a steady terminal velocity.
Stokes lawThree forces on a falling sphereTerminal velocity
15 min0%
→
4
Terminal Velocity at Work
Terminal velocity explains why clouds float and drizzle drifts, why raindrops do not hurt, why a bubble rises steadily and why merged drops fall faster. Timing a falling ball measures a viscosity, and Millikan used the same idea to weigh a single electron's charge.
Raindrops and drizzleRising bubblesDrops that mergeThe falling ball viscometer and millikan
13 min0%
→
5
Poiseuille's Law
The flow of a viscous liquid through a narrow tube is \(Q = \pi Pr^4/8\eta l\). Because of the fourth power, halving the radius cuts the flow to a sixteenth. Tubes resist flow the way resistors resist current, and the law holds only for slow, streamline flow.
Poiseuilles equationThe fourth power of the radiusFluid resistance in series and side by sideWhen poiseuille holds
14 min0%
→

Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

0%
The Workbook
Questions to solve on paper, marked by you.
Open →

Glossary

The key words of this chapter, with the meaning this chapter gives them.

coefficient of viscosity
The constant \(\eta\) in Newton's law of viscosity, \(F = \eta A\,dv/dy\): the tangential stress per unit velocity gradient. A property of the fluid that depends on its temperature. Often simply called the viscosity.
falling-ball viscometer
A tall tube of liquid in which a small ball is timed between two marks at its terminal velocity. The viscosity then follows from \(\eta = 2r^2(\rho - \sigma)g/9v_t\).
fluid resistance
The ratio \(R_f = P/Q\) of the pressure difference across a tube to the flow rate through it. For streamline flow, \(R_f = 8\eta l/\pi r^4\).
pascal second
The SI unit of viscosity, \(\text{Pa s}\), equal to \(\text{N s/m}^2\). Water at room temperature has about \(10^{-3}\,\text{Pa s}\).
poise
The CGS unit of viscosity, one dyne second per square centimetre. \(1\,\text{poise} = 0.1\,\text{Pa s}\).
Reynolds number
The pure number \(R_e = \rho v D/\eta\), which compares a fluid's inertia with its viscosity. In a pipe, flow is streamline below about 2000 and turbulent above about 3000.
Stokes' law
The viscous drag on a small sphere of radius \(r\) moving slowly at speed \(v\) through a fluid of viscosity \(\eta\) is \(6\pi\eta r v\).
streamline flow
A steady flow in which the fluid moves in smooth layers that slide past one another without mixing. Also called laminar flow.
terminal velocity
The constant speed a body finally reaches when falling or rising through a fluid. By then the drag has grown to balance the other forces on it.
turbulent flow
A flow in which the fluid breaks into irregular swirls and eddies, so that the velocity at a point keeps changing.
velocity gradient
The rate \(dv/dy\) at which the speed of the layers of a flowing fluid changes with distance at right angles to the flow. Its unit is \(\text{s}^{-1}\).
viscosity
The friction between layers of a fluid that slide past one another. It makes a fluid resist flowing, and it turns the energy of the flow into heat.
viscous drag
The force a fluid exerts on a body moving through it, against the body's velocity relative to the fluid.
volume flow rate
The volume of fluid passing through a section of a pipe each second, \(Q\), in m³/s.